Longest subsequence such that difference between adjacents is one
Last Updated :
11 Sep, 2023
Given an array of n size, the task is to find the longest subsequence such that difference between adjacents is one.
Examples:
Input : arr[] = {10, 9, 4, 5, 4, 8, 6}
Output : 3
As longest subsequences with difference 1 are, "10, 9, 8",
"4, 5, 4" and "4, 5, 6"
Input : arr[] = {1, 2, 3, 2, 3, 7, 2, 1}
Output : 7
As longest consecutive sequence is "1, 2, 3, 2, 3, 2, 1"
This problem is based upon the concept of Longest Increasing Subsequence Problem.
Let arr[0..n-1] be the input array and
dp[i] be the length of the longest subsequence (with
differences one) ending at index i such that arr[i]
is the last element of the subsequence.
Then, dp[i] can be recursively written as:
dp[i] = 1 + max(dp[j]) where 0 < j < i and
[arr[j] = arr[i] -1 or arr[j] = arr[i] + 1]
dp[i] = 1, if no such j exists.
To find the result for a given array, we need
to return max(dp[i]) where 0 < i < n.
Following is a Dynamic Programming based implementation. It follows the recursive structure discussed above.
C++
#include <bits/stdc++.h>
using namespace std;
int longestSubseqWithDiffOne( int arr[], int n)
{
int dp[n];
for ( int i = 0; i < n; i++)
dp[i] = 1;
for ( int i = 1; i < n; i++) {
for ( int j = 0; j < i; j++) {
if ((arr[i] == arr[j] + 1) || (arr[i] == arr[j] - 1))
dp[i] = max(dp[i], dp[j] + 1);
}
}
int result = 1;
for ( int i = 0; i < n; i++)
if (result < dp[i])
result = dp[i];
return result;
}
int main()
{
int arr[] = { 1, 2, 3, 4, 5, 3, 2 };
int n = sizeof (arr) / sizeof (arr[0]);
cout << longestSubseqWithDiffOne(arr, n);
return 0;
}
|
Java
import java.io.*;
class GFG {
static int longestSubseqWithDiffOne( int arr[],
int n)
{
int dp[] = new int [n];
for ( int i = 0 ; i < n; i++)
dp[i] = 1 ;
for ( int i = 1 ; i < n; i++) {
for ( int j = 0 ; j < i; j++) {
if ((arr[i] == arr[j] + 1 ) || (arr[i] == arr[j] - 1 ))
dp[i] = Math.max(dp[i], dp[j] + 1 );
}
}
int result = 1 ;
for ( int i = 0 ; i < n; i++)
if (result < dp[i])
result = dp[i];
return result;
}
public static void main(String[] args)
{
int arr[] = { 1 , 2 , 3 , 4 , 5 , 3 , 2 };
int n = arr.length;
System.out.println(longestSubseqWithDiffOne(
arr, n));
}
}
|
Python3
def longestSubseqWithDiffOne(arr, n):
dp = [ 1 for i in range (n)]
for i in range (n):
for j in range (i):
if ((arr[i] = = arr[j] + 1 ) or (arr[i] = = arr[j] - 1 )):
dp[i] = max (dp[i], dp[j] + 1 )
result = 1
for i in range (n):
if (result < dp[i]):
result = dp[i]
return result
arr = [ 1 , 2 , 3 , 4 , 5 , 3 , 2 ]
n = len (arr)
print (longestSubseqWithDiffOne(arr, n))
|
C#
using System;
class GFG {
static int longestSubseqWithDiffOne( int [] arr,
int n)
{
int [] dp = new int [n];
for ( int i = 0; i < n; i++)
dp[i] = 1;
for ( int i = 1; i < n; i++) {
for ( int j = 0; j < i; j++) {
if ((arr[i] == arr[j] + 1) || (arr[i] == arr[j] - 1))
dp[i] = Math.Max(dp[i], dp[j] + 1);
}
}
int result = 1;
for ( int i = 0; i < n; i++)
if (result < dp[i])
result = dp[i];
return result;
}
public static void Main()
{
int [] arr = { 1, 2, 3, 4, 5, 3, 2 };
int n = arr.Length;
Console.Write(
longestSubseqWithDiffOne(arr, n));
}
}
|
PHP
<?php
function longestSubseqWithDiffOne( $arr , $n )
{
$dp [ $n ] = 0;
for ( $i = 0; $i < $n ; $i ++)
$dp [ $i ] = 1;
for ( $i = 1; $i < $n ; $i ++)
{
for ( $j = 0; $j < $i ; $j ++)
{
if (( $arr [ $i ] == $arr [ $j ] + 1) ||
( $arr [ $i ] == $arr [ $j ] - 1))
$dp [ $i ] = max( $dp [ $i ],
$dp [ $j ] + 1);
}
}
$result = 1;
for ( $i = 0 ; $i < $n ; $i ++)
if ( $result < $dp [ $i ])
$result = $dp [ $i ];
return $result ;
}
$arr = array (1, 2, 3, 4, 5, 3, 2);
$n = sizeof( $arr );
echo longestSubseqWithDiffOne( $arr , $n );
?>
|
Javascript
<script>
function longestSubseqWithDiffOne(arr, n)
{
let dp = [];
for (let i = 0; i < n; i++)
dp[i] = 1;
for (let i = 1; i < n; i++)
{
for (let j = 0; j < i; j++)
{
if ((arr[i] == arr[j] + 1) ||
(arr[i] == arr[j] - 1))
dp[i] = Math.max(dp[i], dp[j] + 1);
}
}
let result = 1;
for (let i = 0; i < n; i++)
if (result < dp[i])
result = dp[i];
return result;
}
let arr = [1, 2, 3, 4, 5, 3, 2];
let n = arr.length;
document.write(longestSubseqWithDiffOne(arr, n));
</script>
|
Time Complexity: O(n2)
Auxiliary Space: O(n)
Efficient Approach
C++
#include<bits/stdc++.h>
using namespace std;
int longestSubsequence( int n, int arr[])
{
if (n==1)
return 1;
unordered_map< int , int > mapp;
int res = 1;
for ( int i=0;i<n;i++){
if (mapp.count(arr[i]+1) >0 || mapp.count(arr[i]-1)>0){
mapp[arr[i]]=1+max(mapp[arr[i]+1],mapp[arr[i]-1]);
}
else
mapp[arr[i]]=1;
res = max(res, mapp[arr[i]]);
}
return res;
}
int main()
{
int arr[] = {1, 2, 3, 4, 5, 3, 2};
int n = sizeof (arr)/ sizeof (arr[0]);
cout << longestSubsequence(n, arr);
return 0;
}
|
Java
import java.lang.Math;
import java.util.*;
class GFG {
static int longestSubsequence( int n, int arr[])
{
if (n == 1 )
return 1 ;
Integer dp[] = new Integer[n];
HashMap<Integer, Integer> mapp = new HashMap<>();
dp[ 0 ] = 1 ;
mapp.put(arr[ 0 ], 0 );
for ( int i = 1 ; i < n; i++) {
if (Math.abs(arr[i] - arr[i - 1 ]) == 1 )
dp[i] = dp[i - 1 ] + 1 ;
else {
if (mapp.containsKey(arr[i] + 1 )
|| mapp.containsKey(arr[i] - 1 )) {
dp[i] = 1
+ Math.max(mapp.getOrDefault(
arr[i] + 1 , 0 ),
mapp.getOrDefault(
arr[i] - 1 , 0 ));
}
else
dp[i] = 1 ;
}
mapp.put(arr[i], dp[i]);
}
return Collections.max(Arrays.asList(dp));
}
public static void main(String[] args)
{
int arr[] = { 1 , 2 , 3 , 4 , 5 , 3 , 2 };
int n = arr.length;
System.out.println(longestSubsequence(n, arr));
}
}
|
Python3
def longestSubsequence(A, N):
L = [ 1 ] * N
hm = {}
for i in range ( 1 ,N):
if abs (A[i] - A[i - 1 ]) = = 1 :
L[i] = 1 + L[i - 1 ]
elif hm.get(A[i] + 1 , 0 ) or hm.get(A[i] - 1 , 0 ):
L[i] = 1 + max (hm.get(A[i] + 1 , 0 ), hm.get(A[i] - 1 , 0 ))
hm[A[i]] = L[i]
return max (L)
A = [ 1 , 2 , 3 , 4 , 5 , 3 , 2 ]
N = len (A)
print (longestSubsequence(A, N))
|
C#
using System;
using System.Collections.Generic;
using System.Linq;
class GFG {
static int longestSubsequence( int n, int [] arr)
{
if (n == 1)
return 1;
int [] dp = new int [n];
Dictionary< int , int > mapp = new Dictionary< int , int >();
dp[0] = 1;
mapp.Add(arr[0], 0);
for ( int i = 1; i < n; i++)
{
if (Math.Abs(arr[i] - arr[i - 1]) == 1)
dp[i] = dp[i - 1] + 1;
else
{
if (mapp.ContainsKey(arr[i] + 1)
|| mapp.ContainsKey(arr[i] - 1))
{
dp[i] = 1
+ Math.Max(mapp.ContainsKey(arr[i] + 1) ? mapp[arr[i] + 1] : 0,
mapp.ContainsKey(arr[i] - 1) ? mapp[arr[i] - 1] : 0);
}
else
dp[i] = 1;
}
mapp[arr[i]] = dp[i];
}
return dp.Max();
}
public static void Main( string [] args)
{
int [] arr = { 1, 2, 3, 4, 5, 3, 2 };
int n = arr.Length;
Console.WriteLine(longestSubsequence(n, arr));
}
}
|
Javascript
function longestSubsequence(n, arr) {
var L = Array(n).fill(1);
hm = {};
for ( var i = 1; i < n; i++) {
if (Math.abs(arr[i] - arr[i - 1]) == 1) L[i] = 1 + L[i - 1];
else if (get(hm, arr[i] + 1, 0) || get(hm, arr[i] - 1, 0)) {
L[i] = 1 + Math.max(get(hm, arr[i] + 1, 0), get(hm, arr[i] - 1, 0));
}
hm[arr[i]] = L[i];
}
return Math.max(...L);
}
function get(object, key, default_value) {
var result = object[key];
return typeof result !== "undefined" ? result : default_value;
}
var arr = [1, 2, 3, 4, 5, 3, 2];
var n = arr.length;
console.log(longestSubsequence(n, arr));
|
Time Complexity : O(n)
Space Complexity : O(n)
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