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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Runtime.CompilerServices;
using static System.Math;
using System.Text;

namespace Program
{
    public static class ABC102D
    {
        static public void Solve()
        {
            var N = NN;
            var Alist = NNList(N);
            var ruiseki = new long[N + 1];
            for (var i = 1; i <= N; i++)
            {
                ruiseki[i] = ruiseki[i - 1] + Alist[i - 1];
            }

            var left = new long[N + 1];
            left[2] = 1;
            var right = new long[N + 1];
            right[N - 2] = N - 1;
            for (var i = 3; i <= N; i++)
            {
                var diffl = Abs(ruiseki[i] - ruiseki[left[i - 1]] * 2);
                if (diffl > Abs(ruiseki[i] - ruiseki[left[i - 1] + 1] * 2))
                {
                    left[i] = left[i - 1] + 1;
                }
                else
                {
                    left[i] = left[i - 1];
                }
                var diffr = Abs(ruiseki[N] - ruiseki[right[N - i + 1]] * 2 + ruiseki[N - i + 1]);
                if (diffr > Abs(ruiseki[N] - ruiseki[right[N - i + 1] - 1] * 2 + ruiseki[N - i + 1]))
                {
                    right[N - i] = right[N - i + 1] - 1;
                }
                else
                {
                    right[N - i] = right[N - i + 1];
                }
            }
            Console.WriteLine(Range(2, N - 1).Min(e =>
            {
                var P = ruiseki[left[e]];
                var Q = ruiseki[e] - ruiseki[left[e]];
                var R = ruiseki[right[e - 1]] - ruiseki[e];
                var S = ruiseki[N] - ruiseki[right[e - 1]];
                var ary = new[] { P, Q, R, S };
                return ary.Max() - ary.Min();
            }));

        }

        //_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/
        //                    TEST                    _/
        //_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/
        class BT<T> where T : IComparable
        {
            private class Node
            {
                public Node left;
                public Node right;
                public T val;
                public bool isBlack;
            }
            private Comparison<T> c;
            private Node root;
            private bool change;
            public BT(Comparison<T> _c) { c = _c; }
            public BT() : this((x, y) => x.CompareTo(y)) { }
            private bool Red(Node n) => n != null && !n.isBlack;
            private bool Black(Node n) => n != null && n.isBlack;
            private Node RotateL(Node n)
            {
                Node m = n.right, t = m.left;
                m.left = n; n.right = t;
                return m;
            }
            private Node RotateR(Node n)
            {
                Node m = n.left, t = m.right;
                m.right = n; n.left = t;
                return m;
            }
            private Node RotateLR(Node n)
            {
                n.left = RotateL(n.left);
                return RotateR(n);
            }
            private Node RotateRL(Node n)
            {
                n.right = RotateR(n.right);
                return RotateL(n);
            }
            public void Add(T x)
            {
                root = Add(root, x);
                root.isBlack = true;
            }
            private Node Add(Node n, T x)
            {
                if (n == null)
                {
                    change = true;
                    return new Node() { val = x };
                }
                var r = c(x, n.val);
                if (r < 0)
                {
                    n.left = Add(n.left, x);
                    return Balance(n);
                }
                if (r > 0)
                {
                    n.right = Add(n.right, x);
                    return Balance(n);
                }
                change = false;
                return n;
            }

            private Node Balance(Node n)
            {
                if (!change) return n;
                if (!Black(n)) return n;
                if (Red(n.left) && Red(n.left.left))
                {
                    n = RotateR(n);
                    n.left.isBlack = true;
                }
                else if (Red(n.left) && Red(n.left.right))
                {
                    n = RotateLR(n);
                    n.left.isBlack = true;
                }
                else if (Red(n.right) && Red(n.right.left))
                {
                    n = RotateRL(n);
                    n.right.isBlack = true;
                }
                else if (Red(n.right) && Red(n.right.right))
                {
                    n = RotateL(n);
                    n.right.isBlack = true;
                }
                else
                {
                    change = false;
                }
                return n;
            }

            public void Remove(T x)
            {
                root = Remove(root, x);
                if (root != null) root.isBlack = true;
            }
            private Node Remove(Node n, T x)
            {
                if (n == null)
                {
                    change = false;
                    return n;
                }
                var r = c(x, n.val);
                if (r < 0)
                {
                    n.left = Remove(n.left, x);
                    return BalanceL(n);
                }
                if (r > 0)
                {
                    n.right = Remove(n.right, x);
                    return BalanceR(n);
                }
                if (n.left == null)
                {
                    change = n.isBlack;
                    return n.right;
                }
                n.left = RemoveMax(n.left);
                n.val = lmax;
                return BalanceL(n);
            }
            private T lmax;
            private Node RemoveMax(Node n)
            {
                if (n.right != null)
                {
                    n.right = RemoveMax(n.right);
                    return BalanceR(n);
                }
                lmax = n.val;
                change = n.isBlack;
                return n.left;
            }
            private Node BalanceL(Node n)
            {
                if (!change) return n;
                if (Black(n.right) && Red(n.right.left))
                {
                    var b = n.isBlack;
                    n = RotateRL(n);
                    n.isBlack = b;
                    n.left.isBlack = true;
                    change = false;
                }
                else if (Black(n.right) && Red(n.right.right))
                {
                    var b = n.isBlack;
                    n = RotateL(n);
                    n.isBlack = b;
                    n.right.isBlack = true;
                    n.left.isBlack = true;
                    change = false;
                }
                else if (Black(n.right))
                {
                    change = n.isBlack;
                    n.isBlack = true;
                    n.right.isBlack = false;
                }
                else
                {
                    n = RotateL(n);
                    n.isBlack = true;
                    n.left.isBlack = false;
                    n.left = BalanceL(n.left);
                    change = false;
                }
                return n;
            }
            private Node BalanceR(Node n)
            {
                if (!change) return n;
                if (Black(n.left) && Red(n.left.right))
                {
                    var b = n.isBlack;
                    n = RotateLR(n);
                    n.isBlack = b;
                    n.right.isBlack = true;
                    change = false;
                }
                else if (Black(n.left) && Red(n.left.left))
                {
                    var b = n.isBlack;
                    n = RotateR(n);
                    n.isBlack = b;
                    n.left.isBlack = true;
                    n.right.isBlack = true;
                    change = false;
                }
                else if (Black(n.left))
                {
                    change = n.isBlack;
                    n.isBlack = true;
                    n.left.isBlack = false;
                }
                else
                {
                    n = RotateR(n);
                    n.isBlack = true;
                    n.right.isBlack = false;
                    n.right = BalanceR(n.right);
                    change = false;
                }
                return n;
            }
            public Tuple<bool, T> FindUpper(T x)
            {
                var v = FindUpper(root, x);
                if (v == null) return Tuple.Create(false, default(T));
                return v;
            }
            private Tuple<bool, T> FindUpper(Node n, T x)
            {
                if (n == null) return null;
                var r = c(x, n.val);
                if (r < 0)
                {
                    var v = FindUpper(n.left, x);
                    if (v == null) return Tuple.Create(true, n.val);
                    return v;
                }
                else if (r > 0) return FindUpper(n.right, x);
                else return Tuple.Create(true, x);
            }
            public Tuple<bool, T> FindLower(T x)
            {
                var v = FindLower(root, x);
                if (v == null) return Tuple.Create(false, default(T));
                return v;
            }
            private Tuple<bool, T> FindLower(Node n, T x)
            {
                if (n == null) return null;
                var r = c(x, n.val);
                if (r < 0) return FindLower(n.left, x);
                else if (r > 0)
                {
                    var v = FindLower(n.right, x);
                    if (v == null) return Tuple.Create(true, n.val);
                    return v;
                }
                else return Tuple.Create(true, x);
            }
            public T Min()
            {
                Node n = root, p = null;
                while (n != null)
                {
                    p = n;
                    n = n.left;
                }
                return p == null ? default(T) : p.val;
            }
            public T Max()
            {
                Node n = root, p = null;
                while (n != null)
                {
                    p = n;
                    n = n.right;
                }
                return p == null ? default(T) : p.val;
            }
            public bool Any() => root != null;
            public int CountSlow() => List(root).Count();
            public IEnumerable<T> List() => List(root);
            private IEnumerable<T> List(Node n)
            {
                if (n == null) yield break;
                foreach (var i in List(n.left)) yield return i;
                yield return n.val;
                foreach (var i in List(n.right)) yield return i;
            }
        }
        //_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/
        //                  TEST END                  _/
        //_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/

        static public void Main(string[] args)
        {
            if (args.Length == 0) { var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; Console.SetOut(sw); }
            Solve();
            Console.Out.Flush();
        }
        static Random rand = new Random();
        static class Console_
        {
            private static Queue<string> param = new Queue<string>();
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public static string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        }
        static long NN => long.Parse(Console_.NextString());
        static double ND => double.Parse(Console_.NextString());
        static string NS => Console_.NextString();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => x.OrderBy(_ => rand.Next());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void RevSort<T>(T[] l) where T : IComparable { Array.Sort(l, (x, y) => y.CompareTo(x)); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void RevSort<T>(T[] l, Comparison<T> comp) where T : IComparable { Array.Sort(l, (x, y) => comp(y, x)); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<long> Primes(long x) { if (x < 2) yield break; yield return 2; var halfx = x / 2; var table = new bool[halfx + 1]; var max = (long)(Math.Sqrt(x) / 2); for (long i = 1; i <= max; ++i) { if (table[i]) continue; var add = 2 * i + 1; yield return add; for (long j = 2 * i * (i + 1); j <= halfx; j += add) table[j] = true; } for (long i = max + 1; i <= halfx; ++i) if (!table[i] && 2 * i + 1 <= x) yield return 2 * i + 1; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<long> Factors(long x) { if (x < 2) yield break; while (x % 2 == 0) { x /= 2; yield return 2; } var max = (long)Math.Sqrt(x); for (long i = 3; i <= max; i += 2) while (x % i == 0) { x /= i; yield return i; } if (x != 1) yield return x; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static IEnumerable<long> Divisor(long x) { if (x < 1) yield break; var max = (long)Math.Sqrt(x); for (long i = 1; i <= max; ++i) { if (x % i != 0) continue; yield return i; if (i != x / i) yield return x / i; } }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static long GCD(long a, long b) { while (b > 0) { var tmp = b; b = a % b; a = tmp; } return a; }
        static long LCM(long a, long b) => a * b / GCD(a, b);
        class PQ<T> where T : IComparable
        {
            private List<T> h;
            private Comparison<T> c;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, Comparison<T> c, bool asc = true) { h = new List<T>(cap); this.c = asc ? c : (x, y) => c(y, x); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(Comparison<T> c, bool asc = true) { h = new List<T>(); this.c = asc ? c : (x, y) => c(y, x); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Push(T v) { var i = h.Count; h.Add(v); while (i > 0) { var ni = (i - 1) / 2; if (c(v, h[ni]) >= 0) break; h[i] = h[ni]; i = ni; } h[i] = v; }
            public T Peek => h[0];
            public int Count => h.Count;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public T Pop() { var r = h[0]; var v = h[h.Count - 1]; h.RemoveAt(h.Count - 1); if (h.Count == 0) return r; var i = 0; while (i * 2 + 1 < h.Count) { var i1 = i * 2 + 1; var i2 = i * 2 + 2; if (i2 < h.Count && c(h[i1], h[i2]) > 0) i1 = i2; if (c(v, h[i1]) <= 0) break; h[i] = h[i1]; i = i1; } h[i] = v; return r; }
        }
        class PQ<TKey, TValue> where TKey : IComparable
        {
            private PQ<Tuple<TKey, TValue>> q;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, Comparison<TKey> c, bool asc = true) { q = new PQ<Tuple<TKey, TValue>>(cap, (x, y) => c(x.Item1, y.Item1), asc); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(Comparison<TKey> c, bool asc = true) { q = new PQ<Tuple<TKey, TValue>>((x, y) => c(x.Item1, y.Item1), asc); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(int cap, bool asc = true) : this(cap, (x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public PQ(bool asc = true) : this((x, y) => x.CompareTo(y), asc) { }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public void Push(TKey k, TValue v) => q.Push(Tuple.Create(k, v));
            public Tuple<TKey, TValue> Peek => q.Peek;
            public int Count => q.Count;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Tuple<TKey, TValue> Pop() => q.Pop();
        }
        struct Mod : IEquatable<object>
        {
            static public long _mod = 1000000007;
            private long _val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Mod(long x) { if (x < _mod && x >= 0) _val = x; else if ((_val = x % _mod) < 0) _val += _mod; }
            static public implicit operator Mod(long x) => new Mod(x);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public implicit operator long(Mod x) => x._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator +(Mod x, Mod y) => x._val + y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator -(Mod x, Mod y) => x._val - y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator *(Mod x, Mod y) => x._val * y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod operator /(Mod x, Mod y) => x._val * Inverse(y._val);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator ==(Mod x, Mod y) => x._val == y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public bool operator !=(Mod x, Mod y) => x._val != y._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public long Inverse(long x) { long b = _mod, r = 1, u = 0, t = 0; while (b > 0) { var q = x / b; t = u; u = r - q * u; r = t; t = b; b = x - q * b; x = t; } return r < 0 ? r + _mod : r; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            bool IEquatable<object>.Equals(object obj) => obj == null ? false : Equals((Mod)obj);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public override bool Equals(object obj) => obj == null ? false : Equals((Mod)obj);
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public bool Equals(Mod obj) => obj == null ? false : _val == obj._val;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public override int GetHashCode() => _val.GetHashCode();
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public override string ToString() => _val.ToString();
            static private List<Mod> _fact = new List<Mod>() { 1 };
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static private void Build(long n) { if (n >= _fact.Count) for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mod Comb(long n, long k) { Build(n); if (n == 0 && k == 0) return 1; if (n < k || n < 0) return 0; return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k]; }
        }
        struct Mat<T>
        {
            private T[,] m;
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            public Mat(T[,] v) { m = (T[,])v.Clone(); }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public implicit operator Mat<T>(T[,] v) => new Mat<T>(v);
            public T this[int r, int c] { get { return m[r, c]; } set { m[r, c] = value; } }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator +(Mat<T> a, T x) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] += (dynamic)x; return tm; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator +(Mat<T> a, Mat<T> b) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] += (dynamic)b[r, c]; return tm; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator -(Mat<T> a, T x) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] -= (dynamic)x; return tm; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator -(Mat<T> a, Mat<T> b) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] -= (dynamic)b[r, c]; return tm; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator *(Mat<T> a, T x) { var tm = (T[,])a.m.Clone(); for (int r = 0; r < tm.GetLength(0); ++r) for (int c = 0; c < tm.GetLength(1); ++c) tm[r, c] *= (dynamic)x; return tm; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> operator *(Mat<T> a, Mat<T> b) { var nr = a.m.GetLength(0); var nc = b.m.GetLength(1); var tm = new T[nr, nc]; for (int i = 0; i < nr; ++i) for (int j = 0; j < nc; ++j) tm[i, j] = (dynamic)0; for (int r = 0; r < nr; ++r) for (int c = 0; c < nc; ++c) for (int i = 0; i < a.m.GetLength(1); ++i) tm[r, c] += a[r, i] * (dynamic)b[i, c]; return tm; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            static public Mat<T> Pow(Mat<T> x, long y) { var n = x.m.GetLength(0); var t = (Mat<T>)new T[n, n]; for (int i = 0; i < n; ++i) for (int j = 0; j < n; ++j) t[i, j] = (dynamic)(i == j ? 1 : 0); while (y != 0) { if ((y & 1) == 1) t *= x; x *= x; y >>= 1; } return t; }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static Mat<T> Pow<T>(Mat<T> x, long y) => Mat<T>.Pow(x, y);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static T Pow<T>(T x, long y) { T a = (dynamic)1; while (y != 0) { if ((y & 1) == 1) a *= (dynamic)x; x *= (dynamic)x; y >>= 1; } return a; }
        static List<long> _fact = new List<long>() { 1 };
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static void _Build(long n) { if (n >= _fact.Count) for (int i = _fact.Count; i <= n; ++i) _fact.Add(_fact[i - 1] * i); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static long Comb(long n, long k) { _Build(n); if (n == 0 && k == 0) return 1; if (n < k || n < 0) return 0; return _fact[(int)n] / _fact[(int)(n - k)] / _fact[(int)k]; }
    }
}