Leetcode #1862: Sum of Floored Pairs
In this guide, we solve Leetcode #1862 Sum of Floored Pairs 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 an integer array nums, return the sum of floor(nums[i] / nums[j]) for all pairs of indices 0 <= i, j < nums.length in the array. Since the answer may be too large, return it modulo 109 + 7.
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: Array, Math, Binary Search, Prefix Sum
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: nums = [2,5,9]
Output: 10
Explanation:
floor(2 / 5) = floor(2 / 9) = floor(5 / 9) = 0
floor(2 / 2) = floor(5 / 5) = floor(9 / 9) = 1
floor(5 / 2) = 2
floor(9 / 2) = 4
floor(9 / 5) = 1
We calculate the floor of the division for every pair of indices in the array then sum them up.
Python Solution
class Solution:
def sumOfFlooredPairs(self, nums: List[int]) -> int:
mod = 10**9 + 7
cnt = Counter(nums)
mx = max(nums)
s = [0] * (mx + 1)
for i in range(1, mx + 1):
s[i] = s[i - 1] + cnt[i]
ans = 0
for y in range(1, mx + 1):
if cnt[y]:
d = 1
while d * y <= mx:
ans += cnt[y] * d * (s[min(mx, d * y + y - 1)] - s[d * y - 1])
ans %= mod
d += 1
return ans
Complexity
The time complexity is , and the space complexity is . The space complexity is .
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.