Chapter 5. Recursion. Data Structures and Algorithms in Java
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1 Chapter 5 Recursion Data Structures and Algorithms in Java
2 Objectives Discuss the following topics: Recursive Definitions Method Calls and Recursion Implementation Anatomy of a Recursive Call Tail Recursion Nontail Recursion Indirect Recursion Data Structures and Algorithms in Java 2
3 Objectives (continued) Discuss the following topics: Nested Recursion Excessive Recursion Backtracking Case Study: A Recursive Descent Interpreter Data Structures and Algorithms in Java 3
4 Recursive Definitions Recursive definitions are programming concepts that define themselves A recursive definition consists of two parts: The anchor or ground case, the basic elements that are the building blocks of all other elements of the set Rules that allow for the construction of new objects out of basic elements or objects that have already been constructed Data Structures and Algorithms in Java 4
5 Recursive Definitions (continued) Recursive definitions serve two purposes: Generating new elements Testing whether an element belongs to a set Recursive definitions are frequently used to define functions and sequences of numbers Data Structures and Algorithms in Java 5
6 Recursive Functions The factorial function 1 if n = 0( anchor) n! = n.( n 1)! if n > 0( inductive step) Power of 2 function n 1 if n = 0 gn ( ) = 2 = 2. gn ( 1) ifn> 0 Data Structures and Algorithms in Java 6
7 Recursive Factorial Function int factorial (int num) { if (num == 0) return 1; else return num * factorial (num 1); } Data Structures and Algorithms in Java 7
8 Recursive Factorial Trace Data Structures and Algorithms in Java 8
9 Recursive Implementation: Largest Value in Array public static int largest(int list[], int lowerindex, int upperindex) { int max; if(lowerindex == upperindex) //the size of the sublist is 1 return list[lowerindex]; else { max = largest(list, lowerindex + 1, upperindex); if(list[lowerindex] >= max) return list[lowerindex]; else return max; } } Data Structures and Algorithms in Java 9
![1 return list[lowerindex]; else { max = largest(list, lowerindex + 1, upperindex);](/docs-images/50/15120200/images/page_9.jpg "if(list[lowerindex] >= max) return list[lowerindex]; else return max; } } Data Structures and")
10 Execution of largest (list, 0, 3) Data Structures and Algorithms in Java 10
11 Recursive Fibonacci public static int rfibnum(int a, int b, int n) { if(n == 1) return a; else if(n == 2) return b; else return rfibnum(a, b, n - 1) + rfibnum(a, b, n - 2); } Data Structures and Algorithms in Java 11
12 Execution of rfibonacci(2,3,5) Data Structures and Algorithms in Java 12
13 Towers of Hanoi Problem with Three Disks Data Structures and Algorithms in Java 13
14 Towers of Hanoi: Three Disk Solution Data Structures and Algorithms in Java 14
15 Towers of Hanoi: Three Disk Solution Data Structures and Algorithms in Java 15
16 Towers of Hanoi: Recursive Algorithm public static void movedisks(int count, int needle1, int needle3, int needle2) { if(count > 0) { movedisks(count - 1, needle1, needle2, needle3); System.out.println("Move disk + count + from + needle1 + to + needle3 + ". ); movedisks(count - 1, needle2, needle3, needle1); } } Data Structures and Algorithms in Java 16
17 Method Calls and Recursion Implementation Activation records contain the following: Values for all parameters to the method, location of the first cell if an array is passed or a variable is passed by reference, and copies of all other data items Local (automatic) variables that can be stored elsewhere The return address to resume control by the caller, the address of the caller s instruction immediately following the call Data Structures and Algorithms in Java 17
18 Method Calls and Recursion Implementation (continued) A dynamic link, which is a pointer to the caller s activation record The returned value for a method not declared as void Data Structures and Algorithms in Java 18
19 Method Calls and Recursion Implementation (continued) Figure 5-1 Contents of the run-time stack when main() calls method f1(), f1() calls f2(), and f2() calls f3() Data Structures and Algorithms in Java 19
20 Power Function x n 1 if n = 0 = n 1 x. x if n> 0 A Java method for x n double power { double x, int n) { if (n = = 0) return 1.0; // else return x. power (x, n-1); Data Structures and Algorithms in Java 20
21 Trace of recursive calls Static public void main ( String args[ ]) { y = power ( 5.6, 2); } Call 1 power(5.6, 2) Call 2 power(5.6,1) Call 3 power(5.6,0) Call 3 1 Call Call Data Structures and Algorithms in Java 21
22 Anatomy of a Recursive Call Figure 5-2 Changes to the run-time stack during execution of power(5.6,2) Data Structures and Algorithms in Java 22
23 Tail Recursion Tail recursion is characterized by the use of only one recursive call at the very end of a method implementation void tail (int i) { if (i > 0) { System.out.print (i + ""); tail(i-1); } } Data Structures and Algorithms in Java 23
24 Tail Recursion Example of nontail recursion: void nontail (int i) { if (i > 0) { nontail(i-1); System.out.print (i + ""); nontail(i-1); } } Data Structures and Algorithms in Java 24
25 Nontail Recursion /*200 */ void reverse( ) { /*201 */ char ch = getchar( ) ; /*202 */ if (ch!= `\n` ) { /*203 */ reverse( ) ; /*204 */ System.out.print( ch ); } } Data Structures and Algorithms in Java 25
26 Nontail Recursion (continued) Figure 5-3 Changes on the run-time stack during the execution of reverse() Data Structures and Algorithms in Java 26
27 Nontail Recursion (continued) Figure 5-4 Examples of von Koch snowflakes 1. Divide an interval side into three even parts 2. Move one-third of side in the direction specified by angle Data Structures and Algorithms in Java 27
28 Nontail Recursion (continued) 3. Turn to the right 60 (i.e., turn 60 ) and go forward one-third of side 4. Turn to the left 120 and proceed forward one-third of side 5. Turn right 60 and again draw a line one-third of side long Data Structures and Algorithms in Java 28
29 Nontail Recursion (continued) Figure 5-5 The process of drawing four sides of one segment of the von Koch snowflake Data Structures and Algorithms in Java 29
30 Nontail Recursion (continued) drawfourlines (side, level) if (level = 0) draw a line; else drawfourlines(side/3, level-1); turn left 60 ; drawfourlines(side/3, level-1); turn right 120 ; drawfourlines(side/3, level-1); turn left 60 ; drawfourlines(side/3, level-1); Data Structures and Algorithms in Java 30
31 Nontail Recursion (continued) Figure 5-6 Recursive implementation of the von Koch snowflake Data Structures and Algorithms in Java 31
32 Nontail Recursion (continued) Figure 5-6 Recursive implementation of the von Koch snowflake (continued) Data Structures and Algorithms in Java 32
33 Nontail Recursion (continued) Figure 5-6 Recursive implementation of the von Koch snowflake (continued) Data Structures and Algorithms in Java 33
34 Nontail Recursion (continued) Figure 5-6 Recursive implementation of the von Koch snowflake (continued) Data Structures and Algorithms in Java 34
35 Nontail Recursion (continued) Figure 5-6 Recursive implementation of the von Koch snowflake (continued) Data Structures and Algorithms in Java 35
36 Indirect Recursion receive() decode() store() receive() decode() receive(buffer) while buffer is not filled up if information is still incoming get a character and store it in buffer; else exit(); decode(buffer); decode(buffer) decode information in buffer; store(buffer); store(buffer) transfer information from buffer to file; receive(buffer); Data Structures and Algorithms in Java 36
37 Indirect Recursion (continued) 2 x 3 tan x 3 sin( x) = sin. 3 2 x 1+ tan 3 sin( x) tan( x) = cos( x) x cos( x) = 1 sin 2 Data Structures and Algorithms in Java 37
38 Indirect Recursion (continued) Figure 5-7 A tree of recursive calls for sin (x) Data Structures and Algorithms in Java 38
39 Nested Recursion Ackermann Function m+ 1 if n= 0 Anm (, ) = An ( 1,1) ifn> 0, m= 0 An ( 1, Anm (, 1) otherwise A A m+ 3 (3, m) = (4, m) = 2 3 Even A 16 = = (4,1) Data Structures and Algorithms in Java 39
40 Excessive Recursion Figure 5-8 The tree of calls for Fib(6) Data Structures and Algorithms in Java 40
41 Excessive Recursion (continued) Figure 5-9 Number of addition operations and number of recursive calls to calculate Fibonacci numbers Data Structures and Algorithms in Java 41
42 Excessive Recursion (continued) Figure 5-10 Comparison of iterative and recursive algorithms for calculating Fibonacci numbers Data Structures and Algorithms in Java 42
43 Backtracking Backtracking is a technique for returning to a given position (e.g., entry point) after trying other avenues that are unsuccessful in solving a particular problem Data Structures and Algorithms in Java 43
44 Backtracking (continued) Figure 5-11 The eight queens problem Data Structures and Algorithms in Java 44
45 Backtracking (continued) putqueen(row) for every position col on the same row if position col is available place the next queen in position col; if (row < 8) putqueen(row+1); else success; remove the queen from position col; Data Structures and Algorithms in Java 45
46 Backtracking (continued) Figure 5-12 A 4 x 4 chessboard Data Structures and Algorithms in Java 46
47 Backtracking (continued) Figure 5-13 Eight queens problem implementation Data Structures and Algorithms in Java 47
48 Backtracking (continued) Figure 5-13 Eight queens problem implementation (continued) Data Structures and Algorithms in Java 48
49 Backtracking (continued) Figure 5-13 Eight queens problem implementation (continued) Data Structures and Algorithms in Java 49
50 Backtracking (continued) Figure 5-14 Steps leading to the first successful configuration of four queens as found by the method putqueen() Data Structures and Algorithms in Java 50
51 Backtracking (continued) Figure 5-15 Changes in the four arrays used by method putqueen() Data Structures and Algorithms in Java 51
52 Backtracking (continued) Figure 5-16 Changes on the run-time stack for the first successful completion of putqueen() Data Structures and Algorithms in Java 52
53 Backtracking (continued) Figure 5-17 Changes to the chessboard leading to the first successful configuration Data Structures and Algorithms in Java 53
54 Backtracking (continued) Figure 5-18 Trace of calls to putqueen() to place four queens (continued) Data Structures and Algorithms in Java 54
55 Case Study: A Recursive Descent Interpreter The process of translating one executable statement at a time and immediately executing it is called interpretation Translating the entire program first and then executing it is called compilation Data Structures and Algorithms in Java 55
56 Case Study: A Recursive Descent Interpreter (continued) Figure 5-19 Diagrams of methods used by the recursive descent interpreter Data Structures and Algorithms in Java 56
57 Case Study: A Recursive Descent Interpreter (continued) Figure 5-19 Diagrams of methods used by the recursive descent interpreter (continued) Data Structures and Algorithms in Java 57
58 Case Study: A Recursive Descent Interpreter (continued) term() f1 = factor(); while current token is either / or * f2 = factor(); f1 = f1 * f2 or f1 / f2; return f1; factor() process all +s and s preceding a factor; if current token is an identifier return value assigned to the identifier; else if current token is a number return the number; else if current token is ( e = expression(); if current token is ) return e; Data Structures and Algorithms in Java 58
59 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter Data Structures and Algorithms in Java 59
60 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 60
61 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 61
62 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 62
63 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 63
64 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 64
65 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 65
66 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 66
67 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 67
68 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 68
69 Case Study: A Recursive Descent Interpreter (continued) Figure 5-20 Implementation of a simple language interpreter (continued) Data Structures and Algorithms in Java 69
70 Summary Recursive definitions are programming concepts that define themselves Recursive definitions serve two purposes: Generating new elements Testing whether an element belongs to a set Recursive definitions are frequently used to define functions and sequences of numbers Data Structures and Algorithms in Java 70
71 Summary (continued) Tail recursion is characterized by the use of only one recursive call at the very end of a method implementation. Backtracking is a technique for returning to a given position (e.g., entry point) after trying other avenues that are unsuccessful in solving a particular problem. The process of translating one executable statement at a time and immediately executing it is called interpretation. Data Structures and Algorithms in Java 71
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