Javascript Program to Find the Longest Bitonic Subsequence | DP-15

Last Updated : 18 Sep, 2024

Given an array arr[0 … n-1] containing n positive integers, a subsequence of arr[] is called Bitonic if it is first increasing, then decreasing. Write a function that takes an array as argument and returns the length of the longest bitonic subsequence.
A sequence, sorted in increasing order is considered Bitonic with the decreasing part as empty. Similarly, decreasing order sequence is considered Bitonic with the increasing part as empty.

Examples:

Input: arr[] = {1, 11, 2, 10, 4, 5, 2, 1};
Output: 6 (A Longest Bitonic Subsequence of length 6 is 1, 2, 10, 4, 2, 1)

Input: arr[] = {12, 11, 40, 5, 3, 1}
Output: 5 (A Longest Bitonic Subsequence of length 5 is 12, 11, 5, 3, 1)

Input: arr[] = {80, 60, 30, 40, 20, 10}
Output: 5 (A Longest Bitonic Subsequence of length 5 is 80, 60, 30, 20, 10)

Source: Microsoft Interview Question

This problem is a variation of standard Longest Increasing Subsequence (LIS) problem. Let the input array be arr[] of length n. We need to construct two arrays lis[] and lds[] using Dynamic Programming solution of LIS problem. lis[i] stores the length of the Longest Increasing subsequence ending with arr[i]. lds[i] stores the length of the longest Decreasing subsequence starting from arr[i]. Finally, we need to return the max value of lis[i] + lds[i] – 1 where i is from 0 to n-1.

Following is the implementation of the above Dynamic Programming solution.

JavaScript

/* Dynamic Programming implementation in JavaScript for longest bitonic 
   subsequence problem */

/* lbs() returns the length of the Longest Bitonic Subsequence in 
arr[] of size n. The function mainly creates two temporary arrays 
lis[] and lds[] and returns the maximum lis[i] + lds[i] - 1. 
 
lis[i] ==> Longest Increasing subsequence ending with arr[i] 
lds[i] ==> Longest decreasing subsequence starting with arr[i] 
*/
function lbs(arr, n) {
    let i, j;
    /* Allocate memory for LIS[] and initialize LIS values as 1 for 
        all indexes */
    let lis = new Array(n)
    for (i = 0; i < n; i++)
        lis[i] = 1;

    /* Compute LIS values from left to right */
    for (i = 1; i < n; i++)
        for (j = 0; j < i; j++)
            if (arr[i] > arr[j] && lis[i] < lis[j] + 1)
                lis[i] = lis[j] + 1;

    /* Allocate memory for lds and initialize LDS values for 
        all indexes */
    let lds = new Array(n);
    for (i = 0; i < n; i++)
        lds[i] = 1;

    /* Compute LDS values from right to left */
    for (i = n - 2; i >= 0; i--)
        for (j = n - 1; j > i; j--)
            if (arr[i] > arr[j] && lds[i] < lds[j] + 1)
                lds[i] = lds[j] + 1;


    /* Return the maximum value of lis[i] + lds[i] - 1*/
    let max = lis[0] + lds[0] - 1;
    for (i = 1; i < n; i++)
        if (lis[i] + lds[i] - 1 > max)
            max = lis[i] + lds[i] - 1;

    return max;
}
let arr = [0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15]
let n = arr.length;
console.log("Length of LBS is " + lbs(arr, n));

Output

Length of LBS is 7

Complexity Analysis:

Time Complexity: O(n^2)
Auxiliary Space: O(n)

Please refer complete article on Longest Bitonic Subsequence | DP-15 for more details!

Printing Longest Bitonic Subsequence

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