Queries to check if any pair exists in an array having values at most equal to the given pair

Given a vector of pairs arr[] and Q queries in the form of pairs in an array Queries[], the task for each query is to check if there exists any pair with both the values smaller than those in the pair of the current query. If found to be true, then print “Yes”. Otherwise, print “No”.

Examples:

Input: arr[][] = {{3, 5}, {2, 7}, {2, 3}, {4, 9}}, Queries[][] = {{3, 4}, {3, 2}, {4, 1}, {3, 7}}
Output:
Yes
No
No
Yes
Explanation:
Query 1: Pair {2, 3} exists in arr[] which has both the values smaller than {3, 4}.
Query 2: No valid pair found in arr[] for {3, 2}.
Query 3: No valid pair found in arr[] for {4, 1}.
Query 4: Pair {2, 7} exists in arr[] for {3, 7}.

Input: arr[][] = {{2, 1}, {4, 2}, {4, 4}, {7, 2}}, Queries[][] = {{2, 1}, {1, 1}}
Output:
Yes
No

Naive Approach: The simplest approach is to traverse the array Queries[][] and for each pair, traverse the given array of pairs and check if there exists any such pair whose corresponding values is greater than or equal to the pair {p1, p2} then print “Yes”. Otherwise, print “No”. 

Time Complexity: O(N*K)
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach, the idea is to use Binary Search. Follow the steps below to solve this problem:

• Sort the array of pairs with respect to first element in the pairs in increasing order. If there exist 2 pairs whose first values are the same, then pairs are arranged on the basis of the second element of the pair.
• After sorting, traverse the array of pairs, and for all pairs having the same first value, replace the second value with the minimum of all the pairs having the same first value.
• Now, traverse the given Queries[] arrays and perform binary search on the array arr[] for each pair in it.
• If the pairs obtained from the above steps is smaller than the current pair in the Query[], then print “Yes” else print “No”.

Below is the implementation of the above approach:

C++14

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function that performs binary search
// to find value less than or equal to
// first value of the pair
int binary_search(vector<pair<int, int> > vec,
                  int n, int a)
{
    int low, high, mid;
    low = 0;
    high = n - 1;

    // Perform binary search
    while (low < high) {
        // Find the mid
        mid = low + (high - low + 1) / 2;

        // Update the high
        if (vec[mid].first > a) {
            high = mid - 1;
        }

        // Else update low
        else if (vec[mid].first <= a) {
            low = mid;
        }
    }

    // Return the low index
    return low;
}

// Function to modify the second
// value of each pair
void modify_vec(
    vector<pair<int, int> >& v, int n)
{
    // start from index 1
    for (int i = 1; i < n; i++) {
        v[i].second
            = min(v[i].second,
                  v[i - 1].second);
    }
}

// Function to evaluate each query
int evaluate_query(vector<pair<int, int> > v,
                   int n, int m1,
                   int m2)
{
    // Find value less than or equal to
    // the first value of pair
    int temp = binary_search(v, n, m1);

    // check if we got the required
    // pair or not
    if ((v[temp].first <= m1)
        && (v[temp].second <= m2)) {
        return 1;
    }

    return 0;
}

// Function to find a pair whose values is
// less than the given pairs in query
void checkPairs(vector<pair<int, int> >& v,
                vector<pair<int, int> >& queries)
{

    // Find the size of the vector
    int n = v.size();

    // sort the vector based on
    // the first value
    sort(v.begin(), v.end());

    // Function Call to modify the
    // second value of each pair
    modify_vec(v, n);

    int k = queries.size();

    // Traverse each queries
    for (int i = 0; i < k; i++) {
        int m1 = queries[i].first;
        int m2 = queries[i].second;

        // Evaluate each query
        int result
            = evaluate_query(v, n, m1, m2);

        // Print the result
        if (result > 0)
            cout << "Yes\n";
        else
            cout << "No\n";
    }
}

// Driver Code
int main()
{
    vector<pair<int, int> > arr
        = { { 3, 5 }, { 2, 7 }, { 2, 3 }, { 4, 9 } };

    vector<pair<int, int> > queries
        = { { 3, 4 }, { 3, 2 }, { 4, 1 }, { 3, 7 } };

    // Function Call
    checkPairs(arr, queries);

    return 0;
}

Java

// Java program for above approach
import java.util.*;
import java.lang.*;
class GFG{

  // Function that performs binary search
  // to find value less than or equal to
  // first value of the pair
  static int binary_search(int[][] vec,
                           int n, int a)
  {
    int low, high, mid;
    low = 0;
    high = n - 1;

    // Perform binary search
    while (low < high)
    {

      // Find the mid
      mid = low + (high - low + 1) / 2;

      // Update the high
      if (vec[mid][0] > a)
      {
        high = mid - 1;
      }

      // Else update low
      else if (vec[mid][1] <= a)
      {
        low = mid;
      }
    }

    // Return the low index
    return low;
  }

  // Function to modify the second
  // value of each pair
  static void modify_vec(
    int[][] v, int n)
  {
    // start from index 1
    for (int i = 1; i < n; i++)
    {
      v[i][1] =  Math.min(v[i][1],
                          v[i - 1][1]);
    }
  }

  // Function to evaluate each query
  static int evaluate_query(int[][] v,
                            int n, int m1,
                            int m2)
  {

    // Find value less than or equal to
    // the first value of pair
    int temp = binary_search(v, n, m1);

    // check if we got the required
    // pair or not
    if ((v[temp][0] <= m1)
        && (v[temp][1] <= m2))
    {
      return 1;
    }

    return 0;
  }

  // Function to find a pair whose values is
  // less than the given pairs in query
  static void checkPairs(int[][] v,
                         int[][] queries)
  {

    // Find the size of the vector
    int n = v.length;

    // sort the vector based on
    // the first value
    Arrays.sort(v, (a, b)->a[0]-b[0]);

    // Function Call to modify the
    // second value of each pair
    modify_vec(v, n);
    int k = queries.length;

    // Traverse each queries
    for (int i = 0; i < k; i++)
    {
      int m1 = queries[i][0];
      int m2 = queries[i][1];

      // Evaluate each query
      int result
        = evaluate_query(v, n, m1, m2);

      // Print the result
      if (result > 0)
        System.out.println("Yes");
      else
        System.out.println("No");
    }
  }


  // Driver function
  public static void main (String[] args)
  {
    int[][] arr
      = { { 3, 5 }, { 2, 7 }, { 2, 3 }, { 4, 9 } };

    int[][] queries
      = { { 3, 4 }, { 3, 2 }, { 4, 1 }, { 3, 7 } };

    // Function Call
    checkPairs(arr, queries);
  }
}

// This code is contributed by offbeat

Python3

# Python3 program for the above approach

# Function that performs binary search
# to find value less than or equal to
# first value of the pair
def binary_search(vec, n, a):
    low, high, mid = 0, 0, 0
    low = 0
    high = n - 1

    # Perform binary search
    while (low < high):

      # Find the mid
        mid = low + (high - low + 1) // 2

        # Update the high
        if (vec[mid][0] > a):
            high = mid - 1

        # Else update low
        elif vec[mid][0] <= a:
            low = mid

    # Return the low index
    return low

# Function to modify the second
# value of each pair
def modify_vec(v, n):

    # start from index 1
    for i in range(n):
        v[i][1] = min(v[i][1], v[i - 1][1])

    return v

# Function to evaluate each query
def evaluate_query(v, n, m1, m2):

    # Find value less than or equal to
    # the first value of pair
    temp = binary_search(v, n, m1)

    # check if we got the required
    # pair or not
    if ((v[temp][0] <= m1)
        and (v[temp][1] <= m2)):
        return 1

    return 0

# Function to find a pair whose values is
# less than the given pairs in query
def checkPairs(v, queries):

    # Find the size of the vector
    n = len(v)

    # sort the vector based on
    # the first value
    v = sorted(v)

    # Function Call to modify the
    # second value of each pair
    v = modify_vec(v, n)

    k = len(queries)

    # Traverse each queries
    for i in range(k):
        m1 = queries[i][0]
        m2 = queries[i][1]

        # Evaluate each query
        result = evaluate_query(v, n, m1, m2)

        # Print the result
        if (result > 0):
            print("Yes")
        else:
            print("No")

# Driver Code
if __name__ == '__main__':
    arr= [ [ 3, 5 ], [ 2, 7 ], [ 2, 3 ], [ 4, 9 ] ]

    queries = [ [ 3, 4 ], [ 3, 2 ], [ 4, 1 ], [ 3, 7 ] ]

    # Function Call
    checkPairs(arr, queries)

# This code is contributed by mohit kumar 29

C#

// C# program for above approach
using System;
using System.Collections.Generic;

class GFG{


  // Function that performs binary search
  // to find value less than or equal to
  // first value of the pair
  static int binary_search(int[][] vec,
                           int n, int a)
  {
    int low, high, mid;
    low = 0;
    high = n - 1;

    // Perform binary search
    while (low < high)
    {

      // Find the mid
      mid = low + (high - low + 1) / 2;

      // Update the high
      if (vec[mid][0] > a)
      {
        high = mid - 1;
      }

      // Else update low
      else if (vec[mid][1] <= a)
      {
        low = mid;
      }
    }

    // Return the low index
    return low;
  }

  // Function to modify the second
  // value of each pair
  static void modify_vec(
    int[][] v, int n)
  {
    // start from index 1
    for (int i = 1; i < n; i++)
    {
      v[i][1] =  Math.Min(v[i][1],
                          v[i - 1][1]);
    }
  }

  // Function to evaluate each query
  static int evaluate_query(int[][] v,
                            int n, int m1,
                            int m2)
  {

    // Find value less than or equal to
    // the first value of pair
    int temp = binary_search(v, n, m1);

    // check if we got the required
    // pair or not
    if ((v[temp][0] <= m1)
        && (v[temp][1] <= m2))
    {
      return 1;
    }

    return 0;
  }


  static int compare(int[] a, int[] b)
  {
    if (a[0] < b[0])
      return -1;
    if (a[0] == b[0])
      return 0;
    return 1;
  }
  // Function to find a pair whose values is
  // less than the given pairs in query
  static void checkPairs(int[][] v,
                         int[][] queries)
  {

    // Find the size of the vector
    int n = v.Length;

    // sort the vector based on
    // the first value
    Array.Sort(v, compare);

    // Function Call to modify the
    // second value of each pair
    modify_vec(v, n);
    int k = queries.Length;

    // Traverse each queries
    for (int i = 0; i < k; i++)
    {
      int m1 = queries[i][0];
      int m2 = queries[i][1];

      // Evaluate each query
      int result
        = evaluate_query(v, n, m1, m2);

      // Print the result
      if (result > 0)
        Console.WriteLine("Yes");
      else
        Console.WriteLine("No");
    }
  }


  // Driver function
  public static void Main (string[] args)
  {
    int[][] arr
      = { new int[] { 3, 5 }, new int[]  { 2, 7 }, new int[] { 2, 3 }, new int[] { 4, 9 } };

    int[][] queries
      = { new int[] { 3, 4 }, new int[] { 3, 2 }, new int[] { 4, 1 }, new int[] { 3, 7 } };

    // Function Call
    checkPairs(arr, queries);
  }
}

// This code is contributed by phasing17

Javascript

<script>
// Javascript program for above approach

// Function that performs binary search
  // to find value less than or equal to
  // first value of the pair
function binary_search(vec,a,n)
{
    let low, high, mid;
    low = 0;
    high = n - 1;

    // Perform binary search
    while (low < high)
    {

      // Find the mid
      mid = low + Math.floor((high - low + 1) / 2);

      // Update the high
      if (vec[mid][0] > a)
      {
        high = mid - 1;
      }

      // Else update low
      else if (vec[mid][1] <= a)
      {
        low = mid;
      }
    }

    // Return the low index
    return low;
}

 // Function to modify the second
  // value of each pair
function modify_vec(v,n)
{
    // start from index 1
    for (let i = 1; i < n; i++)
    {
      v[i][1] =  Math.min(v[i][1],
                          v[i - 1][1]);
    }
}

// Function to evaluate each query
function evaluate_query(v,n,m1,m2)
{
    // Find value less than or equal to
    // the first value of pair
    let temp = binary_search(v, n, m1);

    // check if we got the required
    // pair or not
    if ((v[temp][0] <= m1)
        && (v[temp][1] <= m2))
    {
      return 1;
    }

    return 0;
}

 // Function to find a pair whose values is
  // less than the given pairs in query
function checkPairs(v, queries)
{
    // Find the size of the vector
    let n = v.length;

    // sort the vector based on
    // the first value
    v.sort(function(a, b){return a[0]-b[0]});

    // Function Call to modify the
    // second value of each pair
    modify_vec(v, n);
    let k = queries.length;

    // Traverse each queries
    for (let i = 0; i < k; i++)
    {
      let m1 = queries[i][0];
      let m2 = queries[i][1];

      // Evaluate each query
      let result
        = evaluate_query(v, n, m1, m2);

      // Print the result
      if (result > 0)
        document.write("Yes<br>");
      else
        document.write("No<br>");
    }
}

// Driver function
let arr=[[ 3, 5 ], [ 2, 7 ], [ 2, 3 ], [ 4, 9 ]];

let queries=[[ 3, 4 ], [ 3, 2 ], [ 4, 1 ], [ 3, 7 ]];

// Function Call
checkPairs(arr, queries);

// This code is contributed by unknown2108
</script>

Output:

Yes
No
No
Yes

Time Complexity: O(Q*log N)
Auxiliary Space: O(1)