Rare
0/8

# Meet In The Middle

Author: Chongtian Ma

Problems involving dividing the search space into two.

## Meet In The Middle

Focus Problem – try your best to solve this problem before continuing!

Resources
CPH
Errichto

### Pro Tip

Meet in the Middle technique can take advantage of the smaller constraint and calculate a naive solution in two halves. Therefore the constraints tend to be doubled.

### Naive Solution

Loop through all subsets in the array and if the sum is equal to $x$, then increase our answer. Worst case this does about $2^{40}$ operations, which is way too slow.

### Meet in the Middle Solution

We can divide the given array into two separate arrays. Let's say that the $\texttt{left}$ array runs from indexes $0$ to $\frac{n}{2}-1$, and the $\texttt{right}$ array runs from indexes $\frac{n}{2}$ to $n-1$. Both arrays will have at most $20$ elements, so we can loop through all subsets of these two arrays in at most $2^{21}$ operations, which is perfectly fine.

Now that we've got the subset sums of these two separate arrays, we need to recombine them to search for our answer. For every $\texttt{sum}$ in the $\texttt{left}$, we can simply check how many elements of $x - \texttt{sum}$ there are in $\texttt{right}$. This can be done using simple binary search.

Time Complexity: $O(N\cdot 2^{N/2})$

C++

#include <bits/stdc++.h>using namespace std;using ll = long long;
int main() {	int n, x;	cin >> n >> x;	vector<int> a(n);	for (int i = 0; i < n; i++) { cin >> a[i]; }


Java

### Warning: Tight Time Limit

CSES has very tight time constraints for java, so the following code will barely pass in around 1 second. It is not guaranteed to pass on first submits, so you might need to submit multiple times.

import java.io.*;import java.util.*;
public class Main {	public static void main(String[] args) {		Kattio io = new Kattio();
int n = io.nextInt(), x = io.nextInt();		int[] a = new int[n];		for (int i = 0; i < n; i++) { a[i] = io.nextInt(); }

### Problems

StatusSourceProblem NameDifficultyTags
CFEasy
Show TagsBinary Search, Meet in the Middle
SilverEasy
Show Tags2P, Meet in the Middle
CFEasy
Show TagsBinary Search, DFS, Meet in the Middle
PrepBytesNormal
Show TagsDP, Meet in the Middle
YSNormal
Show TagsBitmasks, DP, Meet in the Middle
CFHard
Show TagsDFS, Meet in the Middle, NT
CFHard
Show TagsBinary Search, DFS, Meet in the Middle

### Join the USACO Forum!

Stuck on a problem, or don't understand a module? Join the USACO Forum and get help from other competitive programmers!