Sum and Update Queries on 3D Matrix in log(N) time

Given a 3D matrix mat[][][] of size NXNXN with all its elements initialized to 0, the task is to answer Q queries. The queries are given in the form of 2D matrix queries[][], where each queries[i] can be one of the following types:

Type 1 [1, x, y, z, ele]: Update the value of cell(x, y, z) to Val.

Type 2 [2, x, y, z, X, Y, Z]: Print the sum of the submatrix (x, y, z) to (X, Y, Z)

Print the answer to all the queries of type 2. The matrix and queries follow 0-based indexing.

Examples:

Input: N = 2, Q = 2, queries[][] = {{1, 0, 0, 0, 1}, {2, 0, 0, 0, 1, 1, 1}}
Output: 1
Explanation: In the type-2 query, the sum of subarray from (0, 0, 0) to (1, 1, 1) is 1.

Input: N = 10, Q = 6, queries[][] = {{1, 0, 0, 6, 6}, {1, 9, 9, 9, 10}, {1, 8, 5, 9, 5}, {2, 3, 4, 5, 9, 10, 9}, {1, 6, 6, 1, 23}, {2, 0, 0, 0, 8, 9, 10}}
Output: 6 10
Explanation: In the first type-2 query, the sum of subarray from (3, 4, 5) to (9, 10, 9) is 6. In the subsequent type-2 query, the sum of subarray from (0,0,0) to (8,9,10) is 10.

Approach: The problem can be solved using the following approach:

The approach is to implement a 3D binary indexed tree (also known as a Fenwick Tree) to store the Partial Sums of ranges and efficiently update and query ranges in a 3D space. The program takes a series of queries, where each query is either an update or a range query.

Steps to solve the problem:

• Use a 3D array BIT[][][] to store Partial Sums of mat[][][].
• Define a function updateQuery(x, y, z, val) to update the BIT with a given value at position (x, y, z) by updating the values at positions where Partial sum of submatrix[0][0][0] to [x][y][z] is stored.
• It uses three nested loops to iterate over all positions in the BIT that need to be updated based on the given position (x, y, z).
• Define a function sumQuery(x, y, z) to calculate the cumulative sum of the values in the BIT up to position (x, y, z), that is the sum of submatrix[0, 0, 0] to [x, y, z].
• It uses three nested loops similar to updateQuery() but only sums up the values instead of updating them.
• Define a function matrixSumQuery(x, y, z, X, Y, Z) to calculate the sum of values in the given 3D submatrix defined by two opposite corners (x, y, z) and (X, Y, Z).
• It uses the cumulative sum calculated by reading at the eight corners of the cube defined by the given range and get its sum.

Below is the implementation of the above approach:

C++

#include <bits/stdc++.h>
using namespace std;
#define ll long long int

// 3D Binary Indexed Tree to store Partial Sums
ll BIT[102][102][102];
int N;

// Function to handle updates
void updateQuery(int x, int y, int z, ll val)
{
    int i, j, k;
    // Update all the cells that store the partial sum of
    // cell (x, y, z)
    for (i = x; i <= N; i += i & -i) {
        for (j = y; j <= N; j += j & -j) {
            for (k = z; k <= N; k += k & -k) {
                BIT[i][j][k] += val;
            }
        }
    }
}

// Function to get the prefix sum from (0, 0, 0) to
// (x, y, z)
ll sumQuery(int x, int y, int z)
{
    ll sum = 0;
    int i, j, k;
    // Calculate prefix sum by summing up all the cells
    // storing the partial sums
    for (i = x; i > 0; i -= i & -i) {
        for (j = y; j > 0; j -= j & -j) {
            for (k = z; k > 0; k -= k & -k) {
                sum += BIT[i][j][k];
            }
        }
    }
    return sum;
}

// Function to get the submatrix sum from (x, y, z) to (X,
// Y, Z)
ll matrixSumQuery(int x, int y, int z, int X, int Y, int Z)
{
    return sumQuery(X, Y, Z) - sumQuery(x, Y, Z)
        - sumQuery(X, y, Z) - sumQuery(X, Y, z)
        + sumQuery(x, y, Z) + sumQuery(x, Y, z)
        + sumQuery(X, y, z) - sumQuery(x, y, z);
}

int main()
{
    int Q, type, x, y, z, val, X, Y, Z;

    // Predefined input values
    N = 10;
    Q = 6;
    ll queries[Q][7]
        = { { 1, 0, 0, 6, 6 }, { 1, 9, 9, 9, 10 },
            { 1, 8, 5, 9, 5 }, { 2, 3, 4, 5, 9, 10, 9 },
            { 1, 6, 6, 1, 23 }, { 2, 0, 0, 0, 8, 9, 10 } };

    for (int i = 0; i < Q; i++) {
        type = queries[i][0];
        if (type == 1) {
            x = queries[i][1];
            y = queries[i][2];
            z = queries[i][3];
            val = queries[i][4];
            updateQuery(x + 1, y + 1, z + 1, val);
        }
        else {
            x = queries[i][1];
            y = queries[i][2];
            z = queries[i][3];
            X = queries[i][4];
            Y = queries[i][5];
            Z = queries[i][6];
            cout << matrixSumQuery(x, y, z, X + 1, Y + 1,
                                Z + 1)
                << "\n";
        }
    }
    return 0;
}

Java

import java.util.Scanner;

public class BIT3D {
    static long[][][] BIT;
    static int N;

    // Function to handle updates
    static void updateQuery(int x, int y, int z, long val) {
        // Update all the cells that store the partial sum of cell (x, y, z)
        for (int i = x; i <= N + 1; i += i & -i) {
            for (int j = y; j <= N + 1; j += j & -j) {
                for (int k = z; k <= N + 1; k += k & -k) {
                    BIT[i][j][k] += val;
                }
            }
        }
    }

    // Function to get the prefix sum from (0, 0, 0) to (x, y, z)
    static long sumQuery(int x, int y, int z) {
        long sum = 0;
        for (int i = x; i > 0; i -= i & -i) {
            for (int j = y; j > 0; j -= j & -j) {
                for (int k = z; k > 0; k -= k & -k) {
                    sum += BIT[i][j][k];
                }
            }
        }
        return sum;
    }

    // Function to get the submatrix sum from (x, y, z) to (X, Y, Z)
    static long matrixSumQuery(int x, int y, int z, int X, int Y, int Z) {
        return sumQuery(X, Y, Z) - sumQuery(x - 1, Y, Z) - sumQuery(X, y - 1, Z) - sumQuery(X, Y, z - 1)
                + sumQuery(x - 1, y - 1, z - 1) - sumQuery(x - 1, Y, z - 1) - sumQuery(X, y - 1, z - 1)
                + sumQuery(x - 1, y - 1, Z) + sumQuery(x - 1, Y, z) + sumQuery(X, y - 1, z);
    }

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);

        // Predefined input values
        N = 10;
        int Q = 6;
        BIT = new long[N + 2][N + 2][N + 2];

        long[][] queries = { { 1, 0, 0, 6, 6 }, { 1, 9, 9, 9, 10 }, { 1, 8, 5, 9, 5 }, { 2, 3, 4, 5, 9, 10, 9 },
                { 1, 6, 6, 1, 23 }, { 2, 0, 0, 0, 8, 9, 10 } };

        for (int i = 0; i < Q; i++) {
            int type = (int) queries[i][0];
            if (type == 1) {
                int x = (int) queries[i][1];
                int y = (int) queries[i][2];
                int z = (int) queries[i][3];
                long val = queries[i][4];
                updateQuery(x + 1, y + 1, z + 1, val);
            } else {
                int x = (int) queries[i][1];
                int y = (int) queries[i][2];
                int z = (int) queries[i][3];
                int X = (int) queries[i][4];
                int Y = (int) queries[i][5];
                int Z = (int) queries[i][6];
                System.out.println(matrixSumQuery(x, y, z, X + 1, Y + 1, Z + 1));
            }
        }

        sc.close();
    }
}

Python3

# Python Implementation
# 3D Binary Indexed Tree to store Partial Sums
BIT = [[[0 for _ in range(102)] for _ in range(102)] for _ in range(102)]
N = 0

# Function to handle updates
def updateQuery(x, y, z, val):
    global BIT
    # Update all the cells that store the partial sum of
    # cell (x, y, z)
    i, j, k = x, y, z
    while i <= N:
        j = y
        while j <= N:
            k = z
            while k <= N:
                BIT[i][j][k] += val
                k += k & -k
            j += j & -j
        i += i & -i

# Function to get the prefix sum from (0, 0, 0) to
# (x, y, z)
def sumQuery(x, y, z):
    global BIT
    sum = 0
    i, j, k = x, y, z
    # Calculate prefix sum by summing up all the cells
    # storing the partial sums
    while i > 0:
        j = y
        while j > 0:
            k = z
            while k > 0:
                sum += BIT[i][j][k]
                k -= k & -k
            j -= j & -j
        i -= i & -i
    return sum

# Function to get the submatrix sum from (x, y, z) to (X,
# Y, Z)
def matrixSumQuery(x, y, z, X, Y, Z):
    return sumQuery(X, Y, Z) - sumQuery(x, Y, Z) - sumQuery(X, y, Z) - sumQuery(X, Y, z) + sumQuery(x, y, Z) + sumQuery(x, Y, z) + sumQuery(X, y, z) - sumQuery(x, y, z)

# Predefined input values
N = 10
Q = 6
queries = [[1, 0, 0, 6, 6], [1, 9, 9, 9, 10], [1, 8, 5, 9, 5], [2, 3, 4, 5, 9, 10, 9], [1, 6, 6, 1, 23], [2, 0, 0, 0, 8, 9, 10]]

for i in range(Q):
    type = queries[i][0]
    if type == 1:
        x = queries[i][1]
        y = queries[i][2]
        z = queries[i][3]
        val = queries[i][4]
        updateQuery(x + 1, y + 1, z + 1, val)
    else:
        x = queries[i][1]
        y = queries[i][2]
        z = queries[i][3]
        X = queries[i][4]
        Y = queries[i][5]
        Z = queries[i][6]
        print(matrixSumQuery(x, y, z, X + 1, Y + 1, Z + 1))
# This code is contributed by Sakshi

C#

using System;

class MainClass
{
    const int N = 10;
    const int Q = 6;

    // 3D Binary Indexed Tree to store Partial Sums
    static int[][][] BIT = new int[102][][];

    static void Main()
    {
        // Initialize BIT array
        for (int i = 0; i < BIT.Length; i++)
        {
            BIT[i] = new int[102][];
            for (int j = 0; j < BIT[i].Length; j++)
            {
                BIT[i][j] = new int[102];
            }
        }

        int[][] queries = new int[][] {
            new int[] {1, 0, 0, 6, 6},
            new int[] {1, 9, 9, 9, 10},
            new int[] {1, 8, 5, 9, 5},
            new int[] {2, 3, 4, 5, 9, 10, 9},
            new int[] {1, 6, 6, 1, 23},
            new int[] {2, 0, 0, 0, 8, 9, 10},
        };

        for (int i = 0; i < Q; i++)
        {
            int type = queries[i][0];
            int x, y, z, val, X, Y, Z;
            if (type == 1)
            {
                x = queries[i][1];
                y = queries[i][2];
                z = queries[i][3];
                val = queries[i][4];
                UpdateQuery(x + 1, y + 1, z + 1, val);
            }
            else
            {
                x = queries[i][1];
                y = queries[i][2];
                z = queries[i][3];
                X = queries[i][4];
                Y = queries[i][5];
                Z = queries[i][6];
                Console.WriteLine(MatrixSumQuery(x, y, z, X + 1, Y + 1, Z + 1));
            }
        }
    }

    // Function to handle updates
    static void UpdateQuery(int x, int y, int z, int val)
    {
        // Update all the cells that store the partial sum of cell (x, y, z)
        for (int i = x; i <= N; i += i & -i)
        {
            for (int j = y; j <= N; j += j & -j)
            {
                for (int k = z; k <= N; k += k & -k)
                {
                    BIT[i][j][k] += val;
                }
            }
        }
    }

    // Function to get the prefix sum from (0, 0, 0) to (x, y, z)
    static int SumQuery(int x, int y, int z)
    {
        int sum = 0;
        // Calculate prefix sum by summing up all the cells storing the partial sums
        for (int i = x; i > 0; i -= i & -i)
        {
            for (int j = y; j > 0; j -= j & -j)
            {
                for (int k = z; k > 0; k -= k & -k)
                {
                    sum += BIT[i][j][k];
                }
            }
        }
        return sum;
    }

    // Function to get the submatrix sum from (x, y, z) to (X, Y, Z)
    static int MatrixSumQuery(int x, int y, int z, int X, int Y, int Z)
    {
        return (
            SumQuery(X, Y, Z) - SumQuery(x, Y, Z) - SumQuery(X, y, Z) - SumQuery(X, Y, z) +
            SumQuery(x, y, Z) + SumQuery(x, Y, z) + SumQuery(X, y, z) - SumQuery(x, y, z)
        );
    }
}

Javascript

const N = 10;
const Q = 6;

// 3D Binary Indexed Tree to store Partial Sums
const BIT = new Array(102).fill(null).map(() =>
  new Array(102).fill(null).map(() => new Array(102).fill(0))
);

// Function to handle updates
function updateQuery(x, y, z, val) {
    // Update all the cells that store the partial sum of
    // cell (x, y, z)
    for (let i = x; i <= N; i += i & -i) {
        for (let j = y; j <= N; j += j & -j) {
            for (let k = z; k <= N; k += k & -k) {
                BIT[i][j][k] += val;
            }
        }
    }
}

// Function to get the prefix sum from (0, 0, 0) to
// (x, y, z)
function sumQuery(x, y, z) {
    let sum = 0;
    // Calculate prefix sum by summing up all the cells
    // storing the partial sums
    for (let i = x; i > 0; i -= i & -i) {
        for (let j = y; j > 0; j -= j & -j) {
            for (let k = z; k > 0; k -= k & -k) {
                sum += BIT[i][j][k];
            }
        }
    }
    return sum;
}

// Function to get the submatrix sum from (x, y, z) to (X,
// Y, Z)
function matrixSumQuery(x, y, z, X, Y, Z) {
    return (
        sumQuery(X, Y, Z) - sumQuery(x, Y, Z) - sumQuery(X, y, Z) - sumQuery(X, Y, z) +
        sumQuery(x, y, Z) + sumQuery(x, Y, z) + sumQuery(X, y, z) - sumQuery(x, y, z)
    );
}

const queries = [
    [1, 0, 0, 6, 6],
    [1, 9, 9, 9, 10],
    [1, 8, 5, 9, 5],
    [2, 3, 4, 5, 9, 10, 9],
    [1, 6, 6, 1, 23],
    [2, 0, 0, 0, 8, 9, 10],
];

for (let i = 0; i < Q; i++) {
    const type = queries[i][0];
    let x, y, z, val, X, Y, Z;
    if (type === 1) {
        x = queries[i][1];
        y = queries[i][2];
        z = queries[i][3];
        val = queries[i][4];
        updateQuery(x + 1, y + 1, z + 1, val);
    }
    else {
        x = queries[i][1];
        y = queries[i][2];
        z = queries[i][3];
        X = queries[i][4];
        Y = queries[i][5];
        Z = queries[i][6];
        console.log(matrixSumQuery(x, y, z, X + 1, Y + 1, Z + 1));
    }
}

Output

15
34

Time Complexity: O(Q * log(N) * log(N) * log(N))

• Both updateQuery(x, y, z, val) function and matrixSumQuery(x, y, z, X, Y, Z) function takes O(log(N)*log(N)*log(N)) time.
• So, the overall time complexity for Q queries is O(Q*log(N)*log(N)*log(N)).

Auxiliary Space: O(N * N * N)