最大子数组问题为何具备最优子结构？关于动态规划适用条件的困惑咨询

Hey there! Let's clear up your confusion about optimal substructure—you're already on the right track, just mixing up two different ideas here.

First, let's rephrase the core of your question: you noticed that the maximum length-3 subarray in A = [1, 6, -3, 1, 5, -1] is [1,5,-1] (sum 5), while the maximum length-2 subarray is [1,6] (sum 7), and they don't share any elements. You're wondering how this fits with the definition of optimal substructure, which says a problem's optimal solution can be built from optimal solutions of its subproblems.

The Key Misconception #

You're comparing two separate problems here: "find the maximum length-3 subarray" and "find the maximum length-2 subarray". These are distinct tasks, so their optimal solutions don't need to overlap or relate to each other at all.

Optimal substructure isn't about different-sized problems having linked optimal solutions—it's about how a single problem's optimal solution can be constructed using optimal solutions of its own nested subproblems.

Let's Break Down the Correct Application #

Let's take your length-k maximum subarray problem to see optimal substructure in action. Let's focus on the problem: "Find the maximum sum of any length-3 subarray in A".

To solve this, we can frame it using subproblems:

• For each position i in the array (starting from index 2, 0-based), the sum of the length-3 subarray ending at i is equal to the sum of the length-2 subarray ending at i-1 plus A[i].
• To find the overall maximum length-3 subarray, we just need to track the maximum value of these sums across all valid i.

In your example, the optimal length-3 subarray ends at index 5 (element -1). Its corresponding subproblem is the length-2 subarray ending at index 4 (elements 1,5, sum 6)—adding -1 gives 5, which is the sum of our optimal length-3 subarray.

Notice: this subproblem isn't the global maximum length-2 subarray ([1,6]), but a local subproblem directly tied to building our current solution. That's the key! Optimal substructure relies on using the optimal solutions of the subproblems that directly contribute to the current problem's solution, not unrelated global optimal solutions of other problem sizes.

Recapping the Definition #

To restate the Wikipedia definition in plain terms:

A problem has optimal substructure if, when you find the best solution to the whole problem, that solution is made up of the best solutions to smaller pieces of the same problem that are required to build it.

It doesn't say anything about optimal solutions of different problem sizes needing to overlap. Your confusion came from comparing two separate problem's results, not looking at how a single problem's solution is built from its own subproblems.

内容的提问来源于stack exchange，提问作者JustJohn