# Submatrix Sum Queries

Given a matrix of size M x N, there are large number of queries to find submatrix sums. Inputs to queries are left top and right bottom indexes of submatrix whose sum is to find out.

How to preprocess the matrix so that submatrix sum queries can be performed in O(1) time.

Example :

tli :  Row number of top left of query submatrix
tlj :  Column number of top left of query submatrix
rbi :  Row number of bottom right of query submatrix
rbj :  Column number of bottom right of query submatrix

Input: mat[M][N] = {{1, 2, 3, 4, 6},
{5, 3, 8, 1, 2},
{4, 6, 7, 5, 5},
{2, 4, 8, 9, 4} };
Query1: tli = 0, tlj = 0, rbi = 1, rbj = 1
Query2: tli = 2, tlj = 2, rbi = 3, rbj = 4
Query3: tli = 1, tlj = 2, rbi = 3, rbj = 3;

Output:
Query1: 11  // Sum between (0, 0) and (1, 1)
Query2: 38  // Sum between (2, 2) and (3, 4)
Query3: 38  // Sum between (1, 2) and (3, 3)

We strongly recommend you to minimize your browser and try this yourself first.

The idea is to first create an auxiliary matrix aux[M][N] such that aux[i][j] stores sum of elements in submatrix from (0,0) to (i,j). Once aux[][] is constructed, we can compute sum of submatrix between (tli, tlj) and (rbi, rbj) in O(1) time. We need to consider aux[rbi][rbj] and subtract all unncessary elements. Below is complete expression to compute submatrix sum in O(1) time.

Sum between (tli, tlj) and (rbi, rbj) is,
aux[rbi][rbj] - aux[tli-1][rbj] -
aux[rbi][tlj-1] + aux[tli-1][tlj-1]

The submatrix aux[tli-1][tlj-1] is added because
elements of it are subtracted twice.

Illustration:

mat[M][N] = {{1, 2, 3, 4, 6},
{5, 3, 8, 1, 2},
{4, 6, 7, 5, 5},
{2, 4, 8, 9, 4} };

We first preprocess the matrix and build
following aux[M][N]
aux[M][N] = {1,  3,   6,  10, 16}
{6,  11,  22, 27,  35},
{10, 21,  39, 49,  62},
{12, 27,  53, 72,  89} }

Query : tli = 2, tlj = 2, rbi = 3, rbj = 4

Sum between (2, 2) and (3, 4) = 89 - 35 - 27 + 11
= 38

How to build aux[M][N]?
1. Copy first row of mat[][] to aux[][]
2. Do column wise sum of the matrix and store it.
3. Do the row wise sum of updated matrix aux[][] in step 2.

Below is the program based on above idea.

 // C++ program to compute submatrix query sum in O(1) // time #include using namespace std; #define M 4 #define N 5    // Function to preprcess input mat[M][N].  This function // mainly fills aux[M][N] such that aux[i][j] stores sum // of elements from (0,0) to (i,j) int preProcess(int mat[M][N], int aux[M][N]) {    // Copy first row of mat[][] to aux[][]    for (int i=0; i 0)        res = res - aux[tli-1][rbj];        // Remove elements between (0, 0) and (rbi, tlj-1)     if (tlj > 0)        res = res - aux[rbi][tlj-1];        // Add aux[tli-1][tlj-1] as elements between (0, 0)     // and (tli-1, tlj-1) are subtracted twice     if (tli > 0 && tlj > 0)        res = res + aux[tli-1][tlj-1];        return res; }    // Driver program int main() {    int mat[M][N] = {{1, 2, 3, 4, 6},                     {5, 3, 8, 1, 2},                     {4, 6, 7, 5, 5},                     {2, 4, 8, 9, 4} };    int aux[M][N];       preProcess(mat, aux);       int tli = 2, tlj = 2, rbi = 3, rbj = 4;    cout << "\nQuery1: " << sumQuery(aux, tli, tlj, rbi, rbj);       tli = 0, tlj = 0, rbi = 1, rbj = 1;    cout << "\nQuery2: " << sumQuery(aux, tli, tlj, rbi, rbj);       tli = 1, tlj = 2, rbi = 3, rbj = 3;    cout << "\nQuery3: " << sumQuery(aux, tli, tlj, rbi, rbj);       return 0; }

 // Java program to compute submatrix query  // sum in O(1) time class GFG {            static final int M = 4;     static final int N = 5;            // Function to preprcess input mat[M][N].      // This function mainly fills aux[M][N]      // such that aux[i][j] stores sum of      // elements from (0,0) to (i,j)     static int preProcess(int mat[][], int aux[][])     {                    // Copy first row of mat[][] to aux[][]         for (int i = 0; i < N; i++)             aux[0][i] = mat[0][i];                    // Do column wise sum         for (int i = 1; i < M; i++)             for (int j = 0; j < N; j++)                 aux[i][j] = mat[i][j] +                                  aux[i-1][j];                    // Do row wise sum         for (int i = 0; i < M; i++)             for (int j = 1; j < N; j++)                 aux[i][j] += aux[i][j-1];                            return 0;     }            // A O(1) time function to compute sum      // of submatrix between (tli, tlj) and      // (rbi, rbj) using aux[][] which is      // built by the preprocess function     static int sumQuery(int aux[][], int tli,                      int tlj, int rbi, int rbj)     {                    // result is now sum of elements          // between (0, 0) and (rbi, rbj)         int res = aux[rbi][rbj];                // Remove elements between (0, 0)          // and (tli-1, rbj)         if (tli > 0)             res = res - aux[tli-1][rbj];                // Remove elements between (0, 0)          // and (rbi, tlj-1)         if (tlj > 0)             res = res - aux[rbi][tlj-1];                // Add aux[tli-1][tlj-1] as elements          // between (0, 0) and (tli-1, tlj-1)          // are subtracted twice         if (tli > 0 && tlj > 0)             res = res + aux[tli-1][tlj-1];                return res;     }            // Driver code     public static void main (String[] args)     {         int mat[][] = {{1, 2, 3, 4, 6},                        {5, 3, 8, 1, 2},                        {4, 6, 7, 5, 5},                        {2, 4, 8, 9, 4}};                                   int aux[][] = new int[M][N];                    preProcess(mat, aux);                    int tli = 2, tlj = 2, rbi = 3, rbj = 4;         System.out.print("\nQuery1: "             + sumQuery(aux, tli, tlj, rbi, rbj));                    tli = 0; tlj = 0; rbi = 1; rbj = 1;         System.out.print("\nQuery2: "             + sumQuery(aux, tli, tlj, rbi, rbj));                    tli = 1; tlj = 2; rbi = 3; rbj = 3;         System.out.print("\nQuery3: "              + sumQuery(aux, tli, tlj, rbi, rbj));     } }    // This code is contributed by Anant Agarwal.

 # Python 3 program to compute submatrix  # query sum in O(1) time    M = 4 N = 5    # Function to preprcess input mat[M][N].  # This function mainly fills aux[M][N] # such that aux[i][j] stores sum # of elements from (0,0) to (i,j) def preProcess(mat, aux):            # Copy first row of mat[][] to aux[][]     for i in range(0, N, 1):         aux[0][i] = mat[0][i]        # Do column wise sum     for i in range(1, M, 1):         for j in range(0, N, 1):             aux[i][j] = mat[i][j] + aux[i - 1][j]        # Do row wise sum     for i in range(0, M, 1):         for j in range(1, N, 1):             aux[i][j] += aux[i][j - 1]    # A O(1) time function to compute sum of submatrix # between (tli, tlj) and (rbi, rbj) using aux[][] # which is built by the preprocess function def sumQuery(aux, tli, tlj, rbi, rbj):            # result is now sum of elements      # between (0, 0) and (rbi, rbj)     res = aux[rbi][rbj]        # Remove elements between (0, 0)     # and (tli-1, rbj)     if (tli > 0):         res = res - aux[tli - 1][rbj]        # Remove elements between (0, 0)     # and (rbi, tlj-1)     if (tlj > 0):         res = res - aux[rbi][tlj - 1]        # Add aux[tli-1][tlj-1] as elements     # between (0, 0) and (tli-1, tlj-1)     # are subtracted twice     if (tli > 0 and tlj > 0):         res = res + aux[tli - 1][tlj - 1]        return res    # Driver Code if __name__ == '__main__':     mat = [[1, 2, 3, 4, 6],            [5, 3, 8, 1, 2],            [4, 6, 7, 5, 5],            [2, 4, 8, 9, 4]] aux = [[0 for i in range(N)]            for j in range(M)]    preProcess(mat, aux)    tli = 2 tlj = 2 rbi = 3 rbj = 4 print("Query1:", sumQuery(aux, tli, tlj, rbi, rbj))    tli = 0 tlj = 0 rbi = 1 rbj = 1 print("Query2:", sumQuery(aux, tli, tlj, rbi, rbj))    tli = 1 tlj = 2 rbi = 3 rbj = 3 print("Query3:", sumQuery(aux, tli, tlj, rbi, rbj))    # This code is contributed by # Shashank_Sharma

 // C# program to compute submatrix  // query sum in O(1) time using System;    class GFG {      static int M = 4;     static int N = 5;            // Function to preprcess input mat[M][N].      // This function mainly fills aux[M][N]      // such that aux[i][j] stores sum of      // elements from (0,0) to (i,j)     static int preProcess(int [,]mat, int [,]aux)     {         // Copy first row of mat[][] to aux[][]         for (int i = 0; i < N; i++)             aux[0,i] = mat[0,i];                    // Do column wise sum         for (int i = 1; i < M; i++)             for (int j = 0; j < N; j++)                 aux[i,j] = mat[i,j] + aux[i-1,j];                    // Do row wise sum         for (int i = 0; i < M; i++)             for (int j = 1; j < N; j++)                 aux[i,j] += aux[i,j-1];                            return 0;     }            // A O(1) time function to compute sum      // of submatrix between (tli, tlj) and      // (rbi, rbj) using aux[][] which is      // built by the preprocess function     static int sumQuery(int [,]aux, int tli,                          int tlj, int rbi, int rbj)     {         // result is now sum of elements          // between (0, 0) and (rbi, rbj)         int res = aux[rbi,rbj];                // Remove elements between (0, 0)          // and (tli-1, rbj)         if (tli > 0)             res = res - aux[tli-1,rbj];                // Remove elements between (0, 0)          // and (rbi, tlj-1)         if (tlj > 0)             res = res - aux[rbi,tlj-1];                // Add aux[tli-1][tlj-1] as elements          // between (0, 0) and (tli-1, tlj-1)          // are subtracted twice         if (tli > 0 && tlj > 0)             res = res + aux[tli-1,tlj-1];                return res;     }            // Driver code     public static void Main ()     {         int [,]mat = {{1, 2, 3, 4, 6},                       {5, 3, 8, 1, 2},                       {4, 6, 7, 5, 5},                       {2, 4, 8, 9, 4}};                                    int [,]aux = new int[M,N];                    preProcess(mat, aux);                    int tli = 2, tlj = 2, rbi = 3, rbj = 4;                    Console.Write("\nQuery1: " +                        sumQuery(aux, tli, tlj, rbi, rbj));                    tli = 0; tlj = 0; rbi = 1; rbj = 1;                    Console.Write("\nQuery2: " +                       sumQuery(aux, tli, tlj, rbi, rbj));                    tli = 1; tlj = 2; rbi = 3; rbj = 3;                    Console.Write("\nQuery3: " +                        sumQuery(aux, tli, tlj, rbi, rbj));     } }    // This code is contributed by Sam007.

 0)     \$res = \$res - \$aux[\$tli - 1][\$rbj];        // Remove elements between (0, 0)     // and (rbi, tlj-1)     if (\$tlj > 0)     \$res = \$res - \$aux[\$rbi][\$tlj - 1];        // Add aux[tli-1][tlj-1] as elements between (0, 0)     // and (tli-1, tlj-1) are subtracted twice     if (\$tli > 0 && \$tlj > 0)     \$res = \$res + \$aux[\$tli - 1][\$tlj - 1];        return \$res; }    // Driver Code \$mat = array(array(1, 2, 3, 4, 6),              array(5, 3, 8, 1, 2),              array(4, 6, 7, 5, 5),              array(2, 4, 8, 9, 4));    preProcess(\$mat, \$aux);    \$tli = 2; \$tlj = 2; \$rbi = 3; \$rbj = 4; echo ("Query1: "); echo(sumQuery(\$aux, \$tli, \$tlj, \$rbi, \$rbj));    \$tli = 0; \$tlj = 0; \$rbi = 1; \$rbj = 1; echo ("\nQuery2: "); echo(sumQuery(\$aux, \$tli, \$tlj, \$rbi, \$rbj));       \$tli = 1; \$tlj = 2; \$rbi = 3; \$rbj = 3; echo ("\nQuery3: "); echo(sumQuery(\$aux, \$tli, \$tlj, \$rbi, \$rbj));    // This code is contributed by Shivi_Aggarwal ?>

Output:

Query1: 38
Query2: 11
Query3: 38