动态规划应用—硬币组合问题

/*
* Filename :coins.cpp
* Description: solve coin combinations using dynamic programing
* Complier: g++
* Author: python27
*/
#include <iostream>
#include <string>
#include <cmath>
#include <vector>

using namespace std;

/****************************************************************
* coin Combinations: using dynamic programming
*
* Basic idea:
* dp[i][j] = sum(dp[i-1][j-k*coins[i-1]]) for k = 1,2,..., j/coins[i-1]
* dp[0][j] = 1 for j = 0, 1, 2, ..., sum
*
* Input:
* coins[] - array store all values of the coins
* coinKinds - how many kinds of coins there are
* sum - the number you want to construct using coins
*
* Output:
* the number of combinations using coins construct sum
*
* Usage:
* c[3] = {1, 2, 5};
* int result = coinCombinations(c, 3, 10);
*
****************************************************************/
int coinCombinations(int coins[], int coinKinds, int sum)
{
// 2-D array using vector: is equal to: dp[coinKinds+1][sum+1] = {0};
vector<vector<int> > dp(coinKinds + 1);
for (int i = 0; i <= coinKinds; ++i)
{
dp[i].resize(sum + 1);
}
for (int i = 0; i <= coinKinds; ++i)
{
for (int j = 0; j <= sum; ++j)
{
dp[i][j] = 0;
}
}

//init: dp[i][0] = 1; i = 0, 1, 2 ..., coinKinds
//Notice: dp[0][0] must be 1, althongh it make no sense that
//using 0 kinds of coins construct 0 has one way. but it the foundation
//of iteration. without it everything based on it goes wrong
for (int i = 0; i <= coinKinds; ++i)
{
dp[i][0] = 1;
}

// iteration: dp[i][j] = sum(dp[i-1][j - k*coins[i-1]])
// k = 0, 1, 2, ... , j / coins[i-1]
for (int i = 1; i <= coinKinds; ++i)
{
for (int j = 1; j <= sum; ++j)
{
dp[i][j] = 0;
for (int k = 0; k <= j / coins[i-1]; ++k)
{
dp[i][j] += dp[i-1][j - k * coins[i-1]];
}
}
}

return dp[coinKinds][sum];
}

int main()
{
int coins[8] = {1, 2, 5, 10, 20, 50, 100, 200};
int sum = 200;
//int coins[2] = {2, 3};
//int sum = 11;
int result = coinCombinations(coins, 8, sum);
cout << "combinations are: " << endl;
cout << result << endl;

system("pause");
return 0;
}