Given two arrays firstArr[], consisting of distinct elements only, and secondArr[], the task is to find the length of LCS between these 2 arrays.
Examples:
Input: firstArr[] = {3, 5, 1, 8}, secondArr[] = {3, 3, 5, 3, 8}
Output: 3.
Explanation: LCS between these two arrays is {3, 5, 8}.Input : firstArr[] = {1, 2, 1}, secondArr[] = {3}
Output: 0
Naive Approach: Follow the steps below to solve the problem using the simplest possible approach:
- Initialize an array dp[][] such that dp[i][j] stores longest common subsequence of firstArr[ :i] and secondArr[ :j].
- Traverse the array firstArr[] and for every array element of the array firstArr[], traverse the array secondArr[].
- If firstArr[i] = secondArr[j]: Set dp[i][j] = dp[i – 1][j – 1] + 1.
- Otherwise: Set dp[i][j] = max(dp[ i – 1][j], dp[i][j – 1]).
Time Complexity: O(N * M), where N and M are the sizes of the arrays firstArr[] and secondArr[] respectively.Â
Auxiliary Space: O(N * M)
Efficient Approach: To optimize the above approach, follow the steps below:Â
- Initialize a Map, say mp, to store the mappings map[firstArr[i]] = i, i.e. map elements of the first array to their respective indices.
- Since the elements which are present in secondArr[] but not in the firstArr[] are not useful at all, as they can never be a part of a common subsequence, traverse the array secondArr[] andfor each array element, check if it is present in the Map or not.
- If found to be true, push map[secondArr[i]] into a temporary Array. Otherwise ignore it.
- Find the Longest Increasing Subsequence (LIS) of the obtained temporary array and print its length as the required answer.
Illustration:Â
firstArr[] = {3, 5, 1, 8}Â
secondArr={3, 3, 4, 5, 3, 8}Â
Mapping: 3->0, 5->1, 1->2, 8->3 (From firstArr)Â
tempArr[] = {0, 0, 1, 0, 3}Â
Therefore, length of LIS of tempArr[] = 3 ({0, 1, 3})Â
Below is the implementation of the above approach:Â
C++
// C++ program to implement// the above approachÂ
#include <bits/stdc++.h>using namespace std;Â
// Function to find the Longest Common// Subsequence between the two arraysint LCS(vector<int>& firstArr,        vector<int>& secondArr){Â
    // Maps elements of firstArr[]    // to their respective indices    unordered_map<int, int> mp;Â
    // Traverse the array firstArr[]    for (int i = 0; i < firstArr.size(); i++) {        mp[firstArr[i]] = i + 1;    }Â
    // Stores the indices of common elements    // between firstArr[] and secondArr[]    vector<int> tempArr;Â
    // Traverse the array secondArr[]    for (int i = 0; i < secondArr.size(); i++) {Â
        // If current element exists in the Map        if (mp.find(secondArr[i]) != mp.end()) {Â
            tempArr.push_back(mp[secondArr[i]]);        }    }Â
    // Stores lIS from tempArr[]    vector<int> tail;Â
    tail.push_back(tempArr[0]);Â
    for (int i = 1; i < tempArr.size(); i++) {Â
        if (tempArr[i] > tail.back())            tail.push_back(tempArr[i]);Â
        else if (tempArr[i] < tail[0])            tail[0] = tempArr[i];Â
        else {            auto it = lower_bound(tail.begin(),                                  tail.end(),                                  tempArr[i]);            *it = tempArr[i];        }    }    return (int)tail.size();}Â
// Driver Codeint main(){Â Â Â Â vector<int> firstArr = { 3, 5, 1, 8 };Â Â Â Â vector<int> secondArr = { 3, 3, 5, 3, 8 };Â Â Â Â cout << LCS(firstArr, secondArr);Â Â Â Â return 0;} |
Java
// Java program to implement// the above approachimport java.util.*;class GFG{Â
// Function to find the Longest Common// Subsequence between the two arraysstatic int LCS(int[] firstArr,int[] secondArr){Â
    // Maps elements of firstArr[]    // to their respective indices    HashMap<Integer,Integer> mp = new HashMap<Integer,Integer>();Â
    // Traverse the array firstArr[]    for (int i = 0; i < firstArr.length; i++)     {        mp.put(firstArr[i], i + 1);     }Â
    // Stores the indices of common elements    // between firstArr[] and secondArr[]    Vector<Integer> tempArr = new Vector<>();Â
    // Traverse the array secondArr[]    for (int i = 0; i < secondArr.length; i++)     {Â
        // If current element exists in the Map        if (mp.containsKey(secondArr[i]))         {            tempArr.add(mp.get(secondArr[i]));        }    }Â
    // Stores lIS from tempArr[]    Vector<Integer> tail = new Vector<>();    tail.add(tempArr.get(0));Â
    for (int i = 1; i < tempArr.size(); i++)    {        if (tempArr.get(i) > tail.lastElement())            tail.add(tempArr.get(i));        else if (tempArr.get(i) < tail.get(0))            tail.add(0, tempArr.get(i));         }    return (int)tail.size();}Â
// Driver Codepublic static void main(String[] args){Â Â Â Â int[] firstArr = { 3, 5, 1, 8 };Â Â Â Â int[] secondArr = { 3, 3, 5, 3, 8 };Â Â Â Â System.out.print(LCS(firstArr, secondArr));}}Â
// This code is contributed by gauravrajput1 |
Python3
# Python3 program to implement# the above approachfrom bisect import bisect_leftÂ
# Function to find the Longest Common# Subsequence between the two arraysdef LCS(firstArr, secondArr):Â
    # Maps elements of firstArr[]    # to their respective indices    mp = {}Â
    # Traverse the array firstArr[]    for i in range(len(firstArr)):        mp[firstArr[i]] = i + 1Â
    # Stores the indices of common elements    # between firstArr[] and secondArr[]    tempArr = []Â
    # Traverse the array secondArr[]    for i in range(len(secondArr)):Â
        # If current element exists in the Map        if (secondArr[i] in mp):            tempArr.append(mp[secondArr[i]])Â
    # Stores lIS from tempArr[]    tail = []    tail.append(tempArr[0])    for i in range(1, len(tempArr)):        if (tempArr[i] > tail[-1]):            tail.append(tempArr[i])        elif (tempArr[i] < tail[0]):            tail[0] = tempArr[i]        else :            it = bisect_left(tail, tempArr[i])            it = tempArr[i]    return len(tail) Â
# Driver Codeif __name__ == '__main__':Â Â Â Â firstArr = [3, 5, 1, 8 ]Â Â Â Â secondArr = [3, 3, 5, 3, 8 ]Â Â Â Â print (LCS(firstArr, secondArr))Â
# This code is contributed by mohit kumar 29 |
C#
// C# program to implement// the above approachusing System;using System.Collections.Generic;public class GFG{Â
// Function to find the longest Common// Subsequence between the two arraysstatic int LCS(int[] firstArr,int[] secondArr){Â
    // Maps elements of firstArr[]    // to their respective indices    Dictionary<int,int> mp = new Dictionary<int,int>();Â
    // Traverse the array firstArr[]    for (int i = 0; i < firstArr.Length; i++)     {        mp.Add(firstArr[i], i + 1);     }Â
    // Stores the indices of common elements    // between firstArr[] and secondArr[]    List<int> tempArr = new List<int>();Â
    // Traverse the array secondArr[]    for (int i = 0; i < secondArr.Length; i++)     {Â
        // If current element exists in the Map        if (mp.ContainsKey(secondArr[i]))         {            tempArr.Add(mp[secondArr[i]]);        }    }Â
    // Stores lIS from tempArr[]    List<int> tail = new List<int>();    tail.Add(tempArr[0]);Â
    for (int i = 1; i < tempArr.Count; i++)    {        if (tempArr[i] > tail[tail.Count-1])            tail.Add(tempArr[i]);        else if (tempArr[i] < tail[0])            tail.Insert(0, tempArr[i]);         }    return (int)tail.Count;}Â
// Driver Codepublic static void Main(String[] args){Â Â Â Â int[] firstArr = { 3, 5, 1, 8 };Â Â Â Â int[] secondArr = { 3, 3, 5, 3, 8 };Â Â Â Â Console.Write(LCS(firstArr, secondArr));}}Â
// This code is contributed by Rajput-Ji. |
Javascript
<script>Â
// Javascript program to implement// the above approachÂ
// Function to find the longest Common// Subsequence between the two arraysfunction LCS(firstArr, secondArr){         // Maps elements of firstArr[]    // to their respective indices    let mp = new Map()Â
    // Traverse the array firstArr[]    for(let i = 0; i < firstArr.length; i++)    {        mp.set(firstArr[i], i + 1);    }Â
    // Stores the indices of common elements    // between firstArr[] and secondArr[]    let tempArr = [];Â
    // Traverse the array secondArr[]    for(let i = 0; i < secondArr.length; i++)    {                 // If current element exists in the Map        if (mp.has(secondArr[i]))        {            tempArr.push(mp.get(secondArr[i]));        }    }Â
    // Stores lIS from tempArr[]    let tail = [];    tail.push(tempArr[0]);Â
    for(let i = 1; i < tempArr.length; i++)    {        if (tempArr[i] > tail[tail.length - 1])            tail.push(tempArr[i]);                     else if (tempArr[i] < tail[0])            tail.unshift(0, tempArr[i]);       }    return tail.length;}Â
// Driver Codelet firstArr = [ 3, 5, 1, 8 ];let secondArr = [ 3, 3, 5, 3, 8 ];Â
document.write(LCS(firstArr, secondArr));Â
// This code is contributed by gfgkingÂ
</script> |
Output:Â
3
Â
Time Complexity: O(NlogN)
Auxiliary Space: O(N)
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