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Adding approach change #2859

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This pull request adds an approach for the "Change" exercise in the Java track, based on discussions with @kahgoh during mentoring.

I was unsure how to generate a uuid for the approach, so I've left it empty for now. I would greatly appreciate guidance on how to generate one.


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@jagdish-15
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There's only one issue causing the unsuccessful check i.e. the uuid.

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Sorry it took me a while to get to this one @jagdish-15. Looking at the content, it looks the structure and formatting differ quite a bit from the existing ones, so the comments are mostly trying to become more consistent with the existing ones.

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# 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.
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Looking at how we have written the content for some of other approaches (like boolean chain for leap, bit shifting for grains and if statements for Bob, I'd suggest moving the code up here, just under the heading. The description of the approach then begins just under it.

Also, we generally use one sentence per line in our Markdown content. For example:

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.

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Sure, I will move the code to the top and start the explanation afterward. I have also used new lines for every sentence separately, except when it is in a bullet point.


- **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**:
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Could you please remove this section on alternative approaches? This should just focus on the dynamic programming approach as this will become the dynamic programming page. Alternatively, we could move it to a Which approach to use? section in the introduction.md (similar to the isogram or robot name approaches), but we probably should briefly describe the Greedy Approach if we do (and we could possibly move the point about the efficiency there too).

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I have removed both the "Efficiency" and "Alternative Approach" sections entirely. I agree that it makes more sense to include these points in the introduction file under a section like "Which Approach to Use?" However, I believe this should be done only after we add another approach to compare. For now, with just a single approach, these sections are unnecessary.

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# Introduction to Change Calculator
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This suggestion is to make the heading consistent with our other existing approaches.

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# Introduction to Change Calculator
# Introduction

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Indeed!!


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
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I don't think we need to repeat the exercise here. The other approaches generally have a "General guidance" section that talks about the "key things" to solve the exercise - things that all approaches are likely to have to. For example, in leap, it is as simple as knowing whether a year is divisible by 400, 100 and 4. For grains, its knowing that each square is doubled.

For Change, it could be that not all totals can be reached and somehow figuring out which ones can be reached and how.

Suggested change
## Problem Overview
## General guidance

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Sure, I've changed it.


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
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The other approaches generally have a short listing of all the approaches here, with a link to the specific approach's page just underneath. For example, this would normally be a code listing for your dynamic programming approach followed by a link to the page generated from dynamic-programming/content.md (the URL would be https://exercism.org/tracks/java/exercises/change/approaches/dynamic-programming).

Suggested change
## Approach Overview
## Approach: Dynamic programming

- **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).
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In regards to the approach file link, there is a preference to use reference linking to make maintenance easier.

I also suggest changing the link text to "Dynamic Programming approach" as students will see a web page based on your content.md (and not the content.md "file").

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I agree, and I've made the changes accordingly.

"jagdish-15"
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"contributors": [
"kagoh"
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Suggested change
"kagoh"
"kahgoh"

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I am sorry for the oversight.

@jagdish-15
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Thank you so much, @kahgoh, for all these suggestions. Please let me know if there's anything else I should improve, change, or remove.

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Thanks @jagdish-15, that looks better!

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.
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I find this sentence redundant because all the approaches will need to do this to successfully solve the exercise.

Suggested change
This approach ensures that we find the most efficient way to make change and handles edge cases where no solution exists.

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@jagdish-15 jagdish-15 Nov 18, 2024

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Yep, I can't deny! I will remove this sentence altogether.

- 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
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I think the core of this document should be focusing on the specifics of how the approach works. From that point of view, I think the Key Points heading/section is confusing as Time Complexity and Space Complexity isn't a core part of the implementation, but I do think we can certainly mention them. Perhaps we could remove the heading.

Suggested change
## Key Points

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Sure, I've removed this heading.


- **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**:
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Suggest moving the points from the edge cases to be with your explanation as they are part of the implementation details. Idalso suggest separating them to follow the order of the code to make it easier to follow (i.e putting the point about checking if grandTotal is negative to the top and one about if no exact total to the bottom of the explanation).

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I've moved the first point to the start of the explanation as you suggested and removed the second point since it's already covered at the end, just before the 'Time Complexity' section.

## 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.
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I think we could remove this conclusion section. Similar to my earlier comment, I think the first sentence is redundant as all approaches have find the optimal solution to be valid. The second sentence would probably make more sense in a Which approach to use? section if we have more approaches.

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I agree! I've removed the conclusion section altogether.

@jagdish-15
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Sorry for all the back-and-forth! I've submitted it now—please take a look and let me know your thoughts.

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No worries, thanks for preserevering! I think we're almost there!

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.

## Explanation
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With the other heading removed, this is the only one left. I wonder if it is worth replacing this heading with a bit of text to lead into this explanation, (e.g. "The steps taken by this approach are described below:" ) . Alternatively, we could try using a different heading (for example. "Time and Space Complexity") for the time and space complexity points.

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@jagdish-15 jagdish-15 Nov 19, 2024

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I've made some changes, please take a look now.

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