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 Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.io.IOException;
import java.io.InputStream;
import java.sql.Array;
import java.util.*;

public class Main {
    public static void main(String[] args) {

        int h = sc.nextInt();
        int w = sc.nextInt();
        int k = sc.nextInt();
        int[][] s = new int[h+1][w+1];
        for (int i=1;i<=h;i++){
            String S = sc.next();
            for (int j=1;j<=w;j++)s[i][j]=S.charAt(j-1)-'0';
        }
        for (int i=2;i<=w;i++)s[1][i]+=s[1][i-1];
        for (int i=2;i<=h;i++)s[i][1]+=s[i-1][1];
        for (int i=2;i<=h;i++)for (int j=2;j<=w;j++)s[i][j]+=s[i-1][j]+s[i][j-1]-s[i-1][j-1];
        int ans = Integer.MAX_VALUE;
        for (int i=0;i<Math.pow(2,h-1);i++){
            String bb = Integer.toBinaryString(i);
            while (bb.length()<h-1)bb="0"+bb;
            HashSet<int[]> set = new HashSet<>();
            int pre = 0;
            int base = 0;
            for (int y=0;y<h-1;y++){
                if (bb.charAt(y)=='1'){
                    set.add(new int[]{pre,y+1});
                    pre=y+1;
                    base++;
                }
            }
            set.add(new int[]{pre,h});
            int pp = 0;
            int count = 0;
            for (int e=0;e<w;e++){
                boolean over = false;
                for (int[] v : set){
                    int white = s[v[1]][e+1]-s[v[0]][e+1]-s[v[1]][pp]+s[v[0]][pp];
                    if (white > k) {
                        over = true;
                        break;
                    }
                }
                if (over){
                    pp=e;
                    count++;
                }
            }
            ans=Math.min(count+base,ans);
        }
        System.out.println(ans);

    }



    public static long MOD = 1000000007;
    public static int tenE9 = 1000000000;
    static Basic basic = new Basic();
    static FastScanner sc = new FastScanner();
    public static class FastScanner {
        private final InputStream in = System.in;
        private final byte[] buffer = new byte[1024];
        private int ptr = 0;
        private int buflen = 0;
        private boolean hasNextByte() {
            if (ptr < buflen) {
                return true;
            }else{
                ptr = 0;
                try {
                    buflen = in.read(buffer);
                } catch (IOException e) {
                    e.printStackTrace();
                }
                if (buflen <= 0) {
                    return false;
                }
            }
            return true;
        }
        private int readByte() { if (hasNextByte()) return buffer[ptr++]; else return -1;}
        private static boolean isPrintableChar(int c) { return 33 <= c && c <= 126;}
        public boolean hasNext() { while(hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++; return hasNextByte();}
        public String next() {
            if (!hasNext()) throw new NoSuchElementException();
            StringBuilder sb = new StringBuilder();
            int b = readByte();
            while(isPrintableChar(b)) {
                sb.appendCodePoint(b);
                b = readByte();
            }
            return sb.toString();
        }
        public long nextLong() {
            if (!hasNext()) throw new NoSuchElementException();
            long n = 0;
            boolean minus = false;
            int b = readByte();
            if (b == '-') {
                minus = true;
                b = readByte();
            }
            if (b < '0' || '9' < b) {
                throw new NumberFormatException();
            }
            while(true){
                if ('0' <= b && b <= '9') {
                    n *= 10;
                    n += b - '0';
                }else if(b == -1 || !isPrintableChar(b)){
                    return minus ? -n : n;
                }else{
                    throw new NumberFormatException();
                }
                b = readByte();
            }
        }
        public int nextInt() {
            long nl = nextLong();
            if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) throw new NumberFormatException();
            return (int) nl;
        }
        public double nextDouble() { return Double.parseDouble(next());}
    }
    public static class Basic {
        //math
        private final int MAX = 510000;
        private final long MOD = Main.MOD;
        private final long[] fac = new long[MAX];
        private final long[] finv = new long[MAX];
        private final long[] inv = new long[MAX];
        public long factorial(long num) {
            if (num < 2) return 1;
            else return num * factorial(num - 1);
        }
        public int arraySum(int[] array) {
            int ans = 0;
            for (int value : array) ans += value;
            return ans;
        }
        public double log(double base, double antilogarithm) {
            return Math.log(antilogarithm) / Math.log(base);
        }
        public long gcd(long x, long y) {
            if (y == 0) return x;
            else return gcd(y, x % y);
        }
        public long lcm(long x, long y) {
            return x / gcd(x, y) * y;
        }
        public HashMap<Long, Long> factorization(long num) {
            HashMap<Long, Long> hash = new HashMap<>();
            long n = num;
            long count = 2;
            while (n > 1) {
                while (n % count == 0) {
                    n /= count;
                    if (hash.containsKey(count)) hash.put(count, hash.get(count) + 1);
                    else hash.put(count, 1L);
                }
                count++;
            }
            return hash;
        }
        public int sum(int[] num) {
            int ans = 0;
            for (int i = 0; i < num.length; i++) {
                ans += num[i];
            }
            return ans;
        }
        public int[] reduce(int small, int big) {
            int ins = 2;
            while (ins <= small) {
                if (small % ins == 0 && big % ins == 0) {
                    small /= ins;
                    big /= ins;
                    ins = 2;
                } else {
                    ins++;
                }
            }
            return new int[]{small, big};
        }
        public int reduceCount(int small, int big) {
            int ins = 2;
            int ans = 0;
            while (ins <= small) {
                if (small % ins == 0 && big % ins == 0) {
                    small /= ins;
                    big /= ins;
                    ins = 2;
                    ans++;
                } else {
                    ins++;
                }
            }
            return ans;
        }
        public int binarySearch(int[] array, int target) {
            int pos = -1;
            int left = 0;
            int right = array.length - 1;
            int middle;
            while (pos == -1 && left <= right) {
                middle = (left + right) / 2;
                if (array[middle] == target) pos = middle;
                else if (array[middle] > target) right = middle - 1;
                else left = middle + 1;
            }
            return pos;
        }
        public long modPow(long a, long n, long mod) {
            long res = 1;
            while (n > 0) {
                if ((n & 1) != 0) res = res * a % mod;
                a = a * a % mod;
                n >>= 1;
            }
            return res;
        }
        public long modinv(long a, long mod) {
            return modPow(a, mod - 2, mod);
        }
        public void COMinit() {
            fac[0] = 1;
            fac[1] = 1;
            finv[0] = 1;
            finv[1] = 1;
            inv[1] = 1;
            for (int i = 2; i < MAX; i++) {
                fac[i] = fac[i - 1] * i % MOD;
                inv[i] = MOD - inv[(int)(MOD % i)] * (MOD / i) % MOD;
                finv[i] = finv[i - 1] * inv[i] % MOD;
            }
        }
        public long COM(int n, int k) {
            if (n < k) return 0;
            if (n < 0 || k < 0) return 0;
            return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
        }
        public long PER(int n, int k){
            return fac[n]*finv[n-k]%MOD;
        }
        public long HCOM(int n, int k) {
            return COM(n + k - 1, k);
        }

        //array
        public int[] remove(int[] ar, int pos) {
            int[] ret = new int[ar.length - 1];
            for (int i = 0; i < pos; i++) ret[i] = ar[i];
            for (int i = pos + 1; i < ar.length; i++) ret[i - 1] = ar[i];
            return ret;
        }
        public int[] nextArray(int size,int cons){
            int[] ret = new int[size];
            for (int i=0;i<size;i++)ret[i]=sc.nextInt()+cons;
            return  ret;
        }
        public int[] reverse(int[] ar){
            int[] ret = new int[ar.length];
            for (int i=0;i<ar.length;i++){
                ret[i]=ar[ar.length-1-i];
            }
            return ret;
        }
        //graph
        public void warshall_floyd(int[][] d) {
            int n = d.length;
            for (int k = 0; k < n; k++)
                for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) d[i][j] = Math.min(d[i][j], d[i][k] + d[k][j]);
        }//隣接行列が引数。辺が無い場合、INFを入力。
        public int[] bellman_ford(int[][] d, int size, int start){
            int e = d.length;
            int[] ans = new int[size];
            Arrays.fill(ans,Integer.MAX_VALUE);
            ans[start]=0;
            for (int i=0;i<size;i++){
                for (int[] a : d) {
                    if (ans[a[1]] > ans[a[0]] + a[2]) {
                        ans[a[1]] = ans[a[0]] + a[2];
                        if (i == size - 1) {
                            System.out.println("negative loop");
                            break;
                        }
                    }
                }
            }
            return ans;
        }//辺を配列の要素として管理。

    }
    public static class Queue<T>{
        private final int max = 1000000;
        private final T[] queue = (T[]) new Object[max];
        private int tail = 0; private int head = 0;
        public void init(){this.head = 0; tail = 0;}
        public boolean isEmpty(){return (head==tail);}
        public boolean isFull(){return (head == (tail+1)%max);}
        public void enqueue(T v){
            if (isFull()){
                System.out.println("error: queue is full.");
                return;
            }
            queue[tail++] = v;
            if (tail == max) tail = 0;
        }
        public T dequeue(){
            if (isEmpty()){
                System.out.println("error: queue is empty");
                return null;
            }
            T res = queue[head];
            ++head;
            if (head == max) head = 0;
            return res;
        }
    }
    public static class Stack<T>{
        private final int max = 100000;
        private final T[] stack =(T[]) new Object[max];
        private int top = 0;
        public boolean isEmpty(){return (top==0);}
        public boolean isFull(){return (top==max);}
        public void init(){top=0;}
        public void push(T v){
            if (isFull()){
                System.out.println("error: stack is full.");
                return;
            }
            stack[top++] = v;
        }
        public T pop(){
            if (isEmpty()){
                System.out.println("error: stack is empty");
                return null;
            }
            return stack[--top];
        }
    }
    public static class Pair<K,V>{
        private K A;
        private V B;
        Pair(K a,V b){
            A=a;
            B=b;
        }
        public void changeA(K a){A=a;}
        public void changeB(V b){B=b;}
        public void changeAB(K a,V b){A=a;B=b;}
        public static Comparator numberSortWithA = new Comparator() {
            @Override
            public int compare(Object o1, Object o2) {
                Pair p = (Pair)o1;
                Pair q = (Pair)o2;
                if ((int)p.A<(int)q.A)return -1;
                else if ((int)p.A==(int)q.A){
                    if ((int)p.B<(int)q.B)return -1;
                    else if ((int)p.B==(int)q.B)return 0;
                    else return 1;
                }
                else return 1;
            }
        };
        public static Comparator numberSortWithB = new Comparator() {
            @Override
            public int compare(Object o1, Object o2) {
                Pair p = (Pair)o1;
                Pair q = (Pair)o2;
                if ((int)p.B<(int)q.B)return -1;
                else if ((int)p.B==(int)q.B){
                    if ((int)p.A<(int)q.A)return -1;
                    else if ((int)p.A==(int)q.A)return 0;
                    else return 1;
                }
                else return 1;
            }
        };
    }
    public static class Graph{
        private final HashMap<Integer,HashSet<Integer>> graph = new HashMap<>();
        private final int Size;
        private int[] Depth;
        Graph(int size){
            for (int i=0;i<size;i++)graph.put(i,new HashSet<>());
            Size = size;
        }

        public void addEdge(int node1,int node2){
            if (graph.containsKey(node1)&&graph.containsKey(node2)){
                graph.get(node1).add(node2);
                graph.get(node2).add(node1);
            }
        }
        public void addDirectEdge(int from,int to){
            if (graph.containsKey(from)&&graph.containsKey(to)){
                graph.get(from).add(to);
            }
        }
        public int[] Depth(int root){
            Queue<Integer> q = new Queue<>();
            q.enqueue(root);
            boolean[] seen = new boolean[Size];
            Arrays.fill(seen, false);
            seen[root]=true;
            int[] depth = new int[Size];
            depth[root]=0;
            while (!q.isEmpty()){
                int v = q.dequeue();
                for (int u : graph.get(v)){
                    if (!seen[u]){
                        seen[u]=true;
                        q.enqueue(u);
                        depth[u]=depth[v]+1;
                    }
                }
            }
            Depth = depth.clone();
            return depth;
        }
    }
    public static class WeightedGraph{
        private final HashMap<Integer,HashMap<Integer,Integer>> graph = new HashMap<>();
        private final int Size;
        WeightedGraph(int size){
            for (int i=0;i<size;i++)graph.put(i,new HashMap<>());
            Size = size;
        }

        public void addEdge(int node1,int node2,int weight){
            if (graph.containsKey(node1)&&graph.containsKey(node2)){
                graph.get(node1).put(node2,weight);
                graph.get(node2).put(node1,weight);
            }
        }
        public void addDirectEdge(int from,int to,int weight){
            if (graph.containsKey(from)&&graph.containsKey(to)){
                graph.get(from).put(to,weight);
            }
        }
        public int[] dijkstra(int root){
            Comparator<int[]> c = new Comparator<int[]>() {
                @Override
                public int compare(int[] o1, int[] o2) {
                    return Integer.compare(o1[1], o2[1]);
                }
            };
            PriorityQueue<int[]> q = new PriorityQueue<>(c);
            q.add(new int[]{root,0});
            int[] ret = new int[Size];
            Arrays.fill(ret,Integer.MAX_VALUE);
            ret[root]=0;
            while (!q.isEmpty()){
                int[] v = q.poll();
                if (v[1]<=ret[v[0]]){
                    HashMap<Integer,Integer> h = graph.get(v[0]);
                    for (int u : h.keySet()){
                        if (ret[u]>v[1]+h.get(u)){
                            ret[u]=v[1]+h.get(u);
                            q.add(new int[]{u,ret[u]});
                        }
                    }
                }
            }
            return ret;
        }

    }
    public static class UnionFindTree{
        int[] parent;
        int[] rank;
        int[] Size;
        int Number;
        public UnionFindTree(int size){
            this.parent = new int[size];
            this.rank = new int[size];
            this.Size = new int[size];
            makeSet();
        }
        public void makeSet(){
            for (int i=0;i<Size.length;i++){
                parent[i]=i;
                rank[i]=0;
                Size[i]=1;
            }
            Number= Size.length;
        }
        public void unite(int x, int y){
            int xRoot = find(x);
            int yRoot = find(y);
            if (rank[xRoot]>rank[yRoot]){
                parent[yRoot] = xRoot;
                Size[xRoot]+=Size[yRoot];
                Number--;
            }else if (rank[xRoot]<rank[yRoot]){
                parent[xRoot] = yRoot;
                Size[yRoot]+=Size[xRoot];
                Number--;
            }else if (xRoot!=yRoot){
                parent[yRoot] = xRoot;
                rank[xRoot]++;
                Size[xRoot]+=Size[yRoot];
                Number--;
            }
        }
        public int find(int i){
            if (i!= parent[i]){
                parent[i]=find(parent[i]);
            }
            return parent[i];
        }
        public boolean same(int x, int y){
            return find(x) == find(y);
        }
        public int getSize(int i){return Size[find(i)];}
        boolean isConnected(){
            return Number<2;
        }
    }
    public static class BitSearch<T>{
        T[] Array;
        int Width;
        int Length;
        BitSearch(T[] t, int width){
            Array = t;
            Width = width;
            Length = Array.length;
        }
        void start(){
            long max = (long)Math.pow(Width,Length);
            for (long i=0;i<max;i++){
                StringBuilder s = new StringBuilder(Long.toString(i, Width));
                while (s.length()<Length) s.insert(0, "0");
                run(s);
            }
        }
        void run(StringBuilder s){

        }
    }

}