CSG713 Advanced Algorithms Dynamic Programming Example
Fall 2004 September 27, 2004
Dynamic Programming Solution to the
Coin Changing Problem
(1) Characterize the Structure of an Optimal Solution. The Coin Changing problem exhibits opti-
mal substructure in the following manner. Consider any optimal solution to making change for ncents using
coins of denominations d1, d2, . . . , dk. Now consider breaking that solution into two different pieces along
any coin boundary. Suppose that the “left-half ” of the solution amounts to bcents and the “right-half” of
the solution amounts to n−bcents, as shown below.
b
z }| {
d1d3d1d2
n−b
z }| {
d4d5d2
Claim 1 The left-half of the solution must be an optimal way to make change for bcents using coins of
denominations d1, d2, . . . , dk, and the right-half of the solution must be an optimal way to make change for
n−bcents using coins of denominations d1, d2, . . . , dk.
Proof: By contradiction, suppose that there was a better solution to making change for bcents 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 ncents 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. 2
Thus, the optimal solution to the coin changing problem is composed of optimal solutions to smaller
subproblems.
(2) Recursively Define the Value of the Optimal Solution. First, we define in English the quantity
we shall later define recursively. Let C[p] b e the minimum number of coins of denominations d1, d2, . . . , dk
needed to make change for pcents. In the optimal solution to making change for pcents, there must exist
some first coin di, where di≤p. Furthermore, the remaining coins in the optimal solution must themselves
be the optimal solution to making change for p−dicents, since coin changing exhibits optimal substructure
as proven above. Thus, if diis the first coin in the optimal solution to making change for pcents, then
C[p] = 1 + C[p−di]; i.e., one dicoin plus C[p−di] coins to optimally make change for p−dicents. We don’t
know which coin diis the first coin in the optimal solution to making change for pcents; however, we may
check all ksuch possibilities (subject to the constraint that di≤p), and the value of the optimal solution
must correspond to the minimum value of 1 + C[p−di], by definition. Furthermore, when making change
for 0 cents, the value of the optimal solution is clearly 0 coins. We thus have the following recurrence.
Claim 2 C[p] = 0if p= 0
mini:di≤p{1 + C[p−di]}if p > 0
Proof: The correctness of this recursive definition is embodied in the paragraph which precedes it. 2
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