Subsequence Weighting


Problem Statement :


A subsequence of a sequence is a sequence which is obtained by deleting zero or more elements from the sequence. 

You are given a sequence A in which every element is a pair of integers  i.e  A = [(a1, w1), (a2, w2),..., (aN, wN)].

For a subseqence B = [(b1, v1), (b2, v2), ...., (bM, vM)] of the given sequence : 

We call it increasing if for every i (1 <= i < M ) , bi < bi+1.
Weight(B) = v1 + v2 + ... + vM.
Task:
Given a sequence, output the maximum weight formed by an increasing subsequence.

Input:
The first line of input contains a single integer T. T test-cases follow. The first line of each test-case contains an integer N. The next line contains a1, a2 ,... , aN separated by a single space. The next line contains w1, w2, ..., wN separated by a single space.

Output:
For each test-case output a single integer: The maximum weight of increasing subsequences of the given sequence.

Constraints:
1 <= T <= 5
1 <= N <= 150000
1 <= ai <= 109, where i ∈ [1..N]
1 <= wi <= 109, where i ∈ [1..N]

Sample Input:

2  
4  
1 2 3 4  
10 20 30 40  
8  
1 2 3 4 1 2 3 4  
10 20 30 40 15 15 15 50
Sample Output:

100  
110



Solution :



title-img


                            Solution in C :

In  C++ :





#include <vector>
#include <list>
#include <map>
#include <set>
#include <deque>
#include <queue>
#include <stack>
#include <bitset>
#include <algorithm>
#include <functional>
#include <numeric>
#include <utility>
#include <sstream>
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <ctime>
#include <cstring>
#include <climits>

using namespace std;

#define GI ({int new_input;scanf("%d",&new_input);new_input;})
typedef unsigned long long ll;



ll Tree[800000];
void updateTree(int b, int e, int p, ll  val, int idx=1) {
	if(p < b || p > e) return ;
	if(p == b && p == e){ 
		Tree[idx] = max(Tree[idx],val);
		return ;
	}
	int mid = (b+e)/2;
	int lt = (idx<<1);
	int rt = ((idx<<1)+1);
	updateTree(b, mid, p, val, lt);
	updateTree(mid+1, e, p, val, rt);
	Tree[idx] = max(Tree[lt], Tree[rt]);
	return ;
}
ll query(int b,int e,int start,int end,int node){
	if(e<start || b>end)return 0;
	if(b<=start && e>=end)return Tree[node];
	int mid=(start+end)>>1;
	return max(query(b,e,start,mid,node*2),query(b,e,mid+1,end,node*2+1));	
}
ll input[200000];
ll w[200000];
map<ll,int>m;
set<ll>s;
int main() {
    int t=GI;
    while(t--){
		m.clear();s.clear();
		s.empty();
		memset(Tree,0,sizeof Tree);
		int n=GI;
		for(int i=0;i<n;i++){
			scanf("%lld",&input[i]);
			s.insert(input[i]);
		}
		for(int i=0;i<n;i++){
			scanf("%lld",&w[i]);
		}
		int in=1;
		set<ll>::iterator it;
		for(it=s.begin();it!=s.end();it++){
			m[*it]=in;
			in++;
		}in--;
		ll ans=0;
		for(int i=0;i<n;i++){
			int mapped=m[input[i]];
			if(mapped==1){
				updateTree(1,in,mapped,w[i],1);
				ans=max(ans,w[i]);
			}
			else{
				ll get=query(1,mapped-1,1,in,1);
				ans=max(ans,get+w[i]);
				updateTree(1,in,mapped,w[i]+get,1);
			}
		}
		cout<<ans<<endl;
	}
    return  0;
}







In Java :







import java.util.Map.Entry;
import java.util.ArrayList;
import java.util.List;
import java.util.Scanner;
import java.util.SortedMap;
import java.util.TreeMap;

public class Solution {

	public static void main(String[] args) {

		Scanner sc = new Scanner(System.in);
		int  nProb = sc.nextInt();
		for(int k = 1; k <= nProb; ++k) {
			int n = sc.nextInt();
			int[] a = new int[n];
			int[] w = new int[n];
			for(int i = 0; i < n; ++i)
				a[i] = sc.nextInt();
			for(int i = 0; i < n; ++i)
				w[i] = sc.nextInt();
			long bestW = solve(a, w);
			System.out.println(bestW);
		}
	}

	private static long solve(int[] a, int[] w) {
		int n = a.length;
		long best = 0;
		TreeMap<Integer, Long> map = new TreeMap<Integer, Long>();
		for(int k = 0; k < n; ++k) {
			Entry<Integer, Long> e = map.lowerEntry(a[k]);
			long b = (e == null ? 0 : e.getValue()) + w[k];
			SortedMap<Integer, Long> tail = map.tailMap(a[k]);
			List<Integer> del = new ArrayList<Integer>();
			for(Entry<Integer, Long> x : tail.entrySet()) {
				if(x.getValue().longValue() > b)
					break;
				del.add(x.getKey());
			}
			for(Integer i : del) {
				map.remove(i);
			}
			if(!map.containsKey(a[k]))
				map.put(a[k], b);
			if(best < b)
				best = b;
		}
		return best;
	}
}









In C :






#include <stdio.h>
#include <stdlib.h>
#include <time.h>

typedef struct treap {
    int x, p;
    long long y;
    struct treap *l, *r;
}* Treap;

Treap td = NULL;

Treap newTreap(int x, long long y) {
    Treap t;
    if (td) {
        t = td;
        td = td->r;
    }
    else
        t = (Treap) malloc(sizeof(struct treap));
    t->x = x;
    t->y = y;
    t->p = rand();
    t->l = t->r = NULL;
    return t;
}

void dump(Treap t) {
    if (t) {
        dump(t->l);
        dump(t->r);
        t->r = td;
        td = t;
    }
}

Treap merge(Treap l, Treap r) {
    if (!l)
        return r;
    if (!r)
        return l;
    if (l->p > r->p) {
        l->r = merge(l->r, r);
        return l;
    }
    r->l = merge(l, r->l);
    return r;
}

void split(Treap t, Treap *l, Treap *r, long long v, int d) {
    if (!t)
        *l = *r = NULL;
    else if (d ? (t->x < v) : (t->y <= v)) {
        split(t->r, &t->r, r, v, d);
        *l = t;
    }
    else {
        split(t->l, l, &t->l, v, d);
        *r = t;
    }
}

Treap rightmost(Treap t) {
    if (t) {
        while (t->r)
            t = t->r;
    }
    return t;
}

Treap leftmost(Treap t) {
    if (t) {
        while (t->l)
            t = t->l;
    }
    return t;
}

long long solve() {
    int n, i;
    long long v;
    scanf("%d", &n);
    int a[n], w[n];
    for (i = -1; ++i < n; scanf("%d", a + i));
    for (i = -1; ++i < n; scanf("%d", w + i));
    Treap r = newTreap(0, 0), l, m;
    for (i = -1; ++i < n;) {
        split(r, &l, &r, a[i], 1);
        m = rightmost(l);
        v = w[i] + m->y;
        split(r, &m, &r, v, 0);
        if (m) {
            dump(m);
            m = newTreap(a[i], v);
        }
        else {
            m = leftmost(r);
            if (!m || m->x > a[i])
                m = newTreap(a[i], v);
            else
                m = NULL;
        }
        l = merge(l, m);
        r = merge(l, r);
    }
    v = rightmost(r)->y;
    dump(r);
    return v;
}

int main() {
    srand(time(NULL));
    int t;
    scanf("%d", &t);
    while (t--)
        printf("%lld\n", solve());
    return 0;
}









In Python3 :






import os
import sys
import bisect
# Complete the solve function below.
def solve(a, w):
    b = [[0,0],[10000000000,10000000000]]
    for i in range(len(a)):
        g = [a[i],w[i]]
        bisect.insort(b,g)
        ind = b.index(g)
        if b[ind+1][0] != b[ind][0] and b[ind-1][0] != b[ind][0]:
            b[ind][1]+=b[ind-1][1]
            for j in range(ind+1,len(b)):
                if b[j][1] >b[ind][1]:
                    break
            b = b[:ind+1] + b[j:]
        elif b[ind+1][0] == b[ind][0]:
            b[ind][1]+=b[ind-1][1]
            if b[ind+1][1]>=b[ind][1]:
                b.remove(b[ind])
            else:
                b.remove(b[ind+1])
                for j in range(ind+1,len(b)):
                    if b[j][1]>b[ind][1]:
                        break
                b = b[: ind+1] + b[j: ]
        elif b[ind-1][0] ==b[ind][0]:
            b[ind][1] += b[ind-2][1]
            if b[ind-1][1] >= b[ind][1]:
                b.remove(b[ind])
            else:
                for j in range(ind+1,len(b)):
                     if b[j][1]>b[ind][1]:
                        break
                b = b[: ind+1] + b[j: ]
                b.remove(b[ind-1])
    return b[-2][1]
if __name__ == '__main__':
    fptr = open(os.environ['OUTPUT_PATH'], 'w')
    t = int(input())
    for t_itr in range(t):
        n = int(input())
        a = list(map(int, input().rstrip().split()))
        w = list(map(int, input().rstrip().split()))
        result = solve(a, w)
        fptr.write(str(result) + '\n')
    fptr.close()
                        








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