Find the Longest Increasing Subsequence in Circular manner

Given an array, the task is to find to LIS (Longest Increasing Subsequence) in a circular way.

Examples :

Input : arr[] = {5, 4, 3, 2, 1}
Output : 2
Although there is no LIS in a given array 
but in a circular form there can be
{1, 5}, {2, 5}, ...... 

Input : arr[]= {5, 6, 7, 1, 2, 3}
Output : 6
{1, 2, 3, 5, 6, 7} will be the LIS in the
circular manner.



  1. Append the same elements(i.e. whole array) with the given array.
  2. For every window of size n(no. of elements in the given array), perform LIS.
  3. Return maximum length.
For example : Given array is {1, 4, 6, 2, 3}

After appending elements resultant array
will be {1, 4, 6, 2, 3, 1, 4, 6, 2, 3}.

Now for every consecutive n elements perform LIS.
1- {1, 4, 6, 2, 3} --3 is length of LIS.
2- {4, 6, 2, 3, 1} --2 is length of LIS.
3- {6, 2, 3, 1, 4} --3
4- {2, 3, 1, 4, 6}-- 4 {2, 3, 4, 6}
5- {3, 1, 4, 6, 2} --3.
6- {1, 4, 6, 2, 3} Original list.

So, maximum length of LIS in circular manner is 4. 

As in the last window we will have the same elements as in the given array which we don’t need to compute again, so we can append only n-1 elements to reduce the number of operations.

C++

// C++ implementation to find LIS in circular way
#include<bits/stdc++.h>
using namespace std;
  
// Utility function to find LIS using Dynamic programming
int computeLIS(int circBuff[], int start, int end, int n)
{
    int LIS[end-start];
  
    /* Initialize LIS values for all indexes */
    for (int i = start; i < end; i++)
        LIS[i] = 1;
  
    /* Compute optimized LIS values in bottom up manner */
    for (int i = start + 1; i < end; i++)
  
        // Set j on the basis of current window
        // i.e. first element of the current window
        for (int j = start; j < i; j++ )
            if (circBuff[i] > circBuff[j] && LIS[i] < LIS[j] + 1)
                LIS[i] = LIS[j] + 1;
  
    /* Pick maximum of all LIS values */
    int res = INT_MIN;
    for (int i = start; i < end; i++)
        res = max(res, LIS[i]);
  
    return res;
}
  
// Function to find Longest Increasing subsequence in
// Circular manner
int LICS(int arr[], int n)
{
    // Make a copy of given array by appending same
    // array elements  to itself
    int circBuff[2 * n];
    for (int i = 0; i<n; i++)
        circBuff[i] = arr[i];
    for (int i = n; i < 2*n; i++)
        circBuff[i] = arr[i-n];
  
    // Perform LIS for each window of size n
    int res = INT_MIN;
    for (int i=0; i<n; i++)
        res = max(computeLIS(circBuff, i, i + n, n), res);
  
    return res;
}
  
/* Driver program to test above function */
int main()
{
    int arr[] = { 1, 4, 6, 2, 3 };
    int n = sizeof(arr)/sizeof(arr[0]);
  
    cout << "Length of LICS is " << LICS( arr, n );
    return 0;
}

Java

// Java implementation to find LIS in circular way
  
class Test
{
    // Utility method to find LIS using Dynamic programming
    static int computeLIS(int circBuff[], int start, int end, int n)
    {
        int LIS[] = new int[n+end-start];
       
        /* Initialize LIS values for all indexes */
        for (int i = start; i < end; i++)
            LIS[i] = 1;
       
        /* Compute optimized LIS values in bottom up manner */
        for (int i = start + 1; i < end; i++)
       
            // Set j on the basis of current window
            // i.e. first element of the current window
            for (int j = start; j < i; j++ )
                if (circBuff[i] > circBuff[j] && LIS[i] < LIS[j] + 1)
                    LIS[i] = LIS[j] + 1;
       
        /* Pick maximum of all LIS values */
        int res = Integer.MIN_VALUE;
        for (int i = start; i < end; i++)
            res = Math.max(res, LIS[i]);
       
        return res;
    }
       
    // Function to find Longest Increasing subsequence in
    // Circular manner
    static int LICS(int arr[], int n)
    {
        // Make a copy of given array by appending same
        // array elements  to itself
        int circBuff[] = new int[2 * n];
        for (int i = 0; i<n; i++)
            circBuff[i] = arr[i];
        for (int i = n; i < 2*n; i++)
            circBuff[i] = arr[i-n];
       
        // Perform LIS for each window of size n
        int res = Integer.MIN_VALUE;
        for (int i=0; i<n; i++)
            res = Math.max(computeLIS(circBuff, i, i + n, n), res);
       
        return res;
    }
  
    // Driver method
    public static void main(String args[])
    {
         int arr[] = { 1, 4, 6, 2, 3 };
         System.out.println("Length of LICS is " + LICS( arr, arr.length));
    }
}

C#

// C# implementation to find 
// LIS in circular way
using System;
  
class Test
{
    // Utility method to find LIS
    // using Dynamic programming
    static int computeLIS(int []circBuff, int start, 
                                     int end, int n)
    {
        int []LIS = new int[n+end-start];
      
        /* Initialize LIS values for all indexes */
        for (int i = start; i < end; i++)
            LIS[i] = 1;
      
        /* Compute optimized LIS values 
           in bottom up manner */
        for (int i = start + 1; i < end; i++)
      
            // Set j on the basis of current window
            // i.e. first element of the current window
            for (int j = start; j < i; j++ )
                if (circBuff[i] > circBuff[j] && 
                            LIS[i] < LIS[j] + 1)
                    LIS[i] = LIS[j] + 1;
      
        /* Pick maximum of all LIS values */
        int res = int.MinValue;
        for (int i = start; i < end; i++)
            res = Math.Max(res, LIS[i]);
      
        return res;
    }
      
    // Function to find Longest Increasing 
    // subsequence in Circular manner
    static int LICS(int []arr, int n)
    {
        // Make a copy of given array by 
        // appending same array elements to itself
        int []circBuff = new int[2 * n];
          
        for (int i = 0; i<n; i++)
            circBuff[i] = arr[i];
        for (int i = n; i < 2*n; i++)
            circBuff[i] = arr[i-n];
      
        // Perform LIS for each window of size n
        int res = int.MinValue;
        for (int i=0; i<n; i++)
            res = Math.Max(computeLIS(circBuff, i, i + n, n), res);
      
        return res;
    }
  
    // Driver method
    public static void Main()
    {
        int []arr = {1, 4, 6, 2, 3};
        Console.Write("Length of LICS is "
                    LICS( arr, arr.Length));
    }
}
  
// This code is contributed by nitin mittal.

PHP

<?php
// PHP implementation to
// find LIS in circular way
  
// Utility function to find 
// LIS using Dynamic programming
function computeLIS($circBuff
                    $start
                    $end, $n)
{
    $LIS = Array();
  
    /* Initialize LIS values
    for all indexes */
    for ($i = $start; $i < $end; $i++)
        $LIS[$i] = 1;
  
    /* Compute optimized LIS 
    values in bottom up manner */
    for ($i = $start + 1; $i < $end; $i++)
  
        // Set j on the basis of
        // current window
        // i.e. first element of
        // the current window
        for ( $j = $start; $j < $i; $j++ )
            if ($circBuff[$i] > $circBuff[$j] && 
                     $LIS[$i] < $LIS[$j] + 1)
                $LIS[$i] = $LIS[$j] + 1;
  
    /* Pick maximum of
    all LIS values */
    $res = PHP_INT_MIN;
    for ($i = $start; $i < $end; $i++)
        $res = max($res, $LIS[$i]);
  
    return $res;
}
  
// Function to find LIS 
// in Circular manner
function LICS($arr, $n)
{
    // Make a copy of given array
    // by appending same array
    // elements to itself
    for ($i = 0; $i < $n; $i++)
        $circBuff[$i] = $arr[$i];
    for ($i = $n; $i < 2 * $n; $i++)
        $circBuff[$i] = $arr[$i - $n];
  
    // Perform LIS for each 
    // window of size n
    $res = PHP_INT_MIN;
    for ($i = 0; $i < $n; $i++)
        $res = max(computeLIS($circBuff, $i
                              $i + $n, $n), 
                              $res);
  
    return $res;
}
  
// Driver Code
$arr = array(1, 4, 6, 2, 3);
$n = sizeof($arr);
  
echo "Length of LICS is "
            LICS($arr, $n);
      
// This code is contributed by aj_36
?>


Output :

Length of LICS is 4

Time complexity of above solution is O(n3). It can be reduced O(n2 Log n) using O(n Log n) algorithm to find LIS.



Reference :
https://www.careercup.com/question?id=5942735794077696

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Improved By : nitin mittal, jit_t