# Count pair of indices in Array having equal Prefix-MEX and Suffix-MEX

• Last Updated : 27 Feb, 2023

Given an array arr[] of N elements, the task is to find the number of pairs of indices such that the Prefix-MEX of the one index is equal to the Suffix-MEX of the other index. Order of indices in pair matters.

MEX of an array refers to the smallest missing non-negative integer of the array. For the given problems (i, j) and (j, i) are considered different pairs.

Examples:

Input: A[] = {1, 0, 2, 0, 1}
Output: 11
Explanation: In array A,   Prefix-MEX for each element are as {0, 2, 3, 3, 3}, similarly, Suffix-MEX for each element are {3, 3, 3, 2, 0} and all possible pairs of indexes such that prefix-MEX of one is equal to Suffix-MEX of other are (1-based indexing):
(1, 5): prefix of 1 is equal to the suffix of 5
(2, 4):  prefix of 2 is equal to the suffix of 4
(3, 3):  prefix of 3 is equal to the suffix of 3
(3, 2):  prefix of 3 is equal to the suffix of 2
(3, 1):  prefix of 3 is equal to the suffix of 1
(4, 3):  prefix of 4 is equal to the suffix of 3
(4, 2):  prefix of 4 is equal to the suffix of 2
(4, 1):  prefix of 4 is equal to the suffix of 1
(5, 3):  prefix of 5 is equal to the suffix of 3
(5, 2):  prefix of 5 is equal to the suffix of 2
(5, 1):  prefix of 5 is equal to the suffix of 1

Input: A[] = {1, 2, 3}
Output: 9

Naive Approach

The most straightforward way will be to find each element’s PrefixMEX and suffixMEX and count a total number of possible pairs and increase the count. And for each element, we have to traverse the complete array again and again therefore complexity will be of order N2.

Time Complexity: O(N^2)
Auxiliary Space: O(1)

Efficient Approach: In this approach, we will pre-compute and store Prefix-MEX and Suffix-MEX for each element.

Follow the steps below to calculate the Prefix-MEX array for the given array:

• Find the maximum element in the given array arr.
• Create a set of integers and store the numbers from 0 to the max element in the set.
• Traverse through the array from i = 0 to N-1:
• For each element, erase that element from the set.
• Now find the smallest element remaining in the set.
• Store the value of this smallest element remaining in the set in the resultant array.
• Return the resultant array as the required answer.

Follow the steps below to implement the idea:

• Create a variable count and initialize it to 0 to store the number of indexes.
• Calculate a Prefix-MEX for arr as P for a given array.
• Reverse the given array arr and calculate the Prefix-MEX array for this revered array as S and reverse S.
• Create a map mp to count the frequency of each element in P.
• Traverse through P increment frequency of current element.
• Traverse through S and check if the current element exists in mp.
• if the current element does not exist in mp then continue.
• else add the frequency of the current element in the variable count.
• print the value of count as a required result.

Below is the implementation of the above approach:

## C++

 `// C++ code to implement the approach` `#include ``using` `namespace` `std;` `// Function to find the prefix MEX``// for each array element taking vector``// and its size as parameter``vector<``int``> Prefix_MEX(vector<``int``>& A, ``int` `n)``{` `    ``// Maximum element in vector A``    ``int` `mx_element = *max_element(A.begin(), A.end());` `    ``// Set to store all elements for``    ``// 0 to mx_element``    ``set<``int``> s;` `    ``// Vector to store Prefix-MEX for``    ``// given input array``    ``vector<``int``> B(n);` `    ``// Store all number from 0``    ``// to maximum element + 1 in a set``    ``for` `(``int` `i = 0; i <= mx_element + 1; i++) {``        ``// inserting elements in set``        ``s.insert(i);``    ``}` `    ``// Loop to calculate MEX for each``    ``// index in the array``    ``for` `(``int` `i = 0; i < n; i++) {` `        ``// Checking if A[i] is present``        ``// in set``        ``auto` `it = s.find(A[i]);` `        ``// If present then we erase``        ``// that element``        ``if` `(it != s.end())``            ``s.erase(it);` `        ``// Store the first element of set``        ``// in vector B as Mex of``        ``// prefix vector``      ``//cout<<*s.begin()<<" ";``        ``B[i] = *s.begin();``    ``}` `    ``// Return the vector B``    ``return` `B;``}` `void` `countPairs(vector<``int``>& arr, ``int` `N)``{` `    ``// Count variable to store number of``    ``// indices whose prefix-mex and``    ``// suffix-mex are equal``    ``int` `count = 0;` `    ``// Vector P stores the Prefix-MEX``    ``// for given array``    ``vector<``int``> P = Prefix_MEX(arr, N);` `    ``// Reversing given array``    ``reverse(arr.begin(), arr.end());` `    ``// vector S stores the reverse of``    ``// Suffix-MEX for given array` `    ``vector<``int``> S = Prefix_MEX(arr, N);` `    ``// Reversing vector S will give``    ``// suffix-MEX for given array``    ``reverse(S.begin(), S.end());` `    ``// map to count frequency of each``    ``// element in arr1``    ``map<``int``, ``int``> mp;` `    ``// Counting frequency of each``    ``// element of arr1``    ``for` `(``int` `i = 0; i < N; i++) {``        ``mp[P[i]]++;``    ``}` `    ``// Traversal through arr S``    ``for` `(``int` `i = 0; i < N; i++) {` `        ``// Check if element does not exist``        ``// in map``        ``if` `(mp.find(S[i]) == mp.end()) {``            ``continue``;``        ``}` `        ``// if exist add it's frequency``        ``// to count``        ``else` `{``            ``count += mp[S[i]];``        ``}``    ``}``    ``cout << count << endl;``}` `// Driver code``int` `main()``{``    ``vector<``int``> arr = { 1, 0, 2, 0, 1 };``    ``int` `N = arr.size();` `    ``// Function Call``    ``countPairs(arr, N);``    ``arr = { 1, 2, 3 };``    ``N = arr.size();` `    ``// Function Call``    ``countPairs(arr, N);` `    ``return` `0;``}`

## Java

 `import` `java.util.*;` `public` `class` `Main {``  ``static` `List prefixMex(List A) {``    ``int` `mxElement = Collections.max(A);` `    ``Set s = ``new` `HashSet<>();``    ``for` `(``int` `i = ``0``; i < mxElement + ``2``; i++) {``      ``s.add(i);``    ``}` `    ``List B = ``new` `ArrayList<>(Collections.nCopies(A.size(), ``0``));` `    ``for` `(``int` `i = ``0``; i < A.size(); i++) {``      ``s.remove(A.get(i));``      ``B.set(i, Collections.min(s));``    ``}` `    ``return` `B;``  ``}` `  ``static` `int` `countPairs(List arr) {``    ``int` `n = arr.size();` `    ``List P = prefixMex(arr);``    ``List S = ``new` `ArrayList<>(arr);``    ``Collections.reverse(S);``    ``S = prefixMex(S);``    ``Collections.reverse(S);` `    ``Map mp = ``new` `HashMap<>();``    ``for` `(``int` `p : P) {``      ``mp.put(p, mp.getOrDefault(p, ``0``) + ``1``);``    ``}` `    ``int` `count = ``0``;``    ``for` `(``int` `s : S) {``      ``count += mp.getOrDefault(s, ``0``);``    ``}` `    ``return` `count;``  ``}` `  ``public` `static` `void` `main(String[] args) {``    ``List arr = Arrays.asList(``1``, ``0``, ``2``, ``0``, ``1``);``    ``System.out.println(countPairs(arr));` `    ``arr = Arrays.asList(``1``, ``2``, ``3``);``    ``System.out.println(countPairs(arr));``  ``}``}`

## Python3

 `from` `typing ``import` `List``, ``Tuple` `def` `prefix_mex(A: ``List``[``int``]) ``-``> ``List``[``int``]:``    ``mx_element ``=` `max``(A)` `    ``s ``=` `set``(``range``(mx_element ``+` `2``))` `    ``B ``=` `[``0``] ``*` `len``(A)` `    ``for` `i, a ``in` `enumerate``(A):``        ``s.discard(a)``        ``B[i] ``=` `min``(s)` `    ``return` `B` `def` `count_pairs(arr: ``List``[``int``]) ``-``> ``int``:``    ``n ``=` `len``(arr)` `    ``P ``=` `prefix_mex(arr)``    ``S ``=` `prefix_mex(arr[::``-``1``])[::``-``1``]` `    ``mp ``=` `{}``    ``for` `p ``in` `P:``        ``mp[p] ``=` `mp.get(p, ``0``) ``+` `1` `    ``count ``=` `0``    ``for` `s ``in` `S:``        ``count ``+``=` `mp.get(s, ``0``)` `    ``return` `count` `arr ``=` `[``1``, ``0``, ``2``, ``0``, ``1``]``print``(count_pairs(arr))` `arr ``=` `[``1``, ``2``, ``3``]``print``(count_pairs(arr))`

## C#

 `// C# code to implement the approach``using` `System;``using` `System.Collections.Generic;``using` `System.Linq;` `public` `class` `GFG {``    ` `    ``// Function to find the prefix MEX``    ``// for each array element taking vector``    ``// and its size as parameter``    ``static` `List<``int``> Prefix_MEX(List<``int``> A, ``int` `n)``    ``{``        ``// Maximum element in List A``        ``int` `mxElement = A.Max();` `        ``// Set to store all elements for``        ``// 0 to mx_element``        ``HashSet<``int``> s = ``new` `HashSet<``int``>();``        ` `        ``// Store all number from 0``        ``// to maximum element + 1 in a set``        ``for` `(``int` `i = 0; i < mxElement + 2; i++) {``            ``// inserting elements in set``            ``s.Add(i);``        ``}` `        ``// List to store Prefix-MEX for``        ``// given input array``        ``List<``int``> B = ``new` `List<``int``>(``new` `int``[A.Count]);` `        ``// Loop to calculate MEX for each``        ``// index in the array``        ``for` `(``int` `i = 0; i < n; i++) {``            ``// Checking if A[i] is present in set``            ``// If present then we remove that element``            ``s.Remove(A[i]);``            ` `            ``// Store the first element of set``            ``// in vector B as Mex of``            ``B[i] = s.Min();``        ``}` `        ``return` `B;``    ``}` `    ``static` `int` `CountPairs(List<``int``> arr, ``int` `n)``    ``{``        ` `        ``// Vector P stores the Prefix-MEX``        ``// for given array``        ``List<``int``> P = Prefix_MEX(arr, n);``        ` `        ``// vector S stores the reverse of``        ``// Suffix-MEX for given array``        ``List<``int``> S = ``new` `List<``int``>(arr);``        ` `        ``// Reversing given array``        ``S.Reverse();``        ``S = Prefix_MEX(S, n);``        ` `        ``// Reversing vector S will give``        ``// suffix-MEX for given array``        ``S.Reverse();` `         ``// map to count frequency of each``        ``// element in arr1``        ``Dictionary<``int``, ``int``> mp``            ``= ``new` `Dictionary<``int``, ``int``>();``            ` `        ``// Counting frequency of each``        ``// element of arr1``        ``foreach``(``int` `p ``in` `P) {``            ``if` `(mp.ContainsKey(p)) {``                ``mp[p]++;``            ``}``            ``else` `{``                ``mp[p] = 1;``            ``}``        ``}``    ` `        ``// Count variable to store number of``        ``// indices whose prefix-mex and``        ``// suffix-mex are equal``        ``int` `count = 0;``        ` `        ``// Traversal through arr S``        ``foreach``(``int` `s ``in` `S) {``            ``if` `(mp.ContainsKey(s)) {``                ``count += mp[s];``            ``}``        ``}``        ``return` `count;``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ``List<``int``> arr = ``new` `List<``int``>{ 1, 0, 2, 0, 1 };``        ``int` `N = arr.Count;``        ` `        ``// Function Call``        ``Console.WriteLine(CountPairs(arr, N));` `        ``arr = ``new` `List<``int``>{ 1, 2, 3 };``        ``N = arr.Count;``        ` `        ``// Function Call``        ``Console.WriteLine(CountPairs(arr, N));``    ``}``}` `// This Code is Contributed by Prasad Kandekar(prasad264)`

## Javascript

 `// javascript code to implement the approach` `// Function to find the prefix MEX``// for each array element taking vector``// and its size as parameter``function` `Prefix_MEX(A, n)``{` `    ``// Maximum element in vector A``    ``let mx_element = Number.MIN_SAFE_INTEGER;``    ``for``(let i=0; i

Output

```11
9```

Time Complexity: O(N * log N), log N for inserting and deleting elements from the set, and O(N) for traversing the array.
Auxiliary Space:  O(N) for storing Prefix-MEX array.

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