Longest Increasing Odd Even Subsequence
Last Updated :
27 Jul, 2022
Given an array of size n. The problem is to find the length of the subsequence in the given array such that all the elements of the subsequence are sorted in increasing order and also they are alternately odd and even.
Note that the subsequence could start either with the odd number or with the even number.
Examples:
Input : arr[] = {5, 6, 9, 4, 7, 8}
Output : 6
{5, 6, 9,4,7,8} is the required longest
increasing odd even subsequence which is the array itself in this case
Input : arr[] = {1, 12, 2, 22, 5, 30, 31, 14, 17, 11}
{1,12,5,30,31,14,17} is the required longest
increasing odd even subsequence
Output : 7
Naive Approach: Consider all subsequences and select the ones with alternate odd even numbers in increasing order. Out of them select the longest one. This has an exponential time complexity.
Efficient Approach:
Let L(i) be the length of the LIOES (Longest Increasing Odd Even Subsequence) ending at index i such that arr[i] is the last element of the LIOES.
Then, L(i) can be recursively written as:
- L(i) = 1 + max( L(j) ) where 0 < j < i and (arr[j] < arr[i]) and (arr[i]+arr[j])%2 != 0; or
- L(i) = 1, if no such j exists.
To find the LIOES for a given array, we need to return max(L(i)) where 0 < i < n.
Implementation: A dynamic programming approach has been implemented below for the above mentioned recursive relation.
C++
// C++ implementation to find the longest
// increasing odd even subsequence
#include <bits/stdc++.h>
using namespace std;
// function to find the longest
// increasing odd even subsequence
int longOddEvenIncSeq(int arr[], int n)
{
// lioes[i] stores longest increasing odd
// even subsequence ending at arr[i]
int lioes[n];
// to store the length of longest increasing
// odd even subsequence
int maxLen = 0;
// Initialize LIOES values for all indexes
for (int i = 0; i < n; i++)
lioes[i] = 1;
// Compute optimized LIOES values
// in bottom up manner
for (int i = 1; i < n; i++)
for (int j = 0; j < i; j++)
if (arr[i] > arr[j] &&
(arr[i] + arr[j]) % 2 != 0
&& lioes[i] < lioes[j] + 1)
lioes[i] = lioes[j] + 1;
// Pick maximum of all LIOES values
for (int i = 0; i < n; i++)
if (maxLen < lioes[i])
maxLen = lioes[i];
// required maximum length
return maxLen;
}
// Driver program to test above
int main()
{
int arr[] = { 1, 12, 2, 22, 5, 30,
31, 14, 17, 11 };
int n = sizeof(arr) / sizeof(n);
cout << "Longest Increasing Odd Even "
<< "Subsequence: "
<< longOddEvenIncSeq(arr, n);
return 0;
}
// Java implementation to find the longest
// increasing odd even subsequence
import java.util.*;
import java.lang.*;
public class GfG{
// function to find the longest
// increasing odd even subsequence
public static int longOddEvenIncSeq(int arr[],
int n)
{
// lioes[i] stores longest increasing odd
// even subsequence ending at arr[i]
int[] lioes = new int[n];
// to store the length of longest
// increasing odd even subsequence
int maxLen = 0;
// Initialize LIOES values for all indexes
for (int i = 0; i < n; i++)
lioes[i] = 1;
// Compute optimized LIOES values
// in bottom up manner
for (int i = 1; i < n; i++)
for (int j = 0; j < i; j++)
if (arr[i] > arr[j] &&
(arr[i] + arr[j]) % 2 != 0
&& lioes[i] < lioes[j] + 1)
lioes[i] = lioes[j] + 1;
// Pick maximum of all LIOES values
for (int i = 0; i < n; i++)
if (maxLen < lioes[i])
maxLen = lioes[i];
// required maximum length
return maxLen;
}
// driver function
public static void main(String argc[]){
int[] arr = new int[]{ 1, 12, 2, 22,
5, 30, 31, 14, 17, 11 };
int n = 10;
System.out.println("Longest Increasing Odd"
+ " Even Subsequence: "
+ longOddEvenIncSeq(arr, n));
}
}
/* This code is contributed by Sagar Shukla */
# Python3 implementation to find the longest
# increasing odd even subsequence
# function to find the longest
# increasing odd even subsequence
def longOddEvenIncSeq( arr , n ):
# lioes[i] stores longest increasing odd
# even subsequence ending at arr[i]
lioes = list()
# to store the length of longest increasing
# odd even subsequence
maxLen = 0
# Initialize LIOES values for all indexes
for i in range(n):
lioes.append(1)
# Compute optimized LIOES values
# in bottom up manner
i=1
for i in range(n):
for j in range(i):
if (arr[i] > arr[j] and
(arr[i] + arr[j]) % 2 != 0 and
lioes[i] < lioes[j] + 1):
lioes[i] = lioes[j] + 1
# Pick maximum of all LIOES values
for i in range(n):
if maxLen < lioes[i]:
maxLen = lioes[i]
# required maximum length
return maxLen
# Driver to test above
arr = [ 1, 12, 2, 22, 5, 30, 31, 14, 17, 11 ]
n = len(arr)
print("Longest Increasing Odd Even " +
"Subsequence: ",longOddEvenIncSeq(arr, n))
# This code is contributed by "Sharad_Bhardwaj".
// C# implementation to find the longest
// increasing odd even subsequence
using System;
class GFG {
// function to find the longest
// increasing odd even subsequence
public static int longOddEvenIncSeq(int[] arr,
int n)
{
// lioes[i] stores longest increasing odd
// even subsequence ending at arr[i]
int[] lioes = new int[n];
// to store the length of longest
// increasing odd even subsequence
int maxLen = 0;
// Initialize LIOES values for all indexes
for (int i = 0; i < n; i++)
lioes[i] = 1;
// Compute optimized LIOES values
// in bottom up manner
for (int i = 1; i < n; i++)
for (int j = 0; j < i; j++)
if (arr[i] > arr[j] &&
(arr[i] + arr[j]) % 2 != 0 &&
lioes[i] < lioes[j] + 1)
lioes[i] = lioes[j] + 1;
// Pick maximum of all LIOES values
for (int i = 0; i < n; i++)
if (maxLen < lioes[i])
maxLen = lioes[i];
// required maximum length
return maxLen;
}
// driver function
public static void Main()
{
int[] arr = new int[]{ 1, 12, 2, 22,
5, 30, 31, 14, 17, 11 };
int n = 10;
Console.Write("Longest Increasing Odd"
+ " Even Subsequence: "
+ longOddEvenIncSeq(arr, n));
}
}
// This code is contributed by Sam007
<?php
// PHP implementation to find the longest
// increasing odd even subsequence
// function to find the longest
// increasing odd even subsequence
function longOddEvenIncSeq(&$arr, $n)
{
// lioes[i] stores longest increasing
// odd even subsequence ending at arr[i]
$lioes= array_fill(0, $n, NULL);
// to store the length of longest
// increasing odd even subsequence
$maxLen = 0;
// Initialize LIOES values for
// all indexes
for ($i = 0; $i < $n; $i++)
$lioes[$i] = 1;
// Compute optimized LIOES values
// in bottom up manner
for ($i = 1; $i < $n; $i++)
for ($j = 0; $j < $i; $j++)
if ($arr[$i] > $arr[$j] &&
($arr[$i] + $arr[$j]) % 2 != 0 &&
$lioes[$i] < $lioes[$j] + 1)
$lioes[$i] = $lioes[$j] + 1;
// Pick maximum of all LIOES values
for ($i = 0; $i < $n; $i++)
if ($maxLen < $lioes[$i])
$maxLen = $lioes[$i];
// required maximum length
return $maxLen;
}
// Driver Code
$arr = array( 1, 12, 2, 22, 5, 30,
31, 14, 17, 11) ;
$n = sizeof($arr);
echo "Longest Increasing Odd Even ".
"Subsequence: " . longOddEvenIncSeq($arr, $n);
// This code is contributed
// by ChitraNayal
?>
<script>
// javascript program to find the longest
// increasing odd even subsequence
// function to find the longest
// increasing odd even subsequence
function longOddEvenIncSeq(arr, n)
{
// lioes[i] stores longest increasing odd
// even subsequence ending at arr[i]
let lioes = [];
// to store the length of longest
// increasing odd even subsequence
let maxLen = 0;
// Initialize LIOES values for all indexes
for (let i = 0; i < n; i++)
lioes[i] = 1;
// Compute optimized LIOES values
// in bottom up manner
for (let i = 1; i < n; i++)
for (let j = 0; j < i; j++)
if (arr[i] > arr[j] &&
(arr[i] + arr[j]) % 2 != 0
&& lioes[i] < lioes[j] + 1)
lioes[i] = lioes[j] + 1;
// Pick maximum of all LIOES values
for (let i = 0; i < n; i++)
if (maxLen < lioes[i])
maxLen = lioes[i];
// required maximum length
return maxLen;
}
// Driver code
let arr = [ 1, 12, 2, 22,
5, 30, 31, 14, 17, 11 ];
let n = 10;
document.write("Longest Increasing Odd"
+ " Even Subsequence: "
+ longOddEvenIncSeq(arr, n));
// This code is contributed by sanjoy_62.
</script>
OutputLongest Increasing Odd Even Subsequence: 5
Time Complexity: O(n2).
Auxiliary Space: O(n).
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