Submatrix Sum Queries (original) (raw)

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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)

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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 unnecessary 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++ `

// C++ program to compute submatrix query sum in O(1) // time #include using namespace std; #define M 4 #define N 5

// Function to preprocess 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<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]; }

// 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 int sumQuery(int aux[M][N], 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 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

// Java program to compute submatrix query // sum in O(1) time import java.util.; import java.io.;

class GFG {

static final int M = 4;
static final int N = 5;

// Function to preprocess 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.

Python3

Python 3 program to compute submatrix

query sum in O(1) time

M = 4 N = 5

Function to preprocess 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#

// 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 preprocess 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.

PHP

i<i < i<N; $i++) aux[0][aux[0][aux[0][i] = mat[0][mat[0][mat[0][i]; // Do column wise sum for ($i = 1; i<i < i<M; $i++) for ($j = 0; j<j < j<N; $j++) aux[aux[aux[i][$j] = mat[mat[mat[i][$j] + aux[aux[aux[i - 1][$j]; // Do row wise sum for ($i = 0; i<i < i<M; $i++) for ($j = 1; j<j < j<N; $j++) aux[aux[aux[i][$j] += aux[aux[aux[i][$j - 1]; } // A O(1) time function to compute sum of // submatrix between (tli, tlj) and // (rbi, rbj) using aux[][] which is built // by the preprocess function function sumQuery(&$aux, tli,tli, tli,tlj, rbi,rbi,rbi,rbj) { // result is now sum of elements // between (0, 0) and (rbi, rbj) res=res = res=aux[$rbi][$rbj]; // Remove elements between (0, 0) // and (tli-1, rbj) if ($tli > 0) res=res = res=res - aux[aux[aux[tli - 1][$rbj]; // Remove elements between (0, 0) // and (rbi, tlj-1) if ($tlj > 0) res=res = res=res - aux[aux[aux[rbi][$tlj - 1]; // Add aux[tli-1][tlj-1] as elements between (0, 0) // and (tli-1, tlj-1) are subtracted twice if ($tli > 0 && $tlj > 0) res=res = res=res + aux[aux[aux[tli - 1][$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,tli, tli,tlj, rbi,rbi, rbi,rbj)); $tli = 0; $tlj = 0; $rbi = 1; $rbj = 1; echo ("\nQuery2: "); echo(sumQuery($aux, tli,tli, tli,tlj, rbi,rbi, rbi,rbj)); $tli = 1; $tlj = 2; $rbi = 3; $rbj = 3; echo ("\nQuery3: "); echo(sumQuery($aux, tli,tli, tli,tlj, rbi,rbi, rbi,rbj)); // This code is contributed by Shivi_Aggarwal ?>

JavaScript

`

Output

Query1: 38 Query2: 11 Query3: 38

Time complexity: O(M x N).
Auxiliary Space: O(M x N), since M x N extra space has been taken.

Source: https://www.geeksforgeeks.org/amazon-interview-experience-set-241-1-5-years-experience/