Problem description:

Given a positive integer n, find the least number of perfect square numbers (for example, 1, 4, 9, 16, …) which sum to n.

Example 1:

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Input: n = 12
Output: 3 
Explanation: 12 = 4 + 4 + 4.

Example 2:

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Input: n = 13
Output: 2
Explanation: 13 = 4 + 9.

Solution:

In this question, we can treat every number as combination of two numbers. In the following graph, we can see that we can use a number i and a square number j*j to get the final number i+j*j.

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class Solution {
public:
    int numSquares(int n) {
        if(n <=0) return 0;
        vector<int> dp(n+1, INT_MAX);
        
        dp[0] = 0;
        for(int i= 0; i*i<=n; ++i)  //pre-process, setup all the square number 
            dp[i*i] = 1;
        
        for(int i= 1; i<= n; ++i){      //pick first number i
            for(int j=1; i+j*j<=n; ++j){    //pick another number j*j  
                dp[i+j*j] = min(dp[i]+1, dp[i+j*j]);
            }
        }
        return dp[n];
    }
};

time complexity: $O(n)$
space complexity: $O(n)$
reference:
https://goo.gl/4R5avB