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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;

namespace Contest
{
    class Program
    {
        static void Main(string[] args)
        {
            ARC100.D();
        }


        class ARC100
        {
            public static void C()
            {
                var N = Read.Int();
                var A = Read.LongN();
                A = A.Select((x, i) => x - i).OrderBy(x => x).ToArray();
                var median = A[(N - 1) / 2];
                Console.WriteLine(A.Select(x => Math.Abs(x - median)).Sum());
            }
            public static void D()
            {
                var N = Read.Int();
                var A = Read.LongN();
                var ASum = Util.ASum(A);
                var S = new Util.Set<long>();
                foreach(var a in ASum) { S.Insert(a); }

                long min = (long)1e18;
                for(int i = 2; i < N - 1; ++i)
                {
                    var part1 = ASum[i];
                    var part2 = ASum[N] - ASum[i];
                    var idx1 = S.LowerBound(part1 / 2);
                    long v1 = ASum[idx1];
                    long v2 = part1 - v1;
                    if (Math.Abs(v2 - v1) > Math.Abs(part1 - ASum[idx1 - 1] * 2))
                    {
                        v1 = ASum[idx1 - 1];
                        v2 = part1 - v1;
                    }
                    var idx2 = S.LowerBound(part1 + part2 / 2);
                    long v3 = ASum[idx2] - part1;
                    long v4 = part2 - v3;
                    if (Math.Abs(v4 - v3) > Math.Abs(part2 - (ASum[idx2 - 1] - part1) * 2))
                    {
                        v3 = ASum[idx2 - 1] - part1;
                        v4 = part2 - v3;
                    }
                    var array = new long[] { v1, v2, v3, v4 };
                    min = Math.Min(min, array.Max() - array.Min());
                }
                Console.WriteLine(min);
            }
            public static void E()
            {

            }
            public static void F()
            {

            }
        }


        class Util
        {
            public static long[] ASum(long[] array)
            {
                var res = new long[array.Length + 1];
                for (int i = 0; i < array.Length; ++i)
                {
                    res[i + 1] = res[i] + array[i];
                }
                return res;
            }
            public class BST<T> where T : IComparable
            {
                public class Node
                {
                    public T V;
                    public Node L, R;
                    public int Count;
                    public Node(T v) { V = v; Count = 1; }
                }
                static Random _rnd = new Random();
                public static int Count(Node t) { return t == null ? 0 : t.Count; }
                static Node Update(Node t) { t.Count = Count(t.L) + Count(t.R) + 1; return t; }
                public static Node Merge(Node l, Node r)
                {
                    if (l == null || r == null) return l == null ? r : l;
                    if ((double)Count(l) / (double)(Count(l) + Count(r)) > _rnd.NextDouble())
                    { l.R = Merge(l.R, r); return Update(l); }
                    else { r.L = Merge(l, r.L); return Update(r); }
                }
                public static Tuple<Node, Node> Split(Node t, int k)
                {
                    if (t == null) return new Tuple<Node, Node>(null, null);
                    if (k <= Count(t.L))
                    {
                        var s = Split(t.L, k);
                        t.L = s.Item2;
                        return new Tuple<Node, Node>(s.Item1, Update(t));
                    }
                    else
                    {
                        var s = Split(t.R, k - Count(t.L) - 1);
                        t.R = s.Item1;
                        return new Tuple<Node, Node>(Update(t), s.Item2);
                    }
                }
                public static Node Remove(Node t, T v)
                {
                    if (Find(t, v) == null) return t;
                    return RemoveAt(t, LowerBound(t, v));
                }
                public static Node RemoveAt(Node t, int k)
                {
                    var s = Split(t, k);
                    var s2 = Split(s.Item2, 1);
                    return Merge(s.Item1, s2.Item2);
                }
                public static bool Contains(Node t, T v) { return Find(t, v) != null; }
                public static Node Find(Node t, T v)
                {
                    while (t != null)
                    {
                        var cmp = t.V.CompareTo(v);
                        if (cmp > 0) t = t.L;
                        else if (cmp < 0) t = t.R;
                        else break;
                    }
                    return t;
                }

                public static Node FindByIndex(Node t, int idx)
                {
                    if (t == null) return null;
                    var currentIdx = Count(t) - Count(t.R) - 1;
                    while (t != null)
                    {
                        if (currentIdx == idx) return t;
                        if (currentIdx > idx)
                        { t = t.L; currentIdx -= (Count(t == null ? null : t.R) + 1); }
                        else { t = t.R; currentIdx += (Count(t == null ? null : t.L) + 1); }
                    }
                    return null;
                }
                public static int UpperBound(Node t, T v)
                {
                    var torg = t;
                    if (t == null) return -1;

                    var ret = Int32.MaxValue;
                    var idx = Count(t) - Count(t.R) - 1;
                    while (t != null)
                    {
                        var cmp = t.V.CompareTo(v);

                        if (cmp > 0)
                        {
                            ret = Math.Min(ret, idx);
                            t = t.L;
                            idx -= (Count(t == null ? null : t.R) + 1);
                        }
                        else if (cmp <= 0)
                        {
                            t = t.R;
                            idx += (Count(t == null ? null : t.L) + 1);
                        }
                    }
                    return ret == Int32.MaxValue ? Count(torg) : ret;
                }
                public static int LowerBound(Node t, T v)
                {
                    var torg = t;
                    if (t == null) return -1;

                    var idx = Count(t) - Count(t.R) - 1;
                    var ret = Int32.MaxValue;
                    while (t != null)
                    {
                        var cmp = t.V.CompareTo(v);
                        if (cmp >= 0)
                        {
                            if (cmp == 0) ret = Math.Min(ret, idx);
                            t = t.L;
                            if (t == null) ret = Math.Min(ret, idx);
                            idx -= t == null ? 0 : (Count(t.R) + 1);
                        }
                        else if (cmp < 0)
                        {
                            t = t.R;
                            idx += (Count(t == null ? null : t.L) + 1);
                            if (t == null) return idx;
                        }
                    }
                    return ret == Int32.MaxValue ? Count(torg) : ret;
                }
                public static Node Insert(Node t, T v)
                {
                    var ub = LowerBound(t, v);
                    return InsertByIdx(t, ub, v);
                }
                static Node InsertByIdx(Node t, int k, T v)
                {
                    var s = Split(t, k);
                    return Merge(Merge(s.Item1, new Node(v)), s.Item2);
                }
                public static IEnumerable<T> Enumerate(Node t)
                {
                    var ret = new List<T>();
                    Enumerate(t, ret);
                    return ret;
                }
                static void Enumerate(Node t, List<T> ret)
                {
                    if (t == null) return;
                    Enumerate(t.L, ret);
                    ret.Add(t.V);
                    Enumerate(t.R, ret);
                }
            }
            public class Set<T> where T : IComparable
            {
                protected BST<T>.Node _root;
                public T this[int idx] { get { return ElementAt(idx); } }
                public int Count() { return BST<T>.Count(_root); }
                public virtual void Insert(T v)
                {
                    if (_root == null) _root = new BST<T>.Node(v);
                    else
                    {
                        if (BST<T>.Find(_root, v) != null) return;
                        _root = BST<T>.Insert(_root, v);
                    }
                }
                public void Clear() { _root = null; }
                public void Remove(T v) { _root = BST<T>.Remove(_root, v); }
                public bool Contains(T v) { return BST<T>.Contains(_root, v); }
                public T ElementAt(int k)
                {
                    var node = BST<T>.FindByIndex(_root, k);
                    if (node == null) throw new IndexOutOfRangeException();
                    return node.V;
                }
                public int Count(T v) { return BST<T>.UpperBound(_root, v) - BST<T>.LowerBound(_root, v); }
                public int LowerBound(T v) { return BST<T>.LowerBound(_root, v); }
                public int UpperBound(T v) { return BST<T>.UpperBound(_root, v); }
                public Tuple<int, int> EqualRange(T v)
                {
                    if (!Contains(v)) return new Tuple<int, int>(-1, -1);
                    return new Tuple<int, int>(BST<T>.LowerBound(_root, v), BST<T>.UpperBound(_root, v) - 1);
                }
                public List<T> ToList() { return new List<T>(BST<T>.Enumerate(_root)); }
            }
        }

        static class Read
        {
            static Queue<string> queue = new Queue<string>();
            static void Enqueue() { var ss = Console.ReadLine().Split(' '); foreach (var s in ss) { queue.Enqueue(s); } }
            public static int Int() { if (queue.Count == 0) { Enqueue(); } return Convert.ToInt32(queue.Dequeue()); }
            public static long Long() { if (queue.Count == 0) { Enqueue(); } return Convert.ToInt64(queue.Dequeue()); }
            public static string Str() { if (queue.Count == 0) { Enqueue(); } return queue.Dequeue(); }
            public static int[] IntN() { return Console.ReadLine().Split(' ').Select(s => Convert.ToInt32(s)).ToArray(); }
            public static long[] LongN() { return Console.ReadLine().Split(' ').Select(s => Convert.ToInt64(s)).ToArray(); }
            public static string[] StrN() { return Console.ReadLine().Split(' '); }
            public static Tuple<int, int>[] TupleInt(int n) { List<Tuple<int, int>> list = new List<Tuple<int, int>>(); for (int i = 0; i < n; ++i) { var ab = Read.IntN(); list.Add(new Tuple<int, int>(ab[0], ab[1])); } return list.ToArray(); }
            public static Tuple<long, long>[] TupleLong(int n) { List<Tuple<long, long>> list = new List<Tuple<long, long>>(); for (int i = 0; i < n; ++i) { var ab = Read.LongN(); list.Add(new Tuple<long, long>(ab[0], ab[1])); } return list.ToArray(); }
            public static Tuple<string, string>[] TupleStr(int n) { List<Tuple<string, string>> list = new List<Tuple<string, string>>(); for (int i = 0; i < n; ++i) { var ab = Read.StrN(); list.Add(new Tuple<string, string>(ab[0], ab[1])); } return list.ToArray(); }
            public static Tuple<double, double>[] TupleDouble(int n) { List<Tuple<double, double>> list = new List<Tuple<double, double>>(); for (int i = 0; i < n; ++i) { var ab = Read.LongN(); list.Add(new Tuple<double, double>(ab[0], ab[1])); } return list.ToArray(); }
            public static long[,] LongMatrix(int r, int c)
            {
                var mat = new long[r, c];
                for (int i = 0; i < r; ++i)
                {
                    var x = Read.LongN();
                    for (int j = 0; j < c; ++j)
                    {
                        mat[i, j] = x[j];
                    }
                }
                return mat;
            }
            public static Dictionary<int, HashSet<int>> AdjacencyList(int N, int M)
            {
                Dictionary<int, HashSet<int>> dict = new Dictionary<int, HashSet<int>>();
                for (int i = 0; i <= N; ++i)
                {
                    dict.Add(i, new HashSet<int>());
                }
                for (int i = 0; i < M; ++i)
                {
                    var ab = Read.IntN();
                    dict[ab[0]].Add(ab[1]);
                    dict[ab[1]].Add(ab[0]);
                }
                return dict;
            }
            public static Dictionary<int, Dictionary<int, long>> AdjacencyListWithWeight(int N, int M)
            {
                Dictionary<int, Dictionary<int, long>> dict = new Dictionary<int, Dictionary<int, long>>();
                for (int i = 0; i <= N; ++i)
                {
                    dict.Add(i, new Dictionary<int, long>());
                }
                for (int i = 0; i < M; ++i)
                {
                    var ab = Read.IntN();
                    dict[ab[0]].Add(ab[1], ab[2]);
                    dict[ab[1]].Add(ab[0], ab[2]);
                }
                return dict;
            }
        }
    }
}