动态规划入门教程：用Logo语言探索算法之美
动态规划（Dynamic Programming，简称DP）是一种在数学、管理科学、计算机科学、经济学和生物信息学等领域广泛应用的算法设计方法。它通过将复杂问题分解为更小的子问题，并存储这些子问题的解，从而避免重复计算，提高算法效率。本文将使用Logo语言，一种简单易学的编程语言，来介绍动态规划的基本概念和入门技巧。
一、什么是Logo语言？
Logo语言是一种面向对象的编程语言，最初由Wally Feurzeig和 Seymour Papert在1970年代开发，用于教育目的。它以其独特的turtle图形绘制功能而闻名，通过控制一个名为turtle的虚拟小海龟在屏幕上移动来绘制图形。
Logo语言的特点包括：
- 简单易学：Logo语言的语法简单，适合初学者。
- 图形化编程：通过turtle绘制图形，使编程过程更加直观。
- 面向对象：Logo语言支持面向对象编程，可以创建对象和类。
二、动态规划的基本概念
动态规划的核心思想是将一个复杂问题分解为若干个相互重叠的子问题，并存储这些子问题的解，以避免重复计算。动态规划通常具有以下特点：
- 最优化原理：子问题的解构成了原问题的最优解。
- 子问题重叠：子问题之间具有重叠性，即子问题的解会被多次使用。
- 无后效性：一旦某个子问题的解被确定，它就不会被改变。
三、Logo语言中的动态规划实现
下面我们将通过一个简单的例子来展示如何在Logo语言中实现动态规划。
3.1 斐波那契数列
斐波那契数列是一个经典的动态规划问题，其定义如下：
- F(0) = 0
- F(1) = 1
- F(n) = F(n-1) + F(n-2) （n > 1）
下面是使用Logo语言实现的斐波那契数列动态规划：
logo
to fib :n
ifelse :n < 2
[ :n ]
[ fib :n - 1 + fib :n - 2 ]
end
在这个例子中，我们定义了一个名为`fib`的Logo程序，它接受一个参数`n`，并返回斐波那契数列的第`n`项。程序使用递归的方式计算斐波那契数列，其中`ifelse`语句用于判断是否到达了基本情况（即`n`小于2）。
3.2 最长公共子序列
最长公共子序列（Longest Common Subsequence，简称LCS）是另一个经典的动态规划问题。给定两个序列`X`和`Y`，我们需要找到它们的最长公共子序列。
下面是使用Logo语言实现的LCS动态规划：
```logo
to lcs :x :y
localmake "m" (length :x) (length :y)
let [[row col] = [length :x - 1 length :y - 1]]
if :x = empty or :y = empty
[ empty ]
if :x = first :y
[ first :x lcs rest :x rest :y ]
if :x = first :y
[ first :y lcs rest :x rest :y ]
let [[m row col] = [item row col m]]
if :x = first :y
[ item row col lcs rest :x rest :y ]
[ item row col lcs rest :x rest :y ]
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[ item row col lcs rest :x rest :y ]
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