Leetcode #805: Split Array With Same Average
In this guide, we solve Leetcode #805 Split Array With Same Average 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
You are given an integer array nums. You should move each element of nums into one of the two arrays A and B such that A and B are non-empty, and average(A) == average(B).
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: Bit Manipulation, Array, Math, Dynamic Programming, Bitmask
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: nums = [1,2,3,4,5,6,7,8]
Output: true
Explanation: We can split the array into [1,4,5,8] and [2,3,6,7], and both of them have an average of 4.5.
Python Solution
class Solution:
def splitArraySameAverage(self, nums: List[int]) -> bool:
n = len(nums)
if n == 1:
return False
s = sum(nums)
for i, v in enumerate(nums):
nums[i] = v * n - s
m = n >> 1
vis = set()
for i in range(1, 1 << m):
t = sum(v for j, v in enumerate(nums[:m]) if i >> j & 1)
if t == 0:
return True
vis.add(t)
for i in range(1, 1 << (n - m)):
t = sum(v for j, v in enumerate(nums[m:]) if i >> j & 1)
if t == 0 or (i != (1 << (n - m)) - 1 and -t in vis):
return True
return False
Complexity
The time complexity is , which will time out. 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.