Queries for the minimum element in an array excluding the given index range

Given an array arr[] of N integers and Q queries where each query consists of an index range [L, R]. For each query, the task is to find the minimum element in the array excluding the elements from the given index range.

Examples:

Input: arr[] = {3, 2, 1, 4, 5}, Q[][] = {{1, 2}, {2, 3}}
Output:
3
2
Query 1: min(arr[0], arr[3..4]) = min(3, 4, 5) = 3
Query 2: min(arr[0..1], arr[4]) = min(3, 2, 5) = 2



Input: arr[] = {1, 2, 3, 4, 5}, Q[][] = {{0, 2}}
Output:
4

Approach: In case of multiple queries a segment tree can be built to find the minimum element in any index range. Now, for every query [L, R] the minimum element excluding this range will be min(min(arr[0…L-1]), min(arr[R+1…N-1]))

Below is the implementation of the above approach:

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// A utility function to get minimum of two numbers
int minVal(int x, int y) { return (x < y) ? x : y; }
  
// A utility function to get the
// middle index from corner indexes.
int getMid(int s, int e) { return s + (e - s) / 2; }
  
/* A recursive function to get the 
minimum value in a given range 
of array indexes. The following 
are parameters for this function. 
  
    st --> Pointer to segment tree 
    index --> Index of current node in the 
        segment tree. Initially 0 is 
        passed as root is always at index 0 
    ss & se --> Starting and ending indexes 
                of the segment represented 
                by current node, i.e., st[index] 
    qs & qe --> Starting and ending indexes of query range */
int RMQUtil(int* st, int ss, int se, int qs, int qe, int index)
{
    // If segment of this node is a part
    // of given range, then return
    // the min of the segment
    if (qs <= ss && qe >= se)
        return st[index];
  
    // If segment of this node
    // is outside the given range
    if (se < qs || ss > qe)
        return INT_MAX;
  
    // If a part of this segment
    // overlaps with the given range
    int mid = getMid(ss, se);
    return minVal(RMQUtil(st, ss, mid, qs, qe, 2 * index + 1),
                  RMQUtil(st, mid + 1, se, qs, qe, 2 * index + 2));
}
  
// Return minimum of elements in range
// from index qs (quey start) to
// qe (query end). It mainly uses RMQUtil()
int RMQ(int* st, int n, int qs, int qe)
{
    // Check for erroneous input values
    if (qs < 0 || qe > n - 1 || qs > qe) {
        cout << "Invalid Input";
        return -1;
    }
  
    return RMQUtil(st, 0, n - 1, qs, qe, 0);
}
  
// A recursive function that constructs
// Segment Tree for array[ss..se].
// si is index of current node in segment tree st
int constructSTUtil(int arr[], int ss, int se,
                    int* st, int si)
{
    // If there is one element in array,
    // store it in current node of
    // segment tree and return
    if (ss == se) {
        st[si] = arr[ss];
        return arr[ss];
    }
  
    // If there are more than one elements,
    // then recur for left and right subtrees
    // and store the minimum of two values in this node
    int mid = getMid(ss, se);
    st[si] = minVal(constructSTUtil(arr, ss, mid, st, si * 2 + 1),
                    constructSTUtil(arr, mid + 1, se, st, si * 2 + 2));
    return st[si];
}
  
/* Function to construct segment tree 
from given array. This function allocates 
memory for segment tree and calls constructSTUtil() to 
fill the allocated memory */
int* constructST(int arr[], int n)
{
    // Allocate memory for segment tree
  
    // Height of segment tree
    int x = (int)(ceil(log2(n)));
  
    // Maximum size of segment tree
    int max_size = 2 * (int)pow(2, x) - 1;
  
    int* st = new int[max_size];
  
    // Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0);
  
    // Return the constructed segment tree
    return st;
}
  
// Driver program to test above functions
int main()
{
    int arr[] = { 3, 2, 1, 4, 5 };
    int n = sizeof(arr) / sizeof(arr[0]);
  
    int queries[][2] = { { 1, 2 }, { 2, 3 } };
    int q = sizeof(queries) / sizeof(queries[0]);
  
    // Build segment tree from given array
    int* st = constructST(arr, n);
  
    // Perform queries
    for (int i = 0; i < q; i++) {
  
        // Current index range to be exluded
        int L = queries[i][0];
        int R = queries[i][1];
  
        // Minimum in the left part
        int left = ((L - 1) < 0)
                       ? INT_MAX
                       : RMQ(st, n, 0, L - 1);
  
        // Minimum in the right part
        int right = ((R + 1) >= n)
                        ? INT_MAX
                        : RMQ(st, n, R + 1, n - 1);
  
        // Minimum in the array excluding the given range
        int minn = min(left, right);
  
        // Complete array has been excluded
        if (minn == INT_MAX)
            cout << -1 << endl;
        else
            cout << minn << endl;
    }
  
    return 0;
}

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Output:

3
2


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Second year Department of Information Technology Jadavpur University

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