Divide array into increasing and decreasing subsequence without changing the order

Given a merged sequence which consists of two sequences which got merged, one of them was strictly increasing and the other was strictly decreasing. Elements of increasing sequence were inserted between elements of the decreasing one without changing the order.

Sequences [1, 3, 4] and [10, 4, 2] can produce the following resulting sequences: 
[10, 1, 3, 4, 2, 4], [1, 3, 4, 10, 4, 2]. 
The following sequence cannot be the result of these insertions: 
[1, 10, 4, 4, 3, 2] because the order of elements in the increasing sequence was changed. 

Given a merged sequence, the task is to find any two suitable initial sequences, one of them should be strictly increasing, and another should be strictly decreasing. 
Note: An empty sequence and the sequence consisting of one element can be considered as increasing or decreasing.
Examples:

Input: arr[] = {5, 1, 3, 6, 8, 2, 9, 0, 10} 
Output: [1, 3, 6, 8, 9, 10] [5, 2, 0]

Input: arr[] = {1, 2, 4, 0, 2} 
Output: -1 
No such sequences possible. 

Method 1: We can modify Longest Increasing Sequence) and solve the required problem. It will take O(nlogn) time.

Method 2: We can also solve this problem only in a single traversal. The Idea used here is that maintain two sorted arrays. 
For a new element x, 

• If it can be appended to only one of the arrays then append it.
• If it can be appended to neither, then the answer is -1.
• If it can be appended to both then check the next element y, if y > x then append x to the increasing one otherwise append x to the decreasing one.

Now when we encounter an element which can be included in both the sequence we need to decide based on the next element’s value. Let’s consider a situation where we need to iterate over the remaining value x,y,z where ( x < z < y) and we have already the last element of the increasing and decreasing sequence as inc and dec respectively from the visited portion of the array. 

Case 1 : x<y and inc<x<dec

so we can include x in any sequence. 

If we include it in decreasing sequence then dec will become x. And then for y we have only one choice i.e. to include it in increasing sequence as y>dec and inc becomes y. If we do this we cannot insert z in any sequence as z>dec and z<inc. 

But if we include x to increasing sequence (inc becomes x) and y to decreasing sequence (dec becomes y) following the same logic then we can place z in any sequence and get an answer.

Case 2 : x>=y and inc<x<dec

it follows the same logic as above.

Below is the implementation of the above approach: 

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;

// Function to print strictly increasing and
// strictly decreasing sequence if possible
void Find_Sequence(int arr[], int n)
{
    // Arrays to store strictly increasing and
    // decreasing sequence
    vector<int> inc_arr, dec_arr;

    // Initializing last element of both sequence
    int flag = 0;
    long inc = -1, dec = 1e7;

    // Iterating through the array
    for (int i = 0; i < n; i++)
    {
        // If current element can be appended
        // to both the sequences
        if (inc < arr[i] && arr[i] < dec)
        {
            // If next element is greater than
            // the current element
            // Then append it to the strictly
            // increasing array
            if (arr[i] < arr[i + 1])
            {
                inc = arr[i];
                inc_arr.emplace_back(arr[i]);
            }

            // Otherwise append it to the
            // strictly decreasing array
            else
            {
                dec = arr[i];
                dec_arr.emplace_back(arr[i]);
            }
        }

        // If current element can be appended
        // to the increasing sequence only
        else if (inc < arr[i])
        {
            inc = arr[i];
            inc_arr.emplace_back(arr[i]);
        }

        // If current element can be appended
        // to the decreasing sequence only
        else if (dec > arr[i])
        {
            dec = arr[i];
            dec_arr.emplace_back(arr[i]);
        }

        // Else we can not make such sequences
        // from the given array
        else
        {
            cout << -1 << endl;
            flag = 1;
            break;
        }
    }

    // Print the required sequences
    if (!flag)
    {
        for (auto i = inc_arr.begin();
                  i != inc_arr.end(); i++)
            cout << *i << " ";
        cout << endl;

        for (auto i = dec_arr.begin();
                  i != dec_arr.end(); i++)
            cout << *i << " ";
        cout << endl;
    }
}

// Driver code
int main()
{
    int arr[] = { 5, 1, 3, 6, 8, 2, 9, 0, 10 };
    int n = sizeof(arr) / sizeof(arr[0]);
    Find_Sequence(arr, n);
}

// This code is contributed by sanjeev2552

Java

// Java implementation of the approach
import java.util.*;

class GFG
{

    // Function to print strictly increasing and
    // strictly decreasing sequence if possible
    static void Find_Sequence(int[] arr, int n)
    {

        // Arrays to store strictly increasing and
        // decreasing sequence
        Vector<Integer> inc_arr = new Vector<>(),
                        dec_arr = new Vector<>();

        // Initializing last element of both sequence
        int flag = 0;
        long inc = -1, dec = (long) 1e7;

        // Iterating through the array
        for (int i = 0; i < n; i++)
        {

            // If current element can be appended
            // to both the sequences
            if (inc < arr[i] && arr[i] < dec)
            {

                // If next element is greater than
                // the current element
                // Then append it to the strictly
                // increasing array
                if (arr[i] < arr[i + 1])
                {
                    inc = arr[i];
                    inc_arr.add(arr[i]);
                }

                // Otherwise append it to the
                // strictly decreasing array
                else
                {
                    dec = arr[i];
                    dec_arr.add(arr[i]);
                }
            }

            // If current element can be appended
            // to the increasing sequence only
            else if (inc < arr[i])
            {
                inc = arr[i];
                inc_arr.add(arr[i]);
            }

            // If current element can be appended
            // to the decreasing sequence only
            else if (dec > arr[i])
            {
                dec = arr[i];
                dec_arr.add(arr[i]);
            }

            // Else we can not make such sequences
            // from the given array
            else
            {
                System.out.println(-1);
                flag = 1;
                break;
            }
        }

        // Print the required sequences
        if (flag == 0)
        {
            for (int i : inc_arr)
                System.out.print(i + " ");
            System.out.println();

            for (int i : dec_arr)
                System.out.print(i + " ");
            System.out.println();
        }
    }

    // Driver Code
    public static void main(String[] args)
    {
        int[] arr = { 5, 1, 3, 6, 8, 2, 9, 0, 10 };
        int n = arr.length;
        Find_Sequence(arr, n);
    }
}

// This code is contributed by
// sanjeev2552

Python3

# Python3 implementation of the approach

# Function to print strictly increasing and
# strictly decreasing sequence if possible
def Find_Sequence(array, n):

    # Arrays to store strictly increasing and
    # decreasing sequence
    inc_arr, dec_arr =[], []

    # Initializing last element of both sequence
    inc, dec = -1, 1e7

    # Iterating through the array
    for i in range(n):

        # If current element can be appended
        # to both the sequences
        if inc < array[i] < dec:

            # If next element is greater than
            # the current element
            # Then append it to the strictly
            # increasing array
            if array[i] < array[i + 1]:
                inc = array[i]
                inc_arr.append(array[i])

            # Otherwise append it to the
            # strictly decreasing array
            else:
                dec = array[i]
                dec_arr.append(array[i])

        # If current element can be appended
        # to the increasing sequence only
        elif inc < array[i]:
            inc = array[i]
            inc_arr.append(array[i])

        # If current element can be appended
        # to the decreasing sequence only
        elif dec > array[i]:
            dec = array[i]
            dec_arr.append(array[i])

        # Else we can not make such sequences
        # from the given array
        else:
            print('-1')
            break

    # Print the required sequences
    else:
        print(inc_arr, dec_arr)

# Driver code
arr = [5, 1, 3, 6, 8, 2, 9, 0, 10]
n = len(arr)
Find_Sequence(arr, n)

C#

// C# implementation of the approach
using System;
using System.Collections;
using System.Collections.Generic;

class GFG{

// Function to print strictly increasing and
// strictly decreasing sequence if possible
static void Find_Sequence(int[] arr, int n)
{

    // Arrays to store strictly increasing and
    // decreasing sequence
    ArrayList inc_arr = new ArrayList();
    ArrayList dec_arr = new ArrayList();

    // Initializing last element of both sequence
    int flag = 0;
    long inc = -1, dec = (long)1e7;

    // Iterating through the array
    for(int i = 0; i < n; i++)
    {

        // If current element can be appended
        // to both the sequences
        if (inc < arr[i] && arr[i] < dec)
        {

            // If next element is greater than
            // the current element
            // Then append it to the strictly
            // increasing array
            if (arr[i] < arr[i + 1])
            {
                inc = arr[i];
                inc_arr.Add(arr[i]);
            }

            // Otherwise append it to the
            // strictly decreasing array
            else
            {
                dec = arr[i];
                dec_arr.Add(arr[i]);
            }
        }

        // If current element can be appended
        // to the increasing sequence only
        else if (inc < arr[i])
        {
            inc = arr[i];
            inc_arr.Add(arr[i]);
        }

        // If current element can be appended
        // to the decreasing sequence only
        else if (dec > arr[i])
        {
            dec = arr[i];
            dec_arr.Add(arr[i]);
        }

        // Else we can not make such sequences
        // from the given array
        else
        {
            Console.Write(-1);
            flag = 1;
            break;
        }
    }

    // Print the required sequences
    if (flag == 0)
    {
        foreach(int i in inc_arr)
            Console.Write(i + " ");

        Console.Write('\n');

        foreach(int i in dec_arr)
            Console.Write(i + " ");

        Console.Write('\n');
    }
}

// Driver Code
public static void Main(string[] args)
{
    int[] arr = { 5, 1, 3, 6, 8,
                  2, 9, 0, 10 };
    int n = arr.Length;

    Find_Sequence(arr, n);
}
}

// This code is contributed by rutvik_56

PHP

<?php
// Php implementation of the approach

// Function to print strictly increasing and
// strictly decreasing sequence if possible
function Find_Sequence($arr, $n)
{

    // Arrays to store strictly increasing and
    // decreasing sequence
    $inc_arr = array(); $dec_arr = array();

    // Initializing last element of both sequence
    $inc = -1; $dec = 1e7;

    // Iterating through the array
    for ($i = 0; $i < $n ; $i++)
    {

        // If current element can be appended
        // to both the sequences
        if ($inc < $arr[$i] && $arr[$i] < $dec)
        {

            // If next element is greater than
            // the current element
            // Then append it to the strictly
            // increasing array
            if ($arr[$i] < $arr[$i + 1])
            {
                $inc = $arr[$i];
                array_push($inc_arr, $arr[$i]);
            }

            // Otherwise append it to the
            // strictly decreasing array
            else
            {
                $dec = $arr[$i];
                array_push($dec_arr, $arr[$i]);
            }
        }

        // If current element can be appended
        // to the increasing sequence only
        else if ($inc < $arr[$i])
        {
            $inc = $arr[$i];
            array_push($inc_arr, $arr[$i]);
        }

        // If current element can be appended
        // to the decreasing sequence only
        else if($dec > $arr[$i])
        {
            $dec = $arr[$i];
            array_push($dec_arr, $arr[$i]);
        }

        // Else we can not make such sequences
        // from the given array
        else
        {
            echo '-1';
            break;
        }
    }

    // Print the required sequences
    print_r($inc_arr);
    print_r($dec_arr);
}

// Driver code
$arr = array(5, 1, 3, 6, 8, 2, 9, 0, 10);
$n = count($arr);
Find_Sequence($arr, $n);

// This code is contributed by Ryuga
?>

Javascript

<script>

// Javascript implementation of the approach

    // Function to print strictly increasing and
    // strictly decreasing sequence if possible
    function Find_Sequence(arr, n)
    {

        // Arrays to store strictly increasing and
        // decreasing sequence
        let inc_arr =[],
                        dec_arr = [];

        // Initializing last element of both sequence
        let flag = 0;
        let inc = -1, dec = 1e7;

        // Iterating through the array
        for (let i = 0; i < n; i++)
        {

            // If current element can be appended
            // to both the sequences
            if (inc < arr[i] && arr[i] < dec)
            {

                // If next element is greater than
                // the current element
                // Then append it to the strictly
                // increasing array
                if (arr[i] < arr[i + 1])
                {
                    inc = arr[i];
                    inc_arr.push(arr[i]);
                }

                // Otherwise append it to the
                // strictly decreasing array
                else
                {
                    dec = arr[i];
                    dec_arr.push(arr[i]);
                }
            }

            // If current element can be appended
            // to the increasing sequence only
            else if (inc < arr[i])
            {
                inc = arr[i];
                inc_arr.push(arr[i]);
            }

            // If current element can be appended
            // to the decreasing sequence only
            else if (dec > arr[i])
            {
                dec = arr[i];
                dec_arr.push(arr[i]);
            }

            // Else we can not make such sequences
            // from the given array
            else
            {
               document.write(-1);
                flag = 1;
                break;
            }
        }

        // Print the required sequences
        if (flag == 0)
        {
             document.write("[");
            for (let i in inc_arr)
                document.write( inc_arr[i] + " ");
            document.write("] ");

             document.write("[");
            for (let i in dec_arr)
                document.write( dec_arr[i] + " ");
            document.write("]");
        }
    }

// Driver Code

        let arr = [ 5, 1, 3, 6, 8, 2, 9, 0, 10 ];
        let n = arr.length;
        Find_Sequence(arr, n);

// This code is contributed by target_2.
</script>

Output: 

[1, 3, 6, 8, 9, 10] [5, 2, 0]

Time Complexity : O(n) ,where n is size of given array.

Space Complexity : O(n) ,extra space required to store strictly increasing and decreasing sequence.