Problem:
Huge Knapsack Problem

You have N items that you want to put them into a knapsack. Item i has value vi and weight wi.

You want to find a subset of items to put such that:

The total value of the items is as large as possible.

The items have combined weight at most W, that is capacity of the knapsack.

Find the maximum total value of items in the knapsack.

Input

N W
v1 w1
v2 w2
:
vN wN

The first line consists of the integers N and W. In the following N lines, the value and weight of the i-th item are given.

Output

Print the maximum total values of the items in a line.

Constraints

1 ≤ N ≤ 40

1 ≤ vi ≤ 1015

1 ≤ wi ≤ 1015

1 ≤ W ≤ 1015

Sample Input 1

4 5
4 2
5 2
2 1
8 3

Sample Output 1

13

Sample Input 2

2 20
5 9
4 10

Sample Output 2

9

Solution in JAVA (judge verdict: Wrong Answer)

import java.io.IOException;
import java.io.InputStream;
import java.util.ArrayList;
import java.util.Collections;
import java.util.NoSuchElementException;
 
public class Main {
	private static FastScanner sc = new FastScanner();
	
	public static void main(String[] args) {
		int N = sc.nextInt();
		long W = sc.nextLong();
		long[] v = new long[N];
		long[] w = new long[N];
		for(int i=0; i<N; i++) {
			v[i] = sc.nextLong();
			w[i] = sc.nextLong();
		}
		
		if(N <= 20) {
			long max = 0;
			for(int i=0; i<Math.pow(2, N); i++) {
				long vsum = 0;
				long wsum = 0;
				for(int j=0; j<N; j++) {
					if((i >> j & 1) == 1) {
						vsum += v[j];
						wsum += w[j];
					}
				}
				if(wsum <= W && vsum > max) {
					max = vsum;
				}
			}
			System.out.println(max);
		} else {
			ArrayList<Set> set1 = new ArrayList<>();
			for(int i=0; i<Math.pow(2, 20); i++) {
				long vsum = 0;
				long wsum = 0;
				for(int j=0; j<20; j++) {
					if((i >> j & 1) == 1) {
						vsum += v[j];
						wsum += w[j];
					}
				}
				set1.add(new Set(vsum, wsum));
			}
			ArrayList<Set> set2 = new ArrayList<>();
			for(int i=0; i<Math.pow(2, N-20); i++) {
				long vsum = 0;
				long wsum = 0;
				for(int j=0; j<N-20; j++) {
					if((i >> j & 1) == 1) {
						vsum += v[j+20];
						wsum += w[j+20];
					}
				}
				set2.add(new Set(vsum, wsum));
			}
			Collections.sort(set2, (a, b) -> a.wsum > b.wsum ? -1 : 1);
			long temp = 0;
			for(int i=set2.size()-1; i>=0; i--) {
				if(set2.get(i).vsum > temp) {
					temp = set2.get(i).vsum;
				} else if(set2.get(i).vsum < temp) {
					set2.get(i).vsum = temp;
				}
			}
			long max = 0;
			for(int i=0; i<set1.size(); i++) {
				long wrest = W - set1.get(i).wsum;
				int l = 0;
				int r = set2.size() - 1;
				int m = (l + r) / 2;
				while(l < r) {
					if(set2.get(m).wsum > wrest) {
						r = m - 1;
					} else if(set2.get(m).wsum < wrest) {
						if(set2.get(m+1).wsum > wrest) {
							break;
						} else {
							l = m + 1;
						}
					} else {
						while(true) {
							if(set2.get(m+1).wsum == set2.get(m).wsum) {
								m++;
							} else {
								break;
							}
						}
						break;
					}
					m = (l + r) / 2;
				}
				if(set1.get(i).vsum + set2.get(m).vsum > max) {
					max = set1.get(i).vsum + set2.get(m).vsum;
				}
			}
			System.out.println(max);
		}
	}
	
	static class Set {
		long vsum;
		long wsum;
		Set(long vsum, long wsum) {
			this.vsum = vsum;
			this.wsum = wsum;
		}
	}
	
	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;}
        private void skipUnprintable() { while(hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;}
        public boolean hasNext() { skipUnprintable(); 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() {
            return Long.parseLong(next());
        }
        public int nextInt(){
            return Integer.parseInt(next());
        }
        public double nextDouble(){
            return Double.parseDouble(next());
        }
    }
}