Coin change

Coin Change is the problem of finding the number of ways of making changes for a particular amount of cents, $$n$$, using a given set of denominations $$d_1 \ldots d_m$$. It is a general case of Integer Partition, and can be solved with dynamic programming.

Overview
The problem is typically asked as: If we want to make change for $$n$$ cents, and we have infinite supply of each of $$S = \{ S_1, S_2, \ldots, S_m \}$$ valued coins, how many ways can we make the change? (For simplicity's sake, the order does not matter.)

It is more precisely defined as:

Given an integer $$N$$ and a set of integers $$S = \{ S_1, S_2, \ldots, S_m \}$$, how many ways can one express $$n$$ as a linear combination of $$S = \{ S_1, S_2, \ldots, S_m \}$$ with non-negative coefficients?

Mathematically, how many solutions are there to $$N = \sum_{k = 1 \ldots m}{ x_k S_k }$$ where $$x_k \geq 0, k \in \{ 1 \ldots m \} $$

For example, for $$n = 4, S = \{1, 2, 3\}$$, there are four solutions: $$\{1, 1, 1, 1\}, \{1, 1, 2\} , \{2, 2\} , \{1, 3\}$$.

Other common variations on the problem include decision-based question, such as:

Is there a solution for $$N = \sum_{k = 1 \ldots m}{ x_k S_k }$$ where $$x_k \geq 0, k \in \{ 1 \ldots m \} $$ (Is there a solution for integer $$N$$ and a set of integers $$S = \{ S_1, S_2, \ldots, S_m \}$$?)

Is there a solution for $$N = \sum_{k = 1 \ldots m}{ x_k S_k }$$ where $$x_k \geq 0, k \in \{ 1 \ldots m \}, \sum_{k = 1 \ldots m}{x_k} \leq T $$ (Is there a solution for integer $$N$$ and a set of integers $$S = \{ S_1, S_2, \ldots, S_m \}$$ such that $$\sum_{k = 1 \ldots m}{x_k} \leq T $$ - using at most $$T$$ coins)

Recursive Formulation
We are trying to count the number of distinct sets.

Since order does not matter, we will impose that our solutions (sets) are all sorted in non-decreasing order (Thus, we are looking at sorted-set solutions: collections).

For a particular $$N$$ and $$S = \{ S_1, S_2, \ldots, S_m \}$$ (now with the restriction that $$S_1 < S_2 < \ldots < S_m$$, our solutions can be constructed in non-decreasing order), the set of solutions for this problem, $$C( n, m )$$, can be partitioned into two sets:


 * There are those sets that does not contain any $$S_m$$,
 * Those sets that contain at least 1 $$S_m$$,

If a solution does not contain $$S_m$$, then we can solve the subproblem of $$N$$ with $$S = \{ S_1, S_2, \ldots, S_{m-1} \}$$, or the solutions of $$C( n, m - 1 )$$.

If a solution does contain $$S_m$$, then we are using at least one $$S_m$$, thus we are now solving the subproblem of $$N - S_m$$, $$S = \{ S_1, S_2, \ldots, S_m \}$$. This is $$C( n - S_m, m )$$.

Thus, we can formulate the following:

$$C( n, m ) = C( n, m - 1 ) + C( n - S_m, m )$$

with the base cases:


 * $$C( n, m ) = 1, n = 0$$
 * $$C( n, m ) = 0, n < 0$$
 * $$C( n, m ) = 0, n \geq 1, m \leq 0$$

Pseudocode
func count( n, m ) if ( n == 0 ) return 1 if ( n < 0 ) return 0 if ( m <= 0 and n >= 1 ) return 0

return count( n, m - 1 ) + count( n - S_m, m )

Dynamic Programming
Note that the recursion satisfies the weak ordering $$R(n,m) < R(x,y) \iff n \leq x, m \leq y, (n,m) \ne (x,y)$$. As a result satisfies The optimal-substructure property of dynamic programming.

The result can be computed in $$O(nm)$$ time - the above pseudocode can easily be modified to contain memoization. It can be also rewritten as:

func count( n, m ) initialize table with base cases

for i from 0 to n   for j from 0 to m      table[ i, j ] = table[ i - S_j, j ] + table[ i, j - 1 ]

return table[ n, m ]

Special Cases
There are cases where the greedy algorithm is optimal - for example, the US coin system.