Leetcode #2040: Kth Smallest Product of Two Sorted Arrays
In this guide, we solve Leetcode #2040 Kth Smallest Product of Two Sorted Arrays in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
Given two sorted 0-indexed integer arrays nums1 and nums2 as well as an integer k, return the kth (1-based) smallest product of nums1[i] * nums2[j] where 0 <= i < nums1.length and 0 <= j < nums2.length. Example 1: Input: nums1 = [2,5], nums2 = [3,4], k = 2 Output: 8 Explanation: The 2 smallest products are: - nums1[0] * nums2[0] = 2 * 3 = 6 - nums1[0] * nums2[1] = 2 * 4 = 8 The 2nd smallest product is 8.
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: Array, Binary Search
Intuition
The problem structure suggests a monotonic decision, which makes binary search a natural fit.
By halving the search space each step, we reach the answer efficiently.
Approach
Search either directly on a sorted array or on the answer space using a check function.
Each check is fast, and the logarithmic search keeps the overall runtime low.
Steps:
- Define the search bounds.
- Check the mid point condition.
- Narrow the bounds until convergence.
Example
Input: nums1 = [2,5], nums2 = [3,4], k = 2
Output: 8
Explanation: The 2 smallest products are:
- nums1[0] * nums2[0] = 2 * 3 = 6
- nums1[0] * nums2[1] = 2 * 4 = 8
The 2nd smallest product is 8.
Python Solution
class Solution:
def kthSmallestProduct(self, nums1: List[int], nums2: List[int], k: int) -> int:
def count(p: int) -> int:
cnt = 0
n = len(nums2)
for x in nums1:
if x > 0:
cnt += bisect_right(nums2, p / x)
elif x < 0:
cnt += n - bisect_left(nums2, p / x)
else:
cnt += n * int(p >= 0)
return cnt
mx = max(abs(nums1[0]), abs(nums1[-1])) * max(abs(nums2[0]), abs(nums2[-1]))
return bisect_left(range(-mx, mx + 1), k, key=count) - mx
Complexity
The time complexity is , where and are the lengths of and , respectively, and is the maximum absolute value in and . The space complexity is O(1).
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.