Problem:
Score : 500 points

Problem Statement

Given are a sequence of N positive integers A_1, A_2, \ldots, A_N, and a positive integer K.

Find the number of non-empty contiguous subsequences in A such that the remainder when dividing the sum of its elements by K is equal to the number of its elements. We consider two subsequences different if they are taken from different positions, even if they are equal sequences.

Constraints

All values in input are integers.

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

1 \leq K \leq 10^9

1 \leq A_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N K
A_1 A_2 \cdots A_N

Output

Print the number of subsequences that satisfy the condition.

Sample Input 1

5 4
1 4 2 3 5

Sample Output 1

4

Four sequences satisfy the condition: (1), (4,2), (1,4,2), and (5).

Sample Input 2

8 4
4 2 4 2 4 2 4 2

Sample Output 2

7

(4,2) is counted four times, and (2,4) is counted three times.

Sample Input 3

10 7
14 15 92 65 35 89 79 32 38 46

Sample Output 3

8

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

using System;
using System.Collections;
using System.Collections.Generic;
using System.Globalization;
using System.IO;
using System.Linq;

using E = System.Linq.Enumerable;

internal partial class Solver {
    public void Run() {
        var n = ni();
        var k = ni();
        var A = ni(n);
        for (int i = 0; i < n; i++) {
            A[i]--;
            A[i] %= k;
        }
        for (int i = 1; i < n; i++) {
            A[i] += A[i - 1];
            A[i] %= k;
        }
        long ans = 0;

        A = new[] { 0 }.Concat(A).ToArray();

        var hist = new Dictionary<int, int>();

        for (int i = 0; i < A.Length; i++) {
            var x = A[i];
            if (!hist.ContainsKey(x)) hist[x] = 0;

            if (i - k >= 0) {
                hist[A[i - k]]--;
            }
            ans += hist[x]++;
        }

        cout.WriteLine(ans);
    }
}

public partial class MultiSet<T> {
    private readonly LLRBTreeNode<T> end = new LLRBTreeNode<T>(default(T));
    private LLRBTreeNode<T> root = null;
    private readonly IComparer<T> _comparer;


    public MultiSet() : this(Comparer<T>.Default) {
    }

    public MultiSet(IComparer<T> comparer) {
        _comparer = comparer;
        root = end;
    }

    protected virtual bool AllowMultiple { get { return true; } }

    public int Count { get { return root == null ? 0 : root.Count + root.LeftTotalCount + root.RightTotalCount - 1; } }

    private LLRBTreeNode<T> FindNode(T value) {
        var node = root;
        while (node != null) {
            int cmp = Compare(value, node);
            if (cmp == 0) {
                return node;
            }
            node = (cmp < 0) ? node.Left : node.Right;
        }
        return null;
    }

    public void Clear() {
        root = end;
    }

    public bool Contains(T value) {
        var node = FindNode(value);
        return node != null;
    }

    public int CountOf(T value) {
        var node = FindNode(value);
        return node != null ? node.Count : 0;
    }

    public int IndexOfLowerBound(T value) {
        var node = root;
        int index = 0;
        while (node != null) {
            int cmp = Compare(value, node);
            if (cmp <= 0) {
                node = node.Left;
            } else {
                index += node.LeftTotalCount + node.Count;
                node = node.Right;
            }
        }
        return index;
    }

    private int Compare(T value, LLRBTreeNode<T> node) {
        if (node == end) {
            return -1;
        }

        return _comparer.Compare(value, node.Value);
    }

    public int IndexOfUpperBound(T value) {
        var node = root;
        int index = 0;
        while (node != null) {
            int cmp = Compare(value, node);
            if (cmp < 0) {
                node = node.Left;
            } else {
                index += node.LeftTotalCount + node.Count;
                node = node.Right;
            }
        }
        return index;
    }

    public T At(int index) {
        var node = root;
        while (node != null) {
            int leftCount = node.LeftTotalCount;
            if (leftCount <= index && index < leftCount + node.Count) {
                return node.Value;
            }

            if (index < leftCount) {
                node = node.Left;
            } else {
                index -= leftCount + node.Count;
                node = node.Right;
            }
        }
        throw new ArgumentOutOfRangeException("index");
    }

    public LLRBTreeIterator<T> AtIterator(int index) {
        var node = root;
        while (node != null) {
            int leftCount = node.LeftTotalCount;
            if (leftCount <= index && index < leftCount + node.Count) {
                return new LLRBTreeIterator<T>(node, index - leftCount);
            }

            if (index < leftCount) {
                node = node.Left;
            } else {
                index -= leftCount + node.Count;
                node = node.Right;
            }
        }
        throw new ArgumentOutOfRangeException("index");
    }

    public void Add(T value) {
        root = Add(root, value);
        root.Color = LLRBTreeNode<T>.BLACK;
    }

    private LLRBTreeNode<T> Add(LLRBTreeNode<T> h, T value) {
        if (h == null) {
            return new LLRBTreeNode<T>(value);
        }

        int cmp = Compare(value, h);
        if (cmp == 0) {
            if (AllowMultiple) {
                h.Count++;
                FixCount(h);
            }
        } else if (cmp <= 0) {
            h.Left = Add(h.Left, value);
            FixCount(h);
        } else {
            h.Right = Add(h.Right, value);
            FixCount(h);
        }

        if (IsRed(h.Right) && !IsRed(h.Left)) {
            h = RotateLeft(h);
        }

        if (IsRed(h.Left) && IsRed(h.Left.Left)) {
            h = RotateRight(h);
        }

        if (IsRed(h.Left) && IsRed(h.Right)) {
            ColorFlip(h);
        }

        return h;
    }

    private LLRBTreeNode<T> RemoveLeftestNode(LLRBTreeNode<T> h, out LLRBTreeNode<T> leftMost) {
        if (h.Left == null) {
            leftMost = h;
            return null;
        }

        if (!IsRed(h.Left) && !IsRed(h.Left.Left)) {
            h = MoveRedLeft(h);
        }

        h.Left = RemoveLeftestNode(h.Left, out leftMost);
        return FixUp(h);
    }

    private bool TryReduceCount(LLRBTreeNode<T> h, T value, out int num) {
        num = 0;
        if (h == null) {
            return false;
        }

        int cmp = Compare(value, h);
        if (cmp == 0) {
            num = h.Count;
            if (h.Count > 1) {
                h.Count--;
                FixCount(h);
                return true;
            }
            return false;
        }
        if (TryReduceCount((cmp < 0) ? h.Left : h.Right, value, out num)) {
            FixCount(h);
            return true;
        }
        return false;
    }


    public bool Remove(T value) {
        if (AllowMultiple) {
            int num;
            if (TryReduceCount(root, value, out num)) {
                return true;
            }
            if (num == 0) {
                return false;
            }
        } else {
            if (!Contains(value)) {
                return false;
            }
        }

        root = RemoveNode(root, value);
        if (root != null) {
            root.Color = LLRBTreeNode<T>.BLACK;
        }
        return true;
    }

    private LLRBTreeNode<T> RemoveNode(LLRBTreeNode<T> h, T value) {
        if (Compare(value, h) < 0) {
            if (h.Left == null) {
                throw new KeyNotFoundException("value");
            }

            if (!IsRed(h.Left) && !IsRed(h.Left.Left)) {
                h = MoveRedLeft(h);
            }
            h.Left = RemoveNode(h.Left, value);
            FixCount(h);
        } else {
            if (IsRed(h.Left)) {
                h = RotateRight(h);
            }

            if (h.Right == null) {
                if (Compare(value, h) == 0) {
                    return null;
                }

                throw new KeyNotFoundException("value");
            }

            if (!IsRed(h.Right) && !IsRed(h.Right.Left)) {
                h = MoveRedRight(h);
            }

            if (Compare(value, h) == 0) {
                LLRBTreeNode<T> sub;
                h.Right = RemoveLeftestNode(h.Right, out sub);
                h = Substitute(h, sub);
                FixCount(h);
            } else {
                h.Right = RemoveNode(h.Right, value);
                FixCount(h);
            }
        }

        return FixUp(h);
    }

    private LLRBTreeNode<T> Substitute(LLRBTreeNode<T> node, LLRBTreeNode<T> sub) {
        sub.Parent = node.Parent;
        sub.Left = node.Left;
        sub.Right = node.Right;
        sub.TotalCount = node.TotalCount;
        sub.Color = node.Color;

        if (sub.Parent != null) {
            if (sub.Parent.Left == node) {
                sub.Parent.Left = sub;
            }

            if (sub.Parent.Right == node) {
                sub.Parent.Right = sub;
            }
        }
        if (sub.Left != null) {
            sub.Left.Parent = sub;
        }

        if (sub.Right != null) {
            sub.Right.Parent = sub;
        }

        return sub;
    }

    private static LLRBTreeNode<T> MoveRedLeft(LLRBTreeNode<T> h) {
        ColorFlip(h);

        if (IsRed(h.Right.Left)) {
            h.Right = RotateRight(h.Right);
            FixCount(h);
            h = RotateLeft(h);
            ColorFlip(h);
        }

        return h;
    }

    private static LLRBTreeNode<T> MoveRedRight(LLRBTreeNode<T> h) {
        ColorFlip(h);

        if (IsRed(h.Left.Left)) {
            h = RotateRight(h);
            ColorFlip(h);
        }

        return h;
    }

    private static LLRBTreeNode<T> RotateLeft(LLRBTreeNode<T> h) {
        var x = h.Right;
        h.Right = x.Left;
        x.Left = h;

        x.Color = h.Color;
        h.Color = LLRBTreeNode<T>.RED;
        FixCount(h);
        FixCount(x);
        return x;
    }

    private static LLRBTreeNode<T> RotateRight(LLRBTreeNode<T> h) {
        var x = h.Left;
        h.Left = x.Right;
        x.Right = h;

        x.Color = h.Color;
        h.Color = LLRBTreeNode<T>.RED;
        FixCount(h);
        FixCount(x);
        return x;
    }

    private static bool IsRed(LLRBTreeNode<T> h) {
        return h != null && h.Color == LLRBTreeNode<T>.RED;
    }

    private static void ColorFlip(LLRBTreeNode<T> h) {
        h.Color = !h.Color;
        h.Left.Color = !h.Left.Color;
        h.Right.Color = !h.Right.Color;
    }

    private static LLRBTreeNode<T> GetLeftestNode(LLRBTreeNode<T> h) {
        while (h.Left != null) {
            h = h.Left;
        }
        return h;
    }

    private static LLRBTreeNode<T> FixUp(LLRBTreeNode<T> h) {
        if (IsRed(h.Right)) {
            h = RotateLeft(h);
        }

        if (h.Left != null && IsRed(h.Left) && IsRed(h.Left.Left)) {
            h = RotateRight(h);
        }

        if (IsRed(h.Left) && IsRed(h.Right)) {
            ColorFlip(h);
        }
        FixCount(h);
        return h;
    }

    private static void FixCount(LLRBTreeNode<T> node) {
        node.TotalCount = node.Count + node.LeftTotalCount + node.RightTotalCount;
    }

    public void Debug() {
        if (root != null) {
            root.Debug();
        }
    }
}

public class LLRBTreeNode<T> {
    public const bool RED = true;
    public const bool BLACK = false;
    public T Value;
    public bool Color = RED;
    public LLRBTreeNode<T> Parent { get; set; }
    private LLRBTreeNode<T> _left;
    private LLRBTreeNode<T> _right;
    public LLRBTreeNode<T> Left {
        get { return _left; }
        set {
            if (_left == value) {
                return;
            }

            if (_left != null && _left.Parent == this) {
                _left.Parent = null;
            }

            _left = value;
            if (_left != null) {
                _left.Parent = this;
            }
        }
    }
    public LLRBTreeNode<T> Right {
        get { return _right; }
        set {
            if (_right == value) {
                return;
            }

            if (_right != null && _right.Parent == this) {
                _right.Parent = null;
            }

            _right = value;
            if (_right != null) {
                _right.Parent = this;
            }
        }
    }
    public int Count = 1;
    public LLRBTreeNode(T value) {
        Value = value;
    }

    public int TotalCount = 1;
    public int LeftTotalCount { get { return Left == null ? 0 : Left.TotalCount; } }
    public int RightTotalCount { get { return Right == null ? 0 : Right.TotalCount; } }

    public void Debug(string indent = "") {
        Console.WriteLine(indent + ":" + Value + " - " + Count + " - " + TotalCount);
        if (Left != null) {
            Left.Debug(indent + "L");
        }
        if (Right != null) {
            Right.Debug(indent + "R");
        }
    }
}

public struct LLRBTreeIterator<T> {
    private readonly LLRBTreeNode<T> _node;
    private readonly int _index;

    public LLRBTreeIterator(LLRBTreeNode<T> node, int index = 0) {
        _node = node;
        _index = index;
    }

    public T Value { get { return _node.Value; } }

    public static LLRBTreeIterator<T> operator ++(LLRBTreeIterator<T> x) {
        return x.GetSuccessor();
    }

    public static LLRBTreeIterator<T> operator --(LLRBTreeIterator<T> x) {
        return x.GetPredecessor();
    }

    private LLRBTreeIterator<T> GetSuccessor() {
        if (_index + 1 < _node.Count) {
            return new LLRBTreeIterator<T>(_node, _index + 1);
        } else {
            LLRBTreeNode<T> current, parent;
            if (_node.Right == null) {
                for (current = _node, parent = _node.Parent; parent.Right == current; current = parent, parent = current.Parent) {
                    ;
                }

                return new LLRBTreeIterator<T>(parent);
            } else {
                for (parent = _node.Right, current = parent.Left; current != null; parent = current, current = parent.Left) {
                    ;
                }

                return new LLRBTreeIterator<T>(parent);
            }
        }
    }

    private LLRBTreeIterator<T> GetPredecessor() {
        if (0 <= _index - 1) {
            return new LLRBTreeIterator<T>(_node, _index - 1);
        } else {
            LLRBTreeNode<T> current, parent;
            if (_node.Left == null) {
                for (current = _node, parent = _node.Parent; parent.Left == current; current = parent, parent = current.Parent) {
                    ;
                }

                return new LLRBTreeIterator<T>(parent, parent.Count - 1);
            } else {
                for (parent = _node.Left, current = parent.Right; current != null; parent = current, current = parent.Right) {
                    ;
                }

                return new LLRBTreeIterator<T>(parent, parent.Count - 1);
            }
        }
    }
}

// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main(string[] args) {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        Console.SetIn(new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length));
        new Solver(Console.In, Console.Out).Run();
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

internal static class LinqPadExtension {
    public static T Dump<T>(this T obj) {
#if LOCAL
        return LINQPad.Extensions.Dump(obj);
#else
        return obj;
#endif
    }
}

public class Scanner {
    private readonly TextReader Reader;
    private readonly Queue<string> TokenQueue = new Queue<string>();
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return int.Parse(Next(), ci); }
    public long NextLong() { return long.Parse(Next(), ci); }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (TokenQueue.Count == 0) {
            if (!StockTokens()) {
                throw new InvalidOperationException();
            }
        }
        return TokenQueue.Dequeue();
    }

    public bool HasNext() {
        if (TokenQueue.Count > 0) {
            return true;
        }

        return StockTokens();
    }

    private static readonly char[] _separator = new[] { ' ', '\t' };
    private bool StockTokens() {
        while (true) {
            string line = Reader.ReadLine();
            if (line == null) {
                return false;
            }

            string[] tokens = line.Split(_separator, StringSplitOptions.RemoveEmptyEntries);
            if (tokens.Length == 0) {
                continue;
            }

            foreach (string token in tokens) {
                TokenQueue.Enqueue(token);
            }

            return true;
        }
    }
}