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This presentation delves into advanced recursion concepts, focusing on how to break down complex problems into manageable sub-problems. Using the example of finding files in directories and the classic Towers of Hanoi puzzle, we explore recursion's principles, such as wishful thinking and base cases. Participants will learn to implement recursive solutions for both examples, including identifying parameters and using methods to move disks in the Towers of Hanoi. This interactive session is designed to enhance problem-solving skills through advanced recursion techniques.
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CS320n –Visual Programming Advanced Recursion (Slides 8-2) Thanks to Wanda Dann, Steve Cooper, and Susan Rodger for slide ideas.
What We Will Do Today • Look at using recursion to solve other forms of problems Advanced Recursion
A second form of recursion • A second form of recursion is used when the solution to a problem depends on the ability to break a problem down into smaller and smaller sub-problems. • Example • Files are stored on computers in directories. • Directories can store files and other directories (sub directories) • How do you find all the files in a directory? Advanced Recursion
Directory Structure Directories Files How deep can this go? Advanced Recursion
Finding all the Files • Break the big problem up into smaller problems • Find the “immediate” files in this directory • then go to each sub directory and find the files in it • To solve the big problem (find all the files in a directory) we solve smaller problems (find all the files in the sub directories) • The smaller problems are solved the same way as the big problem! Advanced Recursion
Another Example • Towers of Hanoi • A classic puzzle Advanced Recursion
A Towers Problem • The challenge is to move all the disks from the source cone to the target cone. • RULES: • Move 1 disk at a time • A larger disk may never be on top of a smaller disk Source Spare Target (coneFrom) (coneSpare) (coneTo) Advanced Recursion
Initial world • The disks are instances of the Torus class. (A torus is a doughnut shaped object.) • Each cone is positioned exactly 1 meter from its nearest neighbor. • Each disk is positioned exactly 0.1 meter above the disk below it. Advanced Recursion
Identifying the disks • To make it easier to describe our solution, we give each disk an id number and a name. id number name 1 disk1 2 disk2 3 disk3 4 disk4 Advanced Recursion
Solving the problem • Our solution will use the • Principle of “wishful thinking” • assume we could solve a smaller version of the same problem • if we could solve the smaller version, it would make it easier to solve this problem. • Base case – the simplest possible version of this problem, one that can obviously be solved. Advanced Recursion
Wishful thinking in practice Assume I could move 3 of the disks to the spare cone. Then I could move the 4th disk (base case) to the target cone. Advanced Recursion
Storyboard • To solve the towers problem, we need to know howmany disks we have and which cone is the source, the target, and the spare: World.towers: Parameters: howmany, source, target, spare If howmany is equal to 1 move it (the smallest disk) from the source to the target Else (1) call towers to move howmany-1 disks from source to spare (using target as spare) (2) move it (disk # howmany) from the source to the target (3) call towers to move howmany-1 disks from the spare to the target (using the source as the spare) base case – move 1 disk a smaller problem -- recursively move the rest of the disks Advanced Recursion
towers • The base case occurs when howmany equals 1, just move the disk. • Two recursive calls are used to solve the smaller problem (moving howmany-1 disks), which helps us solve the bigger problem (moving howmany disks). Advanced Recursion
Moving a disk • A challenge in this animation is how to move a disk from one tower to another. • In the towers method, we assumed that we had a method named moveIt that would accomplish the task. To write the moveIt method, we need to know: • What are the parameters to send in to our method? • What steps need to occur? • How high to raise the disk object? • How far (forward/back) to move it? Advanced Recursion
moveIt Storyboard • The parameters are: • whichdisk – the disk id number • fromcone – the source cone • tocone – the target cone • A storyboard describing the steps is: moveIt: Parameters: whichdisk, fromcone, tocone Do in order Lift the disk up above the top of the fromcone Move it (forward or back) to a location above the tocone Lower the disk down onto the tocone Advanced Recursion
Nested Ifs • The disk id number is used to determine which disk to • move up • move over • move down • This means that nested Ifs must be used three times! (The code on this slide is for just one move.) Advanced Recursion
Using an expression • We noticed that the distance each disk has to move up (and then back down) is 0.3 meters more than 0.1 * the id number (whichdisk). • We could use an expression to compute the distance for the move up and move down instructions. move the appropriate disk 0.3 + 0.1 *whichdisk Advanced Recursion
Problem • The problem with this nifty math expression is that we need to have the disk's name to write a move instruction. • For example, disk1 move up … must be an object,cannot use the id number here Advanced Recursion
Using a Function • We decided to write a function to convert the disk id number (i) to the disk name. Advanced Recursion
moveIt Advanced Recursion
Demo! Advanced Recursion
