Two Dimensional Binary Indexed Tree or Fenwick Tree
Prerequisite – Fenwick Tree
We know that to answer range sum queries on a 1-D array efficiently, binary indexed tree (or Fenwick Tree) is the best choice (even better than segment tree due to less memory requirements and a little faster than segment tree).
Can we answer sub-matrix sum queries efficiently using Binary Indexed Tree ?
The answer is yes. This is possible using a 2D BIT which is nothing but an array of 1D BIT.
Algorithm:
We consider the below example. Suppose we have to find the sum of all numbers inside the highlighted area:
We assume the origin of the matrix at the bottom – O.Then a 2D BIT exploits the fact that-
Sum under the marked area = Sum(OB) - Sum(OD) -
Sum(OA) + Sum(OC)
In our program, we use the getSum(x, y) function which finds the sum of the matrix from (0, 0) to (x, y).
Hence the below formula :
Sum under the marked area = Sum(OB) - Sum(OD) -
Sum(OA) + Sum(OC)
The above formula gets reduced to,
Query(x1,y1,x2,y2) = getSum(x2, y2) -
getSum(x2, y1-1) -
getSum(x1-1, y2) +
getSum(x1-1, y1-1)
where,
x1, y1 = x and y coordinates of C
x2, y2 = x and y coordinates of B
The updateBIT(x, y, val) function updates all the elements under the region – (x, y) to (N, M) where,
N = maximum X co-ordinate of the whole matrix.
M = maximum Y co-ordinate of the whole matrix.
The rest procedure is quite similar to that of 1D Binary Indexed Tree.
Below is the C++ implementation of 2D indexed tree
C++
/* C++ program to implement 2D Binary Indexed Tree 2D BIT is basically a BIT where each element is another BIT. Updating by adding v on (x, y) means it's effect will be found throughout the rectangle [(x, y), (max_x, max_y)], and query for (x, y) gives you the result of the rectangle [(0, 0), (x, y)], assuming the total rectangle is [(0, 0), (max_x, max_y)]. So when you query and update on this BIT,you have to be careful about how many times you are subtracting a rectangle and adding it. Simple set union formula works here. So if you want to get the result of a specific rectangle [(x1, y1), (x2, y2)], the following steps are necessary: Query(x1,y1,x2,y2) = getSum(x2, y2)-getSum(x2, y1-1) - getSum(x1-1, y2)+getSum(x1-1, y1-1) Here 'Query(x1,y1,x2,y2)' means the sum of elements enclosed in the rectangle with bottom-left corner's co-ordinates (x1, y1) and top-right corner's co-ordinates - (x2, y2) Constraints -> x1<=x2 and y1<=y2 /\ y | | --------(x2,y2) | | | | | | | | | | --------- | (x1,y1) | |___________________________ (0, 0) x--> In this program we have assumed a square matrix. The program can be easily extended to a rectangular one. */ #include<bits/stdc++.h> using namespace std; #define N 4 // N-->max_x and max_y // A structure to hold the queries struct Query { int x1, y1; // x and y co-ordinates of bottom left int x2, y2; // x and y co-ordinates of top right }; // A function to update the 2D BIT void updateBIT( int BIT[][N+1], int x, int y, int val) { for (; x <= N; x += (x & -x)) { // This loop update all the 1D BIT inside the // array of 1D BIT = BIT[x] for ( int yy=y; yy <= N; yy += (yy & -yy)) BIT[x][yy] += val; } return ; } // A function to get sum from (0, 0) to (x, y) int getSum( int BIT[][N+1], int x, int y) { int sum = 0; for (; x > 0; x -= x&-x) { // This loop sum through all the 1D BIT // inside the array of 1D BIT = BIT[x] for ( int yy=y; yy > 0; yy -= yy&-yy) { sum += BIT[x][yy]; } } return sum; } // A function to create an auxiliary matrix // from the given input matrix void constructAux( int mat[][N], int aux[][N+1]) { // Initialise Auxiliary array to 0 for ( int i=0; i<=N; i++) for ( int j=0; j<=N; j++) aux[i][j] = 0; // Construct the Auxiliary Matrix for ( int j=1; j<=N; j++) for ( int i=1; i<=N; i++) aux[i][j] = mat[N-j][i-1]; return ; } // A function to construct a 2D BIT void construct2DBIT( int mat[][N], int BIT[][N+1]) { // Create an auxiliary matrix int aux[N+1][N+1]; constructAux(mat, aux); // Initialise the BIT to 0 for ( int i=1; i<=N; i++) for ( int j=1; j<=N; j++) BIT[i][j] = 0; for ( int j=1; j<=N; j++) { for ( int i=1; i<=N; i++) { // Creating a 2D-BIT using update function // everytime we/ encounter a value in the // input 2D-array int v1 = getSum(BIT, i, j); int v2 = getSum(BIT, i, j-1); int v3 = getSum(BIT, i-1, j-1); int v4 = getSum(BIT, i-1, j); // Assigning a value to a particular element // of 2D BIT updateBIT(BIT, i, j, aux[i][j]-(v1-v2-v4+v3)); } } return ; } // A function to answer the queries void answerQueries(Query q[], int m, int BIT[][N+1]) { for ( int i=0; i<m; i++) { int x1 = q[i].x1 + 1; int y1 = q[i].y1 + 1; int x2 = q[i].x2 + 1; int y2 = q[i].y2 + 1; int ans = getSum(BIT, x2, y2)-getSum(BIT, x2, y1-1)- getSum(BIT, x1-1, y2)+getSum(BIT, x1-1, y1-1); printf ( "Query(%d, %d, %d, %d) = %d\n" , q[i].x1, q[i].y1, q[i].x2, q[i].y2, ans); } return ; } // Driver program int main() { int mat[N][N] = {{1, 2, 3, 4}, {5, 3, 8, 1}, {4, 6, 7, 5}, {2, 4, 8, 9}}; // Create a 2D Binary Indexed Tree int BIT[N+1][N+1]; construct2DBIT(mat, BIT); /* Queries of the form - x1, y1, x2, y2 For example the query- {1, 1, 3, 2} means the sub-matrix- y /\ 3 | 1 2 3 4 Sub-matrix 2 | 5 3 8 1 {1,1,3,2} ---> 3 8 1 1 | 4 6 7 5 6 7 5 0 | 2 4 8 9 | --|------ 0 1 2 3 ----> x | Hence sum of the sub-matrix = 3+8+1+6+7+5 = 30 */ Query q[] = {{1, 1, 3, 2}, {2, 3, 3, 3}, {1, 1, 1, 1}}; int m = sizeof (q)/ sizeof (q[0]); answerQueries(q, m, BIT); return (0); } |
Java
/* Java program to implement 2D Binary Indexed Tree 2D BIT is basically a BIT where each element is another BIT. Updating by adding v on (x, y) means it's effect will be found throughout the rectangle [(x, y), (max_x, max_y)], and query for (x, y) gives you the result of the rectangle [(0, 0), (x, y)], assuming the total rectangle is [(0, 0), (max_x, max_y)]. So when you query and update on this BIT,you have to be careful about how many times you are subtracting a rectangle and adding it. Simple set union formula works here. So if you want to get the result of a specific rectangle [(x1, y1), (x2, y2)], the following steps are necessary: Query(x1,y1,x2,y2) = getSum(x2, y2)-getSum(x2, y1-1) - getSum(x1-1, y2)+getSum(x1-1, y1-1) Here 'Query(x1,y1,x2,y2)' means the sum of elements enclosed in the rectangle with bottom-left corner's co-ordinates (x1, y1) and top-right corner's co-ordinates - (x2, y2) Constraints -> x1<=x2 and y1<=y2 /\ y | | --------(x2,y2) | | | | | | | | | | --------- | (x1,y1) | |___________________________ (0, 0) x--> In this program we have assumed a square matrix. The program can be easily extended to a rectangular one. */ class GFG { static final int N = 4 ; // N-.max_x and max_y // A structure to hold the queries static class Query { int x1, y1; // x and y co-ordinates of bottom left int x2, y2; // x and y co-ordinates of top right public Query( int x1, int y1, int x2, int y2) { this .x1 = x1; this .y1 = y1; this .x2 = x2; this .y2 = y2; } }; // A function to update the 2D BIT static void updateBIT( int BIT[][], int x, int y, int val) { for (; x <= N; x += (x & -x)) { // This loop update all the 1D BIT inside the // array of 1D BIT = BIT[x] for (; y <= N; y += (y & -y)) BIT[x][y] += val; } return ; } // A function to get sum from (0, 0) to (x, y) static int getSum( int BIT[][], int x, int y) { int sum = 0 ; for (; x > 0 ; x -= x&-x) { // This loop sum through all the 1D BIT // inside the array of 1D BIT = BIT[x] for (; y > 0 ; y -= y&-y) { sum += BIT[x][y]; } } return sum; } // A function to create an auxiliary matrix // from the given input matrix static void constructAux( int mat[][], int aux[][]) { // Initialise Auxiliary array to 0 for ( int i = 0 ; i <= N; i++) for ( int j = 0 ; j <= N; j++) aux[i][j] = 0 ; // Construct the Auxiliary Matrix for ( int j = 1 ; j <= N; j++) for ( int i = 1 ; i <= N; i++) aux[i][j] = mat[N - j][i - 1 ]; return ; } // A function to construct a 2D BIT static void construct2DBIT( int mat[][], int BIT[][]) { // Create an auxiliary matrix int [][]aux = new int [N + 1 ][N + 1 ]; constructAux(mat, aux); // Initialise the BIT to 0 for ( int i = 1 ; i <= N; i++) for ( int j = 1 ; j <= N; j++) BIT[i][j] = 0 ; for ( int j = 1 ; j <= N; j++) { for ( int i = 1 ; i <= N; i++) { // Creating a 2D-BIT using update function // everytime we/ encounter a value in the // input 2D-array int v1 = getSum(BIT, i, j); int v2 = getSum(BIT, i, j - 1 ); int v3 = getSum(BIT, i - 1 , j - 1 ); int v4 = getSum(BIT, i - 1 , j); // Assigning a value to a particular element // of 2D BIT updateBIT(BIT, i, j, aux[i][j] - (v1 - v2 - v4 + v3)); } } return ; } // A function to answer the queries static void answerQueries(Query q[], int m, int BIT[][]) { for ( int i = 0 ; i < m; i++) { int x1 = q[i].x1 + 1 ; int y1 = q[i].y1 + 1 ; int x2 = q[i].x2 + 1 ; int y2 = q[i].y2 + 1 ; int ans = getSum(BIT, x2, y2) - getSum(BIT, x2, y1 - 1 ) - getSum(BIT, x1 - 1 , y2) + getSum(BIT, x1 - 1 , y1 - 1 ); System.out.printf( "Query(%d, %d, %d, %d) = %d\n" , q[i].x1, q[i].y1, q[i].x2, q[i].y2, ans); } return ; } // Driver Code public static void main(String[] args) { int mat[][] = { { 1 , 2 , 3 , 4 }, { 5 , 3 , 8 , 1 }, { 4 , 6 , 7 , 5 }, { 2 , 4 , 8 , 9 } }; // Create a 2D Binary Indexed Tree int [][]BIT = new int [N + 1 ][N + 1 ]; construct2DBIT(mat, BIT); /* Queries of the form - x1, y1, x2, y2 For example the query- {1, 1, 3, 2} means the sub-matrix- y /\ 3 | 1 2 3 4 Sub-matrix 2 | 5 3 8 1 {1,1,3,2} --. 3 8 1 1 | 4 6 7 5 6 7 5 0 | 2 4 8 9 | --|------ 0 1 2 3 ---. x | Hence sum of the sub-matrix = 3+8+1+6+7+5 = 30 */ Query q[] = { new Query( 1 , 1 , 3 , 2 ), new Query( 2 , 3 , 3 , 3 ), new Query( 1 , 1 , 1 , 1 )}; int m = q.length; answerQueries(q, m, BIT); } } // This code is contributed by 29AjayKumar |
Python3
'''Python3 program to implement 2D Binary Indexed Tree 2D BIT is basically a BIT where each element is another BIT. Updating by adding v on (x, y) means it's effect will be found throughout the rectangle [(x, y), (max_x, max_y)], and query for (x, y) gives you the result of the rectangle [(0, 0), (x, y)], assuming the total rectangle is [(0, 0), (max_x, max_y)]. So when you query and update on this BIT,you have to be careful about how many times you are subtracting a rectangle and adding it. Simple set union formula works here. So if you want to get the result of a specific rectangle [(x1, y1), (x2, y2)], the following steps are necessary: Query(x1,y1,x2,y2) = getSum(x2, y2)-getSum(x2, y1-1) - getSum(x1-1, y2)+getSum(x1-1, y1-1) Here 'Query(x1,y1,x2,y2)' means the sum of elements enclosed in the rectangle with bottom-left corner's co-ordinates (x1, y1) and top-right corner's co-ordinates - (x2, y2) Constraints -> x1<=x2 and y1<=y2 /\ y | | --------(x2,y2) | | | | | | | | | | --------- | (x1,y1) | |___________________________ (0, 0) x--> In this program we have assumed a square matrix. The program can be easily extended to a rectangular one. ''' N = 4 # N-.max_x and max_y # A structure to hold the queries class Query: def __init__( self , x1,y1,x2,y2): self .x1 = x1; self .y1 = y1; self .x2 = x2; self .y2 = y2; # A function to update the 2D BIT def updateBIT(BIT,x,y,val): while x < = N: # This loop update all the 1D BIT inside the # array of 1D BIT = BIT[x] while y < = N: BIT[x][y] + = val; y + = (y & - y) x + = (x & - x) return ; # A function to get sum from (0, 0) to (x, y) def getSum(BIT,x,y): sum = 0 ; while x > 0 : # This loop sum through all the 1D BIT # inside the array of 1D BIT = BIT[x] while y > 0 : sum + = BIT[x][y]; y - = y& - y x - = x& - x return sum ; # A function to create an auxiliary matrix # from the given input matrix def constructAux(mat,aux): # Initialise Auxiliary array to 0 for i in range (N + 1 ): for j in range (N + 1 ): aux[i][j] = 0 # Construct the Auxiliary Matrix for j in range ( 1 , N + 1 ): for i in range ( 1 , N + 1 ): aux[i][j] = mat[N - j][i - 1 ]; return # A function to construct a 2D BIT def construct2DBIT(mat,BIT): # Create an auxiliary matrix aux = [ None for i in range (N + 1 )] for i in range (N + 1 ) : aux[i] = [ None for i in range (N + 1 )] constructAux(mat, aux) # Initialise the BIT to 0 for i in range ( 1 , N + 1 ): for j in range ( 1 , N + 1 ): BIT[i][j] = 0 ; for j in range ( 1 , N + 1 ): for i in range ( 1 , N + 1 ): # Creating a 2D-BIT using update function # everytime we/ encounter a value in the # input 2D-array v1 = getSum(BIT, i, j); v2 = getSum(BIT, i, j - 1 ); v3 = getSum(BIT, i - 1 , j - 1 ); v4 = getSum(BIT, i - 1 , j); # Assigning a value to a particular element # of 2D BIT updateBIT(BIT, i, j, aux[i][j] - (v1 - v2 - v4 + v3)); return ; # A function to answer the queries def answerQueries(q,m,BIT): for i in range (m): x1 = q[i].x1 + 1 ; y1 = q[i].y1 + 1 ; x2 = q[i].x2 + 1 ; y2 = q[i].y2 + 1 ; ans = getSum(BIT, x2, y2) - \ getSum(BIT, x2, y1 - 1 ) - \ getSum(BIT, x1 - 1 , y2) + \ getSum(BIT, x1 - 1 , y1 - 1 ); print ( "Query (" , q[i].x1, ", " , q[i].y1, ", " , q[i].x2, ", " , q[i].y2, ") = " ,ans, sep = "") return ; # Driver Code mat = [[ 1 , 2 , 3 , 4 ], [ 5 , 3 , 8 , 1 ], [ 4 , 6 , 7 , 5 ], [ 2 , 4 , 8 , 9 ]]; # Create a 2D Binary Indexed Tree BIT = [ None for i in range (N + 1 )] for i in range (N + 1 ): BIT[i] = [ None for i in range (N + 1 )] for j in range (N + 1 ): BIT[i][j] = 0 construct2DBIT(mat, BIT); ''' Queries of the form - x1, y1, x2, y2 For example the query- {1, 1, 3, 2} means the sub-matrix- y /\ 3 | 1 2 3 4 Sub-matrix 2 | 5 3 8 1 {1,1,3,2} --. 3 8 1 1 | 4 6 7 5 6 7 5 0 | 2 4 8 9 | --|------ 0 1 2 3 ---. x | Hence sum of the sub-matrix = 3+8+1+6+7+5 = 30 ''' q = [Query( 1 , 1 , 3 , 2 ), Query( 2 , 3 , 3 , 3 ), Query( 1 , 1 , 1 , 1 )]; m = len (q) answerQueries(q, m, BIT); # This code is contributed by phasing17 |
C#
/* C# program to implement 2D Binary Indexed Tree 2D BIT is basically a BIT where each element is another BIT. Updating by.Adding v on (x, y) means it's effect will be found throughout the rectangle [(x, y), (max_x, max_y)], and query for (x, y) gives you the result of the rectangle [(0, 0), (x, y)], assuming the total rectangle is [(0, 0), (max_x, max_y)]. So when you query and update on this BIT,you have to be careful about how many times you are subtracting a rectangle and.Adding it. Simple set union formula works here. So if you want to get the result of a specific rectangle [(x1, y1), (x2, y2)], the following steps are necessary: Query(x1,y1,x2,y2) = getSum(x2, y2)-getSum(x2, y1-1) - getSum(x1-1, y2)+getSum(x1-1, y1-1) Here 'Query(x1,y1,x2,y2)' means the sum of elements enclosed in the rectangle with bottom-left corner's co-ordinates (x1, y1) and top-right corner's co-ordinates - (x2, y2) Constraints -> x1<=x2 and y1<=y2 /\ y | | --------(x2,y2) | | | | | | | | | | --------- | (x1,y1) | |___________________________ (0, 0) x--> In this program we have assumed a square matrix. The program can be easily extended to a rectangular one. */ using System; class GFG { static readonly int N = 4; // N-.max_x and max_y // A structure to hold the queries public class Query { public int x1, y1; // x and y co-ordinates of bottom left public int x2, y2; // x and y co-ordinates of top right public Query( int x1, int y1, int x2, int y2) { this .x1 = x1; this .y1 = y1; this .x2 = x2; this .y2 = y2; } }; // A function to update the 2D BIT static void updateBIT( int [,]BIT, int x, int y, int val) { for (; x <= N; x += (x & -x)) { // This loop update all the 1D BIT inside the // array of 1D BIT = BIT[x] for (; y <= N; y += (y & -y)) BIT[x,y] += val; } return ; } // A function to get sum from (0, 0) to (x, y) static int getSum( int [,]BIT, int x, int y) { int sum = 0; for (; x > 0; x -= x&-x) { // This loop sum through all the 1D BIT // inside the array of 1D BIT = BIT[x] for (; y > 0; y -= y&-y) { sum += BIT[x, y]; } } return sum; } // A function to create an auxiliary matrix // from the given input matrix static void constructAux( int [,]mat, int [,]aux) { // Initialise Auxiliary array to 0 for ( int i = 0; i <= N; i++) for ( int j = 0; j <= N; j++) aux[i, j] = 0; // Construct the Auxiliary Matrix for ( int j = 1; j <= N; j++) for ( int i = 1; i <= N; i++) aux[i, j] = mat[N - j, i - 1]; return ; } // A function to construct a 2D BIT static void construct2DBIT( int [,]mat, int [,]BIT) { // Create an auxiliary matrix int [,]aux = new int [N + 1, N + 1]; constructAux(mat, aux); // Initialise the BIT to 0 for ( int i = 1; i <= N; i++) for ( int j = 1; j <= N; j++) BIT[i, j] = 0; for ( int j = 1; j <= N; j++) { for ( int i = 1; i <= N; i++) { // Creating a 2D-BIT using update function // everytime we/ encounter a value in the // input 2D-array int v1 = getSum(BIT, i, j); int v2 = getSum(BIT, i, j - 1); int v3 = getSum(BIT, i - 1, j - 1); int v4 = getSum(BIT, i - 1, j); // Assigning a value to a particular element // of 2D BIT updateBIT(BIT, i, j, aux[i,j] - (v1 - v2 - v4 + v3)); } } return ; } // A function to answer the queries static void answerQueries(Query []q, int m, int [,]BIT) { for ( int i = 0; i < m; i++) { int x1 = q[i].x1 + 1; int y1 = q[i].y1 + 1; int x2 = q[i].x2 + 1; int y2 = q[i].y2 + 1; int ans = getSum(BIT, x2, y2) - getSum(BIT, x2, y1 - 1) - getSum(BIT, x1 - 1, y2) + getSum(BIT, x1 - 1, y1 - 1); Console.Write( "Query({0}, {1}, {2}, {3}) = {4}\n" , q[i].x1, q[i].y1, q[i].x2, q[i].y2, ans); } return ; } // Driver Code public static void Main(String[] args) { int [,]mat = { {1, 2, 3, 4}, {5, 3, 8, 1}, {4, 6, 7, 5}, {2, 4, 8, 9} }; // Create a 2D Binary Indexed Tree int [,]BIT = new int [N + 1,N + 1]; construct2DBIT(mat, BIT); /* Queries of the form - x1, y1, x2, y2 For example the query- {1, 1, 3, 2} means the sub-matrix- y /\ 3 | 1 2 3 4 Sub-matrix 2 | 5 3 8 1 {1,1,3,2} --. 3 8 1 1 | 4 6 7 5 6 7 5 0 | 2 4 8 9 | --|------ 0 1 2 3 ---. x | Hence sum of the sub-matrix = 3+8+1+6+7+5 = 30 */ Query []q = { new Query(1, 1, 3, 2), new Query(2, 3, 3, 3), new Query(1, 1, 1, 1)}; int m = q.Length; answerQueries(q, m, BIT); } } // This code is contributed by Rajput-Ji |
Javascript
<script> /* Javascript program to implement 2D Binary Indexed Tree 2D BIT is basically a BIT where each element is another BIT. Updating by adding v on (x, y) means it's effect will be found throughout the rectangle [(x, y), (max_x, max_y)], and query for (x, y) gives you the result of the rectangle [(0, 0), (x, y)], assuming the total rectangle is [(0, 0), (max_x, max_y)]. So when you query and update on this BIT,you have to be careful about how many times you are subtracting a rectangle and adding it. Simple set union formula works here. So if you want to get the result of a specific rectangle [(x1, y1), (x2, y2)], the following steps are necessary: Query(x1,y1,x2,y2) = getSum(x2, y2)-getSum(x2, y1-1) - getSum(x1-1, y2)+getSum(x1-1, y1-1) Here 'Query(x1,y1,x2,y2)' means the sum of elements enclosed in the rectangle with bottom-left corner's co-ordinates (x1, y1) and top-right corner's co-ordinates - (x2, y2) Constraints -> x1<=x2 and y1<=y2 /\ y | | --------(x2,y2) | | | | | | | | | | --------- | (x1,y1) | |___________________________ (0, 0) x--> In this program we have assumed a square matrix. The program can be easily extended to a rectangular one. */ let N = 4; // N-.max_x and max_y // A structure to hold the queries class Query { constructor(x1,y1,x2,y2) { this .x1 = x1; this .y1 = y1; this .x2 = x2; this .y2 = y2; } } // A function to update the 2D BIT function updateBIT(BIT,x,y,val) { for (; x <= N; x += (x & -x)) { // This loop update all the 1D BIT inside the // array of 1D BIT = BIT[x] for (; y <= N; y += (y & -y)) BIT[x][y] += val; } return ; } // A function to get sum from (0, 0) to (x, y) function getSum(BIT,x,y) { let sum = 0; for (; x > 0; x -= x&-x) { // This loop sum through all the 1D BIT // inside the array of 1D BIT = BIT[x] for (; y > 0; y -= y&-y) { sum += BIT[x][y]; } } return sum; } // A function to create an auxiliary matrix // from the given input matrix function constructAux(mat,aux) { // Initialise Auxiliary array to 0 for (let i = 0; i <= N; i++) for (let j = 0; j <= N; j++) aux[i][j] = 0; // Construct the Auxiliary Matrix for (let j = 1; j <= N; j++) for (let i = 1; i <= N; i++) aux[i][j] = mat[N - j][i - 1]; return ; } // A function to construct a 2D BIT function construct2DBIT(mat,BIT) { // Create an auxiliary matrix let aux = new Array(N + 1); for (let i=0;i<(N+1);i++) { aux[i]= new Array(N+1); } constructAux(mat, aux); // Initialise the BIT to 0 for (let i = 1; i <= N; i++) for (let j = 1; j <= N; j++) BIT[i][j] = 0; for (let j = 1; j <= N; j++) { for (let i = 1; i <= N; i++) { // Creating a 2D-BIT using update function // everytime we/ encounter a value in the // input 2D-array let v1 = getSum(BIT, i, j); let v2 = getSum(BIT, i, j - 1); let v3 = getSum(BIT, i - 1, j - 1); let v4 = getSum(BIT, i - 1, j); // Assigning a value to a particular element // of 2D BIT updateBIT(BIT, i, j, aux[i][j] - (v1 - v2 - v4 + v3)); } } return ; } // A function to answer the queries function answerQueries(q,m,BIT) { for (let i = 0; i < m; i++) { let x1 = q[i].x1 + 1; let y1 = q[i].y1 + 1; let x2 = q[i].x2 + 1; let y2 = q[i].y2 + 1; let ans = getSum(BIT, x2, y2) - getSum(BIT, x2, y1 - 1) - getSum(BIT, x1 - 1, y2) + getSum(BIT, x1 - 1, y1 - 1); document.write( "Query (" +q[i].x1+ ", " +q[i].y1+ ", " +q[i].x2+ ", " +q[i].y2+ ") = " +ans+ "<br>" ); } return ; } // Driver Code let mat= [[1, 2, 3, 4], [5, 3, 8, 1], [4, 6, 7, 5], [2, 4, 8, 9]]; // Create a 2D Binary Indexed Tree let BIT = new Array(N + 1); for (let i=0;i<(N+1);i++) { BIT[i]= new Array(N+1); for (let j=0;j<(N+1);j++) { BIT[i][j]=0; } } construct2DBIT(mat, BIT); /* Queries of the form - x1, y1, x2, y2 For example the query- {1, 1, 3, 2} means the sub-matrix- y /\ 3 | 1 2 3 4 Sub-matrix 2 | 5 3 8 1 {1,1,3,2} --. 3 8 1 1 | 4 6 7 5 6 7 5 0 | 2 4 8 9 | --|------ 0 1 2 3 ---. x | Hence sum of the sub-matrix = 3+8+1+6+7+5 = 30 */ let q = [ new Query(1, 1, 3, 2), new Query(2, 3, 3, 3), new Query(1, 1, 1, 1)]; let m = q.length; answerQueries(q, m, BIT); // This code is contributed by rag2127 </script> |
Query(1, 1, 3, 2) = 30 Query(2, 3, 3, 3) = 7 Query(1, 1, 1, 1) = 6
Time Complexity:
- Both updateBIT(x, y, val) function and getSum(x, y) function takes O(log(N)*log(M)) time.
- Building the 2D BIT takes O(NM log(N)*log(M)).
- Since in each of the queries we are calling getSum(x, y) function so answering all the Q queries takes O(Q*log(N)*log(M)) time.
- Hence the overall time complexity of the program is O((NM+Q)*log(N)*log(M)) where,
N = maximum X co-ordinate of the whole matrix.
M = maximum Y co-ordinate of the whole matrix.
Q = Number of queries.
Auxiliary Space: O(NM) to store the BIT and the auxiliary array