Problem¶

You are given an integer array coins representing coins of different denominations and an integer amount representing a total amount of money.

Return the number of combinations that make up that amount. If that amount of money cannot be made up by any combination of the coins, return 0.

You may assume that you have an infinite number of each kind of coin.

The answer is guaranteed to fit into a signed 32-bit integer.

Example 1:

Input: amount = 5, coins = [1,2,5]
Output: 4
Explanation: there are four ways to make up the amount:
5=5
5=2+2+1
5=2+1+1+1
5=1+1+1+1+1

Example 2:

Input: amount = 3, coins = [2]
Output: 0
Explanation: the amount of 3 cannot be made up just with coins of 2.

Example 3:

Input: amount = 10, coins = [10]
Output: 1

Constraints:

• 1 <= coins.length <= 300
• 1 <= coins[i] <= 5000
• All the values of coins are unique.
• 0 <= amount <= 5000

Solve¶

Dynamic programming¶

O(n ** 2)

The ideal start with a update function:

stage_a + coin_value -> stage_b = stage_<a+ coin_value>

Which lead to:

total_way_of_stage_b = total_way_of_stage_a (with coin_value) + <other_way>

With every coins, we loop and adding up the total way to reach each amount.

class Solution {
    public int change(int amount, int[] coins) {
        int[] count = new int[amount + 1];
        count[0] = 1;
        for (int i : coins) {
            for (int c = i; c <= amount; c++) {
                count[c] += count[c-i];
            }
        }
        return count[amount];
    }
}

The for loop order have an impact on how the Calculation run:

If you instead doing this:

for (int c = 0; c <= amount; c++) {
    for (int i : coins) {
        if (c-i > 0) 
            count[c] += count[c-i];
    }
}

• We have total of way adding coin (that count with all possible different position 1 2 1 != 1 1 2) to reach that amount

Comparing¶

I also try some more language to help me familiar with other language array too, it quite surprise to see rust and java beat c in runtime.

java:¶

We have int array automate initiation as 0

c:¶

It keep yelling me about how I can’t using int count[amount+1] = {0}; This just mean I have to use a hard code value to use this feature of the language.

#define MAX_AMOUNT 5001
int count[MAX_AMOUNT] = {0};

To initiation use a variable size, we instead use memset.
Also, C not have for each built-in the language syntax, so I use a normal iterator loop

int change(int amount, int* coins, int coinsSize){
    int count[amount + 1];
    memset(count, 0, sizeof(count));  
    count[0] = 1;
    for (int i = 0; i < coinsSize; i++) {
        for (int c = coins[i]; c <= amount; c++) {
            count[c] += count[c-coins[i]];
        }
    }
    return count[amount];
}

For an un-know reason, but it c quite slow some time when I try to submit result.

python:¶

I can’t seem to know better way to initiation a array in python. Also, I use python to make my first implementation, planning process.

class Solution:
    def change(self, amount: int, coins: List[int]) -> int:
        count = [0] * (amount + 1)
        count[0] = 1
        for i in coins:
            for c in range(amount+1):
                if c-i < 0:
                    continue
                count[c] += count[c-i]

        return count[amount]

rust:¶

While rust have some more similar way to create array like c, it require the variable to be constant example.

const MAX_AMOUNT: usize = 5000;
let mut count = [0; MAX_AMOUNT];

• To create and allocation array, using dynamic size variable, we have to use vec!

impl Solution {
    pub fn change(amount: i32, coins: Vec<i32>) -> i32 {
        let mut count = vec![0; (amount + 1) as usize];
        count[0] = 1;

        for coin in coins {
            for c in coin..=amount {
                count[c as usize] += count[(c - coin) as usize];
            }
        }

        count[amount as usize]
    }
}

Time and memory for comparation:¶