Php Program to Find the Longest Bitonic Subsequence
Last Updated :
21 Dec, 2021
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.Â
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.Â
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
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SolutionÂ
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.Â
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PHP
<?php
function lbs(& $arr , $n )
{
$lis = array_fill (0, $n , NULL);
for ( $i = 0; $i < $n ; $i ++)
$lis [ $i ] = 1;
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;
$lds = array_fill (0, $n , NULL);
for ( $i = 0; $i < $n ; $i ++)
$lds [ $i ] = 1;
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;
$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 ;
}
$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 );
?>
|
Output:Â
Length of LBS is 7
Time Complexity: O(n^2)Â
Auxiliary Space: O(n)
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Please refer complete article on Longest Bitonic Subsequence | DP-15 for more details!
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