Coin Change - Dynamic Programming - How read all solutions from the DP table

Question

I have seen different solutions to the same problem, but none of them seem to use the approach I used. So here I'm trying to solve the classical coin-change problem in a bottom up dynamic programming approach using a DP table.

    int[][] dp = new int[nums.length+1][target+1];
    for (int i = 0; i <= nums.length; ++i)
        dp[i][0] = 1;

    for (int i = 1; i < dp.length; ++i) {
        for (int j = 1; j < dp[i].length; ++j) {
            if (nums[i-1] <= j)
                dp[i][j] = dp[i-1][j] + dp[i][j-nums[i-1]];
            else
                dp[i][j] = dp[i-1][j];
        }
    }

The above code generates the table. For fun if I have: {2,3,5} coins, and want to exchange 8, the table would look like:

1 0 0 0 0 0 0 0 0
1 0 1 0 1 0 1 0 1
1 0 1 1 1 1 2 1 2
1 0 1 1 1 2 2 2 3

For the above the following method seem to be working well:

current <- 4
while (current > 0) do
    i <- current
    j <- 8
    while (j > 0) do
        if (dp[i][j] != dp[i-1][j]) then
            nums[i-1] is part of the current solution
            j <- j-1
        else
            i <- i-1
        end
    end
    current <- current-1
end

Walking through for the above example, we get the following solutions:

1) [5,3]
2) [3,3,2]
3) [2,2,2,2]

Which is great! At least I thought, until I tried: {1,2} with a T=4. For this the table looks like:

1 0 0 0 0
1 1 1 1 1
1 1 2 2 3

With the above algorithm to print all solutions, I only get:

[2,2]
[1,1,1,1]

Which means I won't recover [2,1,1]. So this question is not about the generic how to print the solutions for different approaches to this problem, but how can I read the above DP table to find all solutions.

Answer 1

Well I have one solution but sure... I can see why the other similar answers are using different approaches for this problem. So the algorithm to generate all the possible answers from the above table looks like:

private List<List<Integer>> readSolutions(int[] nums, int[][] dp, int currentRow, int currentColumn, List<Integer> partialSolution) {
    if (currentRow == 0) {
        return new ArrayList<>();
    }

    if (currentColumn == 0) {
        return new ArrayList<List<Integer>>(Collections.singleton(partialSolution));
    }

    List<List<Integer>> solutions = new ArrayList<>();
    if (dp[currentRow][currentColumn] != dp[currentRow-1][currentColumn]) {
        List<Integer> newSolution = new ArrayList<>(partialSolution);
        newSolution.add(nums[currentRow-1]);
        solutions.addAll(readSolutions(nums, dp, currentRow, currentColumn-nums[currentRow-1], newSolution));
        solutions.addAll(readSolutions(nums, dp, currentRow-1, currentColumn, partialSolution));

        return solutions;
    }

    return readSolutions(nums, dp, currentRow-1, currentColumn, partialSolution);
}

The logic on the other hand is simple. Take the above table for example:

  0 1 2 3 4
0 1 0 0 0 0
1 1 1 1 1 1
2 1 1 2 2 3

To get a single solution we start from the bottom right corner. If the value does match with the value directly above us, we move up. If it doesn't we move left by the amount corresponding to the row we're in. To generate all answers on on the other hand from the above table...

• we're at some position (i,j)
• if the value at (i-1,j) is the same as (i,j) we make a recursive call to (i-1,j)
• if the values do not match, we have 2 choices...
• we can use the number corresponding to the current row, and recurse into (i,j-n)
• we can skip the number and check if we can create (i,j) instead by using a recursive call to (i-1,j) instead.
• if we reach the first row, we return an empty list
• if we reach the first column, we return what we have already found, which will have the sum of target.

Looks awful right, but works as expected.