计算子序列和是定值的子序列个数

题目如下：

Counting Subsequences

Time Limit: 5000 MSMemory Limit: 65536 K

Description

"47 is the quintessential random number," states the 47 society. And there might be a grain of truth in that. For example, the first ten digits of the Euler's constant are: 2 7 1 8 2 8 1 8 2 8 And what's their sum? Of course, it is 47. You are given a sequence S of integers we saw somewhere in the nature. Your task will be to compute how strongly does this sequence support the above claims. We will call a continuous subsequence of S interesting if the sum of its terms is equal to 47. E.g., consider the sequence S = (24, 17, 23, 24, 5, 47). Here we have two interesting continuous subsequences: the sequence (23, 24) and the sequence (47). Given a sequence S, find the count of its interesting subsequences.

Input

The first line of the input file contains an integer T(T <= 10) specifying the number of test cases. Each test case is preceded by a blank line. The first line of each test case contains the length of a sequence N(N <= 500000). The second line contains N space-separated integers – the elements of the sequence. Sum of any continuous subsequences will fit in 32 bit signed integers.

Output

For each test case output a single line containing a single integer – the count of interesting subsequences of the given sentence.

Sample Input

2   13 2 7 1 8 2 8 1 8 2 8 4 5 9   7 2 47 10047 47 1047 47 47

Sample Output

3 4

这道题的意思就是给你一个整形的序列，让你计算满足和是47的子序列的个数。序列中的值可能有负数。

所以最直观的方法就是计算所有子序列的和，然后判断是否与47相等。算法的时间复杂度为O(N³)，代码如下：

#include<iostream>
#include<vector>
#include<map>
#define N 47

using namespace std;

template<class type>
int countSubseq(const vector<type> &data)
{
	int count = 0;
	for(size_t i = 0; i < data.size(); ++i)
	{
		int sum = 0;
		for(int j = 0; j <= i; ++j)
		{
			int sum = 0;
			for(int k = j; k <= i; ++k)
			{
				sum += data[k];
			}
			if(sum == N)
			{
				++count;
			}
		}
	}
	return count;
}

int main()
{
	int nCase;
	cin >> nCase;
	for(int i = 0; i < nCase; ++i)
	{
		int n, d;
		cin >> n;
		vector<int> data(n, 0);
		for(int j = 0; j < n; ++j)
		{
			cin >> d;
			data[j] = d;
		}
		cout << countSubseq(data) << endl;
	}
	return 0;
}

通过分析可以看出程序存在许多重复计算，上一个sum可以进行一次加法运算得到下一个sum。经过再次优化使其时间复杂度变为O(N²)，代码如下：

template<class type>
int countSubseq(const vector<type> &data)
{
	int count = 0;
	for(size_t i = 0; i < data.size(); ++i)
	{
		int sum = 0;
		for(int j = i; j < data.size(); ++j)
		{
			
			sum += data[j];
			if(sum == N)
			{
				++count;
			}
		}
	}
	return count;
}

但是程序依然超时，所以要设计出复杂度更低的算法才行。

设SUM[i,j] = A[i] + A[i+1] + ... + A[j] (0 <= i <= j < N)，所以SUM[i, j] = SUM[0, j] - SUM[,0, i-1]，也就是说只要求出所有的SUM[0, K] (K∈[0,N))就能计算出任意SUM[i, j].那么我们每次计算K的一种取值得到的SUM时，查找之前计算的SUM是否有与当前SUM差是47的K的个数，可以使用map来降低查找的复杂度，这样时间复杂度降为O(N)，代码如下：

template<class type>
int countSubseq(const vector<type> &data)
{
	int count = 0, sum = 0;
	map<int, int> sumToSeqCnt;//存储和是sum的序列K的个数
	sumToSeqCnt[0] = 1;
	for(size_t i = 0; i < data.size(); ++i)
	{
		sum += data[i];
		sumToSeqCnt[sum]++;//计算和是sum的序列K的个数
		count += sumToSeqCnt[sum - N];
	}
	return count;
}