Google Interview - Integers as Sum of Squares

给一个整数，表示成n个整数的平方和，n最小，求n。
10017 = 100^2 + 4^2 + 1 --> n = 3

 

这道题的快速解法要牵扯到数学定理了。

 

Lagrange's four-square theorem

Every natural number can be represented as the sum of four integer squares.

 

Legendre's three-square theorem

Let m, n be integers equal to or greater than 0. If N = 4^m(8n+7), then N is NOT expressible as the sum of three squares of integers.
Example:
1584 = 4²(8(12)+3) thus not in the form 4^m(8n+7), so it can be expressed as the sum of 3 squares.
960 = 4³(8(1)+7). Because it is of the form 4^m(8n+7), it cannot be expressed as the sum of 3 squares.

Interesting Note:
For the equation N = 4^m(8n+7), the (8n+7) part is equivalent to 7 mod 8. This stems from the fact that every square of a number is either 0,1, or 4 mod 8. Since this holds true for the squares of numbers, no integer congruent to 7 mod 8 can be represented as the sum of three squares. Therefore, we must have some numbers that need to be represented by the sum of four squares.

 

时间复杂度为O(sqrt(n))。

def solve(n):
    l = [];
    s1 = {};
    for i in xrange(1, n+1):
        if i * i > n:
            break;
        l.append(i);
        s1[i * i] = 1;
    if s1.has_key(n):
        return 1;
    for i in l:
        if s1.has_key(n - i * i):
            return 2;
    while(n % 4 ==0):
        n /= 4;
    if n % 8 != 7:
        return 3;
    return 4;

 

int numOfSquares(int n) {
	if(n < 0) return 0;
	if(n <= 1) return 1;
	vector<int> f(n+1, 4);
	f[0] = f[1] = 1;
	for(int i=2; i<=n; i++) {
		int rt = (int)sqrt(i);
		if(rt*rt == i) {
			f[i] = 1;
			continue;
		}
		for(int j=1; j<=rt; j++) {
			f[i] = min(f[i], f[i-j*j]+1);
			if(f[i] == 2) break; // optional
		}
	}
	return f[n];
}
 
int numOfSquares2(int n) {
	if(n < 0) return 0;
	unordered_set<int> set;
	int rt = (int)sqrt(n);
	if(rt*rt == n) return 1;
	for(int i=1; i<=rt; i++) {
		set.insert(i*i);
	}
	for(int i=1; i<=rt; i++) {
		if(set.count(n-i*i)) return 2;
	}
	while(n % 4 == 0) n /= 4;
	if(n % 8 != 7) return 3;
	return 4;
}

 

Reference:

http://www.1point3acres.com/bbs/thread-17806-1-1.html

http://math453spring2009.wikidot.com/lecture-32:integers-as-sums-of-squares