Coin Change (Dynamic Programming)

May 5th, 2015
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Characterize the Structure of an Optimal Solution. The Coin Changing problem exhibits optimal substructure in the following manner. Consider any optimal solution to making change for n cents using coins of denominations

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Coin Change (Dynamic Programming)Dynamic Programming Solution to theCoin Changing Problem(1) Characterize the Structure of an Optimal Solution. The Coin Changing problem exhibits optimal substructure in the following manner. Consider any optimal solution to making change for n cents using coins of denominations d1 , d2 , . . . , dk . Now consider breaking that solution into two dierent pieces along any coin boundary. Suppose that the left-half of the solution amounts to b cents and the right-half of the solution amounts to n b cents, as shown below.nbbd1d3d1d2d4d5d2Claim 1 The left-half of the solution must be an optimal way to make change for b cents using coins of denominations d1 , d2 , . . . , dk , and the right-half of the solution must be an optimal way to make change for n b cents using coins of denominations d1 , d2 , . . . , dk . Proof: By contradiction, suppose that there was a better solution to making change for b cents than the left-half of the optimal solution shown. Then the left-half of the optimal solution could be replaced with this better solution, yielding a valid solution to making change for n cents with fewer coins than the solution being considered. But this contradicts the supposed optimality of the given solution, . An identical argument applies to the right-half of the solution.2Thus, the optimal solution to the coin changing problem is composed of optimal solutions to smaller

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