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
Solution
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.
PHP
<?php // Dynamic Programming implementation // of 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) { /* Allocate memory for LIS[] and initialize LIS values as 1 for all indexes */ $lis = array_fill(0, $n, NULL); for ($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 */ $lds = array_fill(0, $n, NULL); for ($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*/ $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; } // Driver Code $arr = array(0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15); $n = sizeof($arr); echo "Length of LBS is " . lbs( $arr, $n ); // This code is contributed by ita_c ?> |
Output:
Length of LBS is 7
Time Complexity: O(n^2)
Auxiliary Space: O(n)
Please refer complete article on Longest Bitonic Subsequence | DP-15 for more details!
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