Problem:
Score : 500 points

Problem Statement

We have a chocolate bar partitioned into H horizontal rows and W vertical columns of squares.

The square (i, j) at the i-th row from the top and the j-th column from the left is dark if S_{i,j} is 0, and white if S_{i,j} is 1.

We will cut the bar some number of times to divide it into some number of blocks. In each cut, we cut the whole bar by a line running along some boundaries of squares from end to end of the bar.

How many times do we need to cut the bar so that every block after the cuts has K or less white squares?

Constraints

1 \leq H \leq 10

1 \leq W \leq 1000

1 \leq K \leq H \times W

S_{i,j} is 0 or 1.

Input

Input is given from Standard Input in the following format:

H W K
S_{1,1}S_{1,2}...S_{1,W}
:
S_{H,1}S_{H,2}...S_{H,W}

Output

Print the number of minimum times the bar needs to be cut so that every block after the cuts has K or less white squares.

Sample Input 1

3 5 4
11100
10001
00111

Sample Output 1

2

For example, cutting between the 1-st and 2-nd rows and between the 3-rd and 4-th columns - as shown in the figure to the left - works.

Note that we cannot cut the bar in the ways shown in the two figures to the right.

Sample Input 2

3 5 8
11100
10001
00111

Sample Output 2

0

No cut is needed.

Sample Input 3

4 10 4
1110010010
1000101110
0011101001
1101000111

Sample Output 3

3

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using static System.Math;
using static ABC159.E.Cin;
using static ABC159.E.Util;
using Pair = ABC159.E.VTuple<long, long>;
using Graph = System.Collections.Generic.List<System.Collections.Generic.List<long>>;
using System.Text;

namespace ABC159
{
    public class E
    {
        public static void Main(string[] args)
        {
            long H = rl;
            long W = rl;
            long K = rl;
            long[,] A = new long[H, W];
            for (int i = 0; i < H; i++)
            {
                string tmp = rs;
                for (int j = 0; j < W; j++)
                {
                    A[i, j] = tmp[j]-'0';
                }
            }

            long[,] s = new long[H + 1, W + 1];
            for (int i = 0; i < H; ++i)
            {
                for (int j = 0; j < W; ++j)
                {
                    s[i + 1, j + 1] = s[i, j + 1] + s[i + 1, j] - s[i, j] + A[i, j];
                }
            }

            var memo = new HashMap<VTuple<int, int, int, int>, int>();
            Func<int, int, int, int, int> rec = null;
            rec = (x1, y1, x2, y2) =>
            {
                var t = s[x2, y2] - s[x1, y2] - s[x2, y1] + s[x1, y1];
                if (t <= K)
                {
                    return 0;
                }
                else
                {
                    var key = new VTuple<int, int, int, int>(x1, y1, x2, y2);

                    if (memo.ContainsKey(key))
                    {
                        return memo[key];
                    }
                    int min = int.MaxValue;

                    // horizontal cut
                    for (int h = x1 + 1; h < x2; h++)
                    {
                        var t1 = rec(x1, y1, h, y2);
                        var t2 = rec(h, y1, x2, y2);
                        min = Min(min, t1 + t2 + 1);
                    }

                    // vertical cut
                    for (int w = y1 + 1; w < y2; w++)
                    {
                        var t1 = rec(x1, y1, x2, w);
                        var t2 = rec(x1, w, x2, y2);
                        min = Min(min, t1 + t2 + 1);
                    }

                    memo[key] = min;
                    return min;
                }
            };

            int ans = rec(0, 0, (int)H, (int)W);
            Console.WriteLine(ans);
        }

        public struct Mint:System.IComparable<Mint>,System.IEquatable<Mint>{public static readonly long MOD=(long)1e9+7;public long Value;public Mint(long a){Value=a%MOD;if(Value<0)Value+=MOD;}static Mint i(long a){return new Mint(){Value=a};}public static Mint operator+(Mint a,Mint b){long c=a.Value+b.Value;if(c>MOD)c-=MOD;return i(c);}public static Mint operator-(Mint a,Mint b){long c=a.Value-b.Value;if(c<0)c+=MOD;return i(c);}public static Mint operator*(Mint a,Mint b){long c=a.Value*b.Value;if(c>MOD)c%=MOD;return i(c);}public static Mint operator/(Mint a,Mint b){return a*Inv(b);}public override bool Equals(object a){return a is Mint&&Value==((Mint)a).Value;}public override int GetHashCode(){return Value.GetHashCode();}public static implicit operator Mint(long a){return new Mint(a);}public static explicit operator long(Mint a){return a.Value;}public static Mint Pow(Mint a,long b){if(b==0)return new Mint(1);Mint c=Pow(a,b>>1);c*=c;if((b&1)==1)c*=a;return c;}public static Mint Inv(Mint a){long b=a.Value;long c=MOD;long d=1;long e=0;while(c!=0){long f=b/c;long g=e;e=d-f*e;d=g;long h=b;b=c;c=h-(f*c);}return new Mint(d);}public bool Equals(Mint a){return Value==a.Value;}public int CompareTo(Mint a){return Comparer<long>.Default.Compare(Value,a.Value);}}public struct VTuple<T1,T2>:System.IComparable<VTuple<T1,T2>>,System.IEquatable<VTuple<T1,T2>>{public T1 Item1;public T2 Item2;public T1 First{get{return Item1;}}public T2 Second{get{return Item2;}}public VTuple(T1 a,T2 b){Item1=a;Item2=b;}public override bool Equals(object a){return a is VTuple<T1,T2>&&Equals((VTuple<T1,T2>)a);}public override int GetHashCode(){int a=Item1!=null?Item1.GetHashCode():0;int b=Item2!=null?Item2.GetHashCode():0;return HashHelpers.CombineHashCodes(a,b);}public bool Equals(VTuple<T1,T2>a){return EqualityComparer<T1>.Default.Equals(Item1,a.Item1)&&EqualityComparer<T2>.Default.Equals(Item2,a.Item2);}public int CompareTo(VTuple<T1,T2>a){int b=Comparer<T1>.Default.Compare(Item1,a.Item1);if(b!=0)return b;return Comparer<T2>.Default.Compare(Item2,a.Item2);}}public struct VTuple<T1,T2,T3>:System.IComparable<VTuple<T1,T2,T3>>,System.IEquatable<VTuple<T1,T2,T3>>{public T1 Item1;public T2 Item2;public T3 Item3;public VTuple(T1 a,T2 b,T3 c){Item1=a;Item2=b;Item3=c;}public override bool Equals(object a){return a is VTuple<T1,T2,T3>&&Equals((VTuple<T1,T2,T3>)a);}public override int GetHashCode(){int a=Item1!=null?Item1.GetHashCode():0;int b=Item2!=null?Item2.GetHashCode():0;int c=Item3!=null?Item3.GetHashCode():0;return HashHelpers.CombineHashCodes(a,b,c);}public bool Equals(VTuple<T1,T2,T3>a){return EqualityComparer<T1>.Default.Equals(Item1,a.Item1)&&EqualityComparer<T2>.Default.Equals(Item2,a.Item2)&&EqualityComparer<T3>.Default.Equals(Item3,a.Item3);}public int CompareTo(VTuple<T1,T2,T3>a){int b=Comparer<T1>.Default.Compare(Item1,a.Item1);if(b!=0)return b;b=Comparer<T2>.Default.Compare(Item2,a.Item2);if(b!=0)return b;return Comparer<T3>.Default.Compare(Item3,a.Item3);}}public struct VTuple<T1,T2,T3,T4>:System.IComparable<VTuple<T1,T2,T3,T4>>,System.IEquatable<VTuple<T1,T2,T3,T4>>{public T1 Item1;public T2 Item2;public T3 Item3;public T4 Item4;public VTuple(T1 a,T2 b,T3 c,T4 d){Item1=a;Item2=b;Item3=c;Item4=d;}public override bool Equals(object a){return a is VTuple<T1,T2,T3,T4>&&Equals((VTuple<T1,T2,T3,T4>)a);}public override int GetHashCode(){int a=Item1!=null?Item1.GetHashCode():0;int b=Item2!=null?Item2.GetHashCode():0;int c=Item3!=null?Item3.GetHashCode():0;int d=Item3!=null?Item4.GetHashCode():0;return HashHelpers.CombineHashCodes(a,b,c,d);}public bool Equals(VTuple<T1,T2,T3,T4>a){return EqualityComparer<T1>.Default.Equals(Item1,a.Item1)&&EqualityComparer<T2>.Default.Equals(Item2,a.Item2)&&EqualityComparer<T3>.Default.Equals(Item3,a.Item3)&&EqualityComparer<T4>.Default.Equals(Item4,a.Item4);}public int CompareTo(VTuple<T1,T2,T3,T4>a){int b=Comparer<T1>.Default.Compare(Item1,a.Item1);if(b!=0)return b;b=Comparer<T2>.Default.Compare(Item2,a.Item2);if(b!=0)return b;b=Comparer<T3>.Default.Compare(Item3,a.Item3);if(b!=0)return b;return Comparer<T4>.Default.Compare(Item4,a.Item4);}}public static class HashHelpers{public static readonly int RandomSeed=new System.Random().Next(int.MinValue,int.MaxValue);public static int Combine(int a,int b){uint c=((uint)a<<5)|((uint)a>>27);return((int)c+a)^b;}public static int CombineHashCodes(int a,int b){return Combine(Combine(RandomSeed,a),b);}public static int CombineHashCodes(int a,int b,int c){return Combine(CombineHashCodes(a,b),c);}public static int CombineHashCodes(int a,int b,int c,int d){return Combine(CombineHashCodes(a,b,c),d);}}public class HashMap<K,V>:System.Collections.Generic.Dictionary<K,V>{V i;static Func<V>j=System.Linq.Expressions.Expression.Lambda<Func<V>>(System.Linq.Expressions.Expression.New(typeof(V))).Compile();public HashMap(){}public HashMap(V a){i=a;}new public V this[K a]{get{V b;if(TryGetValue(a,out b))return b;else return base[a]=i!=null?i:j();}set{base[a]=value;}}}public static class Util{public static readonly long INF=(long)1e17;public readonly static long MOD=(long)1e9+7;public readonly static int[]DXY4={0,1,0,-1,0};public readonly static int[]DXY8={1,1,0,1,-1,0,-1,-1,1};public static void DontAutoFlush(){var a=new System.IO.StreamWriter(Console.OpenStandardOutput()){AutoFlush=false};Console.SetOut(a);}public static void Flush(){Console.Out.Flush();}public static T[]Sort<T>(T[]a){Array.Sort<T>(a);return a;}public static T[]SortDecend<T>(T[]a){Array.Sort<T>(a);Array.Reverse(a);return a;}public static void Swap<T>(ref T a,ref T b){T c=a;a=b;b=c;}public static long GCD(long a,long b){while(b!=0){long c=a;a=b;b=c%b;}return a;}public static long LCM(long a,long b){if(a==0||b==0)return 0;return a*b/GCD(a,b);}public static void ChMax(ref long a,long b){if(a<b)a=b;}public static void ChMin(ref long a,long b){if(a>b)a=b;}public static void ChMax(ref int a,int b){if(a<b)a=b;}public static void ChMin(ref int a,int b){if(a>b)a=b;}public static void FillArray<T>(T[]a,T b){int c=a.Length;for(int d=0;d<c;d++)a[d]=b;}public static void FillArray<T>(T[,]a,T b){int c=a.GetLength(0);int d=a.GetLength(1);for(int e=0;e<c;e++)for(int f=0;f<d;f++){a[e,f]=b;}}public static void FillArray<T>(T[,,]a,T b){int c=a.GetLength(0);int d=a.GetLength(1);int e=a.GetLength(2);for(int f=0;f<c;f++)for(int g=0;g<d;g++){for(int h=0;h<e;h++){a[f,g,h]=b;}}}public static long[]Accumulate(long[]a){long[]b=new long[a.Length+1];for(int c=0;c<a.Length;c++)b[c+1]=b[c]+a[c];return b;}public static double[]Accumulate(double[]a){double[]b=new double[a.Length+1];for(int c=0;c<a.Length;c++)b[c+1]=b[c]+a[c];return b;}}public static class Cin{public static int ri{get{return ReadInt();}}public static int[]ria{get{return ReadIntArray();}}public static long rl{get{return ReadLong();}}public static long[]rla{get{return ReadLongArray();}}public static double rd{get{return ReadDouble();}}public static double[]rda{get{return ReadDoubleArray();}}public static string rs{get{return ReadString();}}public static string[]rsa{get{return ReadStringArray();}}public static int ReadInt(){return int.Parse(i());}public static long ReadLong(){return long.Parse(i());}public static double ReadDouble(){return double.Parse(i());}public static string ReadString(){return i();}public static int[]ReadIntArray(){g=null;return Array.ConvertAll(Console.ReadLine().Split(' '),int.Parse);}public static long[]ReadLongArray(){g=null;return Array.ConvertAll(Console.ReadLine().Split(' '),long.Parse);}public static double[]ReadDoubleArray(){g=null;return Array.ConvertAll(Console.ReadLine().Split(' '),double.Parse);}public static string[]ReadStringArray(){g=null;return Console.ReadLine().Split(' ');}public static void ReadCol(out long[]a,long b){a=new long[b];for(int c=0;c<b;c++)a[c]=ReadLong();}public static void ReadCols(out long[]a,out long[]b,long c){a=new long[c];b=new long[c];for(int d=0;d<c;d++){a[d]=ReadLong();b[d]=ReadLong();}}public static void ReadCols(out long[]a,out long[]b,out long[]c,long d){a=new long[d];b=new long[d];c=new long[d];for(int e=0;e<d;e++){a[e]=ReadLong();b[e]=ReadLong();c[e]=ReadLong();}}public static int[,]ReadIntTable(int a,int b){g=null;int[,]c=new int[a,b];for(int d=0;d<a;d++)for(int e=0;e<b;e++){c[d,e]=ReadInt();}return c;}public static long[,]ReadLongTable(long a,long b){g=null;long[,]c=new long[a,b];for(int d=0;d<a;d++)for(int e=0;e<b;e++){c[d,e]=ReadLong();}return c;}public static char[,]ReadCharTable(int a,int b){g=null;char[,]c=new char[a,b];for(int d=0;d<a;d++){var e=ReadString();for(int f=0;f<b;f++)c[d,f]=e[f];}return c;}static string[]g;static int h;static string i(){if(g==null||g.Length<=h){g=Console.ReadLine().Split(' ');h=0;}return g[h++];}}
    }
}