Money Change

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Problem Description

Given a value N, if we want to make change for N cents, and we have an 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.

Bounds Chart

Change-making problemBoundsChart.png

Step Chart

Change-making problemStepChart.png

Improvement Table

Complexity Classes Algorithm Paper Links Lower Bounds Paper Links
Exp/Factorial [Brute force (1940)]
Polynomial > 3 [Dynamic Programming (1953)]

probabilistic convolution tree (2014)

Cubic
Quadratic
nlogn
Linear
logn