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
Approach:
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
// 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
Complexity Analysis:
- Time Complexity: O(n^2)
- Auxiliary Space: O(n)