Given an array A[] consisting of N distinct integers and another array B[] consisting of M integers, the task is to find the minimum number of elements to be added to the array B[] such that the array A[] becomes the subsequence of the array B[].
Examples:
Input: N = 5, M = 6, A[] = {1, 2, 3, 4, 5}, B[] = {2, 5, 6, 4, 9, 12}
Output: 3
Explanation:
Below are the elements that need to be added:
1) Add 1 before element 2 of B[]
2) Add 3 after element 6 of B[]
3) Add 5 in the last position of B[].
Therefore, the resulting array B[] is {1, 2, 5, 6, 3, 4, 9, 12, 5}.
Hence, A[] is the subsequence of B[] after adding 3 elements.Input: N = 5, M = 5, A[] = {3, 4, 5, 2, 7}, B[] = {3, 4, 7, 9, 2}
Output: 2
Explanation:
Below are the elements that need to be added:
1) Add 5 after element 4.
2) Add 2 after element 5.
Therefore, the resulting array B[] is {3, 4, 5, 2, 7, 9, 2}.
Hence, 2 elements are required to be added.
Naive Approach: Refer to the previous post of this article for the simplest approach to solve the problem.
Time Complexity: O(N * 2M)
Auxiliary Space: O(M + N)
Dynamic Programming Approach: Refer to the previous post of this article for the Longest Common Subsequence based approach.
Time Complexity: O(N * M)
Auxiliary Space: O(N * M)
Efficient Approach: The idea is similar to finding the Longest Increasing Subsequence(LIS) from the array B[]. Follow the steps below to solve the problem:
- Consider elements of array B[] which are present in the array A[], and store the indices of each element of the array A[] in a Map
- Then, find the LIS array subseq[] using Binary Search which consists of the indices in increasing order.
- Finally, the minimum number of elements to be inserted into array B[] is equal to N - len(LIS), where len(LIS) is calculated using Binary Search in the above steps.
Below is the implementation of the above approach:
// C++ program for the
// above approach
#include <bits/stdc++.h>
using namespace std;
// Function to return minimum
// element to be added in array
// B so that array A become
// subsequence of array B
int minElements(int A[], int B[],
int N, int M)
{
// Stores indices of the
// array elements
map<int, int> map;
// Iterate over the array
for (int i = 0; i < N; i++)
{
// Store the indices of
// the array elements
map[A[i]] = i;
}
// Stores the LIS
vector<int> subseq;
int l = 0, r = -1;
for (int i = 0; i < M; i++)
{
// Check if element B[i]
// is in array A[]
if (map.find(B[i]) !=
map.end())
{
int e = map[B[i]];
// Perform Binary Search
while (l <= r)
{
// Find the value of
// mid m
int m = l + (r - l) / 2;
// Update l and r
if (subseq[m] < e)
l = m + 1;
else
r = m - 1;
}
// If found better element
// 'e' for pos r + 1
if (r + 1 < subseq.size())
{
subseq[r + 1] = e;
}
// Otherwise, extend the
// current subsequence
else
{
subseq.push_back(e);
}
l = 0;
r = subseq.size() - 1;
}
}
// Return the answer
return N - subseq.size();
}
// Driver code
int main()
{
// Given arrays
int A[] = {1, 2, 3, 4, 5};
int B[] = {2, 5, 6, 4, 9, 12};
int M = sizeof(A) /
sizeof(A[0]);
int N = sizeof(B) /
sizeof(B[0]);
// Function Call
cout << minElements(A, B,
M, N);
return 0;
}
// This code is contributed by divyeshrabadiya07
// Java program for the above approach
import java.util.*;
import java.lang.*;
class GFG {
// Function to return minimum element
// to be added in array B so that array
// A become subsequence of array B
static int minElements(
int[] A, int[] B, int N, int M)
{
// Stores indices of the
// array elements
Map<Integer, Integer> map
= new HashMap<>();
// Iterate over the array
for (int i = 0;
i < A.length; i++) {
// Store the indices of
// the array elements
map.put(A[i], i);
}
// Stores the LIS
ArrayList<Integer> subseq
= new ArrayList<>();
int l = 0, r = -1;
for (int i = 0; i < M; i++) {
// Check if element B[i]
// is in array A[]
if (map.containsKey(B[i])) {
int e = map.get(B[i]);
// Perform Binary Search
while (l <= r) {
// Find the value of
// mid m
int m = l + (r - l) / 2;
// Update l and r
if (subseq.get(m) < e)
l = m + 1;
else
r = m - 1;
}
// If found better element
// 'e' for pos r + 1
if (r + 1 < subseq.size()) {
subseq.set(r + 1, e);
}
// Otherwise, extend the
// current subsequence
else {
subseq.add(e);
}
l = 0;
r = subseq.size() - 1;
}
}
// Return the answer
return N - subseq.size();
}
// Driver Code
public static void main(String[] args)
{
// Given arrays
int[] A = { 1, 2, 3, 4, 5 };
int[] B = { 2, 5, 6, 4, 9, 12 };
int M = A.length;
int N = B.length;
// Function Call
System.out.println(
minElements(A, B, M, N));
}
}
# Python3 program for the above approach
# Function to return minimum element
# to be added in array B so that array
# A become subsequence of array B
def minElements(A, B, N, M):
# Stores indices of the
# array elements
map = {}
# Iterate over the array
for i in range(len(A)):
# Store the indices of
# the array elements
map[A[i]] = i
# Stores the LIS
subseq = []
l = 0
r = -1
for i in range(M):
# Check if element B[i]
# is in array A[]
if B[i] in map:
e = map[B[i]]
# Perform Binary Search
while (l <= r):
# Find the value of
# mid m
m = l + (r - l) // 2
# Update l and r
if (subseq[m] < e):
l = m + 1
else:
r = m - 1
# If found better element
# 'e' for pos r + 1
if (r + 1 < len(subseq)):
subseq[r + 1]= e
# Otherwise, extend the
# current subsequence
else:
subseq.append(e)
l = 0
r = len(subseq) - 1
# Return the answer
return N - len(subseq)
# Driver Code
if __name__ == '__main__':
# Given arrays
A = [ 1, 2, 3, 4, 5 ]
B = [ 2, 5, 6, 4, 9, 12 ]
M = len(A)
N = len(B)
# Function call
print(minElements(A, B, M, N))
# This code is contributed by mohit kumar 29
// C# program for
// the above approach
using System;
using System.Collections.Generic;
class GFG{
// Function to return minimum element
// to be added in array B so that array
// A become subsequence of array B
static int minElements(int[] A, int[] B,
int N, int M)
{
// Stores indices of the
// array elements
Dictionary<int,
int> map = new Dictionary<int,
int>();
// Iterate over the array
for (int i = 0;
i < A.Length; i++)
{
// Store the indices of
// the array elements
map.Add(A[i], i);
}
// Stores the LIS
List<int> subseq = new List<int>();
int l = 0, r = -1;
for (int i = 0; i < M; i++)
{
// Check if element B[i]
// is in array []A
if (map.ContainsKey(B[i]))
{
int e = map[B[i]];
// Perform Binary Search
while (l <= r)
{
// Find the value of
// mid m
int m = l + (r - l) / 2;
// Update l and r
if (subseq[m] < e)
l = m + 1;
else
r = m - 1;
}
// If found better element
// 'e' for pos r + 1
if (r + 1 < subseq.Count)
{
subseq[r + 1] = e;
}
// Otherwise, extend the
// current subsequence
else
{
subseq.Add(e);
}
l = 0;
r = subseq.Count - 1;
}
}
// Return the answer
return N - subseq.Count;
}
// Driver Code
public static void Main(String[] args)
{
// Given arrays
int[] A = {1, 2, 3, 4, 5};
int[] B = {2, 5, 6, 4, 9, 12};
int M = A.Length;
int N = B.Length;
// Function Call
Console.WriteLine(minElements(A, B,
M, N));
}
}
// This code is contributed by Princi Singh
<script>
// Javascript program for the
// above approach
// Function to return minimum
// element to be added in array
// B so that array A become
// subsequence of array B
function minElements(A, B, N, M)
{
// Stores indices of the
// array elements
var map = new Map();
// Iterate over the array
for (var i = 0; i < N; i++)
{
// Store the indices of
// the array elements
map.set(A[i], i);
}
// Stores the LIS
var subseq = [];
var l = 0, r = -1;
for (var i = 0; i < M; i++)
{
// Check if element B[i]
// is in array A[]
if (map.has(B[i]))
{
var e = map.get(B[i]);
// Perform Binary Search
while (l <= r)
{
// Find the value of
// mid m
var m = l + parseInt((r - l) / 2);
// Update l and r
if (subseq[m] < e)
l = m + 1;
else
r = m - 1;
}
// If found better element
// 'e' for pos r + 1
if (r + 1 < subseq.length)
{
subseq[r + 1] = e;
}
// Otherwise, extend the
// current subsequence
else
{
subseq.push(e);
}
l = 0;
r = subseq.length - 1;
}
}
// Return the answer
return N - subseq.length;
}
// Driver code
// Given arrays
var A = [1, 2, 3, 4, 5];
var B = [2, 5, 6, 4, 9, 12];
var M = A.length;
var N = B.length;
// Function Call
document.write( minElements(A, B,
M, N));
</script>
Output:
3
Time Complexity: O(N logN)
Auxiliary Space: O(N)
