Given a value N, if we want to make change for N cents, and we have infinite supply of each of S = { S1, S2, .. , Sm} valued coins, how many ways can we make the change? The order of coins doesn’t matter.
For example, for N = 4 and S = {1,2,3}, there are four solutions: {1,1,1,1},{1,1,2},{2,2},{1,3}. So output should be 4. For N = 10 and S = {2, 5, 3, 6}, there are five solutions: {2,2,2,2,2}, {2,2,3,3}, {2,2,6}, {2,3,5} and {5,5}. So the output should be 5.
- Optimal Substructure
To count total number solutions, we can divide all set solutions in two sets.- Solutions that do not contain mth coin (or Sm).
- Solutions that contain at least one Sm.
Let count(S[], m, n) be the function to count the number of solutions, then it can be written as sum of count(S[], m-1, n) and count(S[], m, n-Sm).
Therefore, the problem has optimal substructure property as the problem can be solved using solutions to subproblems.
[ad type=”banner”]- Overlapping Subproblems
Following is a simple recursive implementation of the Coin Change problem. The implementation simply follows the recursive structure mentioned above.
It should be noted that the above function computes the same subproblems again and again. See the following recursion tree for S = {1, 2, 3} and n = 5.
The function C({1}, 3) is called two times. If we draw the complete tree, then we can see that there are many subproblems being called more than once.
C() --> count() C({1,2,3}, 5) / \ / \ C({1,2,3}, 2) C({1,2}, 5) / \ / \ / \ / \ C({1,2,3}, -1) C({1,2}, 2) C({1,2}, 3) C({1}, 5) / \ / \ / \ / \ / \ / \ C({1,2},0) C({1},2) C({1,2},1) C({1},3) C({1}, 4) C({}, 5) / \ / \ / \ / \ / \ / \ / \ / \ . . . . . . C({1}, 3) C({}, 4) / \ / \ . .
Since same suproblems are called again, this problem has Overlapping Subprolems property. So the Coin Change problem has both properties of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by constructing a temporary array table[][] in bottom up manner.
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Output:
4
Time Complexity: O(mn)
Following is a simplified version of method 2. The auxiliary space required here is O(n) only.
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