Problem:
Score : 100 points

Problem Statement

There are N stones, numbered 1, 2, \ldots, N.
For each i (1 \leq i \leq N), the height of Stone i is h_i.
Here, h_1 < h_2 < \cdots < h_N holds.

There is a frog who is initially on Stone 1.
He will repeat the following action some number of times to reach Stone N:

If the frog is currently on Stone i, jump to one of the following: Stone i + 1, i + 2, \ldots, N. Here, a cost of (h_j - h_i)^2 + C is incurred, where j is the stone to land on.

Find the minimum possible total cost incurred before the frog reaches Stone N.

Constraints

All values in input are integers.

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

1 \leq C \leq 10^{12}

1 \leq h_1 < h_2 < \cdots < h_N \leq 10^6

Input

Input is given from Standard Input in the following format:

N C
h_1 h_2 \ldots h_N

Output

Print the minimum possible total cost incurred.

Sample Input 1

5 6
1 2 3 4 5

Sample Output 1

20

If we follow the path 1 → 3 → 5, the total cost incurred would be ((3 - 1)^2 + 6) + ((5 - 3)^2 + 6) = 20.

Sample Input 2

2 1000000000000
500000 1000000

Sample Output 2

1250000000000

The answer may not fit into a 32-bit integer type.

Sample Input 3

8 5
1 3 4 5 10 11 12 13

Sample Output 3

62

If we follow the path 1 → 2 → 4 → 5 → 8, the total cost incurred would be ((3 - 1)^2 + 5) + ((5 - 3)^2 + 5) + ((10 - 5)^2 + 5) + ((13 - 10)^2 + 5) = 62.

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

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Runtime.CompilerServices;
using System.Text;
using System.Diagnostics;
using System.Numerics;
using static System.Console;
using static System.Convert;
using static System.Math;
using static Template;
using Pi = Pair<int, int>;

class Solver
{
    public void Solve(Scanner sc)
    {
        int N;long C;
        sc.Make(out N, out C);
        var cn = new ConvexHullTrick_Monotone();
        var H = sc.ArrLong;
        var dp = new long[N];
        cn.Add(-2 * H[0], H[0] * H[0] + C);
        for (var i = 1; i < N; i++)
        {
            dp[i] = cn.Query_inc(H[i])+H[i]*H[i];
            cn.Add(-2 * H[i], H[i] * H[i] + C + dp[i]);
        }
        WriteLine(dp[N-1]);
    }
}

class ConvexHullTrick_Monotone
{
    bool isMin;
    Deque<Line> lines;
    public ConvexHullTrick_Monotone(bool isMin = true)
    {
        this.isMin = isMin;
        lines = new Deque<Line>();
    }
    bool CanErase(Line l1, Line l2, Line l3) => (l3.B - l2.B) * (l2.A - l1.A) >= (l2.B - l1.B) * (l3.A - l2.A);
    public void Add(long A, long B)
    {
        if (!isMin) { A *= -1; B *= -1; }
        var L = new Line(A, B);
        if (lines.Count == 0)
        {
            lines.PushHead(L); return;
        }
        if (lines[0].A <= A)
        {
            if (lines[0].A == A)
            {
                if (lines[0].B <= B) return;
                lines.PopHead();
            }
            while (lines.Count >= 2 && CanErase(L, lines[0], lines[1]))
                lines.PopHead();
            lines.PushHead(L);
        }
        else
        {
            if (lines[lines.Count - 1].A == A)
            {
                if (lines[lines.Count - 1].B <= B) return;
                lines.PopTail();
            }
            while (lines.Count >= 2 && CanErase(lines[lines.Count - 2], lines[lines.Count - 1], L))
                lines.PopTail();
            lines.PushTail(L);
        }
    }
    long Calc(int i, long x) => lines[i].A * x + lines[i].B;
    public long Query(long x)
    {
        int r = lines.Count - 1, l = -1;
        while (r - l > 1)
        {
            var m = (r + l) >> 1;
            if (Calc(m, x) <= Calc(m + 1, x)) r = m;
            else l = m;
        }
        return (isMin ? 1 : -1) * Calc(r, x);
    }
    public long Query_inc(long x)
    {
        while (lines.Count >= 2 && Calc(0, x) >= Calc(1, x)) lines.PopHead();
        return (isMin ? 1 : -1) * Calc(0, x);
    }
    public long Query_dec(long x)
    {
        while (lines.Count >= 2 && Calc(lines.Count - 1, x) >= Calc(lines.Count - 2, x)) lines.PopTail();
        return (isMin ? 1 : -1) * Calc(lines.Count - 1, x);
    }
    private struct Line
    {
        public long A, B;
        public Line(long A, long B) { this.A = A; this.B = B; }
        public int CompareTo(Line l2) => A == l2.A ? B.CompareTo(l2.B) : A.CompareTo(l2.A);
    }
}
public class Deque<T>
{
    private T[] data;
    int offset, sz;
    public int Count { get; private set; }
    public Deque() { data = new T[sz = 16]; }
    public T this[int i]
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        get { return data[MOD(i + offset)]; }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        set { data[MOD(i + offset)] = value; }
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int MOD(int v) => v & (sz - 1);
    private void Extend()
    {
        var t = new T[sz << 1];
        for (var i = 0; i < sz; i++)
            t[i] = data[MOD(offset + i)];
        offset = 0;
        data = t;
        sz <<= 1;
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public void PushHead(T item)
    {
        if (Count == sz) Extend();
        data[MOD(--offset)] = item;
        Count++;
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T PopHead()
    {
        if (Count == 0) return default(T);
        Count--;
        return data[MOD(offset++)];
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public void PushTail(T item)
    {
        if (Count == sz) Extend();
        data[MOD(Count++ + offset)] = item;
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T PopTail()
    {
        if (Count == 0) return default(T);
        return data[MOD(--Count + offset)];
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public bool Any() => Count > 0;
    public T[] ToArray()
    {
        var rt = new T[Count];
        for (int i = 0; i < Count; i++)
            rt[i] = this[i];
        return rt;
    }
}
#region Template
public static class Template
{
    static void Main(string[] args)
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver().Solve(new Scanner());
        Console.Out.Flush();
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) == 1) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T a, T b) where T : IComparable<T> { if (a.CompareTo(b) == -1) { a = b; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T a, ref T b) { var t = b; b = a; a = t; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(this IList<T> A, int i, int j) { var t = A[i]; A[i] = A[j]; A[j] = t; }
    public static T[] Shuffle<T>(this IList<T> A) { T[] rt = A.ToArray(); Random rnd = new Random(); for (int i = rt.Length - 1; i >= 1; i--) swap(ref rt[i], ref rt[rnd.Next(i + 1)]); return rt; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T[] Create<T>(int n, Func<T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(); return rt; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T[] Create<T>(int n, Func<int, T> f) { var rt = new T[n]; for (var i = 0; i < rt.Length; ++i) rt[i] = f(i); return rt; }
    public static void Fail<T>(T s) { Console.WriteLine(s); Console.Out.Close(); Environment.Exit(0); }

}
public class Scanner
{
    public string Str => Console.ReadLine().Trim();
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public int[] ArrInt => Str.Split(' ').Select(int.Parse).ToArray();
    public long[] ArrLong => Str.Split(' ').Select(long.Parse).ToArray();
    public char[][] Grid(int n) => Create(n, () => Str.ToCharArray());
    public int[] ArrInt1D(int n) => Create(n, () => Int);
    public long[] ArrLong1D(int n) => Create(n, () => Long);
    public int[][] ArrInt2D(int n) => Create(n, () => ArrInt);
    public long[][] ArrLong2D(int n) => Create(n, () => ArrLong);
    private Queue<string> q = new Queue<string>();
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public T Next<T>() { if (q.Count == 0) foreach (var item in Str.Split(' ')) q.Enqueue(item); return (T)Convert.ChangeType(q.Dequeue(), typeof(T)); }
    public void Make<T1>(out T1 v1) => v1 = Next<T1>();
    public void Make<T1, T2>(out T1 v1, out T2 v2) { v1 = Next<T1>(); v2 = Next<T2>(); }
    public void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3) { Make(out v1, out v2); v3 = Next<T3>(); }
    public void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4) { Make(out v1, out v2, out v3); v4 = Next<T4>(); }
    public void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5) { Make(out v1, out v2, out v3, out v4); v5 = Next<T5>(); }
    public void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6) { Make(out v1, out v2, out v3, out v4, out v5); v6 = Next<T6>(); }
    //public (T1, T2) Make<T1, T2>() { Make(out T1 v1, out T2 v2); return (v1, v2); }
    //public (T1, T2, T3) Make<T1, T2, T3>() { Make(out T1 v1, out T2 v2, out T3 v3); return (v1, v2, v3); }
    //public (T1, T2, T3, T4) Make<T1, T2, T3, T4>() { Make(out T1 v1, out T2 v2, out T3 v3, out T4 v4); return (v1, v2, v3, v4); }
}
public class Pair<T1, T2> : IComparable<Pair<T1, T2>>
{
    public T1 v1;
    public T2 v2;
    public Pair() { }
    public Pair(T1 v1, T2 v2)
    { this.v1 = v1; this.v2 = v2; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2> p)
    {
        var c = Comparer<T1>.Default.Compare(v1, p.v1);
        if (c == 0)
            c = Comparer<T2>.Default.Compare(v2, p.v2);
        return c;
    }
    public override string ToString() => $"{v1.ToString()} {v2.ToString()}";
    public void Deconstruct(out T1 a, out T2 b) { a = v1; b = v2; }
}

public class Pair<T1, T2, T3> : Pair<T1, T2>, IComparable<Pair<T1, T2, T3>>
{
    public T3 v3;
    public Pair() : base() { }
    public Pair(T1 v1, T2 v2, T3 v3) : base(v1, v2)
    { this.v3 = v3; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2, T3> p)
    {
        var c = base.CompareTo(p);
        if (c == 0)
            c = Comparer<T3>.Default.Compare(v3, p.v3);
        return c;
    }
    public override string ToString() => $"{base.ToString()} {v3.ToString()}";
    public void Deconstruct(out T1 a, out T2 b, out T3 c) { Deconstruct(out a, out b); c = v3; }
}
#endregion