From e3cbf382d7f54bc6ceecdfcd9495b648645ada6b Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 22:53:33 +0530 Subject: [PATCH 01/26] Adding approach change --- .../practice/change/.approaches/config,json | 21 +++++ .../dynamic-programming/content.md | 82 +++++++++++++++++++ .../dynamic-programming/snippet.txt | 36 ++++++++ .../change/.approaches/intrduction.md | 25 ++++++ 4 files changed, 164 insertions(+) create mode 100644 exercises/practice/change/.approaches/config,json create mode 100644 exercises/practice/change/.approaches/dynamic-programming/content.md create mode 100644 exercises/practice/change/.approaches/dynamic-programming/snippet.txt create mode 100644 exercises/practice/change/.approaches/intrduction.md diff --git a/exercises/practice/change/.approaches/config,json b/exercises/practice/change/.approaches/config,json new file mode 100644 index 000000000..78950cae5 --- /dev/null +++ b/exercises/practice/change/.approaches/config,json @@ -0,0 +1,21 @@ +{ + "introduction": { + "authors": [ + "jagdish-15" + ] + }, + "approaches": [ + { + "uuid": "", + "slug": "dynamic-programming", + "title": "Dynamic Programming Approach", + "blurb": "Use dynamic programming to find the most efficient change combination.", + "authors": [ + "jagdish-15" + ], + "contributors": [ + "kagoh" + ] + } + ] +} diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md new file mode 100644 index 000000000..38e8376f5 --- /dev/null +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -0,0 +1,82 @@ +# Dynamic Programming Approach + +The **Dynamic Programming (DP)** approach is an efficient way to solve the problem of making change for a given total using a list of available coin denominations. It minimizes the number of coins needed by breaking down the problem into smaller subproblems and solving them progressively. + +This approach ensures that we find the most efficient way to make change and handles edge cases where no solution exists. + +## Explanation + +1. **Initialize Coins Usage Tracker**: + - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. + - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. + +2. **Iterative Dynamic Programming**: + - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. + - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). + - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. + +3. **Result**: + - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. + - If no valid combination exists for `grandTotal`, an exception is thrown. + + +```java +import java.util.List; +import java.util.ArrayList; + +class ChangeCalculator { + private final List currencyCoins; + + ChangeCalculator(List currencyCoins) { + this.currencyCoins = currencyCoins; + } + + List computeMostEfficientChange(int grandTotal) { + if (grandTotal < 0) + throw new IllegalArgumentException("Negative totals are not allowed."); + + List> coinsUsed = new ArrayList<>(grandTotal + 1); + coinsUsed.add(new ArrayList()); + + for (int i = 1; i <= grandTotal; i++) { + List bestCombination = null; + for (int coin: currencyCoins) { + if (coin <= i && coinsUsed.get(i - coin) != null) { + List currentCombination = new ArrayList<>(coinsUsed.get(i - coin)); + currentCombination.add(0, coin); + if (bestCombination == null || currentCombination.size() < bestCombination.size()) + bestCombination = currentCombination; + } + } + coinsUsed.add(bestCombination); + } + + if (coinsUsed.get(grandTotal) == null) + throw new IllegalArgumentException("The total " + grandTotal + " cannot be represented in the given currency."); + + return coinsUsed.get(grandTotal); + } +} +``` + +## Key Points + +- **Time Complexity**: The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. + +- **Space Complexity**: The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. + +- **Edge Cases**: + - If the `grandTotal` is negative, an exception is thrown immediately. + - If there is no way to make the exact total with the given denominations, an exception is thrown with a descriptive message. + +## Trade-offs and Considerations + +- **Efficiency**: This approach is highly efficient in terms of minimizing the number of coins, but it might require significant memory for larger `grandTotal` values, as the space complexity grows linearly with `grandTotal`. + +- **Alternative Approaches**: + - A **Greedy Approach** could be faster for some cases but does not always guarantee the minimum number of coins. + - This dynamic programming approach is best when the goal is to guarantee the fewest coins possible, especially when no simple greedy solution exists. + +## Conclusion + +The dynamic programming approach provides an optimal solution for the change-making problem, ensuring that we minimize the number of coins used while efficiently solving the problem for any `grandTotal`. However, it’s essential to consider the trade-offs in terms of memory usage and the time complexity when dealing with very large inputs. diff --git a/exercises/practice/change/.approaches/dynamic-programming/snippet.txt b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt new file mode 100644 index 000000000..3bbc3ad8c --- /dev/null +++ b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt @@ -0,0 +1,36 @@ +import java.util.List; +import java.util.ArrayList; + +class ChangeCalculator { + private final List currencyCoins; + + ChangeCalculator(List currencyCoins) { + this.currencyCoins = currencyCoins; + } + + List computeMostEfficientChange(int grandTotal) { + if (grandTotal < 0) + throw new IllegalArgumentException("Negative totals are not allowed."); + + List> coinsUsed = new ArrayList<>(grandTotal + 1); + coinsUsed.add(new ArrayList()); + + for (int i = 1; i <= grandTotal; i++) { + List bestCombination = null; + for (int coin: currencyCoins) { + if (coin <= i && coinsUsed.get(i - coin) != null) { + List currentCombination = new ArrayList<>(coinsUsed.get(i - coin)); + currentCombination.add(0, coin); + if (bestCombination == null || currentCombination.size() < bestCombination.size()) + bestCombination = currentCombination; + } + } + coinsUsed.add(bestCombination); + } + + if (coinsUsed.get(grandTotal) == null) + throw new IllegalArgumentException("The total " + grandTotal + " cannot be represented in the given currency."); + + return coinsUsed.get(grandTotal); + } +} \ No newline at end of file diff --git a/exercises/practice/change/.approaches/intrduction.md b/exercises/practice/change/.approaches/intrduction.md new file mode 100644 index 000000000..27aaef140 --- /dev/null +++ b/exercises/practice/change/.approaches/intrduction.md @@ -0,0 +1,25 @@ +## Introduction to Change Calculation + +In the "Change Calculator" exercise, the goal is to determine the minimum number of coins needed to reach a given total using a specific set of coin denominations. This is a classic problem in dynamic programming, where efficient change-making is essential, especially when there are constraints on coin types or large totals. + +### Problem Overview + +Given: +- A list of coin denominations, each representing an available currency unit. +- A total amount (`grandTotal`) we want to reach using the fewest possible coins from the given denominations. + +The solution should find the optimal combination of coins to match the total. If it's impossible to match the total exactly, the solution should indicate this by throwing an exception. + +### Approach Overview + +Our solution uses a **dynamic programming approach**, where we systematically build up the optimal combinations for all totals from `0` up to the target amount (`grandTotal`). For each total, we track the fewest coins needed to make that total, reusing previous results to make the solution efficient. + +This approach ensures that we find the minimum number of coins required in a structured, repeatable way, avoiding the need for complex recursive calls or excessive backtracking. + +### Key Features of the Approach + +- **Efficiency**: By building solutions for each increment up to `grandTotal`, this approach minimizes redundant calculations. +- **Flexibility**: Handles cases where exact change is impossible by checking at each step. +- **Scalability**: Works for various coin denominations and totals, though large inputs may impact performance. + +For a detailed look at the code and logic, see the full explanation in the [approach file](https://exercism.org/tracks/java/exercises/change/approaches/dynamic-programming). From 4a7811f81f30eb1a1f4f43fe2ccf313214817876 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:05:15 +0530 Subject: [PATCH 02/26] Fixing typo in filename --- .../practice/change/.approaches/{config,json => config.json} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename exercises/practice/change/.approaches/{config,json => config.json} (100%) diff --git a/exercises/practice/change/.approaches/config,json b/exercises/practice/change/.approaches/config.json similarity index 100% rename from exercises/practice/change/.approaches/config,json rename to exercises/practice/change/.approaches/config.json From e8d2787928d2c320d3f16a564c0e67502b3b22ba Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:13:50 +0530 Subject: [PATCH 03/26] Fixing style errors --- .../change/.approaches/dynamic-programming/content.md | 4 +++- exercises/practice/change/.approaches/intrduction.md | 3 ++- 2 files changed, 5 insertions(+), 2 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 38e8376f5..2f283a312 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -7,7 +7,7 @@ This approach ensures that we find the most efficient way to make change and han ## Explanation 1. **Initialize Coins Usage Tracker**: - - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. + - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. 2. **Iterative Dynamic Programming**: @@ -66,6 +66,7 @@ class ChangeCalculator { - **Space Complexity**: The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. - **Edge Cases**: + - If the `grandTotal` is negative, an exception is thrown immediately. - If there is no way to make the exact total with the given denominations, an exception is thrown with a descriptive message. @@ -74,6 +75,7 @@ class ChangeCalculator { - **Efficiency**: This approach is highly efficient in terms of minimizing the number of coins, but it might require significant memory for larger `grandTotal` values, as the space complexity grows linearly with `grandTotal`. - **Alternative Approaches**: + - A **Greedy Approach** could be faster for some cases but does not always guarantee the minimum number of coins. - This dynamic programming approach is best when the goal is to guarantee the fewest coins possible, especially when no simple greedy solution exists. diff --git a/exercises/practice/change/.approaches/intrduction.md b/exercises/practice/change/.approaches/intrduction.md index 27aaef140..84531fad3 100644 --- a/exercises/practice/change/.approaches/intrduction.md +++ b/exercises/practice/change/.approaches/intrduction.md @@ -1,10 +1,11 @@ -## Introduction to Change Calculation +# Introduction to Change Calculator In the "Change Calculator" exercise, the goal is to determine the minimum number of coins needed to reach a given total using a specific set of coin denominations. This is a classic problem in dynamic programming, where efficient change-making is essential, especially when there are constraints on coin types or large totals. ### Problem Overview Given: + - A list of coin denominations, each representing an available currency unit. - A total amount (`grandTotal`) we want to reach using the fewest possible coins from the given denominations. From 6152c100a88f280066afd36ae745f39e03b38dee Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:19:55 +0530 Subject: [PATCH 04/26] Fixing style errors --- .../change/.approaches/dynamic-programming/content.md | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 2f283a312..260e52e77 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -6,16 +6,19 @@ This approach ensures that we find the most efficient way to make change and han ## Explanation -1. **Initialize Coins Usage Tracker**: +1. **Initialize Coins Usage Tracker**: + - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. -2. **Iterative Dynamic Programming**: +2. **Iterative Dynamic Programming**: + - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. -3. **Result**: +3. **Result**: + - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - If no valid combination exists for `grandTotal`, an exception is thrown. From 5d398d014afa7b223a92f12044b7dd929c0307e3 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:29:29 +0530 Subject: [PATCH 05/26] Fixing stle errors --- .../change/.approaches/dynamic-programming/content.md | 5 ++--- exercises/practice/change/.approaches/intrduction.md | 6 +++--- 2 files changed, 5 insertions(+), 6 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 260e52e77..cce152a0b 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -14,7 +14,7 @@ This approach ensures that we find the most efficient way to make change and han 2. **Iterative Dynamic Programming**: - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. - - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). + - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. 3. **Result**: @@ -22,7 +22,6 @@ This approach ensures that we find the most efficient way to make change and han - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - If no valid combination exists for `grandTotal`, an exception is thrown. - ```java import java.util.List; import java.util.ArrayList; @@ -77,7 +76,7 @@ class ChangeCalculator { - **Efficiency**: This approach is highly efficient in terms of minimizing the number of coins, but it might require significant memory for larger `grandTotal` values, as the space complexity grows linearly with `grandTotal`. -- **Alternative Approaches**: +- **Alternative Approaches**: - A **Greedy Approach** could be faster for some cases but does not always guarantee the minimum number of coins. - This dynamic programming approach is best when the goal is to guarantee the fewest coins possible, especially when no simple greedy solution exists. diff --git a/exercises/practice/change/.approaches/intrduction.md b/exercises/practice/change/.approaches/intrduction.md index 84531fad3..3834d706d 100644 --- a/exercises/practice/change/.approaches/intrduction.md +++ b/exercises/practice/change/.approaches/intrduction.md @@ -2,7 +2,7 @@ In the "Change Calculator" exercise, the goal is to determine the minimum number of coins needed to reach a given total using a specific set of coin denominations. This is a classic problem in dynamic programming, where efficient change-making is essential, especially when there are constraints on coin types or large totals. -### Problem Overview +## Problem Overview Given: @@ -11,13 +11,13 @@ Given: The solution should find the optimal combination of coins to match the total. If it's impossible to match the total exactly, the solution should indicate this by throwing an exception. -### Approach Overview +## Approach Overview Our solution uses a **dynamic programming approach**, where we systematically build up the optimal combinations for all totals from `0` up to the target amount (`grandTotal`). For each total, we track the fewest coins needed to make that total, reusing previous results to make the solution efficient. This approach ensures that we find the minimum number of coins required in a structured, repeatable way, avoiding the need for complex recursive calls or excessive backtracking. -### Key Features of the Approach +## Key Features of the Approach - **Efficiency**: By building solutions for each increment up to `grandTotal`, this approach minimizes redundant calculations. - **Flexibility**: Handles cases where exact change is impossible by checking at each step. From 115a331e23500913ec0afafd2460bbad3cc25384 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:31:46 +0530 Subject: [PATCH 06/26] Fixing file names --- .../dynamic-programming/snippet.txt | 36 ------------------- .../{intrduction.md => introduction.md} | 0 2 files changed, 36 deletions(-) delete mode 100644 exercises/practice/change/.approaches/dynamic-programming/snippet.txt rename exercises/practice/change/.approaches/{intrduction.md => introduction.md} (100%) diff --git a/exercises/practice/change/.approaches/dynamic-programming/snippet.txt b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt deleted file mode 100644 index 3bbc3ad8c..000000000 --- a/exercises/practice/change/.approaches/dynamic-programming/snippet.txt +++ /dev/null @@ -1,36 +0,0 @@ -import java.util.List; -import java.util.ArrayList; - -class ChangeCalculator { - private final List currencyCoins; - - ChangeCalculator(List currencyCoins) { - this.currencyCoins = currencyCoins; - } - - List computeMostEfficientChange(int grandTotal) { - if (grandTotal < 0) - throw new IllegalArgumentException("Negative totals are not allowed."); - - List> coinsUsed = new ArrayList<>(grandTotal + 1); - coinsUsed.add(new ArrayList()); - - for (int i = 1; i <= grandTotal; i++) { - List bestCombination = null; - for (int coin: currencyCoins) { - if (coin <= i && coinsUsed.get(i - coin) != null) { - List currentCombination = new ArrayList<>(coinsUsed.get(i - coin)); - currentCombination.add(0, coin); - if (bestCombination == null || currentCombination.size() < bestCombination.size()) - bestCombination = currentCombination; - } - } - coinsUsed.add(bestCombination); - } - - if (coinsUsed.get(grandTotal) == null) - throw new IllegalArgumentException("The total " + grandTotal + " cannot be represented in the given currency."); - - return coinsUsed.get(grandTotal); - } -} \ No newline at end of file diff --git a/exercises/practice/change/.approaches/intrduction.md b/exercises/practice/change/.approaches/introduction.md similarity index 100% rename from exercises/practice/change/.approaches/intrduction.md rename to exercises/practice/change/.approaches/introduction.md From d2c24511394e6dd4f2be9cbf1adfabf86aedadf7 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Thu, 14 Nov 2024 23:25:31 +0530 Subject: [PATCH 07/26] Adding uuid for approach of change --- exercises/practice/change/.approaches/config.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/exercises/practice/change/.approaches/config.json b/exercises/practice/change/.approaches/config.json index 78950cae5..f9b602eb2 100644 --- a/exercises/practice/change/.approaches/config.json +++ b/exercises/practice/change/.approaches/config.json @@ -6,7 +6,7 @@ }, "approaches": [ { - "uuid": "", + "uuid": "d0b615ca-3a02-4d66-ad10-e0c513062189", "slug": "dynamic-programming", "title": "Dynamic Programming Approach", "blurb": "Use dynamic programming to find the most efficient change combination.", From b3a98dfc6021b4313ed330daf288471b7fc3bcca Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Thu, 14 Nov 2024 23:27:08 +0530 Subject: [PATCH 08/26] Adding snippet for approach of change --- .../change/.approaches/dynamic-programming/snippet.txt | 8 ++++++++ 1 file changed, 8 insertions(+) create mode 100644 exercises/practice/change/.approaches/dynamic-programming/snippet.txt diff --git a/exercises/practice/change/.approaches/dynamic-programming/snippet.txt b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt new file mode 100644 index 000000000..25f90e6f5 --- /dev/null +++ b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt @@ -0,0 +1,8 @@ +class ChangeCalculator { + private final List currencyCoins; + + ChangeCalculator(List currencyCoins) { + this.currencyCoins = currencyCoins; + } + // computeMostEfficientChange method +} From 94383250b3de52c49fd4b5d06d760331ac6a8ebd Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Fri, 15 Nov 2024 21:42:42 +0530 Subject: [PATCH 09/26] Updating approach files --- .../practice/change/.approaches/config.json | 2 +- .../dynamic-programming/content.md | 57 ++++++++----------- .../change/.approaches/introduction.md | 25 ++++---- 3 files changed, 39 insertions(+), 45 deletions(-) diff --git a/exercises/practice/change/.approaches/config.json b/exercises/practice/change/.approaches/config.json index f9b602eb2..00716db81 100644 --- a/exercises/practice/change/.approaches/config.json +++ b/exercises/practice/change/.approaches/config.json @@ -14,7 +14,7 @@ "jagdish-15" ], "contributors": [ - "kagoh" + "kahgoh" ] } ] diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index cce152a0b..4277c61ff 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -1,27 +1,5 @@ # Dynamic Programming Approach -The **Dynamic Programming (DP)** approach is an efficient way to solve the problem of making change for a given total using a list of available coin denominations. It minimizes the number of coins needed by breaking down the problem into smaller subproblems and solving them progressively. - -This approach ensures that we find the most efficient way to make change and handles edge cases where no solution exists. - -## Explanation - -1. **Initialize Coins Usage Tracker**: - - - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. - - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. - -2. **Iterative Dynamic Programming**: - - - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. - - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). - - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. - -3. **Result**: - - - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - - If no valid combination exists for `grandTotal`, an exception is thrown. - ```java import java.util.List; import java.util.ArrayList; @@ -61,6 +39,29 @@ class ChangeCalculator { } ``` +The **Dynamic Programming (DP)** approach is an efficient way to solve the problem of making change for a given total using a list of available coin denominations. +It minimizes the number of coins needed by breaking down the problem into smaller subproblems and solving them progressively. + +This approach ensures that we find the most efficient way to make change and handles edge cases where no solution exists. + +## Explanation + +1. **Initialize Coins Usage Tracker**: + + - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. + - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. + +2. **Iterative Dynamic Programming**: + + - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. + - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). + - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. + +3. **Result**: + + - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. + - If no valid combination exists for `grandTotal`, an exception is thrown. + ## Key Points - **Time Complexity**: The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. @@ -72,15 +73,7 @@ class ChangeCalculator { - If the `grandTotal` is negative, an exception is thrown immediately. - If there is no way to make the exact total with the given denominations, an exception is thrown with a descriptive message. -## Trade-offs and Considerations - -- **Efficiency**: This approach is highly efficient in terms of minimizing the number of coins, but it might require significant memory for larger `grandTotal` values, as the space complexity grows linearly with `grandTotal`. - -- **Alternative Approaches**: - - - A **Greedy Approach** could be faster for some cases but does not always guarantee the minimum number of coins. - - This dynamic programming approach is best when the goal is to guarantee the fewest coins possible, especially when no simple greedy solution exists. - ## Conclusion -The dynamic programming approach provides an optimal solution for the change-making problem, ensuring that we minimize the number of coins used while efficiently solving the problem for any `grandTotal`. However, it’s essential to consider the trade-offs in terms of memory usage and the time complexity when dealing with very large inputs. +The dynamic programming approach provides an optimal solution for the change-making problem, ensuring that we minimize the number of coins used while efficiently solving the problem for any `grandTotal`. +However, it’s essential to consider the trade-offs in terms of memory usage and the time complexity when dealing with very large inputs. diff --git a/exercises/practice/change/.approaches/introduction.md b/exercises/practice/change/.approaches/introduction.md index 3834d706d..713e5070f 100644 --- a/exercises/practice/change/.approaches/introduction.md +++ b/exercises/practice/change/.approaches/introduction.md @@ -1,19 +1,17 @@ -# Introduction to Change Calculator +# Introduction -In the "Change Calculator" exercise, the goal is to determine the minimum number of coins needed to reach a given total using a specific set of coin denominations. This is a classic problem in dynamic programming, where efficient change-making is essential, especially when there are constraints on coin types or large totals. +There is an idiomatic approach to solving "Change." +You can use [dynamic programming][dynamic-programming] to calculate the minimum number of coins required for a given total. -## Problem Overview +## General guidance -Given: +The key to solving "Change" is understanding that not all totals can be reached with the available coin denominations. +The solution needs to figure out which totals can be achieved and how to combine the coins optimally. -- A list of coin denominations, each representing an available currency unit. -- A total amount (`grandTotal`) we want to reach using the fewest possible coins from the given denominations. +## Approach: Dynamic Programming -The solution should find the optimal combination of coins to match the total. If it's impossible to match the total exactly, the solution should indicate this by throwing an exception. - -## Approach Overview - -Our solution uses a **dynamic programming approach**, where we systematically build up the optimal combinations for all totals from `0` up to the target amount (`grandTotal`). For each total, we track the fewest coins needed to make that total, reusing previous results to make the solution efficient. +Our solution uses a **dynamic programming approach**, where we systematically build up the optimal combinations for all totals from `0` up to the target amount (`grandTotal`). +For each total, we track the fewest coins needed to make that total, reusing previous results to make the solution efficient. This approach ensures that we find the minimum number of coins required in a structured, repeatable way, avoiding the need for complex recursive calls or excessive backtracking. @@ -23,4 +21,7 @@ This approach ensures that we find the minimum number of coins required in a str - **Flexibility**: Handles cases where exact change is impossible by checking at each step. - **Scalability**: Works for various coin denominations and totals, though large inputs may impact performance. -For a detailed look at the code and logic, see the full explanation in the [approach file](https://exercism.org/tracks/java/exercises/change/approaches/dynamic-programming). +For a detailed look at the code and logic, see the full explanation in the [Dynamic Programming Approach][approach-dynamic-programming]. + +[approach-dynamic-programming]: https://exercism.org/tracks/java/exercises/change/approaches/dynamic-programming +[dynamic-programming]: https://en.wikipedia.org/wiki/Dynamic_programming \ No newline at end of file From fde837b7eb8b5a1c835b2563a259e1d4b0f71ead Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Fri, 15 Nov 2024 21:45:08 +0530 Subject: [PATCH 10/26] Fixing styling errors --- exercises/practice/change/.approaches/introduction.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/exercises/practice/change/.approaches/introduction.md b/exercises/practice/change/.approaches/introduction.md index 713e5070f..49100bf51 100644 --- a/exercises/practice/change/.approaches/introduction.md +++ b/exercises/practice/change/.approaches/introduction.md @@ -24,4 +24,4 @@ This approach ensures that we find the minimum number of coins required in a str For a detailed look at the code and logic, see the full explanation in the [Dynamic Programming Approach][approach-dynamic-programming]. [approach-dynamic-programming]: https://exercism.org/tracks/java/exercises/change/approaches/dynamic-programming -[dynamic-programming]: https://en.wikipedia.org/wiki/Dynamic_programming \ No newline at end of file +[dynamic-programming]: https://en.wikipedia.org/wiki/Dynamic_programming From 7ec29607ea117ec671e7840e41fa8e0f5d726461 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 22:53:33 +0530 Subject: [PATCH 11/26] Adding approach change --- .../practice/change/.approaches/config,json | 21 +++++ .../dynamic-programming/content.md | 82 +++++++++++++++++++ .../dynamic-programming/snippet.txt | 36 ++++++++ .../change/.approaches/intrduction.md | 25 ++++++ 4 files changed, 164 insertions(+) create mode 100644 exercises/practice/change/.approaches/config,json create mode 100644 exercises/practice/change/.approaches/dynamic-programming/content.md create mode 100644 exercises/practice/change/.approaches/dynamic-programming/snippet.txt create mode 100644 exercises/practice/change/.approaches/intrduction.md diff --git a/exercises/practice/change/.approaches/config,json b/exercises/practice/change/.approaches/config,json new file mode 100644 index 000000000..78950cae5 --- /dev/null +++ b/exercises/practice/change/.approaches/config,json @@ -0,0 +1,21 @@ +{ + "introduction": { + "authors": [ + "jagdish-15" + ] + }, + "approaches": [ + { + "uuid": "", + "slug": "dynamic-programming", + "title": "Dynamic Programming Approach", + "blurb": "Use dynamic programming to find the most efficient change combination.", + "authors": [ + "jagdish-15" + ], + "contributors": [ + "kagoh" + ] + } + ] +} diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md new file mode 100644 index 000000000..38e8376f5 --- /dev/null +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -0,0 +1,82 @@ +# Dynamic Programming Approach + +The **Dynamic Programming (DP)** approach is an efficient way to solve the problem of making change for a given total using a list of available coin denominations. It minimizes the number of coins needed by breaking down the problem into smaller subproblems and solving them progressively. + +This approach ensures that we find the most efficient way to make change and handles edge cases where no solution exists. + +## Explanation + +1. **Initialize Coins Usage Tracker**: + - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. + - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. + +2. **Iterative Dynamic Programming**: + - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. + - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). + - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. + +3. **Result**: + - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. + - If no valid combination exists for `grandTotal`, an exception is thrown. + + +```java +import java.util.List; +import java.util.ArrayList; + +class ChangeCalculator { + private final List currencyCoins; + + ChangeCalculator(List currencyCoins) { + this.currencyCoins = currencyCoins; + } + + List computeMostEfficientChange(int grandTotal) { + if (grandTotal < 0) + throw new IllegalArgumentException("Negative totals are not allowed."); + + List> coinsUsed = new ArrayList<>(grandTotal + 1); + coinsUsed.add(new ArrayList()); + + for (int i = 1; i <= grandTotal; i++) { + List bestCombination = null; + for (int coin: currencyCoins) { + if (coin <= i && coinsUsed.get(i - coin) != null) { + List currentCombination = new ArrayList<>(coinsUsed.get(i - coin)); + currentCombination.add(0, coin); + if (bestCombination == null || currentCombination.size() < bestCombination.size()) + bestCombination = currentCombination; + } + } + coinsUsed.add(bestCombination); + } + + if (coinsUsed.get(grandTotal) == null) + throw new IllegalArgumentException("The total " + grandTotal + " cannot be represented in the given currency."); + + return coinsUsed.get(grandTotal); + } +} +``` + +## Key Points + +- **Time Complexity**: The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. + +- **Space Complexity**: The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. + +- **Edge Cases**: + - If the `grandTotal` is negative, an exception is thrown immediately. + - If there is no way to make the exact total with the given denominations, an exception is thrown with a descriptive message. + +## Trade-offs and Considerations + +- **Efficiency**: This approach is highly efficient in terms of minimizing the number of coins, but it might require significant memory for larger `grandTotal` values, as the space complexity grows linearly with `grandTotal`. + +- **Alternative Approaches**: + - A **Greedy Approach** could be faster for some cases but does not always guarantee the minimum number of coins. + - This dynamic programming approach is best when the goal is to guarantee the fewest coins possible, especially when no simple greedy solution exists. + +## Conclusion + +The dynamic programming approach provides an optimal solution for the change-making problem, ensuring that we minimize the number of coins used while efficiently solving the problem for any `grandTotal`. However, it’s essential to consider the trade-offs in terms of memory usage and the time complexity when dealing with very large inputs. diff --git a/exercises/practice/change/.approaches/dynamic-programming/snippet.txt b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt new file mode 100644 index 000000000..3bbc3ad8c --- /dev/null +++ b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt @@ -0,0 +1,36 @@ +import java.util.List; +import java.util.ArrayList; + +class ChangeCalculator { + private final List currencyCoins; + + ChangeCalculator(List currencyCoins) { + this.currencyCoins = currencyCoins; + } + + List computeMostEfficientChange(int grandTotal) { + if (grandTotal < 0) + throw new IllegalArgumentException("Negative totals are not allowed."); + + List> coinsUsed = new ArrayList<>(grandTotal + 1); + coinsUsed.add(new ArrayList()); + + for (int i = 1; i <= grandTotal; i++) { + List bestCombination = null; + for (int coin: currencyCoins) { + if (coin <= i && coinsUsed.get(i - coin) != null) { + List currentCombination = new ArrayList<>(coinsUsed.get(i - coin)); + currentCombination.add(0, coin); + if (bestCombination == null || currentCombination.size() < bestCombination.size()) + bestCombination = currentCombination; + } + } + coinsUsed.add(bestCombination); + } + + if (coinsUsed.get(grandTotal) == null) + throw new IllegalArgumentException("The total " + grandTotal + " cannot be represented in the given currency."); + + return coinsUsed.get(grandTotal); + } +} \ No newline at end of file diff --git a/exercises/practice/change/.approaches/intrduction.md b/exercises/practice/change/.approaches/intrduction.md new file mode 100644 index 000000000..27aaef140 --- /dev/null +++ b/exercises/practice/change/.approaches/intrduction.md @@ -0,0 +1,25 @@ +## Introduction to Change Calculation + +In the "Change Calculator" exercise, the goal is to determine the minimum number of coins needed to reach a given total using a specific set of coin denominations. This is a classic problem in dynamic programming, where efficient change-making is essential, especially when there are constraints on coin types or large totals. + +### Problem Overview + +Given: +- A list of coin denominations, each representing an available currency unit. +- A total amount (`grandTotal`) we want to reach using the fewest possible coins from the given denominations. + +The solution should find the optimal combination of coins to match the total. If it's impossible to match the total exactly, the solution should indicate this by throwing an exception. + +### Approach Overview + +Our solution uses a **dynamic programming approach**, where we systematically build up the optimal combinations for all totals from `0` up to the target amount (`grandTotal`). For each total, we track the fewest coins needed to make that total, reusing previous results to make the solution efficient. + +This approach ensures that we find the minimum number of coins required in a structured, repeatable way, avoiding the need for complex recursive calls or excessive backtracking. + +### Key Features of the Approach + +- **Efficiency**: By building solutions for each increment up to `grandTotal`, this approach minimizes redundant calculations. +- **Flexibility**: Handles cases where exact change is impossible by checking at each step. +- **Scalability**: Works for various coin denominations and totals, though large inputs may impact performance. + +For a detailed look at the code and logic, see the full explanation in the [approach file](https://exercism.org/tracks/java/exercises/change/approaches/dynamic-programming). From 404de05c450e885cd94316ed8c119a2fc20dd77a Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:05:15 +0530 Subject: [PATCH 12/26] Fixing typo in filename --- .../practice/change/.approaches/{config,json => config.json} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename exercises/practice/change/.approaches/{config,json => config.json} (100%) diff --git a/exercises/practice/change/.approaches/config,json b/exercises/practice/change/.approaches/config.json similarity index 100% rename from exercises/practice/change/.approaches/config,json rename to exercises/practice/change/.approaches/config.json From 1aa6d9c62356a7038cedaaedaf4906eb2b50cd55 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:13:50 +0530 Subject: [PATCH 13/26] Fixing style errors --- .../change/.approaches/dynamic-programming/content.md | 4 +++- exercises/practice/change/.approaches/intrduction.md | 3 ++- 2 files changed, 5 insertions(+), 2 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 38e8376f5..2f283a312 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -7,7 +7,7 @@ This approach ensures that we find the most efficient way to make change and han ## Explanation 1. **Initialize Coins Usage Tracker**: - - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. + - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. 2. **Iterative Dynamic Programming**: @@ -66,6 +66,7 @@ class ChangeCalculator { - **Space Complexity**: The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. - **Edge Cases**: + - If the `grandTotal` is negative, an exception is thrown immediately. - If there is no way to make the exact total with the given denominations, an exception is thrown with a descriptive message. @@ -74,6 +75,7 @@ class ChangeCalculator { - **Efficiency**: This approach is highly efficient in terms of minimizing the number of coins, but it might require significant memory for larger `grandTotal` values, as the space complexity grows linearly with `grandTotal`. - **Alternative Approaches**: + - A **Greedy Approach** could be faster for some cases but does not always guarantee the minimum number of coins. - This dynamic programming approach is best when the goal is to guarantee the fewest coins possible, especially when no simple greedy solution exists. diff --git a/exercises/practice/change/.approaches/intrduction.md b/exercises/practice/change/.approaches/intrduction.md index 27aaef140..84531fad3 100644 --- a/exercises/practice/change/.approaches/intrduction.md +++ b/exercises/practice/change/.approaches/intrduction.md @@ -1,10 +1,11 @@ -## Introduction to Change Calculation +# Introduction to Change Calculator In the "Change Calculator" exercise, the goal is to determine the minimum number of coins needed to reach a given total using a specific set of coin denominations. This is a classic problem in dynamic programming, where efficient change-making is essential, especially when there are constraints on coin types or large totals. ### Problem Overview Given: + - A list of coin denominations, each representing an available currency unit. - A total amount (`grandTotal`) we want to reach using the fewest possible coins from the given denominations. From 2fbec2a3add6d3d08e9ee6be7191d3b1dcdb45af Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:19:55 +0530 Subject: [PATCH 14/26] Fixing style errors --- .../change/.approaches/dynamic-programming/content.md | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 2f283a312..260e52e77 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -6,16 +6,19 @@ This approach ensures that we find the most efficient way to make change and han ## Explanation -1. **Initialize Coins Usage Tracker**: +1. **Initialize Coins Usage Tracker**: + - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. -2. **Iterative Dynamic Programming**: +2. **Iterative Dynamic Programming**: + - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. -3. **Result**: +3. **Result**: + - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - If no valid combination exists for `grandTotal`, an exception is thrown. From c1d07e591aea8b99bcee452bbcfb71ed407da227 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:29:29 +0530 Subject: [PATCH 15/26] Fixing stle errors --- .../change/.approaches/dynamic-programming/content.md | 5 ++--- exercises/practice/change/.approaches/intrduction.md | 6 +++--- 2 files changed, 5 insertions(+), 6 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 260e52e77..cce152a0b 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -14,7 +14,7 @@ This approach ensures that we find the most efficient way to make change and han 2. **Iterative Dynamic Programming**: - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. - - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). + - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. 3. **Result**: @@ -22,7 +22,6 @@ This approach ensures that we find the most efficient way to make change and han - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - If no valid combination exists for `grandTotal`, an exception is thrown. - ```java import java.util.List; import java.util.ArrayList; @@ -77,7 +76,7 @@ class ChangeCalculator { - **Efficiency**: This approach is highly efficient in terms of minimizing the number of coins, but it might require significant memory for larger `grandTotal` values, as the space complexity grows linearly with `grandTotal`. -- **Alternative Approaches**: +- **Alternative Approaches**: - A **Greedy Approach** could be faster for some cases but does not always guarantee the minimum number of coins. - This dynamic programming approach is best when the goal is to guarantee the fewest coins possible, especially when no simple greedy solution exists. diff --git a/exercises/practice/change/.approaches/intrduction.md b/exercises/practice/change/.approaches/intrduction.md index 84531fad3..3834d706d 100644 --- a/exercises/practice/change/.approaches/intrduction.md +++ b/exercises/practice/change/.approaches/intrduction.md @@ -2,7 +2,7 @@ In the "Change Calculator" exercise, the goal is to determine the minimum number of coins needed to reach a given total using a specific set of coin denominations. This is a classic problem in dynamic programming, where efficient change-making is essential, especially when there are constraints on coin types or large totals. -### Problem Overview +## Problem Overview Given: @@ -11,13 +11,13 @@ Given: The solution should find the optimal combination of coins to match the total. If it's impossible to match the total exactly, the solution should indicate this by throwing an exception. -### Approach Overview +## Approach Overview Our solution uses a **dynamic programming approach**, where we systematically build up the optimal combinations for all totals from `0` up to the target amount (`grandTotal`). For each total, we track the fewest coins needed to make that total, reusing previous results to make the solution efficient. This approach ensures that we find the minimum number of coins required in a structured, repeatable way, avoiding the need for complex recursive calls or excessive backtracking. -### Key Features of the Approach +## Key Features of the Approach - **Efficiency**: By building solutions for each increment up to `grandTotal`, this approach minimizes redundant calculations. - **Flexibility**: Handles cases where exact change is impossible by checking at each step. From afee45ab5bd4871a5903ec32da0c27b43cee0b5d Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 9 Nov 2024 23:31:46 +0530 Subject: [PATCH 16/26] Fixing file names --- .../dynamic-programming/snippet.txt | 36 ------------------- .../{intrduction.md => introduction.md} | 0 2 files changed, 36 deletions(-) delete mode 100644 exercises/practice/change/.approaches/dynamic-programming/snippet.txt rename exercises/practice/change/.approaches/{intrduction.md => introduction.md} (100%) diff --git a/exercises/practice/change/.approaches/dynamic-programming/snippet.txt b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt deleted file mode 100644 index 3bbc3ad8c..000000000 --- a/exercises/practice/change/.approaches/dynamic-programming/snippet.txt +++ /dev/null @@ -1,36 +0,0 @@ -import java.util.List; -import java.util.ArrayList; - -class ChangeCalculator { - private final List currencyCoins; - - ChangeCalculator(List currencyCoins) { - this.currencyCoins = currencyCoins; - } - - List computeMostEfficientChange(int grandTotal) { - if (grandTotal < 0) - throw new IllegalArgumentException("Negative totals are not allowed."); - - List> coinsUsed = new ArrayList<>(grandTotal + 1); - coinsUsed.add(new ArrayList()); - - for (int i = 1; i <= grandTotal; i++) { - List bestCombination = null; - for (int coin: currencyCoins) { - if (coin <= i && coinsUsed.get(i - coin) != null) { - List currentCombination = new ArrayList<>(coinsUsed.get(i - coin)); - currentCombination.add(0, coin); - if (bestCombination == null || currentCombination.size() < bestCombination.size()) - bestCombination = currentCombination; - } - } - coinsUsed.add(bestCombination); - } - - if (coinsUsed.get(grandTotal) == null) - throw new IllegalArgumentException("The total " + grandTotal + " cannot be represented in the given currency."); - - return coinsUsed.get(grandTotal); - } -} \ No newline at end of file diff --git a/exercises/practice/change/.approaches/intrduction.md b/exercises/practice/change/.approaches/introduction.md similarity index 100% rename from exercises/practice/change/.approaches/intrduction.md rename to exercises/practice/change/.approaches/introduction.md From d9694faf8b91e0cfa5e76c1ac18fcc96f4123087 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Thu, 14 Nov 2024 23:25:31 +0530 Subject: [PATCH 17/26] Adding uuid for approach of change --- exercises/practice/change/.approaches/config.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/exercises/practice/change/.approaches/config.json b/exercises/practice/change/.approaches/config.json index 78950cae5..f9b602eb2 100644 --- a/exercises/practice/change/.approaches/config.json +++ b/exercises/practice/change/.approaches/config.json @@ -6,7 +6,7 @@ }, "approaches": [ { - "uuid": "", + "uuid": "d0b615ca-3a02-4d66-ad10-e0c513062189", "slug": "dynamic-programming", "title": "Dynamic Programming Approach", "blurb": "Use dynamic programming to find the most efficient change combination.", From 7aa412ef35d2b4b5a63a4a4a541bcd026eddf5cd Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Thu, 14 Nov 2024 23:27:08 +0530 Subject: [PATCH 18/26] Adding snippet for approach of change --- .../change/.approaches/dynamic-programming/snippet.txt | 8 ++++++++ 1 file changed, 8 insertions(+) create mode 100644 exercises/practice/change/.approaches/dynamic-programming/snippet.txt diff --git a/exercises/practice/change/.approaches/dynamic-programming/snippet.txt b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt new file mode 100644 index 000000000..25f90e6f5 --- /dev/null +++ b/exercises/practice/change/.approaches/dynamic-programming/snippet.txt @@ -0,0 +1,8 @@ +class ChangeCalculator { + private final List currencyCoins; + + ChangeCalculator(List currencyCoins) { + this.currencyCoins = currencyCoins; + } + // computeMostEfficientChange method +} From f3eb007e0340e879aab4c93a8adaa151eefa487c Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Fri, 15 Nov 2024 21:42:42 +0530 Subject: [PATCH 19/26] Updating approach files --- .../practice/change/.approaches/config.json | 2 +- .../dynamic-programming/content.md | 57 ++++++++----------- .../change/.approaches/introduction.md | 25 ++++---- 3 files changed, 39 insertions(+), 45 deletions(-) diff --git a/exercises/practice/change/.approaches/config.json b/exercises/practice/change/.approaches/config.json index f9b602eb2..00716db81 100644 --- a/exercises/practice/change/.approaches/config.json +++ b/exercises/practice/change/.approaches/config.json @@ -14,7 +14,7 @@ "jagdish-15" ], "contributors": [ - "kagoh" + "kahgoh" ] } ] diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index cce152a0b..4277c61ff 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -1,27 +1,5 @@ # Dynamic Programming Approach -The **Dynamic Programming (DP)** approach is an efficient way to solve the problem of making change for a given total using a list of available coin denominations. It minimizes the number of coins needed by breaking down the problem into smaller subproblems and solving them progressively. - -This approach ensures that we find the most efficient way to make change and handles edge cases where no solution exists. - -## Explanation - -1. **Initialize Coins Usage Tracker**: - - - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. - - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. - -2. **Iterative Dynamic Programming**: - - - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. - - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). - - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. - -3. **Result**: - - - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - - If no valid combination exists for `grandTotal`, an exception is thrown. - ```java import java.util.List; import java.util.ArrayList; @@ -61,6 +39,29 @@ class ChangeCalculator { } ``` +The **Dynamic Programming (DP)** approach is an efficient way to solve the problem of making change for a given total using a list of available coin denominations. +It minimizes the number of coins needed by breaking down the problem into smaller subproblems and solving them progressively. + +This approach ensures that we find the most efficient way to make change and handles edge cases where no solution exists. + +## Explanation + +1. **Initialize Coins Usage Tracker**: + + - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. + - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. + +2. **Iterative Dynamic Programming**: + + - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. + - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). + - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. + +3. **Result**: + + - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. + - If no valid combination exists for `grandTotal`, an exception is thrown. + ## Key Points - **Time Complexity**: The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. @@ -72,15 +73,7 @@ class ChangeCalculator { - If the `grandTotal` is negative, an exception is thrown immediately. - If there is no way to make the exact total with the given denominations, an exception is thrown with a descriptive message. -## Trade-offs and Considerations - -- **Efficiency**: This approach is highly efficient in terms of minimizing the number of coins, but it might require significant memory for larger `grandTotal` values, as the space complexity grows linearly with `grandTotal`. - -- **Alternative Approaches**: - - - A **Greedy Approach** could be faster for some cases but does not always guarantee the minimum number of coins. - - This dynamic programming approach is best when the goal is to guarantee the fewest coins possible, especially when no simple greedy solution exists. - ## Conclusion -The dynamic programming approach provides an optimal solution for the change-making problem, ensuring that we minimize the number of coins used while efficiently solving the problem for any `grandTotal`. However, it’s essential to consider the trade-offs in terms of memory usage and the time complexity when dealing with very large inputs. +The dynamic programming approach provides an optimal solution for the change-making problem, ensuring that we minimize the number of coins used while efficiently solving the problem for any `grandTotal`. +However, it’s essential to consider the trade-offs in terms of memory usage and the time complexity when dealing with very large inputs. diff --git a/exercises/practice/change/.approaches/introduction.md b/exercises/practice/change/.approaches/introduction.md index 3834d706d..713e5070f 100644 --- a/exercises/practice/change/.approaches/introduction.md +++ b/exercises/practice/change/.approaches/introduction.md @@ -1,19 +1,17 @@ -# Introduction to Change Calculator +# Introduction -In the "Change Calculator" exercise, the goal is to determine the minimum number of coins needed to reach a given total using a specific set of coin denominations. This is a classic problem in dynamic programming, where efficient change-making is essential, especially when there are constraints on coin types or large totals. +There is an idiomatic approach to solving "Change." +You can use [dynamic programming][dynamic-programming] to calculate the minimum number of coins required for a given total. -## Problem Overview +## General guidance -Given: +The key to solving "Change" is understanding that not all totals can be reached with the available coin denominations. +The solution needs to figure out which totals can be achieved and how to combine the coins optimally. -- A list of coin denominations, each representing an available currency unit. -- A total amount (`grandTotal`) we want to reach using the fewest possible coins from the given denominations. +## Approach: Dynamic Programming -The solution should find the optimal combination of coins to match the total. If it's impossible to match the total exactly, the solution should indicate this by throwing an exception. - -## Approach Overview - -Our solution uses a **dynamic programming approach**, where we systematically build up the optimal combinations for all totals from `0` up to the target amount (`grandTotal`). For each total, we track the fewest coins needed to make that total, reusing previous results to make the solution efficient. +Our solution uses a **dynamic programming approach**, where we systematically build up the optimal combinations for all totals from `0` up to the target amount (`grandTotal`). +For each total, we track the fewest coins needed to make that total, reusing previous results to make the solution efficient. This approach ensures that we find the minimum number of coins required in a structured, repeatable way, avoiding the need for complex recursive calls or excessive backtracking. @@ -23,4 +21,7 @@ This approach ensures that we find the minimum number of coins required in a str - **Flexibility**: Handles cases where exact change is impossible by checking at each step. - **Scalability**: Works for various coin denominations and totals, though large inputs may impact performance. -For a detailed look at the code and logic, see the full explanation in the [approach file](https://exercism.org/tracks/java/exercises/change/approaches/dynamic-programming). +For a detailed look at the code and logic, see the full explanation in the [Dynamic Programming Approach][approach-dynamic-programming]. + +[approach-dynamic-programming]: https://exercism.org/tracks/java/exercises/change/approaches/dynamic-programming +[dynamic-programming]: https://en.wikipedia.org/wiki/Dynamic_programming \ No newline at end of file From 0b5b6f26f1b8950d65e14a54b2b0af3a40b765c0 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Fri, 15 Nov 2024 21:45:08 +0530 Subject: [PATCH 20/26] Fixing styling errors --- exercises/practice/change/.approaches/introduction.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/exercises/practice/change/.approaches/introduction.md b/exercises/practice/change/.approaches/introduction.md index 713e5070f..49100bf51 100644 --- a/exercises/practice/change/.approaches/introduction.md +++ b/exercises/practice/change/.approaches/introduction.md @@ -24,4 +24,4 @@ This approach ensures that we find the minimum number of coins required in a str For a detailed look at the code and logic, see the full explanation in the [Dynamic Programming Approach][approach-dynamic-programming]. [approach-dynamic-programming]: https://exercism.org/tracks/java/exercises/change/approaches/dynamic-programming -[dynamic-programming]: https://en.wikipedia.org/wiki/Dynamic_programming \ No newline at end of file +[dynamic-programming]: https://en.wikipedia.org/wiki/Dynamic_programming From 2758bc3737fb09d4b91f37b61ecf790f6f02c2ef Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Tue, 19 Nov 2024 00:22:45 +0530 Subject: [PATCH 21/26] Updating approaches for Change after feedback --- .../dynamic-programming/content.md | 14 +----- .../change/.approaches/introduction.md | 48 +++++++++++++++---- 2 files changed, 39 insertions(+), 23 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 4277c61ff..c5801342c 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -42,12 +42,11 @@ class ChangeCalculator { The **Dynamic Programming (DP)** approach is an efficient way to solve the problem of making change for a given total using a list of available coin denominations. It minimizes the number of coins needed by breaking down the problem into smaller subproblems and solving them progressively. -This approach ensures that we find the most efficient way to make change and handles edge cases where no solution exists. - ## Explanation 1. **Initialize Coins Usage Tracker**: + - If the `grandTotal` is negative, an exception is thrown immediately. - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. @@ -62,18 +61,7 @@ This approach ensures that we find the most efficient way to make change and han - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - If no valid combination exists for `grandTotal`, an exception is thrown. -## Key Points - **Time Complexity**: The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. - **Space Complexity**: The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. - -- **Edge Cases**: - - - If the `grandTotal` is negative, an exception is thrown immediately. - - If there is no way to make the exact total with the given denominations, an exception is thrown with a descriptive message. - -## Conclusion - -The dynamic programming approach provides an optimal solution for the change-making problem, ensuring that we minimize the number of coins used while efficiently solving the problem for any `grandTotal`. -However, it’s essential to consider the trade-offs in terms of memory usage and the time complexity when dealing with very large inputs. diff --git a/exercises/practice/change/.approaches/introduction.md b/exercises/practice/change/.approaches/introduction.md index 49100bf51..672aae038 100644 --- a/exercises/practice/change/.approaches/introduction.md +++ b/exercises/practice/change/.approaches/introduction.md @@ -10,16 +10,44 @@ The solution needs to figure out which totals can be achieved and how to combine ## Approach: Dynamic Programming -Our solution uses a **dynamic programming approach**, where we systematically build up the optimal combinations for all totals from `0` up to the target amount (`grandTotal`). -For each total, we track the fewest coins needed to make that total, reusing previous results to make the solution efficient. - -This approach ensures that we find the minimum number of coins required in a structured, repeatable way, avoiding the need for complex recursive calls or excessive backtracking. - -## Key Features of the Approach - -- **Efficiency**: By building solutions for each increment up to `grandTotal`, this approach minimizes redundant calculations. -- **Flexibility**: Handles cases where exact change is impossible by checking at each step. -- **Scalability**: Works for various coin denominations and totals, though large inputs may impact performance. +```java +import java.util.List; +import java.util.ArrayList; + +class ChangeCalculator { + private final List currencyCoins; + + ChangeCalculator(List currencyCoins) { + this.currencyCoins = currencyCoins; + } + + List computeMostEfficientChange(int grandTotal) { + if (grandTotal < 0) + throw new IllegalArgumentException("Negative totals are not allowed."); + + List> coinsUsed = new ArrayList<>(grandTotal + 1); + coinsUsed.add(new ArrayList()); + + for (int i = 1; i <= grandTotal; i++) { + List bestCombination = null; + for (int coin: currencyCoins) { + if (coin <= i && coinsUsed.get(i - coin) != null) { + List currentCombination = new ArrayList<>(coinsUsed.get(i - coin)); + currentCombination.add(0, coin); + if (bestCombination == null || currentCombination.size() < bestCombination.size()) + bestCombination = currentCombination; + } + } + coinsUsed.add(bestCombination); + } + + if (coinsUsed.get(grandTotal) == null) + throw new IllegalArgumentException("The total " + grandTotal + " cannot be represented in the given currency."); + + return coinsUsed.get(grandTotal); + } +} +``` For a detailed look at the code and logic, see the full explanation in the [Dynamic Programming Approach][approach-dynamic-programming]. From c0a46c607fadaa0f3c44a40f006f29fd57660534 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Tue, 19 Nov 2024 00:25:44 +0530 Subject: [PATCH 22/26] Fixing styling issues for approache for Change --- .../practice/change/.approaches/dynamic-programming/content.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index c5801342c..4c2ce7c1d 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -46,7 +46,7 @@ It minimizes the number of coins needed by breaking down the problem into smalle 1. **Initialize Coins Usage Tracker**: - - If the `grandTotal` is negative, an exception is thrown immediately. + - If the `grandTotal` is negative, an exception is thrown immediately. - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. @@ -61,7 +61,6 @@ It minimizes the number of coins needed by breaking down the problem into smalle - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - If no valid combination exists for `grandTotal`, an exception is thrown. - - **Time Complexity**: The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. - **Space Complexity**: The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. From f76ad5de74e3c694acec962eddc7ada9b5c76143 Mon Sep 17 00:00:00 2001 From: jagdish-15 Date: Tue, 19 Nov 2024 21:38:08 +0530 Subject: [PATCH 23/26] Update exercises/practice/change/.approaches/dynamic-programming/content.md Co-authored-by: Kah Goh --- .../practice/change/.approaches/dynamic-programming/content.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 4c2ce7c1d..29ff4ee8a 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -61,6 +61,6 @@ It minimizes the number of coins needed by breaking down the problem into smalle - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - If no valid combination exists for `grandTotal`, an exception is thrown. -- **Time Complexity**: The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. +The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. - **Space Complexity**: The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. From 5db8204a406f8b4789192fc6e7ff152005e2be9b Mon Sep 17 00:00:00 2001 From: jagdish-15 Date: Tue, 19 Nov 2024 21:38:31 +0530 Subject: [PATCH 24/26] Update exercises/practice/change/.approaches/dynamic-programming/content.md Co-authored-by: Kah Goh --- .../practice/change/.approaches/dynamic-programming/content.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 29ff4ee8a..0a5ccfddc 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -63,4 +63,4 @@ It minimizes the number of coins needed by breaking down the problem into smalle The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. -- **Space Complexity**: The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. +The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. From 505a92de0242c58c68f31817b576e12b82d70241 Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Tue, 19 Nov 2024 22:14:04 +0530 Subject: [PATCH 25/26] Updating content for dynamic approach of change --- .../change/.approaches/dynamic-programming/content.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 0a5ccfddc..7e264778f 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -61,6 +61,8 @@ It minimizes the number of coins needed by breaking down the problem into smalle - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - If no valid combination exists for `grandTotal`, an exception is thrown. -The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. +## Time and Space Complexity + +- The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. -The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. +- The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. From 2dea9103e6f3c62ad3ca687b2efd5fb0d6a0483a Mon Sep 17 00:00:00 2001 From: Jagdish Prajapati Date: Sat, 23 Nov 2024 20:43:20 +0530 Subject: [PATCH 26/26] Updating content.md of dynamic approach of chnage exercise --- .../dynamic-programming/content.md | 26 +++++++++---------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/exercises/practice/change/.approaches/dynamic-programming/content.md b/exercises/practice/change/.approaches/dynamic-programming/content.md index 7e264778f..640c58a47 100644 --- a/exercises/practice/change/.approaches/dynamic-programming/content.md +++ b/exercises/practice/change/.approaches/dynamic-programming/content.md @@ -44,25 +44,25 @@ It minimizes the number of coins needed by breaking down the problem into smalle ## Explanation -1. **Initialize Coins Usage Tracker**: +### Initialize Coins Usage Tracker - - If the `grandTotal` is negative, an exception is thrown immediately. - - We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. - - The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. +- If the `grandTotal` is negative, an exception is thrown immediately. +- We create a list `coinsUsed`, where each index `i` stores the most efficient combination of coins that sum up to the value `i`. +- The list is initialized with an empty list at index `0`, as no coins are needed to achieve a total of zero. -2. **Iterative Dynamic Programming**: +### Iterative Dynamic Programming - - For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. - - For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). - - If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. +- For each value `i` from 1 to `grandTotal`, we explore all available coin denominations to find the best combination that can achieve the total `i`. +- For each coin, we check if it can be part of the solution (i.e., if `coin <= i` and `coinsUsed[i - coin]` is a valid combination). +- If so, we generate a new combination by adding the current coin to the solution for `i - coin`. We then compare the size of this new combination with the existing best combination and keep the one with fewer coins. -3. **Result**: +### Result - - After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. - - If no valid combination exists for `grandTotal`, an exception is thrown. +- After processing all values up to `grandTotal`, the combination at `coinsUsed[grandTotal]` will represent the most efficient solution. +- If no valid combination exists for `grandTotal`, an exception is thrown. ## Time and Space Complexity -- The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. +The time complexity of this approach is **O(n * m)**, where `n` is the `grandTotal` and `m` is the number of available coin denominations. This is because we iterate over all coin denominations for each amount up to `grandTotal`. -- The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`. +The space complexity is **O(n)** due to the list `coinsUsed`, which stores the most efficient coin combination for each total up to `grandTotal`.