# Change-Making Problem (Change-Making Problem)

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## Description

Given an unlimited amount of coins of denominations $c_1, \ldots, c_n$, and a desired sum $S$, find the minimum number of coins necessary to make $S$.

## Parameters

$n$: number of coin denominations

$S$: sum to be made

## Table of Algorithms

Name | Year | Time | Space | Approximation Factor | Model | Reference |
---|---|---|---|---|---|---|

Brute Force | 1940 | $O(S^n)$ | $O(n)$ | Exact | Deterministic | |

Dynamic Programming | 1953 | $O(Sn)$ | $O(Sn)$ | Exact | Deterministic | Time |

Probabilistic Convolution Tree | 2014 | $O(n \log n)$ | $O(n log n)$ | Deterministic | Time & Space |