APCS-AB: Java Recursion in Java December 12, 2005 week14 1
Check point Double Linked List - extra project grade Must turn in today MBCS - Chapter 1 Installation Exercises Analysis Questions week14 2
Scheme We started off the semester with Scheme what do you guys remember about the way that we solved problems with Scheme? What was a list in Scheme? Was it really just the LinkedList that you guys spent a week coding in Java? Scheme was functional, and we solved programs recursively How can we apply that kind of problem-solving to Java? Can using recursion improve our Java problems? Why didn t we think to use recursion with the LinkedList in Java? week14 3
Recursion in Java We can just as easily do recursion with Java, it s just that for some reason, with OO programming students tend to default to thinking iteratively. However, some problems can t be easily solved iteratively, but recursion gives you a very elegant approach to solve these problems We ll start with the Classics - some simple mathematical problems and the Towers of Hanoi This will give you some easier problems on which to practice thinking recursively again Then next semester we ll move to using recursion to program trees and faster sorting in order to show off recursion in the real world week14 4
Why use recursion? Sometimes a problem can be overwhelming to solve all at once The nice thing about recursion is that it inherently chops a larger problem into smaller ones Especially useful with lists and numbers which can be arbitrarily large Sometimes it can be hard to prove that you have solved a problem correctly In math there s proof by induction (which you may or may not have seen before) So, recursion is just applying those same principles to problem solving with computers Handle the simple cases directly Deal with the general case by breaking it down into operations involving simpler cases (which by induction, we assume we can handle) week14 5
A brief introduction to Proof by induction Prove 1+2+3+4+5+6+.+n = Base Case: Show true for n = 1: 1 = 1(2) 2 =1 n(n +1) 2 Induction Hypothesis (IHOP): Assume claim is true for n Induction Step -- Goal: Show true for n+1 1+ 2 +...+ n + (n +1) = (n +1)((n +1) +1) 2 week14 6
The proof part Start with the original equation, which by the IHOP we assume is true, and add (n+1) to both sides: n(n +1) 1+ 2 +...+ n + (n +1) = + (n +1) 2 n(n +1) 2(n +1) 1+ 2 +...+ n + (n +1) = + 2 2 1+ 2 +...+ n + (n +1) = n 2 + 3n + 2 1+ 2 +...+ n + (n +1) = 2 (n +1)(n + 2) 2 QED week14 7
Why use recursion? Provides an Elegant Solution Translates easily from thought [algorithm] to program Often allows for very short solutions Models some things in the real world much better than alternatives number sequences math functions data structure definitions (eg linked lists) Remember how it came easily in Scheme?? data structure manipulations (eg searching) language definitions week14 8
Recursion: A Reminder Defining something in terms of itself For example defined recursively, a list of numbers is: A (single number) - or A (number, list of numbers) 3 5, {4,5,6,7} Notice with recursion, you will ALWAYS have different cases The Base definition(s) The General definition week14 9
Recursion in Programming Recursive algorithm A solution that is expressed in terms of: a base case (1 or more) Simpler/smaller instances of the same algorithm Note: recursion depends on the fact that the algorithm will eventually call a base case every time it is run The algorithm itself usually depends on applying the right base case (note that you CAN have more than one) What else would happen if you didn t call the base case? It would be infinite recursion and your program would never stop week14 10
Recursion in Math Many mathematical formulas can be expressed recursively. Can you think of/remember any? Factorial 1! = 1 Fact(1) = 1 N! = N*(N-1)! Fact(n) = n*fact(n-1) 5! = 5 * 4! = 5 * 4*3! = 5*4*3*2! = 5*4*3*2*1! = 5*4*3*2*1 Fibonacci numbers Fib(0) = 0 Fib(1) = 1 Fib(n) = Fib(n-1) + Fib(n-2) week14 11
Tracing an Example public int sum(int num){ int result; if(num == 1) result = 1; else result = num + sum(num-1); return result; } week14 12
How the trace works returns 10 sum (4) result = 4 + sum(3) returns 6 sum (3) result = 3 + sum(2) returns 3 sum (2) result = 2 + sum(1) returns 1 sum (1) result = 1 week14 13
How Recursion Works How does the computer keep everything straight: Specifically, how come local variables don't get trashed? The answer is that whenever a recursive method is called, the following will occur: Its local variables are created from unused memory The new recursive call only sees these new memory locations The previous call still is pointing at the other memory locations This process repeats until you hit the base case When each method ends, its variables are deleted from memory the memory become unused again week14 14
Dangers of Recursion Memory Think about how we described the inner workings of recursion The parameter(s) of your recursive method could have tens or thousands of copies Execution Time Executing a method is far slower than running a typical iteration of a loop Because you have to copy variables back and forth and do a variety of other activities So iteration will usually be faster than recursion when executing So often you are trading coding performance for execution performance You can use this fact to get an algorithm coded and tested, and then go in to re-write it iteratively if you need better performance week14 15
Searching and Sorting Searching is one of the most fundamental problems in computer science, and one that is often used Can be implemented by a simple recursive algorithm One kind of search is particularly recursive: binary search Sorting is another task that comes up often in computer science, and some of the best algorithms can be easily implemented recursively Merge Sort Quick Sort We will look at these algorithms soon week14 16
Lab Time: Simple Practice Using recursion write a method to: Calculate the factorial of a number Calculate Fibonacci Numbers of an input Calculate the first perfect square that is equal to or smaller than a number Calculate x to the y power Determine if a number is prime week14 17
Homework Write a program to determine and print the nth line of Pascal s triangle The first line of Pascal s triangle is the single number 1 The second line contains 2 numbers, each a 1 The third line contains 3 numbers. Each of the numbers contains the sum of the numbers (either 1 or two) above it, so it would be 1 2 1 Thus the fourth line would then be 1 3 3 1 Hint: use an array to store the values on each line week14 18
APCS-AB: Java Recursion in Java December 13, 2005 week14 19
Checkpoint DoubleLinkedList - make sure its turned in Recursion Lab and Homework How was the lab? Did we get all the problems Show me your Pascal s Triangle homework for a grade Today: Lecture Featuring: Towers of Hanoi, Mazes and N-Queens Lab: Towers, Mini-Project: N-Queens week14 20
Introducing: The Towers of Hanoi Time for the fun to begin! week14 21
Towers of Hanoi: A Classic in CompSci The Towers of Hanoi puzzle was invented by the French mathematician Edouard Lucas in 1883. There is a legend about an Indian temple which contains a large room with three time-worn posts in it surrounded by 64 golden discs. The priests of Brahma, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the rules of the puzzle. According to the legend, when the last move of the puzzle is completed, the world will end. The puzzle is therefore also known as the Tower of Brahma puzzle. It is not clear whether Lucas invented this legend or was inspired by it. (From Wikipedia) Every computer science student encounters this puzzle at some point - because it is one of the classic recursion and stack problems. And the solution is quite elegant! week14 22
Towers of Hanoi The puzzle is this: There are 3 pegs, resting on these pegs are 64 discs. None of the 64 discs are the same size; in fact, disc 1 is slightly larger in diameter than disc 2, which is slightly larger than disc 3, which is slightly larger than disc 4, etc. The initial configuration of the puzzle has all 64 discs piled in order of size on the first peg with the largest disc on the bottom. The goal is to move all the discs onto the third peg. A large disc must never be placed on a smaller disc and only one disc can be moved at a time. Should the world have ended if the legend is true? How long will it take to move those 64 discs?? week14 23
Using Recursion Given that I just told you Towers of Hanoi can be solved with recursion, how would you do it? I have toys to help! Remember: To formulate a recursive algorithm: Think about the base cases, the small cases where the problem is simple enough to solve. Think about the general case, which you can solve if you can solve the smaller cases. How would we would do this with the Towers? What is the smallest case?? week14 24
Backtracking using Recursion Backtracking is a way to solve hard search problems Go down one path towards a solution, but undo steps if you take a wrong turn Two classics Solving Mazes Placing N-Queens on an N x N board So how do we do this? And where can we use recursion? How do we backtrack with recursion? week14 25
Mazes with recursion are amaze-ing Let s say we had a maze that was represented by a 2D int array with 1 indicating a clear path and 0 a blocked path - How could we use recursion to help solve a maze like: 1110110001111 1011101111001 0000101010100 1110111010111 1010000111001 1011111101111 1000000000000 1111111111111 Top-left is entry point Bottom-right is exit point You can only move down, right, up and left week14 26
Maze Code (Class Handout) How does this work? What happens? For each starting position, we are looking for a solution to the end. (Each problem is a smaller subset of the original problem) When we have a valid move, we try it (and mark it tried so that we don t retrace steps later) We stop recursion for a given path for one of 3 different cases: If the move is invalid because it is out of bounds If the move is invalid because it has been tried before If we have reached our final location (bottom-right) week14 27
The Solution We backtrack each time we try a path that doesn t find a solution (to the bottom-right corner) All the 3 s in the output are places we tried that we had to backtrack from If we find a solution, the grid entry is changed to 7 (the first 7 set is the bottom right corner) 7770110001111 3077707771001 0000707070300 7770777070333 7070000773003 7077777703333 7000000000000 7777777777777 Do we understand how this works?? week14 28
N-Queens N-Queens on an nxn board How can we put n queens on an board so that no two queens attack each other? No Queens are on the same: Row Column Diagonal week14 29
Queens Demo week14 30
Lab & Mini-project Lab: Towers of Hanoi Work on in in class today and tomorrow Mini-project: 8-Queens - for Thursday Solve it for the 8 x 8 case, using code provided The general case is the same, but 8 x 8 definitely has a solution week14 31