No. 26 - Minimal Number of Coins for Change

Problem: Please implement a function which gets the minimal number of coins, whose value is v₁, v₂, …, v_n, to make change for an amount of money with value t. Any coin with value v_i may duplicate for any times to make change.

For example, the minimal number of coins to make change for 15 out of a set of coins with value 1, 3, 9, 10 is 3. We can choose two coins with value 3 and a coin with value 9. The number of coins for other choices should be greater than 3.

Analysis: Firstly let us define a function f(t) which is the minimal number of coins to make change for total value t. If there are n different coins, we have n choices to make change for value t: we can add a coin with value v₁ into a set of coins whose total value is t-v₁. The minimal number of coins to get value t-v₁ is f(t-v₁). Similarly, we can add a coin with value v₂ into a set of coins whose total value is t-v₂. The minimal number of coins to get value t-v₂ is f(t-v₂)…

Therefore, we divide a problem to calculate f(t) into n sub-problems: f(t-v₁), f(t-v₂), …, f(t-v_n). We can get a formal equation for f(t) as the following accordingly:

 

This equation can be implemented with recursion easily. However, the recursive solution will cause serious performance issues since there are overlaps when we divide this problem into n sub-problems. A better solution is to utilize iteration, and store the result of sub-problems into a table (as the Table 1).

In the Table 1, each column except the first one is to denote the number of coins to make change for a specific value. We can calculate the numbers in the Table 1 from left to right, to simulate the iterative process to get the result of f(15).

For instance, there are two numbers 4 and 2 under the column title “6”. We have two alternatives to make change for 6: the first one is to add a coin with value 1 to a set of coins whose total value is 5. Since the minimal number of coins to get value 5 is 3 (highlighted number under the column tile “5”), the number in the first cell under column title “6” is 4 (4=3+1). The second choice is to add a coin with value 3 to a set of coins whose total value is 3. Since the minimal number of coins to get value 3 is 1 (highlighted number under the column tile “3”), the number in the second cell under column title “6” is 2 (2=1+1). We highlight the number 2 in the column under tile 6 because 2 is less than 4.

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	0	1	2	3	2	3	4	3	4	5	2	2	3	3	4	5
3	0	-	-	1	2	3	2	3	4	3	4	5	2	2	3	3
9	0	-	-	-	-	-	-	-	-	1	2	3	2	3	4	3
10	0	-	-	-	-	-	-	-	-	-	1	2	3	2	3	4

Table 1: The iterative process to calculate the minimal number of coins to make changes for 15.

Even though we have a 2-D matrix to show the iterative process, it only requires a 1-D array for coding, because it is only necessary to store the minimal number of coins to make change for each total value. The sample code is shown below:

int GetMinCount(int total, int* coins, int length)

{

    int* counts = new int[total + 1];

    counts[0] = 0;

   

    const int MAX = 0x7FFFFFFF;

    for(int i = 1; i <= total; ++ i)

    {

        int count = MAX;

        for(int j = 0; j < length; ++ j)

        {

            if(i - coins[j] >= 0 && count > counts[i - coins[j]])

                count = counts[i - coins[j]];

        }

        if(count < MAX)

            counts[i] = count + 1;

        else

            counts[i] = MAX;

    }

    int minCount = counts[total];

    delete[] counts;

    return minCount;

}

The discussion about this problem is included in my book <Coding Interviews: Questions, Analysis & Solutions>, with some revisions. You may find the details of this book on Amazon.com, or Apress.

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