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Find longest bitonic sequence such that increasing and decreasing parts are from two different arrays

We are given two arrays, we need to find the longest possible bitonic sequence such that increasing part must be from first array and should be a subsequence of first array. Similarly, decreasing part of must be from second array and should be a subsequence of it.

Examples:

Input : arr1[] = {1, 5, 2, 4, 3, 5}, 
        arr2[] = {8, 6, 4, 7, 3, 2}
Output : 1, 2, 4, 5, 8, 6, 4, 3, 2

Input : arr1[] = {2, 0, 1, 3, 4}, 
        arr2[] = {5, 3, 2, 1}
Output : 0, 1, 2, 3, 4, 5, 3, 2, 1


The idea is to use largest increasing sequence from array1 and largest decreasing sequence from array2 and then combine both to get our result.

// CPP to find largest bitonic sequence such that
#include <bits/stdc++.h>
using namespace std;
  
vector<int> res;
  
// utility Binary search
int GetCeilIndex(int arr[], vector<int>& T, int l, 
                                   int r, int key)
{
    while (r - l > 1) {
        int m = l + (r - l) / 2;
        if (arr[T[m]] >= key)
            r = m;
        else
            l = m;
    }
    return r;
}
  
// function to find LIS in reverse form
void LIS(int arr[], int n)
{
    // Add boundary case, when array n is zero
    // Depend on smart pointers
    vector<int> tailIndices(n, 0); // Initialized with 0
    vector<int> prevIndices(n, -1); // initialized with -1
  
    int len = 1; // it will always point to empty location
    for (int i = 1; i < n; i++) {
  
        // new smallest value
        if (arr[i] < arr[tailIndices[0]])             
            tailIndices[0] = i;
  
        // arr[i] wants to extend largest subsequence
        else if (arr[i] > arr[tailIndices[len - 1]]) {            
            prevIndices[i] = tailIndices[len - 1];
            tailIndices[len++] = i;
        
           
        // arr[i] wants to be a potential candidate of
        // future subsequence
        // It will replace ceil value in tailIndices
        else {
            int pos = GetCeilIndex(arr, tailIndices, -1, 
                                        len - 1, arr[i]);
            prevIndices[i] = tailIndices[pos - 1];
            tailIndices[pos] = i;
        }
    }
  
    // put LIS into vector
    for (int i = tailIndices[len - 1]; i >= 0; i = prevIndices[i])
        res.push_back(arr[i]);
}
  
// function for finding longest bitonic seq
void longestBitonic(int arr1[], int n1, int arr2[], int n2)
{
    // find LIS of array 1 in reverse form
    LIS(arr1, n1);
  
    // reverse res to get LIS of first array
    reverse(res.begin(), res.end());
  
    // reverse array2 and find its LIS
    reverse(arr2, arr2 + n2);
    LIS(arr2, n2);
  
    // print result
    for (int i = 0; i < res.size(); i++)
        cout << res[i] << " ";
}
  
// driver preogram
int main()
{
    int arr1[] = { 1, 2, 4, 3, 2 };
    int arr2[] = { 8, 6, 4, 7, 8, 9 };
    int n1 = sizeof(arr1) / sizeof(arr1[0]);
    int n2 = sizeof(arr2) / sizeof(arr2[0]);
    longestBitonic(arr1, n1, arr2, n2);
    return 0;
}

Output:

1 2 3 8 6 4

Time Complexity : O(n Log n)
Please note that O(n Log n) implementation of LIS is used

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



This article is attributed to GeeksforGeeks.org

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