#1439

Hard

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Find the Kth Smallest Sum of a Matrix With Sorted Rows

Find the kth smallest sum in a matrix with sorted rows using binary search and a heap-based approach.

array binary search heap priority queue matrix

Problem Statement

You are given a matrix mat with m rows and n columns, where each row is sorted in non-decreasing order. You are also given an integer k, and you need to select exactly one element from each row to form an array. Your goal is to return the kth smallest sum of all possible arrays you can form by selecting one element from each row.

For example, given the matrix [[1, 3, 11], [2, 4, 6]] and k = 5, the smallest possible sums are: [1, 2], [1, 4], [3, 2], [3, 4], [1, 6]. The fifth smallest sum is 7. The solution needs to handle large matrix sizes efficiently while finding the correct kth smallest sum.

Examples

Example 1

Input: mat = [[1,3,11],[2,4,6]], k = 5

Output: 7

Choosing one element from each row, the first k smallest sum are: [1,2], [1,4], [3,2], [3,4], [1,6]. Where the 5th sum is 7.

Example 2

Input: mat = [[1,3,11],[2,4,6]], k = 9

Output: 17

Example details omitted.

Example 3

Input: mat = [[1,10,10],[1,4,5],[2,3,6]], k = 7

Output: 9

Choosing one element from each row, the first k smallest sum are: [1,1,2], [1,1,3], [1,4,2], [1,4,3], [1,1,6], [1,5,2], [1,5,3]. Where the 7th sum is 9.

Constraints

m == mat.length
n == mat.length[i]
1 <= m, n <= 40
1 <= mat[i][j] <= 5000
1 <= k <= min(200, nm)
mat[i] is a non-decreasing array.

Solution Approach

Binary Search Over Valid Answer Space

Perform a binary search on the possible sums by selecting the lower and upper bounds. The lower bound is the sum of the smallest elements from each row, and the upper bound is the sum of the largest elements. This approach narrows down the valid answer space and helps pinpoint the kth smallest sum.

Heap for Efficient Sum Generation

Use a heap (priority queue) to track the smallest sums at each step. As you pop the smallest sum, push the next possible sum formed by selecting the next element in any row. This allows efficient generation of the sums while maintaining the order of their sizes.

Avoiding Brute Force

Instead of generating all possible arrays and sorting them (which would be computationally expensive), use a heap to efficiently generate sums in increasing order. This significantly reduces both time and space complexity compared to a brute-force solution.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is O(k log k) for binary search, combined with the heap operations that track the smallest sums. The space complexity depends on the size of the heap and the number of visited sums but is typically much smaller than brute-force approaches.

What Interviewers Usually Probe

Candidate demonstrates understanding of binary search and heap-based approaches for solving matrix sum problems.
The candidate effectively applies binary search over the valid sum range and uses a priority queue for efficient sum management.
Look for a solution where the candidate avoids a brute-force approach and ensures that the algorithm works within the problem's constraints.

Common Pitfalls or Variants

Common pitfalls

Failure to implement binary search correctly, leading to excessive computation.
Not using the heap efficiently to track sums, causing performance issues for larger matrices.
Confusing the problem with typical sum problems that don't involve sorted rows or the heap structure.

Follow-up variants

Consider solving with larger matrices, adjusting the binary search bounds appropriately.
Explore optimization strategies for reducing space complexity when working with very large k values.
Test variations where some rows are empty or have more than one element to ensure robustness.

FAQ

What is the main pattern for solving the 'Find the Kth Smallest Sum of a Matrix With Sorted Rows' problem?

The key pattern is binary search over the valid answer space, combined with a heap to efficiently generate and track the kth smallest sum.

How does binary search apply to this problem?

Binary search is used to narrow down the possible sum range, adjusting the bounds based on the smallest and largest possible sums in the matrix.

Why is using a heap important for this problem?

A heap allows you to track the smallest sums efficiently, ensuring you always generate the next smallest sum without having to sort all possibilities.

What are some common pitfalls when solving this problem?

Common pitfalls include failing to implement binary search correctly, not managing the heap efficiently, or confusing this problem with simpler sum problems.

How can GhostInterview assist with solving this problem?

GhostInterview provides hints on binary search and heap usage, guiding candidates through the problem-solving process and ensuring an efficient solution.

terminal

Solution

Solution 1

#### Python3

1

2

3

4

5

6

class Solution:
    def kthSmallest(self, mat: List[List[int]], k: int) -> int:
        pre = [0]
        for cur in mat:
            pre = sorted(a + b for a in pre for b in cur[:k])[:k]
        return pre[-1]

Continue Practicing

#1337 the k weakest rows in a matrix #1631 path with minimum effort #778 swim in rising water