算法阅读笔记

Longest Common Subsequence

LCS finds the longest sequence of values that appears in both inputs in the same relative order, but not necessarily contiguously.

类型算法分类Sets & Dynamic Programming深度高级典型复杂度O(mn)

简短面试回答

先从这个心智模型开始

LCS finds the longest sequence of values that appears in both inputs in the same relative order, but not necessarily contiguously.

简单解释

它如何工作，不堆砌不必要的术语

dp[i][j] usually means LCS length for prefixes ending before i and j.

Matching characters extend the diagonal state; otherwise take the best of dropping one side.

执行过程

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.

记住一个小例子

Read a tiny input by hand and write down the state after every important operation. For Longest Common Subsequence, the goal is to make the invariant visible rather than to memorize code.

易读的 JavaScript 实现JavaScript

function lcs(a, b) {
  const dp = Array.from({ length: a.length + 1 }, () => Array(b.length + 1).fill(0));
  for (let i = 1; i <= a.length; i++) {
    for (let j = 1; j <= b.length; j++) {
      dp[i][j] = a[i - 1] === b[j - 1]
        ? dp[i - 1][j - 1] + 1
        : Math.max(dp[i - 1][j], dp[i][j - 1]);
    }
  }
  return dp[a.length][b.length];
}

如何阅读这段代码

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。

复杂度

O(mn)

Do not quote this complexity mechanically.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.

何时使用

适用场景

Use DP when subproblems overlap and optimal solutions can be composed from smaller optimal states.

常见错误

通常哪里会出错

Writing recurrence before defining state.
Using DP when state space is larger than the original problem.

面试中如何解释

专业回答结构

Explicitly distinguish subsequence from substring. This is one of the most common interview confusions.

问题→核心思路→不变量 / 状态→Big-O→权衡

快速面试检查

What problem does Longest Common Subsequence solve?

LCS finds the longest sequence of values that appears in both inputs in the same relative order, but not necessarily contiguously.

关于复杂度应该提到什么？

Its typical cost here is O(mn), 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.

继续阅读MemoizationTabulationBacktracking

function lcs(a, b) { const dp = Array.from({ length: a.length + 1 }, () => Array(b.length + 1).fill(0)); for (let i = 1; i <= a.length; i++) { for (let j = 1; j <= b.length; j++) { dp[i][j] = a[i - 1] === b[j - 1] ? dp[i - 1][j - 1] + 1 : Math.max(dp[i - 1][j], dp[i][j - 1]); } } return dp[a.length][b.length]; }