What is the time complexity of Number of Submatrices That Sum to Target?

The optimal solution has a time complexity of O(m^2*n).

What data structures are used to solve Number of Submatrices That Sum to Target?

This problem uses Array, Hash Table, Matrix, Prefix Sum. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Number of Submatrices That Sum to Target?

Insight 1: We can iterate through all possible pairs of rows (r1, r2) to define the top and bottom boundaries of our submatrices. Insight 2: For each pair of rows (r1, r2), we can calculate the column sums between these rows.

Which companies ask LeetCode 1074. Number of Submatrices That Sum to Target in interviews?

This problem is frequently asked at Media.net.

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Number of Submatrices That Sum to Target - LeetCode 1074 Solution

Q: How to solve Number of Submatrices That Sum to Target?

The solution iterates through all possible submatrices by considering every possible pair of starting and ending rows (r1 and r2). For each pair of rows, it calculates a running sum of the columns between those rows. Then, it uses a hash map to store the frequencies of different prefix sums encountered while iterating through the columns. By checking if `current_sum - target` exists in the hash map, we can determine if there's a submatrix ending at the current column whose sum equals the target.

Number of Submatrices That Sum to Target - Complete Solution Guide

Number of Submatrices That Sum to Target is LeetCode problem 1074, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given a matrix and a target , return the number of non-empty submatrices that sum to target . A submatrix x1, y1, x2, y2 is the set of all cells matrix[x][y] with x1 <= x <= x2 and y1 <= y <= y2 . Two submatrices (x1, y1, x2, y2) and (x1', y1', x2', y2') are different if they have some coordinate that is different: for example, if x1 != x1' . Example 1: Input: matrix = [[0,1,0],[1,1,1],[0,1,0]], target = 0 Output: 4 Explanation: The four 1x1 submatrices that only contain 0. Example 2: Input: mat

Detailed Explanation

The problem asks us to find the number of non-empty submatrices within a given matrix whose elements sum up to a specified target value. A submatrix is defined by its top-left and bottom-right corners (x1, y1, x2, y2), and consists of all cells matrix[x][y] where x1 <= x <= x2 and y1 <= y <= y2. We need to count how many different submatrices satisfy this sum condition. The constraints limit the matrix dimensions to a maximum of 100x100, and the values within the matrix and the target value have specific ranges as described.

Solution Approach

The solution iterates through all possible submatrices by considering every possible pair of starting and ending rows (r1 and r2). For each pair of rows, it calculates a running sum of the columns between those rows. Then, it uses a hash map to store the frequencies of different prefix sums encountered while iterating through the columns. By checking if `current_sum - target` exists in the hash map, we can determine if there's a submatrix ending at the current column whose sum equals the target.

Step-by-Step Algorithm

Step 1: Iterate through all possible starting rows (r1) from 0 to m-1.
Step 2: For each starting row r1, iterate through all possible ending rows (r2) from r1 to m-1.
Step 3: Initialize an array `col_sums` to store the sum of elements in each column between rows r1 and r2.
Step 4: Iterate through all columns (c) from 0 to n-1 and calculate `col_sums[c]` by adding matrix[r2][c] to it for each row r2 between r1 and r2 inclusive.
Step 5: Initialize a hash map `prefix_sum_counts` to store the frequency of prefix sums. Set `prefix_sum_counts[0] = 1` to account for the case where the submatrix starting from the first column has a sum equal to the target.
Step 6: Initialize `current_sum` to 0.
Step 7: Iterate through each column sum `s` in `col_sums`.
Step 8: Add `s` to `current_sum`.
Step 9: Check if `current_sum - target` exists as a key in `prefix_sum_counts`. If it does, increment the `count` by the value associated with that key (i.e., the frequency of that prefix sum).
Step 10: Increment the frequency of `current_sum` in `prefix_sum_counts`.
Step 11: Return the final `count`.

Key Insights

Insight 1: We can iterate through all possible pairs of rows (r1, r2) to define the top and bottom boundaries of our submatrices.
Insight 2: For each pair of rows (r1, r2), we can calculate the column sums between these rows.
Insight 3: Once we have the column sums, we can use a hash map (prefix sum counting) to efficiently find the number of submatrices with a sum equal to the target within those columns.

Complexity Analysis

Time Complexity: O(m^2*n)

Space Complexity: O(n)

Topics

This problem involves: Array, Hash Table, Matrix, Prefix Sum.

Companies

Asked at: Media.net.

Number of Submatrices That Sum to Target | LeetCode 1074

Hard

Array Hash Table Matrix Prefix Sum

Time: O(m^2*n)Space: O(n)

Solution Code

Available in:

import java.util.HashMap;
import java.util.Map;

class Solution {
    public int numSubmatrixSumTarget(int[][] matrix, int target) {
        int m = matrix.length;
        int n = matrix[0].length;
        int count = 0;

        for (int r1 = 0; r1 < m; r1++) {
            int[] col_sums = new int[n];
            for (int r2 = r1; r2 < m; r2++) {
                for (int c = 0; c < n; c++) {
                    col_sums[c] += matrix[r2][c];
                }

                Map<Integer, Integer> prefix_sum_counts = new HashMap<>();
                prefix_sum_counts.put(0, 1);
                int current_sum = 0;
                for (int s : col_sums) {
                    current_sum += s;
                    if (prefix_sum_counts.containsKey(current_sum - target)) {
                        count += prefix_sum_counts.get(current_sum - target);
                    }
                    prefix_sum_counts.put(current_sum, prefix_sum_counts.getOrDefault(current_sum, 0) + 1);
                }
            }
        }

        return count;
    }
}

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Problem Info

Description

Problem 1074: Number of Submatrices That Sum to Target - Detailed explanation and solution approaches.

Topics

Array Hash Table Matrix Prefix Sum

Companies

Media.net

Stats

Difficulty:Hard

Languages:4

Companies:1

Problem #:1074

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

11 steps

1
Step 1: Iterate through all possible starting rows (r1) from 0 to m-1.
2
Step 2: For each starting row r1, iterate through all possible ending rows (r2) from r1 to m-1.
3
Step 3: Initialize an array `col_sums` to store the sum of elements in each column between rows r1 and r2.
4
Step 4: Iterate through all columns (c) from 0 to n-1 and calculate `col_sums[c]` by adding matrix[r2][c] to it for each row r2 between r1 and r2 inclusive.
5
Step 5: Initialize a hash map `prefix_sum_counts` to store the frequency of prefix sums. Set `prefix_sum_counts[0] = 1` to account for the case where the submatrix starting from the first column has a sum equal to the target.
6
Step 6: Initialize `current_sum` to 0.
7
Step 7: Iterate through each column sum `s` in `col_sums`.
8
Step 8: Add `s` to `current_sum`.
9
Step 9: Check if `current_sum - target` exists as a key in `prefix_sum_counts`. If it does, increment the `count` by the value associated with that key (i.e., the frequency of that prefix sum).
10
Step 10: Increment the frequency of `current_sum` in `prefix_sum_counts`.
11
Step 11: Return the final `count`.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(m^2*n)

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity is O(m^2 * n) because we have nested loops iterating through all possible pairs of rows (O(m^2)) and for each pair of rows, we iterate through all columns (O(n)) and also perform hash map operations which take O(1) on average. The space complexity is O(n) because we store `col_sums` which requires O(n) space and in worst case, the hash map `prefix_sum_counts` might contain n elements.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Prefix Sum Matrix

Create a prefix sum matrix where `prefix_sum[i][j]` stores the sum of all elements from `matrix[0][0]` to `matrix[i][j]`. Then, iterate through all possible submatrices and calculate their sums using the prefix sum matrix.

Time: O(m^2 * n^2)Space: O(m * n)

Trade-offs: This approach has a higher time complexity than the given solution. The space complexity is also higher. It is not efficient compared to the main solution.

✨

Key Insights

1Insight 1: We can iterate through all possible pairs of rows (r1, r2) to define the top and bottom boundaries of our submatrices.
2Insight 2: For each pair of rows (r1, r2), we can calculate the column sums between these rows.
3Insight 3: Once we have the column sums, we can use a hash map (prefix sum counting) to efficiently find the number of submatrices with a sum equal to the target within those columns.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Hash Table

A data structure that maps keys to values for quick lookup. Like a dictionary!

Edge Cases

Edge case 1: Empty matrix. The code handles this implicitly because the outer loops won't execute if the matrix is empty, and the count will remain 0.
Edge case 2: Target is 0. The initialization of `prefix_sum_counts[0] = 1` correctly handles the case where a submatrix with a sum of 0 starts from the beginning.
Edge case 3: Matrix with all zeros. The prefix sum counting correctly identifies all submatrices with a sum of 0.
Edge case 4: Target is a very large positive or negative value. The `prefix_sum_counts` hashmap can handle negative and large positive sums, assuming there is enough memory, avoiding integer overflow issues.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not initializing `prefix_sum_counts[0] = 1`. This leads to incorrect counts when a submatrix starting from the first column has a sum equal to the target.

✗Common mistake 2: Not considering all possible pairs of rows and columns. The nested loops must iterate through all valid submatrices.

✗Common mistake 3: Integer overflow. If the matrix elements or target values are very large, intermediate sums could overflow. Using appropriate data types (e.g., long in Java/C++) is important for larger values to prevent incorrect results.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Image processing. Finding regions in an image with a specific average pixel intensity can be modeled as finding submatrices with a target sum. For example, identifying specific objects or features based on pixel density.

🏦Real-world scenario 2: Data analysis. Analyzing sales data over different time periods and regions to find subregions and sub-periods with a particular total sales amount. This can help identify successful marketing campaigns or areas needing improvement.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Hard problems often combine multiple techniques. Break the problem into sub-problems.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: November 22, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 14

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 7

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

What are the trade-offs of using a hash table?

Hash tables provide O(1) average-case lookup, insertion, and deletion, making them ideal for frequency counting and duplicate detection. However, they use O(n) extra space and may degrade to O(n) time in worst-case scenarios with hash collisions. Always consider if the space trade-off is acceptable.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Number of Submatrices That Sum to Target"?

For this hard-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether hash table techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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