Recursion Introduction OBJECTIVES

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1 1 1 We must learn to explore all the options and possibilities that confront us in a complex and rapidly changing world. James William Fulbright O thou hast damnable iteration, and art indeed able to corrupt a saint. William Shakespeare Recursion It's a poor sort of memory that only works backwards. Lewis Carroll, Alice in Wonderland Life can only be understood backwards; but it must be lived forwards. Soren Kierkegaard Push on keep moving. Thomas Morton OBJECTIVES In this chapter you will learn: The concept of recursion. How to write and use recursive methods. How to determine the base case and recursion step in a recursive algorithm. How recursive method calls are handled by the system. The differences between recursion and iteration, and when it is appropriate to use each. What geometric shapes called fractals are and how to draw them using recursion. What recursive backtracking is and why it is an effective problem-solving technique. 1.1 Introduction 1. Recursion Concepts 1. Example Using Recursion: Factorials 1. Example Using Recursion: Fibonacci Series 1. Recursion and the Method Call Stack 1. Recursion vs. Iteration 1. Towers of Hanoi 1. Fractals 1.9 Recursive Backtracking 1. Wrap-Up 1.11 Internet and Web Resources 1.1 Introduction Earlier programs structured as methods that call one another in a disciplined, hierarchical manner Recursive methods Call themselves Useful for some problems to define a method to call itself Can be called directly or indirectly through another method Chapter Recursion examples and exercises in this book 1 Factorial Method (Fig. 1. and Fig. 1.) Fibonacci Method (Fig. 1. and Fig. 1.) Towers of Hanoi (Fig. 1.1 and Fig. 1.1) Fractals (Fig. 1.1 and Fig. 1.) What Does This Code Do? (Exercise 1., Exercise 1.1 and Exercise 1.1) Find the Error in the Following Code (Exercise 1.) Raising an Integer to an Integer Power (Exercise 1.9) Visualizing Recursion (Exercise 1.) Greatest Common Divisor (Exercise 1.11) Determine Whether a String Is a Palindrome (Exercise 1.1) Eight Queens (Exercise 1.1) Print an Array (Exercise 1.1) Print an Array Backward (Exercise 1.1) Minimum Value in an Array (Exercise 1.1) Star Fractal (Exercise 1.) Maze Traversal Using Recursive Backtracking (Exercise 1.0) Generating Mazes Randomly (Exercise 1.1) Mazes of Any Size (Exercise 1.) Time Needed to Calculate a Fibonacci Number (Exercise 1.) Fig. 1.1 Summary of the recursion examples and exercises in this text. (Part 1 of ) 1

2 1. Recursion Concepts Chapter Recursion examples and exercises in this book 1 Merge Sort (Fig. 1. and Fig. 1.11) Linear Search (Exercise 1.) Binary Search (Exercise 1.9) Quicksort (Exercise 1.) 1 Binary-Tree Insert (Fig. 1.1) Preorder Traversal of a Binary Tree (Fig. 1.1) Inorder Traversal of a Binary Tree (Fig. 1.1) Postorder Traversal of a Binary Tree (Fig. 1.1) Print a Linked List Backward (Exercise 1.0) Search a Linked List (Exercise 1.1) Recursive problem-solving elements Base case Recursive method capable of solving only simplest case the base case If method is called with base case, method returns result If method is called with more complex problem, problem divided into two pieces a piece the method knows how to do and a piece the method does not know how to do (called recursive call or recursion step) Recursive call/recursion step Must resemble original problem but be slightly simpler or smaller version Method calls fresh copy of itself to work on smaller problem Normally includes statement Indirect recursion Recursive method calls another method that eventually makes call back to recursive method Fig. 1.1 Summary of the recursion examples and exercises in this text. (Part of ) 1. Example Using Recursion: Factorials Factorial of n, or n is the product n (n 1) (n ) 1 With 1 equal to 1 and 0 Defined to be 1. Can be solved recursively or iteratively (nonrecursively) Recursive solution uses following relationship: n n (n 1) Infinite recursion recursive calls are continuously made until memory has been exhausted Caused by either omitting base case or writing recursion step that does not converge on base case 9 Fig. 1. Recursive evaluation of Base case returns 1 9 "# %&#&# Recursion step breaks problem into two parts: one the 11 1 ' ( method knows how to do, one the method does not 1 ) ) 1 1 %(% Portion method Recursive knows how call: to Portion do method does not know how to 1 * do; smaller version of original problem 1 1 %% # #% "# "#% % ,* -.&#./ ) * ) Original call to recursive method Common Programming Error 1.1 Either omitting the base case or writing the recursion step incorrectly so that it does not converge on the base case can cause a logic error known as infinite recursion, where recursive calls are continuously made until memory has been exhausted. This error is analogous to the problem of an infinite loop in an iterative (nonrecursive) solution. 1

3 1 1 %(%, 9 # Calculate and display factorials * 11 ) 1 ) %&# &# &# &# 1&#1 &#% &#% &#%1% 9&#1%% :&#99% %&#99%% 1 1. Example Using Recursion: Fibonacci Series Fibonacci series begins with 0 and 1 and has property that each subsequent Fibonacci number is the sum of previous two Fibonacci numbers. Series occurs in nature, ratio of successive Fibonacci numbers converges on golden ratio or golden mean Fibonacci series defined recursively as: fibonacci(0) 0 fibonacci(1) 1 fibonacci(n) fibonacci(n 1) + fibonacci(n ) Recursive solution for calculating Fibonacci values results in explosion of recursive method calls 1 1 Two base cases 9 ##%;; ;; ## Two recursive calls 11 1 (+ + ( 1 ) ) 1 1 * 1 1 # #%"# "#% % ,* -../ ) * 1 ) Original call to recursive method 1 1, # Calculate and display Fibonacci values 9 * ) 11 ) % % : 1 % Performance Tip 1.1 Avoid Fibonacci-style recursive programs, because they result in an exponential explosion of method calls. Fig. 1. Set of recursive calls for fibonacci( ).

4 1. Recursion and the Method Call Stack 0 Method call stack used to keep track of method calls and local variables within a method call Just as with nonrecursive programming, recursive method calls are placed at the top of the method call stack As recursive method calls return, their activation records are popped off the stack and the previous recursive calls continue executing Current method executing is always method whose activation record is at top of stack Fig. 1. Method calls made within the call fibonacci( ) Recursion vs. Iteration Fig. 1.9 Method calls on the program execution stack. Any problem that can be solved recursively can be solved iteratively Both iteration and recursion use a control statement Iteration uses a repetition statement Recursion uses a selection statement Iteration and recursion both involve a termination test Iteration terminates when the loop-continuation condition fails Recursion terminates when a base case is reached Recursion can be expensive in terms of processor time and memory space, but usually provides a more intuitive solution 1 % < 9 # # Iterative solution uses counter-controlled repetition 11 1 # # 1 '# ) 1 1 %(% * 0 1 %% # #%"# "#% %++ ++,* -.&#. -.&#./- -00 ) * ) 1 %(%, 9 # * 11 ) 1 ) %&# &# &# &# 1&#1 &#% &#% &#%1% 9&#1%% :&#99% %&#99%%

5 Software Engineering Observation 1.1 Any problem that can be solved recursively can also be solved iteratively (nonrecursively). A recursive approach is normally preferred over an iterative approach when the recursive approach more naturally mirrors the problem and results in a program that is easier to understand and debug. A recursive approach can often be implemented with fewer lines of code. Another reason to choose a recursive approach is that an iterative one might not be apparent. Performance Tip 1. Avoid using recursion in situations requiring high performance. Recursive calls take time and consume additional memory. Common Programming Error 1. Accidentally having a nonrecursive method call itself either directly or indirectly through another method can cause infinite recursion. 1. Towers of Hanoi Classic problem Priests in Far East are attempting to move a stack of disks from one peg to another. One disk must be moved at a time, at no time may a larger disk be placed above a smaller disk Recursive solution: Move n 1 disks from peg 1 to peg, using peg as temporary holding area Move the last disk (the largest) from peg 1 to peg Move the n 1 disks from peg to peg, using peg 1 as a temporary holding area Base case: When only one disk needs to be moved no temporary holding area needed, disk is simply moved >? A B B 9 >? B 11 A B# B 1 )>? 1 1 * B @ 1 1 * B B## Base case: Simply display move 0 1,* ) ) 0 Fig. 1.1 Towers of Hanoi for the case with four disks.

B@0 @ B( B@@ * B(0@0@0 @ Move last disk from peg 1 to peg 9 B@ @ 0,* -/...-.-0@0 Move n-1 disks @ from Use peg peg 1 to peg as temporary 1 holding area B( B@ @ B(0@0 @0@ ) ) )>?

6 @ B( B@@ * Move last disk from peg 1 to peg 9 0,* -/ @0 Move n-1 from Use peg peg 1 to peg as temporary 1 holding area B( ) ) )>? Move n-1 disks from Use peg peg to 1 peg as temporary holding area A B# B 1 >? >? # >? A B 1 Make initial call to recursive method 1 B 1 >? A B0@0@0@ 1 ) 1 )>? 1. Fractals 1. Fractals Fractal a geometric figure that often can be generated from a pattern repeated recursively an infinite number of times Pattern applied to each segment of original figure Benoit Mandelbrot introduced term fractal, along with specifics of how fractals are created and their practical applications Help us better understand patterns in nature, the human body and the universe Popular art form Self-similar property fractals have this property in the case that, when subdivided into parts, each resembles a reduced-size copy of the whole If part is exact copy of original, fractal is said to be strictly self similar Each time pattern is applied, fractal is said to be at new level or depth Fractal examples: Koch Curve, Koch Snowflake (a) (b) (c) (d) (e) (f) Fig. 1.1 Koch Curve fractal. Fig. 1.1 Lo fractal at level 0.

7 Fig. 1.1 Determining points C and D for level 1 of Lo fractal. Fig. 1.1 Lo fractal at level 1, with C and D points determined for level. [Note: The fractal at level 0 is included as a dashed line as a reminder of where the line was located in relation to the current fractal.] 9 0 Fig. 1. Lo fractal at level, with dashed lines from level 1 provided. Fig. 1.0 Lo fractal at level. 1 A C* C* D E D C F G F GH 9 F GC F G@ 11 F G 1 1 F FG G 1 1 I<A? I<A?# #1%% 1%% JK< 1?E<J??E<J?# #% % JK< 1 L<MNCEOEC L<MNCEOEC# #% 1 # #HCKE HCKE 0 GHGH0 CGH0 1 CGH GCGC GCGC G@, G@, G@ G@0G@ 1 1 G@# G@ G@C* C* GH# GH -- G@ G@ GH GHD C 9 D C * 0 1 D E #GA ## 9 #HCKE HCKE 0 1, )* D C JK<

8 CGH# GH -AC- 9 G@ CGH 0 CGHD D C 1 D C * D E #,C * Retrieve current level Decrease level 0 1 CGH# GH -<C- -<C- G@ CGH CGHD C D C * D E 9 90 #,C * 9 Retrieve current level Increase level 9 * 0 #L<MNCEOEC L<MNCEOEC 1 GCF -C -C- -+ +,C ) ) * D C 9 Set new level Redraw fractal up to new level 9 * 9 #L<MNCEOEC L<MNCEOEC 9 9 GCF -C ,C 9 99 ) 0 1 ) * * D C Set new level Redraw fractal up to new level GCG@ GC# GC -C%- -C%- G@ GC,# G@ % % 9 1 G@ G@, 111 G@# G@ 11 G@ G@ 11 G@, 11 1 G@ G@ J A F G@ 11 G@G@G 11 G@ G@ F FG@ G@ 9 11, P I<A?0 0?E<J??E<J? PG 11 O *G 1 ) 11, 1 1 # 1 A> GEQ<N>MNC>,E GEQ<N>MNC>,E 1 ) 1 ) I<A? I<A?# #1%% 1%% G@ 1?E<J?E<J??# #1%% 1%% G@ G@ 1 G@ C C 0 #HCKE HCKE P 1 #C HB I?<E P A I<A? I<A?0 0?E<J??E<J? )G@ * 0 FD0 *D0 FH0 FH0 *H0J 9 Coordinates of first point for line 0 Coordinates 1 ##% where fractal of is second being point applied for line where fractal is being applied C FD0*D0FH0*H Base case: Simply draw line, pattern is 0F not applied FD0*D FH0*H F# FD+FH Recursion step: Apply fractal pattern Calculate midpoint *# *D+*H 9 FA0*A 0 FD0*D F0* 1 FA0*A FA#FD+ F(FD FD( *(*D *D *A#*D+ *(*D *D + + F(FD FD * (0FA0*A0FD0*D0 0FA0*A0FD0*D0 (0FA0*A0F0*0 0FA0*A0F0*0 (0FA0*A0FH0*H0 0FA0*A0FH0*H0 9 ) ) 0 ) ) 1 Calculate point to form right triangle Apply pattern to three new lines J 9 0%% %%0 0:% :%0 0:% :%0 0%% %%0 0 0 ) 1 # ) 9 C C C 0 1 #C )C Make first call to recursive method whenever window is repainted

9 C )C 9 )G@ Recursive Backtracking Recursive Backtracking process of using recursion to return to earlier decision point If one set of recursive calls does not result in solution, program backs up to previous decision point and makes different decision, often resulting in another set of recursive calls Examples Maze problem Eight-Queens problem 9

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142 Scope Rules Thus, storage duration does not affect the scope of an identifier. The only identifiers with function-prototype scope are those used in the parameter list of a function prototype. As mentioned