#1911

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Alternating Subsequence Sum

Maximize alternating subsequence sum with dynamic programming and state transitions in an array.

Problem Statement

Given an array nums, your task is to find the maximum alternating sum of any subsequence of nums. A subsequence is a new array derived from the original array by removing some elements without changing the order of the remaining elements.

The alternating sum of a sequence is defined as the sum of elements at even indices minus the sum of elements at odd indices. The problem can be solved efficiently with dynamic programming, focusing on state transitions to track optimal subsequences.

Examples

Example 1

Input: nums = [4,2,5,3]

Output: 7

It is optimal to choose the subsequence [4,2,5] with alternating sum (4 + 5) - 2 = 7.

Example 2

Input: nums = [5,6,7,8]

Output: 8

It is optimal to choose the subsequence [8] with alternating sum 8.

Example 3

Input: nums = [6,2,1,2,4,5]

Output: 10

It is optimal to choose the subsequence [6,1,5] with alternating sum (6 + 5) - 1 = 10.

Constraints

1 <= nums.length <= 105
1 <= nums[i] <= 105

Solution Approach

Dynamic Programming State Transitions

Use two variables to track the current maximum alternating sum: one for when the current element is added to the sum at even indices and another for when it is added at odd indices. This approach reduces the problem to calculating the alternating sum incrementally as you iterate through the array.

Greedy Sequences and State Tracking

Track the sum for both possibilities at each step—whether to include an element in an even or odd indexed subsequence. The greedy approach ensures the subsequence that maximizes the alternating sum is selected without recalculating the sum for every subsequence combination.

Space Optimization

Optimize the space complexity by using only two variables to track the alternating sums for even and odd subsequences. This ensures that the problem is solved in O(n) time with constant space usage.

Complexity Analysis

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

The time complexity is O(n) due to the single traversal through the array, where n is the length of the array. The space complexity is O(1), as we only use a constant amount of space for tracking the alternating sums.

What Interviewers Usually Probe

Look for the ability to optimize both time and space complexity in dynamic programming solutions.
Assess the candidate's understanding of alternating sum and how dynamic programming applies.
Check whether the candidate can correctly handle the state transitions between even and odd indexed subsequences.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the alternating sum definition, which may lead to incorrect subsequences.
Failing to optimize space by using unnecessary arrays for each possible subsequence.
Not handling edge cases, such as very small arrays or arrays with only one element.

Follow-up variants

Consider the scenario where the input array contains a single element.
Explore the impact of large input sizes on performance and how to optimize for such cases.
Challenge the candidate with an extension that requires finding the subsequence with the maximum alternating sum while also considering additional constraints.

FAQ

What is the pattern used to solve the Maximum Alternating Subsequence Sum?

The pattern used is dynamic programming with state transitions to track alternating sums for both even and odd indexed subsequences.

How do you track the alternating sum in this problem?

You track the sum by maintaining two variables—one for even indices and one for odd indices—and iterating through the array to update them accordingly.

What is the time complexity of the Maximum Alternating Subsequence Sum problem?

The time complexity is O(n) because the solution requires a single traversal through the array.

How does GhostInterview assist in solving the Maximum Alternating Subsequence Sum problem?

GhostInterview provides an interactive solver to help you implement dynamic programming and optimize the solution step-by-step.

Are there any edge cases in the Maximum Alternating Subsequence Sum problem?

Yes, edge cases such as arrays with only one element or very small arrays should be handled appropriately.

terminal

Solution

Solution 1

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class Solution:
    def maxAlternatingSum(self, nums: List[int]) -> int:
        n = len(nums)
        f = [0] * (n + 1)
        g = [0] * (n + 1)
        for i, x in enumerate(nums, 1):
            f[i] = max(g[i - 1] - x, f[i - 1])
            g[i] = max(f[i - 1] + x, g[i - 1])
        return max(f[n], g[n])

Solution 2

#### Python3

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class Solution:
    def maxAlternatingSum(self, nums: List[int]) -> int:
        n = len(nums)
        f = [0] * (n + 1)
        g = [0] * (n + 1)
        for i, x in enumerate(nums, 1):
            f[i] = max(g[i - 1] - x, f[i - 1])
            g[i] = max(f[i - 1] + x, g[i - 1])
        return max(f[n], g[n])

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