Maximum Subarray
The maximum subarray problem asks for the contiguous segment with the largest sum. Kadane's algorithm solves it in O(n).
先从这个心智模型开始
The maximum subarray problem asks for the contiguous segment with the largest sum. Kadane's algorithm solves it in O(n).
它如何工作，不堆砌不必要的术语
At each position decide whether to extend the previous segment or start fresh from the current value.
Track both the best segment ending here and the best seen globally.
把算法读成一连串决策
Define the state precisely.
Write the transition/choice from smaller states.
Set base cases, then decide memoization, tabulation or backtracking.
Read a tiny input by hand and write down the state after every important operation. For Maximum Subarray, the goal is to make the invariant visible rather than to memorize code.
function maxSubarray(a) {
let ending = a[0];
let best = a[0];
for (let i = 1; i < a.length; i++) {
ending = Math.max(a[i], ending + a[i]);
best = Math.max(best, ending);
}
return best;
}不要死记语法。跟踪每一行中发生变化的状态，并把它与上面描述的不变量联系起来。
O(n) with Kadane
Do not quote this complexity mechanically. Ask which operation or input dimension it describes, then derive the cost from the loops, recursion depth, data-structure operations and any preprocessing assumptions.
适用场景
- Use DP when subproblems overlap and optimal solutions can be composed from smaller optimal states.
通常哪里会出错
- Writing recurrence before defining state.
- Using DP when state space is larger than the original problem.
专业回答结构
Explain the local decision behind Kadane, not only the two variable names. Also handle the all-negative case correctly.
What problem does Maximum Subarray solve?
The maximum subarray problem asks for the contiguous segment with the largest sum. Kadane's algorithm solves it in O(n).
关于复杂度应该提到什么？
Its typical cost here is O(n) with Kadane, but a complete answer should connect that cost to the representation, input constraints and the dominant operations.
最常见的错误是什么？
Writing recurrence before defining state.