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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.math.BigInteger;
import java.util.ArrayDeque;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.HashMap;
import java.util.HashSet;
import java.util.LinkedList;
import java.util.List;
import java.util.Map;
import java.util.Map.Entry;
import java.util.PriorityQueue;
import java.util.Queue;
import java.util.Scanner;
import java.util.Set;

public class Main {

	static Scanner sc = new Scanner(System.in);

	public static void main(String[] args){

		solve_abc159_e();
	}
  
  	public static void solve_abc159_e(){

		Scanner sc = new Scanner(System.in);

		int H = sc.nextInt();
		int W = sc.nextInt();
		int K = sc.nextInt();

		char[][] chocolateMap = new char[H][W];

		for(int h=0;h<H;h++){
				chocolateMap[h] = sc.next().toCharArray();
		}

		int ans = Integer.MAX_VALUE;
		boolean[] hLine = new boolean[H];
		int numHLine = 0;

		//divide width
		ans = solve_abc159_e_bfs(0, H, W, K, chocolateMap, ans, hLine, numHLine);


		//ans = min line count
		System.out.println(ans);

		sc.close();

	}

	public static int solve_abc159_e_bfs(int h, final int H, final int W, final int K, final char[][] cmap, int ans, boolean[] hLine, int numHLine){

		if(h == H-1){
			int numWLine = solve_abc159_e_width(numHLine, hLine, H, W, K, cmap);
			//System.out.println("ans:" + ans + "\t numHline::" + numHLine + "\t numWLine:" + numWLine);
			
			if(numWLine != Integer.MAX_VALUE)	ans = Math.min(ans, numHLine + numWLine);
		}
		else{
			
			//hLine[h] = true;
			//solve_abc159_e_bfs(h+1, H, ans, hLine)
			hLine[h] = true;
			numHLine += 1;
			ans = solve_abc159_e_bfs(h+1, H, W, K, cmap, ans, hLine, numHLine);

			//hLine[h] = false;
			//solve_abc159_e_bfs(h+1, H, ans, hLine)
			hLine[h] = false;
			numHLine -= 1;
			ans = solve_abc159_e_bfs(h+1, H, W, K, cmap, ans, hLine, numHLine);
			
		}
		return ans;
	}

	public static int solve_abc159_e_width(int numHLine, boolean[] hLine, int H, int W, int K, char[][] cmap){

		int sizeHmap= numHLine + 1;
		int[][] countWC = new int[sizeHmap][W];
		int[] tmpCountWC = new int[sizeHmap];
		boolean addLine = false;
		int countLine = 0;

		int[] idxStart_ = new int[sizeHmap];
		int[] idxEnd_ = new int[sizeHmap];

		int idxHmap_ = 0;
		boolean negative = false;
		
		idxStart_[0] = 0;
		idxEnd_[sizeHmap-1] = H - 1;
		
		for(int h=0;h<H;h++){
			if(hLine[h]){
				idxEnd_[idxHmap_] = h;
				idxStart_[idxHmap_+1] = h + 1;
				idxHmap_++;
			}
		}
		
		/*
		System.out.println("------------------------");
		for(int i=0;i<sizeHmap;i++){
			System.out.println("Start:"+idxStart_[i]+"\t End:"+idxEnd_[i]);
		}
		*/
		
		int idxWPrv = -1;

		for(int w=0;w<W;w++){
			addLine = false;

//			for(Map.Entry<Integer, Integer> ehmap: hmap.entrySet()){
			for(int idxHmap=0;idxHmap<sizeHmap;idxHmap++){

				int idxStart = idxStart_[idxHmap];
				int idxEnd = idxEnd_[idxHmap];

				for(int h=idxStart;h<=idxEnd;h++){
					if(cmap[h][w] == '1') countWC[idxHmap][w]++;
				}
				
//				System.out.println("tmpCountWC:"+ tmpCountWC[idxHmap] + "\t countWC:" + countWC[idxHmap][w]);

				if(tmpCountWC[idxHmap] +  countWC[idxHmap][w] > K && !addLine){
					countLine++;
					addLine = true;
					if(w - idxWPrv == 0) negative = true;
					idxWPrv = w;
				}
				//idxHmap++;
			}
			
			for(int i=0;i<sizeHmap;i++){
				if(addLine){
					tmpCountWC[i] =  countWC[i][w];
				}
				else{
					tmpCountWC[i] +=  countWC[i][w];
				}
			}
		}
		
		if(negative) countLine = Integer.MAX_VALUE;
//		System.out.println("countLine:"+countLine);

		return countLine;
	}
}