Problem:
Score : 600 points

Problem Statement

Snuke has an integer sequence A of length N.

He will make three cuts in A and divide it into four (non-empty) contiguous subsequences B, C, D and E.
The positions of the cuts can be freely chosen.

Let P,Q,R,S be the sums of the elements in B,C,D,E, respectively.
Snuke is happier when the absolute difference of the maximum and the minimum among P,Q,R,S is smaller.
Find the minimum possible absolute difference of the maximum and the minimum among P,Q,R,S.

Constraints

4 \leq N \leq 2 \times 10^5

1 \leq A_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Find the minimum possible absolute difference of the maximum and the minimum among P,Q,R,S.

Sample Input 1

5
3 2 4 1 2

Sample Output 1

2

If we divide A as B,C,D,E=(3),(2),(4),(1,2), then P=3,Q=2,R=4,S=1+2=3.
Here, the maximum and the minimum among P,Q,R,S are 4 and 2, with the absolute difference of 2.
We cannot make the absolute difference of the maximum and the minimum less than 2, so the answer is 2.

Sample Input 2

10
10 71 84 33 6 47 23 25 52 64

Sample Output 2

36

Sample Input 3

7
1 2 3 1000000000 4 5 6

Sample Output 3

999999994

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.util.ArrayList;
import java.util.List;
import java.util.Scanner;
import java.util.StringTokenizer;

public class Main {
    public static void main(String[] args) throws Exception {
        Main main = new Main();
        main.solve();
    }
    public void solve() throws Exception {
        FastScan scan = new FastScan(System.in);
        int N = scan.nextInt();
        long[] a = new long[N];
        for (int i = 0; i < N; i++) {
            a[i] = scan.nextInt();
        }
        long[] sum = new long[N+1];
        for (int i = 0; i < N; i++) {
            sum[i+1] = sum[i] + a[i];
        }
        long ans = Long.MAX_VALUE;
        for (int i = 2; i <= N - 2; i++) {
            int left = 0;
            int right = i;
            while (left + 1 < right) {
                int mid = (left + right) / 2;
                if (sum[mid] < sum[i] - sum[mid]) {
                    left = mid;
                } else {
                    right = right - 1;
                }
            }
            long P1 = sum[left];
            long Q1 = sum[i] - P1;
            long P2 = sum[left+1];
            long Q2 = sum[i] - P2;
            long P = Math.abs(P2-Q2) < Math.abs(P1-Q1) ? P2 : P1;
            long Q = Math.abs(P2-Q2) < Math.abs(P1-Q1) ? Q2 : Q1;
            left = i + 1;
            right = N;
            while (left + 1 < right) {
                int mid = (left + right) / 2;
                if (sum[mid]-sum[i] < sum[N] - sum[mid]) {
                    left = mid;
                } else {
                    right = right - 1;
                }
            }
            long R1 = sum[left] - sum[i];
            long S1 = sum[N] - sum[left];
            long R2 = sum[left+1] - sum[i];
            long S2 = sum[N] - sum[left+1];
            long R = Math.abs(R2-S2) < Math.abs(R1-S1) ? R2 : R1;
            long S = Math.abs(R2-S2) < Math.abs(R1-S1) ? S2 : S1;
            long min = Math.min(P, Math.min(Q, Math.min(R, S)));
            long max = Math.max(P, Math.max(Q, Math.max(R, S)));
            ans = Math.min(ans, Math.abs(max - min));
        }
        System.out.println(ans);
    }
    class FastScan {
        BufferedReader br;
        StringTokenizer st;
        FastScan(InputStream is) {
            InputStreamReader isr = new InputStreamReader(is);
            this.br = new BufferedReader(isr);
        }
        String next() throws IOException {
            while (this.st == null || !this.st.hasMoreTokens()) {
                this.st = new StringTokenizer(br.readLine().trim());
            }
            return st.nextToken();
        }
        long nextLong() throws IOException {
            return Long.parseLong(this.next());
        }
        int nextInt() throws IOException {
            return Integer.parseInt(this.next());
        }
        double scanDouble() throws IOException {
            return Double.parseDouble(this.next());
        }
        char scanCharacter() throws IOException {
            return this.next().charAt(0);
        }
        String nextLine() throws IOException {
            return br.readLine().trim();
        }
    }
}