Optimization of Algorithm [closed]

问题

Here is the link to the problem.

The problem asks the number of solutions to the Diophantine equation of the form 1/x + 1/y = 1/z (where z = n!). Rearranging the given equation clearly tells that the answer is the number of factors of z².

So the problem boils down to finding the number of factors of n!² (n factorial squared).

My algorithm is as follows

Make a Boolean look up table for all primes <= n using Sieve of Eratosthenes algorithm.
Iterate over all primes P <= n and find its exponent in n!. I did this using step function formula. Let the exponent be K, then the exponent of P in n!² will be 2K.
Using step 2 calculate number of factors with the standard formula.

For the worst case input of n = 10⁶, my c code gives answer in 0.056s. I guess the complexity is no way greater than O(n lg n). But when I submitted the code on the site, I could pass only 3/15 test cases with the verdict on the rest as time limit exceeded.

I need some hints for optimizing this algorithm.

Code so far:

#include <stdio.h>
#include <string.h>

#define ULL unsigned long long int
#define MAXN 1000010
#define MOD 1000007

int isPrime[MAXN];

ULL solve(int N) {
    int i, j, f;
    ULL res = 1;
    memset(isPrime, 1, MAXN * sizeof(int));
    isPrime[0] = isPrime[1] = 0;
    for (i = 2; i * i <= N; ++i)
        if (isPrime[i])
            for (j = i * i; j <= N; j += i)
                isPrime[j] = 0;
    for (i = 2; i <= N; ++i) {
        if (isPrime[i]) {
            for (j = i, f = 0; j <= N; j *= i)
                f += N / j;
            f = ((f << 1) + 1) % MOD;
            res = (res * f) % MOD;
        }
    }
    return res % MOD;
}

int main() {
    int N;
    scanf("%d", &N);
    printf("%llu\n", solve(N));
    return 0;
}

回答1:

You have an int overflow here:

for (j = i, f = 0; j <= N; j *= i)

If 46340 < i < 65536 and for many larger i, in the second iteration j will be negative if overflow wraps around (it is undefined behaviour, so anything could happen).

Replace it with e.g.

for(j = N / i, f = 0; j > 0; j /= i) {
    f += j;
}

It is, however, unlikely that the extra iterations due to the overflow would cause a "time limit exceeded", that will likely only cause wrong results.

So the generic advice is

Sieve only odd numbers, perhaps also eliminate multiples of 3 from the sieve. Eliminating the odd numbers from the sieve roughly halves the time needed to sieve.
Don't use an entire int for the flags, use bits or at least chars. That gives better cache locality and a further speedup.

来源：https://stackoverflow.com/questions/10179837/optimization-of-algorithm

标签

c++

algorithm

primes

factors