The K Weakest Rows in a Matrix - Complete Solution Guide

The K Weakest Rows in a Matrix is LeetCode problem 1337, a Easy level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Framing

The K Weakest Rows in a Matrix is a Easy LeetCode problem that rewards careful tracing, edge-case handling, and a clear grasp of Array and Binary Search. The best solutions usually explain why the chosen invariant holds before they optimize for time or space.

Quick Example Mindset

A useful way to test The K Weakest Rows in a Matrix is to start with a tiny input that exposes the boundary conditions, then run the same logic on a slightly larger case to verify the array behavior and the binary search interaction. That second pass is where off-by-one mistakes and missing updates usually appear.

Problem Statement

You are given an m x n binary matrix mat of 1 's (representing soldiers) and 0 's (representing civilians). The soldiers are positioned in front of the civilians. That is, all the 1 's will appear to the left of all the 0 's in each row. A row i is weaker than a row j if one of the following is true: The number of soldiers in row i is less than the number of soldiers in row j . Both rows have the same number of soldiers and i < j . Return the indices of the k weakest rows in the matrix ordered f

Detailed Explanation

This problem challenges us to identify and return the indices of the `k` weakest rows within a given binary matrix. Imagine this matrix as a military deployment: `1`s represent soldiers, and `0`s represent civilians. A crucial structural property is guaranteed for every row: all soldiers are always positioned to the left of all civilians. For instance, `[1,1,1,0,0]` is a valid row, but `[1,0,1,0,0]` is not. This makes counting soldiers straightforward.

Solution Approach

The elegance of the provided solution lies in how it seamlessly leverages Python's built-in sorting capabilities to directly implement the problem's two-tiered 'weakness' definition. For each row, the algorithm first computes the total number of soldiers. Thanks to the problem's guarantee that soldiers are always at the front, a simple `sum(row)` efficiently gives us this count. It then pairs this soldier count with the row's original index `i`, creating tuples like `(soldier_count, i)`. These tuples are then collected into a list. The core insight here is that when you sort a list of tuples in Python, it performs a lexicographical sort. This means it first compares the primary elements (soldier counts). If they differ, that comparison determines the order. Only if the primary elements are identical does it proceed to compare the secondary elements (row indices). This behavior perfectly mirrors our problem's weakness rules: fewer soldiers (the primary sort key) makes a row weaker, and if soldier counts are equal, the smaller row index (the secondary sort key) makes it weaker. After sorting this list of `(soldier_count, row_index)` tuples, the first `k` elements in the sorted list are precisely the `k` weakest rows. We then simply extract their original indices to form the final result.

Step-by-Step Algorithm

1. Step 1: Count the number of soldiers (1s) in each row. In the sorting approaches, this is done iterating through each row and stopping at the first 0 encountered. In the priority queue, it's done similarly within the loop adding to the queue.
2. Step 2: (Sorting Approach) Create a list of pairs, where each pair represents (soldier count, row index). Sort this list based on the soldier count in ascending order. If counts are equal, sort by index.
3. Step 2: (Priority Queue Approach) Insert each (soldier count, row index) pair into a min-heap priority queue. The queue automatically maintains order based on soldier count, with smaller counts having higher priority.
4. Step 3: Extract the row indices of the first `k` elements from the sorted list (sorting approach) or dequeue the top `k` elements from the priority queue (priority queue approach).
5. Step 4: Return the list of `k` row indices.

Key Insights

• **Efficient Soldier Counting with `sum()`**: The problem's constraint that all `1`s (soldiers) appear before all `0`s (civilians) in any row simplifies soldier counting dramatically. Instead of iterating or binary searching for the `0`s boundary, a straightforward `sum(row)` provides the soldier count in `O(N)` time for a row of length `N`.
• **Lexicographical Sort for Dual-Criteria Ranking**: The problem's 'weakness' is defined by two criteria: first by soldier count, then by row index for ties. By creating `(soldier_count, original_row_index)` tuples and applying a standard sort, Python's (and many other languages') default tuple sorting performs a lexicographical comparison. This automatically handles the dual-priority ordering, making the logic surprisingly concise and robust without needing a custom comparator function.
• **Direct Mapping of Problem Logic to Data Structure**: The act of pairing `(soldier_count, row_index)` before sorting directly translates the problem's 'weaker than' definition into a sortable data structure. This `(value, index)` pattern is a common and powerful technique when you need to sort elements based on a computed value while preserving their original position or identity.

Complexity Analysis

Time Complexity: O(m*n + m*log(m))

Space Complexity: O(m)

Topics

This problem involves: Array, Binary Search, Sorting, Heap (Priority Queue), Matrix.

Study Paths

Continue from this problem into the surrounding topic and company clusters to compare how the same pattern appears in other interview settings.

Related topics: Array, Binary Search, Sorting, Heap (Priority Queue)

Frequently Asked Questions