《数据结构》课堂课件5

The same general method is then used for these subproblems, and so recursion continues until the size of the subproblems is reduced to some smallest, base case, where the solution is given directly without using further recursion.
5.1.3
(1) Solving problems using recursion
5.1.3
(1) Solving problems using recursion
(1) Solving problems using recursion

Recursion Versus Iteration


5.1.4
(1) Solving problems using recursion

The idea that gives a solution is to concentrate on the hardest step: moving the bottom disk.
1 2 3
We have to get to this state and then move the bottom disk

分解模式在每一层递归上都有三个步骤：
分解(Divide): 将原问题分解成一系列子问题； 解决(Conquer): 递归地解决各子问题。若子问题足够小， 则直接求解。 合并(Combine): 将子问题的结果合并成原问题的解。

5.1.4
(1) Solving problems using recursion
5.1.4
(1) Solving problems using recursion

Our goal: move(8, 1, 3, 2)
Steps to solve the problem:
move(7, 1, 2, 3); /*move 7 disks from tower 1 to 2 */ cout << “move disk 8 from tower 1 to 3.”<<endl; move(7, 2, 3, 1); /* move 7 disks from tower 2 to 3 */
1. Solving problems using recursion

In Mathematics:
Base case （递归基）
General case that reduces to simpler case （递归步）
5.1.3
The process of recursive call
5.1.3

there are no circuits, that vertex to any other is, no paths starting from a vertex and returning to the same vertex.
B E J F
C
D G H K I
(3) Tree of function calls

The tower of Hanoi （汉诺塔）



关于汉诺塔 在印度，有这么一个古老的传说：在世界中心 贝拿勒斯（在印度北部）的圣庙里，一块黄铜板上插着三根 宝石针。印度教的主神梵天在创造世界的时候，在其中一根 针上从下到上地穿好了由大到小的64片金片，这就是所谓的 汉诺塔。 不论白天黑夜，总有一个僧侣在按照下面的法则移动这些金 片：一次只移动一片，不管在哪根针上，小片必须在大片上 面。 僧侣们预言，当所有的金片都从梵天穿好的那根针上移到另 外一根针上时，世界就将在一声霹雳中消灭，而梵塔、庙宇 和众生也都将同归于尽。
Outline
Introduction to Recursion • Solving problems using recursion • Recursion trees How recursion is implemented on the machine • Stacks and recursion
(1) Solving problems using recursion

Recursive Function
The base case is count == 0
5.1.4
(2) Designing recursive algorithms
5.2.1
(3) Tree of function calls

Rules:
1. Move only one disk at one time; 2. No larger disk can be placed on a smaller disk. Write a computer program that will type out a list of instructions

The Solution
典故：
古印度有一個傳說，婆羅賀摩(Brahma，眾生地方的廟宇安置了一個 含有64個金盤的婆羅賀摩塔，婆羅門教的憎侶們，奉命依遊戲規則 要不停移動這些金盤，傳說中還說，當重排完成時，就是世界末日 的到來，而經過計算，這64個金盤的遊戲至少要經過 264-1次才會 完成，如果移動一次以1秒鐘計算，還要經過 184467440737095551615天才能完成，照這樣看來，這世界還算 蠻安全的。 Our goal: move(8, 1, 3, 2)



There are similarities between recursion and iteration In iteration, a loop repetition condition determines whether to repeat the loop body or exit from the loop In recursion, the condition usually tests for a base case An iterative solution exists to a problem that is solvable by recursion Recursive code may be simpler than an iterative algorithm and thus easier to write, read, and debug
一年大约有
31536926 秒，计算表明移完这些金片需
要5800多亿年，比地球寿命还要长
5.1.4
(1) Solving problems using recursion

Task:
move
64 disks from the tower 1 to tower 3 using tower 2 as temporary storage.
(1) Solving problems using recursion

To obtain the answer to a large problem, a general method is used that reduces the large problem to one or more problems of a similar nature but a smaller size.
5.1.3
(1) Solving problems using recursion
Every recursive process consists of two parts:
1. A smallest, base case that gives a solution; 2. A general method that reduces a particular case to one or more of the smaller cases, reducing the problem all the way to the base case eventually.
5.1.4
(1) Solving problems using recursion

The tower of Hanoi （汉诺塔）
不管这个传说的可信度有多大，如果考虑一下把64
片金片，由一根针上移到另一根针上，并且始终保
持上小下大的顺序。这需要多少次移动呢?
264-1=18446744073709551615 假如每秒钟一次，共需多长时间呢？
CHAPTER 5 RECURSION
Vocabulary

Recursion Factorial Suspend Stack frame
递归 阶乘 中止，挂起 堆栈结构，栈帧
The towers of hanoi
汉诺塔
Chapter Objectives

To understand how to think recursively To learn how to trace a recursive method To learn how to write recursive algorithms To understand how to use recursion to solve the Towers of Hanoi problem
Tree of function calls (函数调用树) is a tool to analyzing the process of function calls. A node(结点) denotes a function call with the function name; A function may invokes other functions, which are denoted as children; Different recursive calls appear as different vertices with the same name; Recursion tree: the recursive part of the tree of function calls.