Longest alternating subsequence

A sequence {x1, x2, .. xn} is alternating sequence if its elements satisfy one of the following relations : 

  x1 < x2 > x3 < x4 > x5 < …. xn or 
  x1 > x2 < x3 > x4 < x5 > …. xn

Examples :

Input: arr[] = {1, 5, 4}
Output: 3
The whole arrays is of the form  x1 < x2 > x3 

Input: arr[] = {1, 4, 5}
Output: 2
All subsequences of length 2 are either of the form 
x1 < x2; or x1 > x2

Input: arr[] = {10, 22, 9, 33, 49, 50, 31, 60}
Output: 6
The subsequences {10, 22, 9, 33, 31, 60} or
{10, 22, 9, 49, 31, 60} or {10, 22, 9, 50, 31, 60}
are longest subsequence of length 6.

This problem is an extension of longest increasing subsequence problem, but requires more thinking for finding optimal substructure property in this.
We will solve this problem by dynamic Programming method, Let A is given array of length n of integers. We define a 2D array las[n][2] such that las[i][0] contains longest alternating subsequence ending at index i and last element is greater than its previous element and las[i][1] contains longest alternating subsequence ending at index i and last element is smaller than its previous element, then we have following recurrence relation between them,  

las[i][0] = Length of the longest alternating subsequence 
          ending at index i and last element is greater
          than its previous element
las[i][1] = Length of the longest alternating subsequence 
          ending at index i and last element is smaller
          than its previous element

Recursive Formulation:
   las[i][0] = max (las[i][0], las[j][1] + 1); 
             for all j < i and A[j] < A[i] 
   las[i][1] = max (las[i][1], las[j][0] + 1); 
             for all j < i and A[j] > A[i]

The first recurrence relation is based on the fact that, If we are at position i and this element has to bigger than its previous element then for this sequence (upto i) to be bigger we will try to choose an element j ( < i) such that A[j] < A[i] i.e. A[j] can become A[i]’s previous element and las[j][1] + 1 is bigger than las[i][0] then we will update las[i][0]. 
Remember we have chosen las[j][1] + 1 not las[j][0] + 1 to satisfy alternate property because in las[j][0] last element is bigger than its previous one and A[i] is greater than A[j] which will break the alternating property if we update. So above fact derives first recurrence relation, similar argument can be made for second recurrence relation also. 

C++

// C++ program to find longest alternating
// subsequence in an array
#include<iostream>
using namespace std;

// Function to return max of two numbers
int max(int a, int b)
{
    return (a > b) ? a : b;
}

// Function to return longest alternating
// subsequence length
int zzis(int arr[], int n)
{

    /*las[i][0] = Length of the longest
        alternating subsequence ending at
        index i and last element is greater
        than its previous element
    las[i][1] = Length of the longest
        alternating subsequence ending
        at index i and last element is
        smaller than its previous element */
    int las[n][2];

    // Initialize all values from 1
    for(int i = 0; i < n; i++)
        las[i][0] = las[i][1] = 1;

    // Initialize result
    int res = 1;

    // Compute values in bottom up manner
    for(int i = 1; i < n; i++)
    {

        // Consider all elements as
        // previous of arr[i]
        for(int j = 0; j < i; j++)
        {

            // If arr[i] is greater, then
            // check with las[j][1]
            if (arr[j] < arr[i] &&
                las[i][0] < las[j][1] + 1)
                las[i][0] = las[j][1] + 1;

            // If arr[i] is smaller, then
            // check with las[j][0]
            if(arr[j] > arr[i] &&
               las[i][1] < las[j][0] + 1)
                las[i][1] = las[j][0] + 1;
        }

        // Pick maximum of both values at index i
        if (res < max(las[i][0], las[i][1]))
            res = max(las[i][0], las[i][1]);
    }
    return res;
}

// Driver code
int main()
{
    int arr[] = { 10, 22, 9, 33,
                  49, 50, 31, 60 };
    int n = sizeof(arr) / sizeof(arr[0]);

    cout << "Length of Longest alternating "
         << "subsequence is " << zzis(arr, n);

    return 0;
}

// This code is contributed by shivanisinghss2110

C

// C program to find longest alternating subsequence in
// an array
#include <stdio.h>
#include <stdlib.h>

// function to return max of two numbers
int max(int a, int b) {  return (a > b) ? a : b; }

// Function to return longest alternating subsequence length
int zzis(int arr[], int n)
{
    /*las[i][0] = Length of the longest alternating subsequence
          ending at index i and last element is greater
          than its previous element
     las[i][1] = Length of the longest alternating subsequence
          ending at index i and last element is smaller
          than its previous element   */
    int las[n][2];

    /* Initialize all values from 1  */
    for (int i = 0; i < n; i++)
        las[i][0] = las[i][1] = 1;

    int res = 1; // Initialize result

    /* Compute values in bottom up manner */
    for (int i = 1; i < n; i++)
    {
        // Consider all elements as previous of arr[i]
        for (int j = 0; j < i; j++)
        {
            // If arr[i] is greater, then check with las[j][1]
            if (arr[j] < arr[i] && las[i][0] < las[j][1] + 1)
                las[i][0] = las[j][1] + 1;

            // If arr[i] is smaller, then check with las[j][0]
            if( arr[j] > arr[i] && las[i][1] < las[j][0] + 1)
                las[i][1] = las[j][0] + 1;
        }

        /* Pick maximum of both values at index i  */
        if (res < max(las[i][0], las[i][1]))
            res = max(las[i][0], las[i][1]);
    }

    return res;
}

/* Driver program */
int main()
{
    int arr[] = { 10, 22, 9, 33, 49, 50, 31, 60 };
    int n = sizeof(arr)/sizeof(arr[0]);
    printf("Length of Longest alternating subsequence is %d\n",
            zzis(arr, n) );
    return 0;
}

Java

// Java program to find longest
// alternating subsequence in an array
import java.io.*;

class GFG {

// Function to return longest
// alternating subsequence length
static int zzis(int arr[], int n)
{
    /*las[i][0] = Length of the longest
        alternating subsequence ending at
        index i and last element is
        greater than its previous element
    las[i][1] = Length of the longest
        alternating subsequence ending at
        index i and last element is
        smaller than its previous
        element */
    int las[][] = new int[n][2];

    /* Initialize all values from 1 */
    for (int i = 0; i < n; i++)
        las[i][0] = las[i][1] = 1;

    int res = 1; // Initialize result

    /* Compute values in bottom up manner */
    for (int i = 1; i < n; i++)
    {
        // Consider all elements as
        // previous of arr[i]
        for (int j = 0; j < i; j++)
        {
            // If arr[i] is greater, then
            // check with las[j][1]
            if (arr[j] < arr[i] &&
                las[i][0] < las[j][1] + 1)
                las[i][0] = las[j][1] + 1;

            // If arr[i] is smaller, then
            // check with las[j][0]
            if( arr[j] > arr[i] &&
              las[i][1] < las[j][0] + 1)
                las[i][1] = las[j][0] + 1;
        }

        /* Pick maximum of both values at
        index i */
        if (res < Math.max(las[i][0], las[i][1]))
            res = Math.max(las[i][0], las[i][1]);
    }

    return res;
}

/* Driver program */
public static void main(String[] args)
{
    int arr[] = { 10, 22, 9, 33, 49,
                  50, 31, 60 };
    int n = arr.length;
    System.out.println("Length of Longest "+
                    "alternating subsequence is " +
                    zzis(arr, n));
}
}
// This code is contributed by Prerna Saini

Python3

# Python3 program to find longest
# alternating subsequence in an array

# Function to return max of two numbers
def Max(a, b):

    if a > b:
        return a
    else:
        return b

# Function to return longest alternating
# subsequence length
def zzis(arr, n):

    """las[i][0] = Length of the longest
        alternating subsequence ending at
        index i and last element is greater
        than its previous element
    las[i][1] = Length of the longest
        alternating subsequence ending
        at index i and last element is
        smaller than its previous element"""
    las = [[0 for i in range(2)]
              for j in range(n)]

    # Initialize all values from 1
    for i in range(n):
        las[i][0], las[i][1] = 1, 1

    # Initialize result
    res = 1

    # Compute values in bottom up manner
    for i in range(1, n):

        # Consider all elements as
        # previous of arr[i]
        for j in range(0, i):

            # If arr[i] is greater, then
            # check with las[j][1]
            if (arr[j] < arr[i] and
             las[i][0] < las[j][1] + 1):
                las[i][0] = las[j][1] + 1

            # If arr[i] is smaller, then
            # check with las[j][0]
            if(arr[j] > arr[i] and
            las[i][1] < las[j][0] + 1):
                las[i][1] = las[j][0] + 1

        # Pick maximum of both values at index i
        if (res < max(las[i][0], las[i][1])):
            res = max(las[i][0], las[i][1])

    return res

# Driver Code
arr = [ 10, 22, 9, 33, 49, 50, 31, 60 ]
n = len(arr)

print("Length of Longest alternating subsequence is" ,
      zzis(arr, n))

# This code is contributed by divyesh072019

C#

// C# program to find longest
// alternating subsequence
// in an array
using System;

class GFG
{

// Function to return longest
// alternating subsequence length
static int zzis(int []arr, int n)
{
    /*las[i][0] = Length of the
        longest alternating subsequence
        ending at index i and last
        element is greater than its
        previous element
    las[i][1] = Length of the longest
        alternating subsequence ending at
        index i and last element is
        smaller than its previous
        element */
    int [,]las = new int[n, 2];

    /* Initialize all values from 1 */
    for (int i = 0; i < n; i++)
        las[i, 0] = las[i, 1] = 1;

    // Initialize result
    int res = 1;

    /* Compute values in
    bottom up manner */
    for (int i = 1; i < n; i++)
    {
        // Consider all elements as
        // previous of arr[i]
        for (int j = 0; j < i; j++)
        {
            // If arr[i] is greater, then
            // check with las[j][1]
            if (arr[j] < arr[i] &&
                las[i, 0] < las[j, 1] + 1)
                las[i, 0] = las[j, 1] + 1;

            // If arr[i] is smaller, then
            // check with las[j][0]
            if( arr[j] > arr[i] &&
            las[i, 1] < las[j, 0] + 1)
                las[i, 1] = las[j, 0] + 1;
        }

        /* Pick maximum of both
        values at index i */
        if (res < Math.Max(las[i, 0],
                           las[i, 1]))
            res = Math.Max(las[i, 0],
                           las[i, 1]);
    }

    return res;
}

// Driver Code
public static void Main()
{
    int []arr = {10, 22, 9, 33,
                 49, 50, 31, 60};
    int n = arr.Length;
    Console.WriteLine("Length of Longest "+
            "alternating subsequence is " +
                             zzis(arr, n));
}
}

// This code is contributed by anuj_67.

PHP

<?php
// PHP program to find longest
// alternating subsequence in
// an array

// Function to return longest
// alternating subsequence length
function zzis($arr, $n)
{
    /*las[i][0] = Length of the
        longest alternating subsequence
        ending at index i and last element
        is greater than its previous element
    las[i][1] = Length of the longest
        alternating subsequence ending at
        index i and last element is
        smaller than its previous element */
    $las = array(array());

    /* Initialize all values from 1 */
    for ( $i = 0; $i < $n; $i++)
        $las[$i][0] = $las[$i][1] = 1;

    $res = 1; // Initialize result

    /* Compute values in
    bottom up manner */
    for ( $i = 1; $i < $n; $i++)
    {
        // Consider all elements
        // as previous of arr[i]
        for ($j = 0; $j < $i; $j++)
        {
            // If arr[i] is greater, then
            // check with las[j][1]
            if ($arr[$j] < $arr[$i] and
                $las[$i][0] < $las[$j][1] + 1)
               $las[$i][0] = $las[$j][1] + 1;

            // If arr[i] is smaller, then
            // check with las[j][0]
            if($arr[$j] > $arr[$i] and
               $las[$i][1] < $las[$j][0] + 1)
                $las[$i][1] = $las[$j][0] + 1;
        }

        /* Pick maximum of both
        values at index i */
        if ($res < max($las[$i][0], $las[$i][1]))
            $res = max($las[$i][0], $las[$i][1]);
    }

    return $res;
}

// Driver Code
$arr = array(10, 22, 9, 33,
             49, 50, 31, 60 );
$n = count($arr);
echo "Length of Longest alternating " .
    "subsequence is ", zzis($arr, $n) ;

// This code is contributed by anuj_67.
?>

Javascript

<script>
    // Javascript program to find longest
    // alternating subsequence in an array

    // Function to return longest
    // alternating subsequence length
    function zzis(arr, n)
    {
        /*las[i][0] = Length of the longest
            alternating subsequence ending at
            index i and last element is
            greater than its previous element
        las[i][1] = Length of the longest
            alternating subsequence ending at
            index i and last element is
            smaller than its previous
            element */
        let las = new Array(n);
        for (let i = 0; i < n; i++)
        {
            las[i] = new Array(2);
            for (let j = 0; j < 2; j++)
            {
                las[i][j] = 0;
            }
        }

        /* Initialize all values from 1 */
        for (let i = 0; i < n; i++)
            las[i][0] = las[i][1] = 1;

        let res = 1; // Initialize result

        /* Compute values in bottom up manner */
        for (let i = 1; i < n; i++)
        {
            // Consider all elements as
            // previous of arr[i]
            for (let j = 0; j < i; j++)
            {
                // If arr[i] is greater, then
                // check with las[j][1]
                if (arr[j] < arr[i] &&
                    las[i][0] < las[j][1] + 1)
                    las[i][0] = las[j][1] + 1;

                // If arr[i] is smaller, then
                // check with las[j][0]
                if( arr[j] > arr[i] &&
                  las[i][1] < las[j][0] + 1)
                    las[i][1] = las[j][0] + 1;
            }

            /* Pick maximum of both values at
            index i */
            if (res < Math.max(las[i][0], las[i][1]))
                res = Math.max(las[i][0], las[i][1]);
        }

        return res;
    }

    let arr = [ 10, 22, 9, 33, 49, 50, 31, 60 ];
    let n = arr.length;
    document.write("Length of Longest "+
                    "alternating subsequence is " +
                    zzis(arr, n));

    // This code is contributed by rameshtravel07.
</script>

Output:

Length of Longest alternating subsequence is 6

Time Complexity: O(n²) 
Auxiliary Space: O(n)

Efficient Solution:
In the above approach, at any moment we are keeping track of two values (Length of the longest alternating subsequence ending at index i, and last element is smaller than or greater than previous element), for every element on array. To optimise space, we only need to store two variables for element at any index i. 

inc = Length of longest alternative subsequence so far with current value being greater than it’s previous value.
dec = Length of longest alternative subsequence so far with current value being smaller than it’s previous value.
The tricky part of this approach is to update these two values. 

“inc” should be increased, if and only if the last element in the alternative sequence was smaller than it’s previous element.
“dec” should be increased, if and only if the last element in the alternative sequence was greater than it’s previous element.

C++

// C++ program for above approach
#include <bits/stdc++.h>
using namespace std;

// Function for finding
// longest alternating
// subsequence
int LAS(int arr[], int n)
{

    // "inc" and "dec" initialized as 1
    // as single element is still LAS
    int inc = 1;
    int dec = 1;

    // Iterate from second element
    for (int i = 1; i < n; i++)
    {

        if (arr[i] > arr[i - 1])
        {

            // "inc" changes iff "dec"
            // changes
            inc = dec + 1;
        }

        else if (arr[i] < arr[i - 1])
        {

            // "dec" changes iff "inc"
            // changes
            dec = inc + 1;
        }
    }

    // Return the maximum length
    return max(inc, dec);
}

// Driver Code
int main()
{
    int arr[] = { 10, 22, 9, 33, 49,
                           50, 31, 60 };
    int n = sizeof(arr) / sizeof(arr[0]);

    // Function Call
    cout << LAS(arr, n) << endl;
    return 0;
}

Java

// Java Program for above approach
public class GFG
{

    // Function for finding
    // longest alternating
    // subsequence
    static int LAS(int[] arr, int n)
    {

        // "inc" and "dec" initialized as 1,
        // as single element is still LAS
        int inc = 1;
        int dec = 1;

        // Iterate from second element
        for (int i = 1; i < n; i++)
        {

            if (arr[i] > arr[i - 1])
            {
                // "inc" changes iff "dec"
                // changes
                inc = dec + 1;
            }
            else if (arr[i] < arr[i - 1])
            {

                // "dec" changes iff "inc"
                // changes
                dec = inc + 1;
            }
        }

        // Return the maximum length
        return Math.max(inc, dec);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int[] arr = { 10, 22, 9, 33, 49,
                               50, 31, 60 };
        int n = arr.length;

        // Function Call
        System.out.println(LAS(arr, n));
    }
}

Python3

# Python3 program for above approach
def LAS(arr, n):

    # "inc" and "dec" initialized as 1
    # as single element is still LAS
    inc = 1
    dec = 1

    # Iterate from second element
    for i in range(1,n):

        if (arr[i] > arr[i-1]):

            # "inc" changes iff "dec"
            # changes
            inc = dec + 1
        else if (arr[i] < arr[i-1]):

            # "dec" changes iff "inc"
            # changes
            dec = inc + 1

    # Return the maximum length
    return max(inc, dec)

# Driver Code
if __name__ == "__main__":
    arr = [10, 22, 9, 33, 49, 50, 31, 60]
    n = len(arr)

    # Function Call
    print(LAS(arr, n))

C#

// C# program for above approach
using System;

class GFG{

// Function for finding
// longest alternating
// subsequence
static int LAS(int[] arr, int n)
{

    // "inc" and "dec" initialized as 1,
    // as single element is still LAS
    int inc = 1;
    int dec = 1;

    // Iterate from second element
    for(int i = 1; i < n; i++)
    {
        if (arr[i] > arr[i - 1])
        {

            // "inc" changes iff "dec"
            // changes
            inc = dec + 1;
        }
        else if (arr[i] < arr[i - 1])
        {

            // "dec" changes iff "inc"
            // changes
            dec = inc + 1;
        }
    }

    // Return the maximum length
    return Math.Max(inc, dec);
}

// Driver code
static void Main()
{
    int[] arr = { 10, 22, 9, 33,
                  49, 50, 31, 60 };
    int n = arr.Length;

    // Function Call
    Console.WriteLine(LAS(arr, n));
}
}

// This code is contributed by divyeshrabadiya07

Javascript

<script>
    // Javascript program for above approach

    // Function for finding
    // longest alternating
    // subsequence
    function LAS(arr, n)
    {

        // "inc" and "dec" initialized as 1
        // as single element is still LAS
        let inc = 1;
        let dec = 1;

        // Iterate from second element
        for (let i = 1; i < n; i++)
        {

            if (arr[i] > arr[i - 1])
            {

                // "inc" changes iff "dec"
                // changes
                inc = dec + 1;
            }

            else if (arr[i] < arr[i - 1])
            {

                // "dec" changes iff "inc"
                // changes
                dec = inc + 1;
            }
        }

        // Return the maximum length
        return Math.max(inc, dec);
    }

    let arr = [ 10, 22, 9, 33, 49, 50, 31, 60 ];
    let n = arr.length;

    // Function Call
    document.write(LAS(arr, n));

     // This code is contributed by mukesh07.
</script>

Output:

6

Time Complexity: O(n) 
Auxiliary Space: O(1)

This article is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.