Chapter 11:  Recursion

	Presentation slides for
	Java Software Solutions
	Foundations of Program Design
	Third Edition
	by John Lewis and William Loftus
	Java Software Solutions is published by Addison-Wesley
	Presentation slides are copyright 2002 by John Lewis and William Loftus. All rights reserved.
	Instructors using the textbook may use and modify these slides for pedagogical purposes.

Recursion

	Recursion is a fundamental programming technique that can provide elegant solutions certain kinds of problems
	Chapter 11 focuses on:
		thinking in a recursive manner
		programming in a recursive manner
		the correct use of recursion
		examples using recursion
		recursion in graphics

Recursive Thinking

	Recursion is a programming technique in which a method can call itself to solve a problem
	A recursive definition is one which uses the word or concept being defined in the definition itself; when defining an English word, a recursive definition usually is not helpful
	But in other situations, a recursive definition can be an appropriate way to express a concept
	Before applying recursion to programming, it is best to practice thinking recursively

Recursive Definitions

	Consider the following list of numbers:
	24, 88, 40, 37
	A list can be defined recursively
	A LIST is a:  number
	or a:  number  comma  LIST
	That is, a LIST is defined to be a single number, or a number followed by a comma followed by a LIST
	The concept of a LIST is used to define itself

Recursive Definitions

	The recursive part of the LIST definition is used several times, ultimately terminating with the non-recursive part:
	number comma LIST
	24     ,   88, 40, 37
	number comma LIST
	88     ,   40, 37
	number comma LIST
	40     ,   37
	number
	37

Infinite Recursion

	All recursive definitions must have a non-recursive part
	If they don't, there is no way to terminate the recursive path
	A definition without a non-recursive part causes infinite recursion
	This problem is similar to an infinite loop with the definition itself causing the infinite “loop”
	The non-recursive part often is called the base case

Recursive Definitions

	Mathematical formulas often are expressed recursively
	N!, for any positive integer N, is defined to be the product of all integers between 1 and N inclusive
	This definition can be expressed recursively as:
	1!  =  1
	N!  =  N * (N-1)!
	The concept of the factorial is defined in terms of another factorial until the base case of 1! is reached

Recursive Definitions

	5!
	5 * 4!
	4 * 3!
	3 * 2!
	2 * 1!
	1

Recursive Programming

	A method in Java can invoke itself;  if set up that way, it is called a recursive method
	The code of a recursive method must be structured to handle both the base case and the recursive case
	Each call to the method sets up a new execution environment, with new parameters and new local variables
	As always, when the method execution completes, control returns to the method that invoked it (which may be an earlier invocation of itself)

Recursive Programming

	Consider the problem of computing the sum of all the numbers between 1 and any positive integer N, inclusive
	This problem can be expressed recursively as:

Recursive Programming

	public int sum (int num)
	{
	int result;
	if (num == 1)
	result = 1;
	else
	result = num + sum (num - 1);
	return result;
	}

Recursive Programming

Recursion vs. Iteration

	Just because we can use recursion to solve a problem, doesn't mean we should
	For instance, we usually would not use recursion to solve the sum of 1 to N problem, because the iterative version is easier to understand;  in fact, there is a formula which is superior to both recursion and iteration!
	You must be able to determine when recursion is the correct technique to use

Recursion vs. Iteration

	Every recursive solution has a corresponding iterative solution
	For example, the sum (or the product) of the numbers between 1 and any positive integer N can be calculated with a for loop
	Recursion has the overhead of multiple method invocations
	Nevertheless, recursive solutions often are more simple and elegant than iterative solutions

Indirect Recursion

	A method invoking itself is considered to be direct recursion
	A method could invoke another method, which invokes another, etc., until eventually the original method is invoked again
	For example, method m1 could invoke m2, which invokes m3, which in turn invokes m1 again until a base case is reached
	This is called indirect recursion, and requires all the same care as direct recursion
	It is often more difficult to trace and debug

Indirect Recursion

Maze Traversal

	We can use recursion to find a path through a maze; a path can be found from any location if a path can be found from any of the location’s neighboring locations
	At each location we encounter, we mark the location as “visited” and we attempt to find a path from that location’s “unvisited” neighbors
	Recursion keeps track of the path through the maze
	The base cases are an prohibited move or arrival at the final destination

Maze Traversal

	See MazeSearch.java (page xxx)
	See Maze.java (page xxx)

MazeSearch.java

Maze.java

Towers of Hanoi

	The Towers of Hanoi is a puzzle made up of three vertical pegs and several disks that slide on the pegs
	The disks are of varying size, initially placed on one peg with the largest disk on the bottom with increasingly smaller disks on top
	The goal is to move all of the disks from one peg to another according to the following rules:
		We can move only one disk at a time
		We cannot place a larger disk on top of a smaller disk
		All disks must be on some peg except for the disk in transit between pegs

Towers of Hanoi

	(Figure 11.5 here)

Towers of Hanoi

	A solution to the three-disk Towers of Hanoi puzzle
	(Figure 11.6 here)

Towers of Hanoi

	To move a stack of N disks from the original peg to the destination peg
		move the topmost N - 1 disks from the original peg to the extra peg
		move the largest disk from the original peg to the destination peg
		move the N-1 disks from the extra peg to the destination peg
		The base case occurs when a “stack” consists of only one disk
	This recursive solution is simple and elegant even though the number of move increases exponentially as the number of disks increases
	The iterative solution to the Towers of Hanoi is much more complex

Towers of Hanoi

	See SolveTowers.java (page xxx)
	See TowersOfHanoi.java (page xxx)

SolveTowers.java

TowersOfHanoi.java

Recursion in Graphics

	Consider the task of repeatedly displaying a set of tiled images in a mosaic in which one of the tiles contains a copy of the entire collage
	The base case is reached when the area for the “remaining” tile shrinks to a certain size
	See TiledPictures.java (page xxx)

TiledPictures.java

Fractals

	A fractal is a geometric shape than can consist of the same pattern repeated in different scales and orientations
	The Koch Snowflake is a particular fractal that begins with an equilateral triangle
	To get a higher order of the fractal, the middle of each edge is replaced with two angled line segments

Fractals

	(Figure 11.7 here)
	See KochSnowflake.java (page xxx)
	See KochPanel.java (page xxx)

KochSnowflake.java

KochPanel.java

Summary

	Chapter 11 has focused on:
		thinking in a recursive manner
		programming in a recursive manner
		the correct use of recursion
		examples using recursion
		recursion in graphics