Hello Everyone,

Given an array of n integers in non-decreasing order. Find the number of occurrences of the most frequent value within a given range.

Examples:

Input : arr[] = {-5, -5, 2, 2, 2, 2, 3, 7, 7, 7}

Query 1: start = 0, end = 9

Query 2: start = 4, end = 9

Output : 4

3

Explanation:

Query 1: ‘2’ occurred the most number of times

with a frequency of 4 within given range.

Query 2: ‘7’ occurred the most number of times

with a frequency of 3 within given range.

Segment Trees can be used to solve this problem efficiently.

The key idea behind this problem is that the given array is in non-decreasing order which means that all occurrences of a number are consecutively placed in the array as the array is in sorted order.

A segment tree can be constructed where each node would store the maximum count of its respective range [i, j]. For that we will build the frequency array and call RMQ (Range Maximum Query) on this array. For e.g.

arr[] = {-5, -5, 2, 2, 2, 2, 3, 7, 7, 7} freq_arr[] = {2, 2, 4, 4, 4, 4, 1, 3, 3, 3} where, freq_arr[i] = frequency(arr[i])

Now there are two cases to be considered,

**Case 1: The value of the numbers at index i and j for the given range are same, i.e. arr[i] = arr[j].**

Solving this case is very easy. Since arr[i] = arr[j], all numbers between these indices are same (since the array is non-decreasing). Hence answer for this case is simply count of all numbers between i and j (inclusive both) i.e. (j – i + 1)

For e.g.

arr[] = {-5, -5, 2, 2, 2, 2, 3, 7, 7, 7} if the given query range is [3, 5], answer would be (5 - 3 + 1) = 3, as 2 occurs 3 times within given range

**Case 2: The value of the numbers at index i and j for the given range are different, i.e. arr[i] != arr[j]**

If arr[i] != arr[j], then there exists an index k where where arr[i] = arr[k] and arr[i] != arr[k + 1]. This may be a case of partial overlap where some occurrences of a particular number lie in the leftmost part of the given range and some lie just before range starts. Here simply calling RMQ would result into an incorrect answer.

For e.g.

arr[] = {-5, -5, 2, 2, 2, 2, 3, 7, 7, 7} freq_arr[] = {2, 2, 4, 4, 4, 4, 1, 3, 3, 3} if the given query is [4, 9], calling RMQ on freq_arr[] will give us 4 as answer which is incorrect as some occurrences of 2 are lying outside the range. Correct answer is 3.

Similar situation can happen at the rightmost part of the given range where some occurrences of a particular number lies inside the range and some lies just after the range ends.

Hence for this case, inside the given range we have to count the leftmost same numbers upto some index say i and rightmost same numbers from index say j to the end of the range. And then calling RMQ (Range Maximum Query) between indices i and j and taking maximum of all these three.

For e.g.

arr[] = {-5, -5, 2, 2, 2, 2, 3, 7, 7, 7} freq_arr[] = {2, 2, 4, 4, 4, 4, 1, 3, 3, 3} if the given query is [4, 7], counting leftmost same numbers i.e 2 which occurs 2 times inside the range and rightmost same numbers i.e. 3 which occur only 1 time and RMQ on [6, 6] is 1. Hence maximum would be 2.

Below is the implementation of the above approach

`// C++ Program to find the occurrence`

`// of the most frequent number within`

`// a given range`

`#include <bits/stdc++.h>`

`using`

`namespace`

`std;`

` `

`// A utility function to get the middle index`

`// from corner indexes.`

`int`

`getMid(`

`int`

`s, `

`int`

`e) { `

`return`

`s + (e - s) / 2; }`

` `

`/* A recursive function to get the maximum value in`

` `

`a given range of array indexes. The following`

` `

`are parameters for this function.`

` `

` `

`st --> Pointer to segment tree`

` `

`index --> Index of current node in the segment `

` `

`tree. Initially 0 is passed as root is`

` `

`always at index 0`

` `

`ss & se --> Starting and ending indexes of the `

` `

`segment represented by current node,`

` `

`i.e., st[index]`

` `

`qs & qe --> Starting and ending indexes of query `

` `

`range */`

`int`

`RMQUtil(`

`int`

`* st, `

`int`

`ss, `

`int`

`se, `

`int`

`qs, `

`int`

`qe, `

` `

`int`

`index)`

`{`

` `

`// If segment of this node is a part of given range,`

` `

`// then return the min of the segment`

` `

`if`

`(qs <= ss && qe >= se)`

` `

`return`

`st[index];`

` `

` `

`// If segment of this node is outside the`

` `

`// given range`

` `

`if`

`(se < qs || ss > qe)`

` `

`return`

`0;`

` `

` `

`// If a part of this segment overlaps `

` `

`// with the given range`

` `

`int`

`mid = getMid(ss, se);`

` `

`return`

`max(RMQUtil(st, ss, mid, qs, qe, 2 * index + 1),`

` `

`RMQUtil(st, mid + 1, se, qs, qe, 2 * index + 2));`

`}`

` `

`// Return minimum of elements in range from`

`// index qs (query start) to`

`// qe (query end). It mainly uses RMQUtil()`

`int`

`RMQ(`

`int`

`* st, `

`int`

`n, `

`int`

`qs, `

`int`

`qe)`

`{`

` `

`// Check for erroneous input values`

` `

`if`

`(qs < 0 || qe > n - 1 || qs > qe) {`

` `

`printf`

`(`

`"Invalid Input"`

`);`

` `

`return`

`-1;`

` `

`}`

` `

` `

`return`

`RMQUtil(st, 0, n - 1, qs, qe, 0);`

`}`

` `

`// A recursive function that constructs Segment Tree`

`// for array[ss..se]. si is index of current node in`

`// segment tree st`

`int`

`constructSTUtil(`

`int`

`arr[], `

`int`

`ss, `

`int`

`se, `

`int`

`* st, `

` `

`int`

`si)`

`{`

` `

`// If there is one element in array, store it in`

` `

`// current node of segment tree and return`

` `

`if`

`(ss == se) {`

` `

`st[si] = arr[ss];`

` `

`return`

`arr[ss];`

` `

`}`

` `

` `

`// If there are more than one elements, then `

` `

`// recur for left and right subtrees and store `

` `

`// the minimum of two values in this node`

` `

`int`

`mid = getMid(ss, se);`

` `

`st[si] = max(constructSTUtil(arr, ss, mid, st, si * 2 + 1),`

` `

`constructSTUtil(arr, mid + 1, se, st, si * 2 + 2));`

` `

`return`

`st[si];`

`}`

` `

`/* Function to construct segment tree from given `

` `

`array. This function allocates memory for segment`

` `

`tree and calls constructSTUtil() to fill the `

` `

`allocated memory */`

`int`

`* constructST(`

`int`

`arr[], `

`int`

`n)`

`{`

` `

`// Allocate memory for segment tree`

` `

` `

`// Height of segment tree`

` `

`int`

`x = (`

`int`

`)(`

`ceil`

`(log2(n)));`

` `

` `

`// Maximum size of segment tree`

` `

`int`

`max_size = 2 * (`

`int`

`)`

`pow`

`(2, x) - 1;`

` `

` `

`int`

`* st = `

`new`

`int`

`[max_size];`

` `

` `

`// Fill the allocated memory st`

` `

`constructSTUtil(arr, 0, n - 1, st, 0);`

` `

` `

`// Return the constructed segment tree`

` `

`return`

`st;`

`}`

` `

`int`

`maximumOccurrence(`

`int`

`arr[], `

`int`

`n, `

`int`

`qs, `

`int`

`qe)`

`{`

` `

`// Declaring a frequency array`

` `

`int`

`freq_arr[n + 1];`

` `

` `

`// Counting frequencies of all array elements.`

` `

`unordered_map<`

`int`

`, `

`int`

`> cnt;`

` `

`for`

`(`

`int`

`i = 0; i < n; i++)`

` `

`cnt[arr[i]]++; `

` `

` `

`// Creating frequency array by replacing the `

` `

`// number in array to the number of times it `

` `

`// has appeared in the array`

` `

`for`

`(`

`int`

`i = 0; i < n; i++)`

` `

`freq_arr[i] = cnt[arr[i]];`

` `

` `

`// Build segment tree from this frequency array`

` `

`int`

`* st = constructST(freq_arr, n);`

` `

` `

`int`

`maxOcc; `

`// to store the answer`

` `

` `

`// Case 1: numbers are same at the starting `

` `

`// and ending index of the query`

` `

`if`

`(arr[qs] == arr[qe])`

` `

`maxOcc = (qe - qs + 1);`

` `

` `

`// Case 2: numbers are different`

` `

`else`

`{`

` `

`int`

`leftmost_same = 0, righmost_same = 0;`

` `

` `

`// Partial Overlap Case of a number with some`

` `

`// occurrences lying inside the leftmost`

` `

`// part of the range and some just before the`

` `

`// range starts`

` `

`while`

`(qs > 0 && qs <= qe && arr[qs] == arr[qs - 1]) {`

` `

`qs++;`

` `

`leftmost_same++;`

` `

`}`

` `

` `

`// Partial Overlap Case of a number with some `

` `

`// occurrences lying inside the rightmost part of `

` `

`// the range and some just after the range ends`

` `

`while`

`(qe >= qs && qe < n - 1 && arr[qe] == arr[qe + 1]) {`

` `

`qe--;`

` `

`righmost_same++;`

` `

`}`

` `

`// Taking maximum of all three`

` `

`maxOcc = max({leftmost_same, righmost_same, `

` `

`RMQ(st, n, qs, qe)});`

` `

`}`

` `

`return`

`maxOcc;`

`}`

` `

`// Driver Code`

`int`

`main()`

`{`

` `

`int`

`arr[] = { -5, -5, 2, 2, 2, 2, 3, 7, 7, 7 };`

` `

`int`

`n = `

`sizeof`

`(arr) / `

`sizeof`

`(arr[0]);`

` `

` `

`int`

`qs = 0; `

`// Starting index of query range`

` `

`int`

`qe = 9; `

`// Ending index of query range`

` `

` `

`// Print occurrence of most frequent number `

` `

`// within given range`

` `

`cout << `

`"Maximum Occurrence in range is = "`

` `

`<< maximumOccurrence(arr, n, qs, qe) << endl;`

` `

` `

`qs = 4; `

`// Starting index of query range`

` `

`qe = 9; `

`// Ending index of query range`

` `

` `

`// Print occurrence of most frequent number`

` `

`// within given range`

` `

`cout << `

`"Maximum Occurrence in range is = "`

` `

`<< maximumOccurrence(arr, n, qs, qe) << endl;`

` `

` `

`return`

`0;`

`}`

**Output:**

Maximum Occurrence in range is = 4 Maximum Occurrence in range is = 3

**Further Optimization:** For the partial overlapping case we have to run a loop to calculate the count of same numbers on both sides. To avoid this loop and perform this operation in O(1), we can store the index of the first occurrence of every number in the given array and hence by doing some precomputation we can find the required count in O(1).

**Time Complexity:**

Time Complexity for tree construction is O(n). Time complexity to query is O(Log n).