Chapter 2 Recursion Objectives Upon completion you will be able to: Explain the difference between iteration and recursion Design a recursive algorithm Determine when an recursion is an appropriate solution Write simple recursive functions 1
How to writing repetitive algorithms Iteration Recursion Is a repetitive process in which an algorithm calls itself. 2
2-1 Factorial - A Case Study We begin the discussion of recursion with a case study and use it to define the concept. This section also presents an iterative and a recursive solution to the factorial algorithm. Recursive Defined Recursive Solution 3
Iterative The definition involves only the algorithm parameter(s) and not the algorithm itself. 4
Recursive A repetitive algorithm use recursion whenever the algorithm appears within the definition itself. 5
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Note that Recursion is a repetitive process in which an algorithm called itself. 7
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Which code is simpler? Which one has not a loop? 10
Calling a recursive algorithm 11
2-2 Designing Recursive Algorithms In this section we present an analytical approach to designing recursive algorithms. We also discuss algorithm designs that are not well suited to recursion. The Design Methodology Limitation of Recusion Design Implemenation 12
The Design Methodology Every recursive call either solves a part of the problem or it reduce the size of the problem. 13
The Design Methodology Base case The statement that solves the problem. Every recursive algorithm must have a base case. General case The rest of the algorithm Contains the logic needed to reduce the size of the problem. 14
Rules for designing a recursive algorithm 1. Determine the base case. 2. Determine the general case. 3. Combine the base case and the general cases into an algorithm. 15
Combine the base case and the general cases into an algorithm Each call must reduce the size of the problem and move it toward the base case. The base case, when reached, must terminate without a call to the recursive algorithms; that is, it must execute a return. 16
Limitations of recursion You should not use recursion if the answer to any of the following questions is no: 1. Is the algorithm or data structure naturally suited to recursion? 2. Is the recursive solution shorter and more understandable? 3. Does the recursive solution run within acceptable time and space limits? 17
Design implementation reverse keyboard input 18
data=6 請注意 print data 在什麼時候執行 data=20 data=14 data=5 19
2-3 Recursive Examples Four recursive programs are developed and analyzed. Only one, the Towers of Hanoi, turns out to be a good application for recursion. Greatest Common Divisor Fiboncci Numbers Prefix to Postfix Conversion The Towers of Honoi Slide 05 20
GCD design a if b 0 gcd( a, b) b if a 0 gcd( b, a mod b) otherwise Greatest Common Divisor Recursive Definition 21
Pseudocode 22
GCD C implementation 23
gcd(10, 25) gcd(25, 10) gcd(10, 5) gcd(5, 0) 24
Another G.C.D. Recursive Definition 25
Fibonacci numbers Mentioned a problem FIBONACCI: A pair of rabbits, a month later able to produce a pair of rabbits, and newborn rabbit have in a month after fertility, but also gave birth to rabbits. Start from a pair of rabbits, a year later there will be a number of rabbits? 26
Fibonacci numbers 27
Fibonacci numbers - Each number is the sum of the previous two numbers. The first few numbers in the Fibonacci series are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 28
Fibonacci numbers 29
Fibonacci numbers 30
Fibonacci numbers an example 31
0 represents.t. 32
(Continued) 33
Analysis 34
We omit Prefix to Postfix Conversion The Towers of Honoi 35
HW2 Write a recursive algorithm to calculate the combination of n objects taken k at a time. Due date : ( 甲 :941012 乙 :941012) 36
2-3 Recursive Examples Four recursive programs are developed and analyzed. Only one, the Towers of Hanoi, turns out to be a good application for recursion. The Towers of Honoi Slide 05 37
Towers of Hanoi Problem: Invented by French mathematician Lucas in 1880s. Original problem set in India in a holy place called Benares. There are 3 diamond needles fixed on a brass plate. One needle contains 64 pure gold disks. Largest resting on the brass plate, other disks of decreasing diameters. Called tower of Brahma. 38
Towers of Hanoi Problem: Priests are supposed to transfer the disks from one needle to the other such that at no time a disk of larger diameter should sit on a disk of smaller diameter. Only one disk can be moved at a time. Later setting shifted to Honoi, but the puzzle and legend remain the same. How much time would it take? Estimate.. 39
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Towers of Hanoi Problem: Today we know that we need to have 2 64-1 41
Recursive Towers of Hanoi Design Find a pattern of moves. Case 1: move one disk from source to destination needle. 42
Move two disks Case 2: Move one disk to auxiliary needle. Move one disk to destination needle. Move one disk to from auxiliary to destination needle. 43
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Move three disks Case 3: Move two disks from source to auxiliary needle. Move one disk from source to destination needle. Move two disks from auxiliary to destination needle. 45
(Continued) Move two disks from source to auxiliary needle. Move two disks from auxiliary to destination needle. 46
Algorithm Tower of Hanoi Towers (numdisks, source, dest, auxiliary) numdisks is number of disks to be moved source is the source tower dest is the destination tower auxiliary is the auxiliary tower 47
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Generalize Tower of Hanoi General case Move n-1 disks from source to auxiliary. Base case Move one disk from source to destination. General case Move n-1 disks from auxiliary to destination. Data Structures: A Pseudocode Approach with C 49
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