Range Queries to count the number of even parity values with updates

• Last Updated : 13 Apr, 2021

Given an array arr[] of N integers, the task is to perform the following two queries:

• query(L, R): Print the number of Even Parity numbers in the subarray from L to R.
• update(i, x): Update the array element reference by index i to x.

Examples:

Input: arr[] = {18, 15, 8, 9, 14, 5}
Query 1: query(L = 0, R = 4)
Query 2: update(i = 3, x = 11)
Query 3: query(L = 0, R = 4)
Output:

Explanation:
Query 1: Subarray is {18, 15, 8, 9, 14}
Binary Representation of these elements –
18 => 10010, Parity = 2
15 => 1111, Parity = 4
8 => 1000, Parity = 1
9 => 1001, Parity = 2
14 => 1110, Parity = 3
Subarray[0-4] have 3 elements with even parity.
Query 2: Update arr[3] = 11
Updated array, {18, 15, 8, 11, 14, 5}
Query 3: Subarray is {18, 15, 8, 11, 14}
Binary Representation of these elements –
18 => 10010, Parity = 2
15 => 1111, Parity = 4
8 => 1000, Parity = 1
11 => 1011, Parity = 3
14 => 1110, Parity = 3
Subarray[0-4] have 2 elements with even parity.

Approach: The idea is to use segment tree to query the count of the even parity elements in a array range and update simultaneously.
We can find the parity for the current value by iterating through each bit of the binary representation of the number and counting the number of set bits. Then check if the parity is even or not. If it has even parity the set it to 1 else to 0.

Building the segment tree:

• Leaf nodes of the segment tree is represented as either 0 (if it is odd parity number) or 1 (if it is even parity number).
• The internal nodes of the segment tree equal to the sum of its child nodes, thus a node represent the total Even Parity numbers in the range from L to R with range [L, R] falling under this node and the sub-tree underneath it.

Handling Queries:

• Query(L, R): Whenever we receive a query from start to end, we can query the segment tree for the sum of nodes in the range from start to end, which in turn represents the number of Even Parity numbers in the range start to end.
• Update(i, x): To perform a update query to update the value at index i to x, we check for the following cases:
• Case 1: If previous value and new value both are Even Parity numbers
Count of Even Parity numbers in the subarray does not change so we just update array and do not modify the segment tree
• Case 2: If previous value and new value both are not Even Parity numbers
Count of Even Parity numbers in the subarray does not change so we just update array and do not modify the segment tree
• Case 3: If previous value is a Even Parity number but new value is not a Even Parity number
Count of Even Parity numbers in the subarray decreases so we update array and add -1 to every range. The index i which is to be updated is a part of in the segment tree
• Case 4: If previous value is not a Even Parity number but new value is a Even Parity number
Count of Even Parity numbers in the subarray increases so we update array and add 1 to every range. The index i which is to be updated is a part of in the segment tree

Below is the implementation of the above approach:

C++

 // C++ implementation to find// number of Even Parity numbers// in a subarray and performing updates #include using namespace std; #define MAX 1000 // Function that returns true if count// of set bits in x is evenbool isEvenParity(int x){    // parity will store the    // count of set bits    int parity = 0;    while (x != 0) {        if (x & 1)            parity++;        x = x >> 1;    }     if (parity % 2 == 0)        return true;    else        return false;} // A utility function to get// the middle indexint getMid(int s, int e){    return s + (e - s) / 2;} // Recursive function to get the number// of Even Parity numbers in a given rangeint queryEvenParityUtil(    int* segmentTree, int segmentStart,    int segmentEnd, int queryStart,    int queryEnd, int index){    // If segment of this node is a part    // of given range, then return    // the number of Even Parity numbers    // in the segment    if (queryStart <= segmentStart        && queryEnd >= segmentEnd)        return segmentTree[index];     // If segment of this node    // is outside the given range    if (segmentEnd < queryStart        || segmentStart > queryEnd)        return 0;     // If a part of this segment    // overlaps with the given range    int mid = getMid(segmentStart, segmentEnd);    return queryEvenParityUtil(               segmentTree, segmentStart, mid,               queryStart, queryEnd, 2 * index + 1)           + queryEvenParityUtil(                 segmentTree, mid + 1, segmentEnd,                 queryStart, queryEnd, 2 * index + 2);} // Recursive function to update// the nodes which have the given// index in their rangevoid updateValueUtil(    int* segmentTree, int segmentStart,    int segmentEnd, int i, int diff, int si){    // Base Case:    if (i < segmentStart || i > segmentEnd)        return;     // If the input index is in range    // of this node, then update the value    // of the node and its children    segmentTree[si] = segmentTree[si] + diff;    if (segmentEnd != segmentStart) {         int mid = getMid(            segmentStart, segmentEnd);        updateValueUtil(            segmentTree, segmentStart,            mid, i, diff, 2 * si + 1);        updateValueUtil(            segmentTree, mid + 1, segmentEnd,            i, diff, 2 * si + 2);    }} // Function to update a value in the// input array and segment treevoid updateValue(int arr[], int* segmentTree,                 int n, int i, int new_val){    // Check for erroneous input index    if (i < 0 || i > n - 1) {        printf("Invalid Input");        return;    }     int diff, oldValue;     oldValue = arr[i];     // Update the value in array    arr[i] = new_val;     // Case 1: Old and new values    // both are Even Parity numbers    if (isEvenParity(oldValue)        && isEvenParity(new_val))        return;     // Case 2: Old and new values    // both not Even Parity numbers    if (!isEvenParity(oldValue)        && !isEvenParity(new_val))        return;     // Case 3: Old value was Even Parity,    // new value is non Even Parity    if (isEvenParity(oldValue)        && !isEvenParity(new_val)) {        diff = -1;    }     // Case 4: Old value was non Even Parity,    // new_val is Even Parity    if (!isEvenParity(oldValue)        && !isEvenParity(new_val)) {        diff = 1;    }     // Update the values of    // nodes in segment tree    updateValueUtil(segmentTree, 0,                    n - 1, i, diff, 0);} // Return number of Even Parity numbersvoid queryEvenParity(int* segmentTree,                     int n, int queryStart,                     int queryEnd){    int EvenParityInRange        = queryEvenParityUtil(            segmentTree, 0, n - 1,            queryStart, queryEnd, 0);     cout << EvenParityInRange << "\n";} // Recursive function that constructs// Segment Tree for the given arrayint constructSTUtil(int arr[],                    int segmentStart,                    int segmentEnd,                    int* segmentTree,                    int si){    // If there is one element in array,    // check if it is Even Parity number    // then store 1 in the segment tree    // else store 0 and return    if (segmentStart == segmentEnd) {         // if arr[segmentStart] is        // Even Parity number        if (isEvenParity(arr[segmentStart]))            segmentTree[si] = 1;        else            segmentTree[si] = 0;         return segmentTree[si];    }     // If there are more than one elements,    // then recur for left and right subtrees    // and store the sum of the    // two values in this node    int mid = getMid(segmentStart,                     segmentEnd);    segmentTree[si]        = constructSTUtil(              arr, segmentStart, mid,              segmentTree, si * 2 + 1)          + constructSTUtil(                arr, mid + 1, segmentEnd,                segmentTree, si * 2 + 2);    return segmentTree[si];} // Function to construct a segment// tree from given arrayint* constructST(int arr[], int n){    // 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* segmentTree = new int[max_size];     // Fill the allocated memory st    constructSTUtil(arr, 0, n - 1,                    segmentTree, 0);     // Return the constructed segment tree    return segmentTree;} // Driver Codeint main(){     int arr[] = { 18, 15, 8, 9, 14, 5 };    int n = sizeof(arr) / sizeof(arr[0]);     // Build segment tree from given array    int* segmentTree = constructST(arr, n);     // Query 1: Query(start = 0, end = 4)    int start = 0;    int end = 4;    queryEvenParity(segmentTree, n, start, end);     // Query 2: Update(i = 3, x = 11),    // i.e Update a[i] to x    int i = 3;    int x = 11;    updateValue(arr, segmentTree, n, i, x);     // Query 3: Query(start = 0, end = 4)    start = 0;    end = 4;    queryEvenParity(        segmentTree, n, start, end);     return 0;}

C#

 // C# implementation to find// number of Even Parity numbers// in a subarray and performing updatesusing System; class GFG{ public static int MAX = 1000; // Function that returns true if count// of set bits in x is evenstatic bool isEvenParity(int x){         // parity will store the    // count of set bits    int parity = 0;         while (x != 0)    {        if ((x & 1) != 0)            parity++;                     x = x >> 1;    }     if (parity % 2 == 0)        return true;    else        return false;} // A utility function to get// the middle indexstatic int getMid(int s, int e){    return s + (e - s) / 2;} // Recursive function to get the number// of Even Parity numbers in a given rangestatic int queryEvenParityUtil(int[] segmentTree,                               int segmentStart,                               int segmentEnd,                               int queryStart,                               int queryEnd, int index){         // If segment of this node is a part    // of given range, then return    // the number of Even Parity numbers    // in the segment    if (queryStart <= segmentStart &&        queryEnd >= segmentEnd)        return segmentTree[index];     // If segment of this node    // is outside the given range    if (segmentEnd < queryStart ||        segmentStart > queryEnd)        return 0;     // If a part of this segment    // overlaps with the given range    int mid = getMid(segmentStart, segmentEnd);    return queryEvenParityUtil(segmentTree, segmentStart,                               mid, queryStart, queryEnd,                               2 * index + 1) +           queryEvenParityUtil(segmentTree, mid + 1,                               segmentEnd, queryStart,                               queryEnd, 2 * index + 2);} // Recursive function to update// the nodes which have the given// index in their rangestatic void updateValueUtil(int[] segmentTree,                            int segmentStart,                            int segmentEnd, int i,                            int diff, int si){         // Base Case:    if (i < segmentStart || i > segmentEnd)        return;     // If the input index is in range    // of this node, then update the value    // of the node and its children    segmentTree[si] = segmentTree[si] + diff;         if (segmentEnd != segmentStart)    {        int mid = getMid(segmentStart, segmentEnd);        updateValueUtil(segmentTree, segmentStart, mid,                        i, diff, 2 * si + 1);        updateValueUtil(segmentTree, mid + 1,                        segmentEnd, i, diff,                        2 * si + 2);    }} // Function to update a value in the// input array and segment treestatic void updateValue(int[] arr, int[] segmentTree,                        int n, int i, int new_val){         // Check for erroneous input index    if (i < 0 || i > n - 1)    {        Console.WriteLine("Invalid Input");        return;    }     int diff = 0, oldValue = 0;     oldValue = arr[i];     // Update the value in array    arr[i] = new_val;     // Case 1: Old and new values    // both are Even Parity numbers    if (isEvenParity(oldValue) &&        isEvenParity(new_val))        return;     // Case 2: Old and new values    // both not Even Parity numbers    if (!isEvenParity(oldValue) &&        !isEvenParity(new_val))        return;     // Case 3: Old value was Even Parity,    // new value is non Even Parity    if (isEvenParity(oldValue) &&       !isEvenParity(new_val))    {        diff = -1;    }     // Case 4: Old value was non Even Parity,    // new_val is Even Parity    if (!isEvenParity(oldValue) &&        !isEvenParity(new_val))    {        diff = 1;    }     // Update the values of    // nodes in segment tree    updateValueUtil(segmentTree, 0, n - 1, i, diff, 0);} // Return number of Even Parity numbersstatic void queryEvenParity(int[] segmentTree, int n,                            int queryStart,                            int queryEnd){    int EvenParityInRange = queryEvenParityUtil(        segmentTree, 0, n - 1, queryStart, queryEnd, 0);     Console.WriteLine(EvenParityInRange);} // Recursive function that constructs// Segment Tree for the given arraystatic int constructSTUtil(int[] arr, int segmentStart,                           int segmentEnd,                           int[] segmentTree, int si){         // If there is one element in array,    // check if it is Even Parity number    // then store 1 in the segment tree    // else store 0 and return    if (segmentStart == segmentEnd)    {                 // if arr[segmentStart] is        // Even Parity number        if (isEvenParity(arr[segmentStart]))            segmentTree[si] = 1;        else            segmentTree[si] = 0;         return segmentTree[si];    }     // If there are more than one elements,    // then recur for left and right subtrees    // and store the sum of the    // two values in this node    int mid = getMid(segmentStart, segmentEnd);    segmentTree[si] = constructSTUtil(arr, segmentStart, mid,                                      segmentTree, si * 2 + 1) +                      constructSTUtil(arr, mid + 1, segmentEnd,                                      segmentTree, si * 2 + 2);    return segmentTree[si];} // Function to construct a segment// tree from given arraystatic int[] constructST(int[] arr, int n){         // Height of segment tree    int x = (int)(Math.Ceiling(Math.Log(n, 2)));     // Maximum size of segment tree    int max_size = 2 * (int)Math.Pow(2, x) - 1;     int[] segmentTree = new int[max_size];     // Fill the allocated memory st    constructSTUtil(arr, 0, n - 1, segmentTree, 0);     // Return the constructed segment tree    return segmentTree;} // Driver Codepublic static void Main(){    int[] arr = { 18, 15, 8, 9, 14, 5 };    int n = arr.Length;     // Build segment tree from given array    int[] segmentTree = constructST(arr, n);     // Query 1: Query(start = 0, end = 4)    int start = 0;    int end = 4;    queryEvenParity(segmentTree, n, start, end);     // Query 2: Update(i = 3, x = 11),    // i.e Update a[i] to x    int i = 3;    int x = 11;    updateValue(arr, segmentTree, n, i, x);     // Query 3: Query(start = 0, end = 4)    start = 0;    end = 4;    queryEvenParity(segmentTree, n, start, end);}} // This code is contributed by ukasp
Output:
3
2

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