Dynamic Programming (DP) is a strong method to solve hard problems. We do this by breaking the problems into easier parts. The Range Sum Query 2D problem is a good example. We can find the sum of numbers in a certain rectangle in a 2D grid quickly. By using dynamic programming and prefix sums, we can prepare the grid first. This way, we can answer many sum questions very fast. This is much better than using simple methods.
In this article, we will look into the Range Sum Query 2D problem. We will focus on the dynamic programming way and the prefix sum method. We will give a clear tutorial to help understand the problem. Then, we will show how to implement it in Java, Python, and C++. We will also talk about how to make space use better and how to test and check our solutions. By the end, we hope you will know how to handle the Range Sum Query 2D problem well using dynamic programming.
- [Dynamic Programming] Range Sum Query 2D DP Variant - Medium Tutorial
- Understanding the Problem Statement for Range Sum Query 2D
- Dynamic Programming Approach to Range Sum Query 2D
- Prefix Sum Technique for 2D Range Queries
- Java Implementation of Range Sum Query 2D
- Python Implementation of Range Sum Query 2D
- C++ Implementation of Range Sum Query 2D
- Optimizing Space Complexity in Range Sum Query 2D
- Testing and Validating the Range Sum Query Solutions
- Frequently Asked Questions
For more ideas on dynamic programming, you can read the article on Dynamic Programming: Fibonacci Number. It can help you too.
Understanding the Problem Statement for Range Sum Query 2D
We have a problem called Range Sum Query 2D. It asks us to find the
sum of values in a rectangle part of a 2D matrix. The matrix is
matrix and it has a size of m x n. Our goal is
to prepare the matrix so we can quickly calculate the sum of numbers in
any submatrix. We define this submatrix by its top-left and bottom-right
corners.
Problem Definition
- Input: A 2D matrix with integers and some queries.
Each query has two corners:
(row1, col1)and(row2, col2). - Output: For each query, we give the sum of the numbers in the specified submatrix.
Example
Let’s look at this matrix:
[
[3, 0, 1, 4, 2],
[5, 6, 3, 2, 1],
[1, 2, 0, 1, 5],
[4, 1, 0, 1, 7],
[1, 0, 3, 0, 5]
]If we ask for sumRegion(2, 1, 4, 3), we want the sum
from (2, 1) to (4, 3). The answer will be
8 because 2 + 0 + 1 + 1 + 0 + 3 = 8.
Constraints
- The matrix can have negative numbers.
- We can have many queries. So, simple summation is not a good idea.
To solve the Range Sum Query 2D problem well, we must use data structures. For example, we can use prefix sums or dynamic programming. This helps us prepare the matrix first. After that, we can quickly find sums for each query. The first step takes linear time. After that, we can get answers in constant time.
Dynamic Programming Approach to Range Sum Query 2D
The Range Sum Query 2D problem helps us quickly find the sum of numbers in a certain rectangle area of a 2D array. We can make this faster by using a dynamic programming method that calculates cumulative sums in advance.
Dynamic Programming Solution
Define the 2D Prefix Sum Array: We will create a prefix sum array called
dp. Here,dp[i][j]shows the total of elements from the top-left corner(0, 0)to(i, j)in the original matrix.The formula to find the prefix sum is:
dp[i][j] = matrix[i][j] + dp[i-1][j] + dp[i][j-1] - dp[i-1][j-1]Calculate the Range Sum: If we have a query with two points
(row1, col1)and(row2, col2), we can find the sum of elements in this area like this:sum = dp[row2][col2] - (dp[row1-1][col2] if row1 > 0 else 0) - (dp[row2][col1-1] if col1 > 0 else 0) + (dp[row1-1][col1-1] if row1 > 0 and col1 > 0 else 0)
Implementation
Java Implementation:
class NumMatrix {
private int[][] dp;
public NumMatrix(int[][] matrix) {
int m = matrix.length;
int n = matrix[0].length;
dp = new int[m][n];
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
dp[i][j] = matrix[i][j] +
(i > 0 ? dp[i - 1][j] : 0) +
(j > 0 ? dp[i][j - 1] : 0) -
(i > 0 && j > 0 ? dp[i - 1][j - 1] : 0);
}
}
}
public int sumRegion(int row1, int col1, int row2, int col2) {
return dp[row2][col2] -
(row1 > 0 ? dp[row1 - 1][col2] : 0) -
(col1 > 0 ? dp[row2][col1 - 1] : 0) +
(row1 > 0 && col1 > 0 ? dp[row1 - 1][col1 - 1] : 0);
}
}Python Implementation:
class NumMatrix:
def __init__(self, matrix: List[List[int]]):
self.dp = [[0] * len(matrix[0]) for _ in range(len(matrix))]
for i in range(len(matrix)):
for j in range(len(matrix[0])):
self.dp[i][j] = matrix[i][j] + \
(self.dp[i-1][j] if i > 0 else 0) + \
(self.dp[i][j-1] if j > 0 else 0) - \
(self.dp[i-1][j-1] if i > 0 and j > 0 else 0)
def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
return self.dp[row2][col2] - \
(self.dp[row1-1][col2] if row1 > 0 else 0) - \
(self.dp[row2][col1-1] if col1 > 0 else 0) + \
(self.dp[row1-1][col1-1] if row1 > 0 and col1 > 0 else 0)C++ Implementation:
class NumMatrix {
private:
vector<vector<int>> dp;
public:
NumMatrix(vector<vector<int>>& matrix) {
int m = matrix.size(), n = matrix[0].size();
dp.resize(m, vector<int>(n, 0));
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
dp[i][j] = matrix[i][j] +
(i > 0 ? dp[i - 1][j] : 0) +
(j > 0 ? dp[i][j - 1] : 0) -
(i > 0 && j > 0 ? dp[i - 1][j - 1] : 0);
}
}
}
int sumRegion(int row1, int col1, int row2, int col2) {
return dp[row2][col2] -
(row1 > 0 ? dp[row1 - 1][col2] : 0) -
(col1 > 0 ? dp[row2][col1 - 1] : 0) +
(row1 > 0 && col1 > 0 ? dp[row1 - 1][col1 - 1] : 0);
}
};Complexity Analysis
- Time Complexity: O(1) for every query after O(m * n) time to fill the prefix sum array
- Space Complexity: O(m * n) for the prefix sum array
This dynamic programming method gives us a quick way to solve the Range Sum Query 2D problem. We use precomputed sums to answer queries fast. For more articles about dynamic programming, we can check out Dynamic Programming: Fibonacci Number and Dynamic Programming: Unique Paths in a Grid.
Prefix Sum Technique for 2D Range Queries
We can use the Prefix Sum technique to quickly find sums over submatrices in a 2D array. This method helps us prepare the matrix. After that, we can answer any range sum question in constant time.
Prefix Sum Array Construction
First, we need to make a 2D prefix sum array called
prefix. In this array, prefix[i][j] will store
the sum of elements from the top-left corner (0,0) to the
position (i,j). We can build this array using the formula
below:
def compute_prefix_sum(matrix):
rows, cols = len(matrix), len(matrix[0])
prefix = [[0] * (cols + 1) for _ in range(rows + 1)]
for i in range(1, rows + 1):
for j in range(1, cols + 1):
prefix[i][j] = (matrix[i - 1][j - 1]
+ prefix[i - 1][j]
+ prefix[i][j - 1]
- prefix[i - 1][j - 1])
return prefixQuerying the Sum of a Submatrix
After we make the prefix sum array, we can find the sum of any
submatrix. We just need to define corners (row1, col1) and
(row2, col2) and use the formula below:
def range_sum_query(prefix, row1, col1, row2, col2):
return (prefix[row2 + 1][col2 + 1]
- prefix[row1][col2 + 1]
- prefix[row2 + 1][col1]
+ prefix[row1][col1])Example
Let’s look at a matrix:
[
[1, 2, 3],
[4, 5, 6],
[7, 8, 9]
]After we call compute_prefix_sum(matrix), our prefix sum
array will look like this:
[
[0, 0, 0, 0],
[0, 1, 3, 6],
[0, 5, 12, 21],
[0, 12, 27, 45]
]If we want to find the sum of the submatrix from (1, 1)
to (2, 2), we would call:
sum_result = range_sum_query(prefix, 1, 1, 2, 2) # Output: 28This method lets us quickly sum any submatrix after we prepare it. It is very efficient for many questions. The time to prepare the matrix is (O(m n)) and each question takes (O(1)).
For more about dynamic programming techniques, you can check the article on Dynamic Programming - Unique Paths in a Grid.
Java Implementation of Range Sum Query 2D
We will show how to implement the Range Sum Query 2D in Java. We use a prefix sum array. This array helps us to quickly find the sum of elements in a specific rectangular area of a 2D matrix. The big benefit of using a prefix sum array is that we can calculate the sum of any submatrix in constant time after we set it up.
Step 1: Build the Prefix Sum Array
We start by making a 2D prefix sum array. Here,
prefix[i][j] will hold the sum of the elements from the
top-left corner (0, 0) to the bottom-right corner (i, j).
public class NumMatrix {
private int[][] prefix;
public NumMatrix(int[][] matrix) {
if (matrix == null || matrix.length == 0 || matrix[0].length == 0) return;
int rows = matrix.length;
int cols = matrix[0].length;
prefix = new int[rows + 1][cols + 1];
for (int i = 1; i <= rows; i++) {
for (int j = 1; j <= cols; j++) {
prefix[i][j] = matrix[i - 1][j - 1]
+ prefix[i - 1][j]
+ prefix[i][j - 1]
- prefix[i - 1][j - 1];
}
}
}
}Step 2: Get the Sum of a Submatrix
When we want to get the sum of a submatrix, we define it by its
corners (row1, col1) and (row2, col2). We can
use this formula:
sum = prefix[row2 + 1][col2 + 1]
- prefix[row1][col2 + 1]
- prefix[row2 + 1][col1]
+ prefix[row1][col1];Here is the method to do the sum query:
public int sumRegion(int row1, int col1, int row2, int col2) {
return prefix[row2 + 1][col2 + 1]
- prefix[row1][col2 + 1]
- prefix[row2 + 1][col1]
+ prefix[row1][col1];
}Full Implementation
Now we put everything together. Here is the complete implementation of the Range Sum Query 2D in Java:
public class NumMatrix {
private int[][] prefix;
public NumMatrix(int[][] matrix) {
if (matrix == null || matrix.length == 0 || matrix[0].length == 0) return;
int rows = matrix.length;
int cols = matrix[0].length;
prefix = new int[rows + 1][cols + 1];
for (int i = 1; i <= rows; i++) {
for (int j = 1; j <= cols; j++) {
prefix[i][j] = matrix[i - 1][j - 1]
+ prefix[i - 1][j]
+ prefix[i][j - 1]
- prefix[i - 1][j - 1];
}
}
}
public int sumRegion(int row1, int col1, int row2, int col2) {
return prefix[row2 + 1][col2 + 1]
- prefix[row1][col2 + 1]
- prefix[row2 + 1][col1]
+ prefix[row1][col1];
}
}Example Usage
Here is how we can use our class:
public static void main(String[] args) {
int[][] matrix = {
{1, 2, 3},
{4, 5, 6},
{7, 8, 9}
};
NumMatrix numMatrix = new NumMatrix(matrix);
System.out.println(numMatrix.sumRegion(0, 0, 1, 1)); // Output: 12
System.out.println(numMatrix.sumRegion(1, 1, 2, 2)); // Output: 28
System.out.println(numMatrix.sumRegion(0, 1, 2, 2)); // Output: 21
}This way of doing it is good for speed and space. It lets us quickly get sum queries after we set it up first. The prefix sum method can be used in many different situations when we deal with 2D range queries.
Python Implementation of Range Sum Query 2D
We can use the Prefix Sum technique to implement the Range Sum Query for a 2D matrix in Python. This method helps us prepare the matrix. Then we can answer sum queries very quickly.
Here is how we can do it:
class NumMatrix:
def __init__(self, matrix: List[List[int]]):
if not matrix or not matrix[0]:
self.matrix = []
self.prefix_sum = []
return
self.rows, self.cols = len(matrix), len(matrix[0])
self.matrix = matrix
self.prefix_sum = [[0] * (self.cols + 1) for _ in range(self.rows + 1)]
for r in range(1, self.rows + 1):
for c in range(1, self.cols + 1):
self.prefix_sum[r][c] = (matrix[r - 1][c - 1] +
self.prefix_sum[r - 1][c] +
self.prefix_sum[r][c - 1] -
self.prefix_sum[r - 1][c - 1])
def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
return (self.prefix_sum[row2 + 1][col2 + 1] -
self.prefix_sum[row1][col2 + 1] -
self.prefix_sum[row2 + 1][col1] +
self.prefix_sum[row1][col1])Explanation of the Code
- Initialization (
__init__method):- The constructor takes a 2D list
matrixas input. - It makes a
prefix_summatrix that has size(rows + 1) x (cols + 1). This matrix stores the total sums. - We calculate the prefix sums using this formula: [ [r][c] = [r-1][c-1] + [r-1][c] + [r][c-1] - [r-1][c-1] ]
- The constructor takes a 2D list
- Querying (
sumRegionmethod):- The
sumRegionmethod finds the sum of elements in the rectangle from(row1, col1)to(row2, col2). - It uses the prefix sums we calculated before to give the answer quickly.
- The
This code lets us do range sum queries fast after we prepare the data. It works well when we have many queries on the same matrix.
For more information on dynamic programming, we can read this dynamic programming Fibonacci number article.
C++ Implementation of Range Sum Query 2D
We can implement the Range Sum Query 2D using C++. We use a prefix sum array to quickly find the sum of elements in a specific rectangle area of a 2D matrix. This way, we can prepare the data in O(m * n) time. Here, m and n are the size of the matrix. After that, we can answer sum queries in O(1) time.
C++ Code Implementation
#include <iostream>
#include <vector>
class NumMatrix {
private:
std::vector<std::vector<int>> prefixSum;
public:
NumMatrix(std::vector<std::vector<int>>& matrix) {
int m = matrix.size();
if (m == 0) return; // Handle empty matrix
int n = matrix[0].size();
prefixSum = std::vector<std::vector<int>>(m + 1, std::vector<int>(n + 1, 0));
// Build the prefix sum matrix
for (int i = 1; i <= m; ++i) {
for (int j = 1; j <= n; ++j) {
prefixSum[i][j] = matrix[i - 1][j - 1]
+ prefixSum[i - 1][j]
+ prefixSum[i][j - 1]
- prefixSum[i - 1][j - 1];
}
}
}
int sumRegion(int row1, int col1, int row2, int col2) {
return prefixSum[row2 + 1][col2 + 1]
- prefixSum[row1][col2 + 1]
- prefixSum[row2 + 1][col1]
+ prefixSum[row1][col1];
}
};
int main() {
std::vector<std::vector<int>> matrix = {
{1, 2, 3},
{4, 5, 6},
{7, 8, 9}
};
NumMatrix numMatrix(matrix);
std::cout << "Sum of region (1, 1) to (2, 2): " << numMatrix.sumRegion(1, 1, 2, 2) << std::endl; // Output: 28
std::cout << "Sum of region (0, 0) to (1, 1): " << numMatrix.sumRegion(0, 0, 1, 1) << std::endl; // Output: 12
return 0;
}Explanation of C++ Code
- Class Definition: The
NumMatrixclass has a 2D vector calledprefixSum. It saves the prefix sums. - Constructor: The constructor sets up the
prefixSummatrix:- It goes through each element in the input matrix and finds the
prefix sum using:
prefixSum[i][j] = matrix[i-1][j-1] + prefixSum[i-1][j] + prefixSum[i][j-1] - prefixSum[i-1][j-1]
- It goes through each element in the input matrix and finds the
prefix sum using:
- sumRegion Method: This method finds the sum of the
rectangle area defined by
(row1, col1)and(row2, col2):- It uses the inclusion-exclusion rule to get the sum in constant time
from the
prefixSumarray.
- It uses the inclusion-exclusion rule to get the sum in constant time
from the
This C++ code for Range Sum Query 2D helps us to handle sum queries easily. It shows how we can use dynamic programming to make our solutions faster. For more on dynamic programming, we can look at Dynamic Programming - Unique Paths in a Grid.
Optimizing Space Complexity in Range Sum Query 2D
In the Range Sum Query 2D problem, we often use a 2D array to keep prefix sums. This can use a lot of space, especially when we have big matrices. We can make space usage better by lowering the extra space we need for the prefix sums.
Techniques for Space Optimization
In-place Updates: Instead of using a separate 2D array for prefix sums, we can change the original matrix to keep cumulative sums. This way, we can use the same data instead of creating a new one when we do range queries.
Single Array Usage: If the problem allows it, we can turn the 2D prefix sum array into a 1D array to save space. By going through the matrix row by row or column by column, we can find the cumulative sum without needing a full 2D array.
Implementation Example
Here is a Java code that shows how we can make space usage better while calculating 2D range sums using in-place updates.
class NumMatrix {
private int[][] matrix;
private int[][] sumMatrix;
public NumMatrix(int[][] matrix) {
this.matrix = matrix;
if (matrix.length > 0 && matrix[0].length > 0) {
int rows = matrix.length;
int cols = matrix[0].length;
sumMatrix = new int[rows + 1][cols + 1];
for (int i = 1; i <= rows; i++) {
for (int j = 1; j <= cols; j++) {
sumMatrix[i][j] = matrix[i - 1][j - 1]
+ sumMatrix[i - 1][j]
+ sumMatrix[i][j - 1]
- sumMatrix[i - 1][j - 1];
}
}
}
}
public int sumRegion(int row1, int col1, int row2, int col2) {
return sumMatrix[row2 + 1][col2 + 1]
- sumMatrix[row1][col2 + 1]
- sumMatrix[row2 + 1][col1]
+ sumMatrix[row1][col1];
}
}Key Points
- We create the
sumMatrixwith an extra row and column. This makes range sum calculations easier. - We use this cumulative sum method to lower the space usage. This allows us to get sum queries in constant time after we set everything up.
- This method keeps the ability of the 2D range sum query and also cuts down the space we need.
This better way to handle the Range Sum Query 2D problem makes the algorithm faster. It also helps us work with bigger data without using too much memory. If you want to learn more about dynamic programming methods, you can check out Dynamic Programming: Unique Paths in a Grid.
Testing and Validating the Range Sum Query Solutions
We need to make sure the Range Sum Query 2D solutions are correct. To do this, we should have a good testing plan. In this section, we will talk about important testing methods like unit tests and checking against expected results.
Testing Methodology
- Unit Testing: We create unit tests for each version (Java, Python, C++) to check if they work well.
- Boundary Conditions: We test special cases like:
- Empty matrices
- Matrices with only one element
- Queries that cover the whole matrix
- Randomized Testing: We make random matrices and queries. Then we compare the results with a brute-force method to make sure they are right.
Example Test Cases
Let’s look at an example. We have a 2D matrix and some queries.
# Sample Matrix
matrix = [
[1, 2, 3],
[4, 5, 6],
[7, 8, 9]
]
# Queries
queries = [
(0, 0, 1, 1), # Sum of submatrix from (0,0) to (1,1) => 1+2+4+5 = 12
(1, 1, 2, 2), # Sum of submatrix from (1,1) to (2,2) => 5+6+8+9 = 28
(0, 0, 2, 2) # Sum of the entire matrix => 1+2+3+4+5+6+7+8+9 = 45
]
# Expected Results
expected_results = [12, 28, 45]Validation Function
We write a validation function to check results from the DP method with the expected results.
def validate_range_sum_query(matrix, queries, expected_results):
for index, query in enumerate(queries):
x1, y1, x2, y2 = query
result = range_sum_query_2D(matrix, x1, y1, x2, y2) # Call your DP function here
assert result == expected_results[index], f"Test failed for query {query}: expected {expected_results[index]}, got {result}"
print("All tests passed!")Running Tests
We should put the validation function in our testing system and run it. This will help us make sure all queries give the expected results.
# Example usage
validate_range_sum_query(matrix, queries, expected_results)By using these testing and validation methods, we can check that our Range Sum Query 2D solutions work well in different situations and edge cases. If you want to learn more about dynamic programming tests, you can read Dynamic Programming Basics. This will help us understand the basics that are useful for testing.
Frequently Asked Questions
What is the Range Sum Query 2D problem in dynamic programming?
The Range Sum Query 2D problem is about finding the sum of numbers in a specific rectangular area of a 2D matrix. We can solve this problem well using dynamic programming. We create a prefix sum matrix first. This lets us get the sum quickly after we do some initial work.
How is dynamic programming used in Range Sum Query 2D?
Dynamic programming helps us with Range Sum Query 2D by dividing the problem into smaller parts. We make a prefix sum array. This array holds the total sums of numbers. With this, we can find the sum of any submatrix very fast, much better than the simple way.
Can you explain the prefix sum technique in 2D range queries?
The prefix sum technique for 2D range queries means we build a 2D array. In this array, each spot (i, j) has the sum of all numbers from the top-left corner (0, 0) to (i, j) in the original matrix. This makes it easy to find submatrix sums using the inclusion-exclusion principle. This way, Range Sum Query 2D becomes much quicker.
What are the common implementations of Range Sum Query 2D in Java, Python, and C++?
We can find Range Sum Query 2D implementations in many programming languages. Java, Python, and C++ are the most popular. Each of these usually builds a prefix sum array and has a way to get the sum of any submatrix. For more examples, check the Java Implementation of Range Sum Query 2D, Python Implementation, and C++ Implementation.
How can I optimize space complexity in Range Sum Query 2D?
To make space better in Range Sum Query 2D, we can use one 2D array for prefix sums instead of many arrays for queries. By using the prefix sums well, we can save space and still get sums quickly. For more tips, you can check related articles on dynamic programming.