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Trionic Array II
Trionic Array II is a challenging problem involving the sum of contiguous subarrays with special index constraints.
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Hard · Trionic Array II core interview pattern
Answer-first summary
Trionic Array II is a challenging problem involving the sum of contiguous subarrays with special index constraints.
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The task is to find the maximum sum of a trionic subarray, a contiguous subarray with specific index constraints. By using dynamic programming, we can efficiently solve this problem even for large input sizes.
Problem Statement
Given an integer array nums of length n, a trionic subarray is a contiguous subarray nums[l...r] (with 0 <= l < r < n) for which there exist indices l < p < q < r such that:
Your goal is to return the maximum sum of any trionic subarray in the given nums array.
Examples
Example 1
Input: nums = [0,-2,-1,-3,0,2,-1]
Output: -4
Pick l = 1 , p = 2 , q = 3 , r = 5 :
Example 2
Input: nums = [1,4,2,7]
Output: 14
Pick l = 0 , p = 1 , q = 2 , r = 3 :
Constraints
- 4 <= n = nums.length <= 105
- -109 <= nums[i] <= 109
- It is guaranteed that at least one trionic subarray exists.
Solution Approach
Dynamic Programming Approach
A dynamic programming approach can be used to maintain subarrays' sums efficiently, ensuring we check possible trionic subarrays within time constraints.
Prefix Sum Optimization
By calculating the prefix sums and storing them in a table, we can quickly evaluate sums for any subarray, improving the algorithm's efficiency.
Sliding Window Technique
Using the sliding window technique, we can maintain a valid trionic subarray window and update the sum progressively, helping to meet the time complexity requirements.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time and space complexities depend on the chosen approach, but dynamic programming, combined with prefix sum or sliding window optimizations, provides a feasible solution with O(n) time complexity and O(n) space complexity.
What Interviewers Usually Probe
- Look for the candidate's understanding of dynamic programming in solving subarray problems.
- Evaluate how well the candidate optimizes their approach using techniques like prefix sum or sliding windows.
- Check if the candidate can handle large input sizes efficiently while maintaining clarity in their solution.
Common Pitfalls or Variants
Common pitfalls
- Not recognizing the need for dynamic programming, leading to inefficient brute force solutions.
- Failing to optimize the solution, resulting in time complexity that does not scale well for larger input sizes.
- Incorrectly identifying or implementing the trionic subarray definition, leading to incorrect results.
Follow-up variants
- Trionic subarrays with additional constraints, such as limiting the sum within a range.
- Non-contiguous trionic subarrays where the separation condition between indices is relaxed.
- Trionic subarray problems where we need to calculate both sum and product within the subarray.
FAQ
What is the core interview pattern for Trionic Array II?
The core interview pattern involves recognizing subarray sum problems that require dynamic programming and optimizations like prefix sums or sliding windows.
How do I efficiently solve Trionic Array II with large input sizes?
You can solve it efficiently by using dynamic programming, optimized with techniques such as prefix sums or the sliding window approach to reduce time complexity.
What is a trionic subarray?
A trionic subarray is a contiguous subarray with specific indices l < p < q < r, where the sum of elements within the subarray must be maximized.
What are the time and space complexities of solving Trionic Array II?
The time complexity can be optimized to O(n) with dynamic programming, and space complexity can also be O(n) with proper storage of intermediate results.
What common mistakes should I avoid when solving Trionic Array II?
Avoid brute force solutions, failing to optimize using dynamic programming, or incorrectly defining the trionic subarray constraints, which may result in incorrect outputs.
Solution
Solution 1: Grouped Loop
We can traverse the array to find all possible maximal trionic subarrays, calculate their sums, and update the maximum value.
class Solution:
def maxSumTrionic(self, nums: List[int]) -> int:
n = len(nums)
i = 0
ans = -inf
while i < n:
l = i
i += 1
while i < n and nums[i - 1] < nums[i]:
i += 1
if i == l + 1:
continue
p = i - 1
s = nums[p - 1] + nums[p]
while i < n and nums[i - 1] > nums[i]:
s += nums[i]
i += 1
if i == p + 1 or i == n or nums[i - 1] == nums[i]:
continue
q = i - 1
s += nums[i]
i += 1
mx = t = 0
while i < n and nums[i - 1] < nums[i]:
t += nums[i]
i += 1
mx = max(mx, t)
s += mx
mx = t = 0
for j in range(p - 2, l - 1, -1):
t += nums[j]
mx = max(mx, t)
s += mx
ans = max(ans, s)
i = q
return ans