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Fall 2007CS 2251 Recursion Chapter 7

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Fall 2007CS 2252 Chapter Objectives To understand how to think recursively To learn how to trace a recursive method To learn how to write recursive algorithms and methods for searching arrays To learn about recursive data structures and recursive methods for a LinkedList class To understand how to use recursion to –solve the Towers of Hanoi problem –to process two-dimensional images –to solve search problems such as finding a path through a maze using backtracking

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Fall 2007CS 2253 Recursive Thinking Recursion is a problem-solving approach that can be used to generate simple solutions to certain kinds of problems that would be difficult to solve in other ways –Recursion provides a different way of thinking about a problem Recursion splits a problem into one or more simpler versions of itself

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Fall 2007CS 2254 Recursive Thinking If there is one figure in the nest process the figure else process the outer figure process the nest inside the outer figure

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Fall 2007CS 2255 When to Use a Recursive Algorithm There must be at least one case (the base case), for a small value of n, that can be solved directly A problem of a given size n can be split into one or more smaller versions of the same problem (recursive case)

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Fall 2007CS 2256 Designing a Recursive Algorithm Recognize the base case and provide a solution to it Devise a strategy to split the problem into smaller versions of itself while making progress toward the base case Combine the solutions of the smaller problems in such a way as to solve the larger problem

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Fall 2007CS 2257 String Length Algorithm if the string is empty the length is 0 else length = 1 plus length of substring that excludes the first character

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Fall 2007CS 2258 Proof by induction Prove the theorem is true for the base case Show that if the theorem is assumed true for n, then it must be true for n+1 See the slides on induction

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Fall 2007CS 2259 Proving that a Recursive Method is Correct Recursive proof is similar to induction –Verify the base case is recognized and solved correctly –Verify that each recursive case makes progress towards the base case –Verify that if all smaller problems are solved correctly, then the original problem is also solved correctly

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Fall 2007CS 22510 Tracing a Recursive Method

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Fall 2007CS 22511 Recursive Definitions of Mathematical Formulas Mathematicians often use recursive definitions of formulas that lead very naturally to recursive algorithms Examples include: –Factorial –Powers –Greatest common divisor If a recursive function never reaches its base case, a stack overflow error occurs

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Fall 2007CS 22512 Recursive Factorial Method Recursive definition n! = 1 if n=0 n * (n - 1)! if n>0 Method public static int factorial( int n) { if (n==0) return 1; else return n * factorial( n-1); }

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Fall 2007CS 22513 Recursive Algorithm for x n Algorithm if n is 0 result is 1 else result is x * x n-1 Method public static double power( double x, int n) { if (n==0) return 1; else return x * power( x, n-1);

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Fall 2007CS 22514 Greatest Common Divisor Euclid's algorithm (assumes m>=n) gcd(m, n) = n if n is a divisor of m gcd(m, n) = gcd( n, m % n)otherwise Method public static double gcd( int m, int n) { if (m%n==0) return n; else if (m<n); return gcd( n, m); else return gcd( n, m%n); }

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Fall 2007CS 22515 Recursion Versus Iteration There are similarities between recursion and iteration In iteration, a loop repetition condition determines whether to repeat the loop body or exit from the loop In recursion, the condition usually tests for a base case You can always write an iterative solution to a problem that is solvable by recursion Recursive code may be simpler than an iterative algorithm and thus easier to write, read, and debug Iteration is usually more efficient

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Fall 2007CS 22516 Iterative Factorial Method public static factorialIter( int n) { int result = 1; for k=1; k<=n; k++) result *= n; return result; }

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Fall 2007CS 22517 Efficiency of Recursion Recursive methods often have slower execution times when compared to their iterative counterparts The overhead for loop repetition is smaller than the overhead for a method call and return If it is easier to conceptualize an algorithm using recursion, then you should code it as a recursive method –The reduction in efficiency does not outweigh the advantage of readable code that is easy to debug

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Fall 2007CS 22518 Recursive Fibonacci Method Recursive definition fib n = 1 for n = 1,2 fib n-1 + fib n-2 for n>2 Method public static int fibonacci( int n) { if (n<=2) return 1; else return fibonacci(n-1) + fibonacci(n-2); }

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Fall 2007CS 22519 Efficiency of Recursion: Exponential Fibonacci Inefficient

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Fall 2007CS 22520 An O(n) Recursive Fibonacci Method To avoid excessive computation, remember the previous two values. Start the computation public static int fibonacciStart( int n) { return fibo( 1, 0, n); } Recursive method private static int fibo( int current, int previous, int n) if (n==1) return current; else return fibo( current + previous, current, n-1); }

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Fall 2007CS 22521 Efficiency of Recursion: O(n) Fibonacci Efficient

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Fall 2007CS 22522 Recursive Array Search Searching an array can be accomplished using recursion Simplest way to search is a linear search –Examine one element at a time starting with the first element and ending with the last Base case for recursive search is an empty array –Result is negative one Another base case would be when the array element being examined matches the target Recursive step is to search the rest of the array, excluding the element just examined

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Fall 2007CS 22523 Algorithm for Recursive Linear Array Search

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Fall 2007CS 22524 Implementation of Recursive Linear Search

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Fall 2007CS 22525 Design of Binary Search Algorithm Binary search can be performed only on an array that has been sorted Stop cases –The array is empty –Element being examined matches the target Checks the middle element for a match with the target Throw away the half of the array that the target cannot lie within

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Fall 2007CS 22526 Binary Search Algorithm

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Fall 2007CS 22527 Trace of a Binary Search

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Fall 2007CS 22528 Implementation of Binary Search

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Fall 2007CS 22529 Implementation of Binary Search

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Fall 2007CS 22530 Efficiency of Binary Search and the Comparable Interface At each recursive call we eliminate half the array elements from consideration –O(log 2 n) Classes that implement the Comparable interface must define a compareTo method that enables its objects to be compared in a standard way –CompareTo allows one to define the ordering of elements for their own classes

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Fall 2007CS 22531 Method Arrays.binarySearch Java API class Arrays contains a binarySearch method –Can be called with sorted arrays of primitive types or with sorted arrays of objects –If the objects in the array are not mutually comparable or if the array is not sorted, the results are undefined –If there are multiple copies of the target value in the array, there is no guarantee which one will be found –Throws ClassCastException if the target is not comparable to the array elements

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Fall 2007CS 22532 Recursive Data Structures Computer scientists often encounter data structures that are defined recursively –Trees (Chapter 8) are defined recursively Linked list can be described as a recursive data structure Recursive methods provide a very natural mechanism for processing recursive data structures The first language developed for artificial intelligence research was a recursive language called LISP

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Fall 2007CS 22533 Recursive Definition of a Linked List A non-empty linked list is a collection of nodes such that each node references another linked list consisting of the nodes that follow it in the list The last node references an empty list A linked list is empty, or it contains a node, called the list head, that stores data and a reference to a linked list

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Fall 2007CS 22534 Recursive Size Method

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Fall 2007CS 22535 Recursive toString Method

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Fall 2007CS 22536 Recursive Replace Method

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Fall 2007CS 22537 Recursive Add Method

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Fall 2007CS 22538 Recursive Remove Method

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Fall 2007CS 22539 Recursive Remove Method (continued)

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Fall 2007CS 22540 Problem Solving with Recursion Will look at two problems –Towers of Hanoi –Counting cells in a blob

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Fall 2007CS 22541 Towers of Hanoi

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Fall 2007CS 22542 Towers of Hanoi (continued)

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Fall 2007CS 22543 Algorithm for Towers of Hanoi

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Fall 2007CS 22544 Algorithm for Towers of Hanoi (continued)

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Fall 2007CS 22545 Algorithm for Towers of Hanoi (continued)

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Fall 2007CS 22546 Recursive Algorithm for Towers of Hanoi

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Fall 2007CS 22547 Implementation of Recursive Towers of Hanoi

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Fall 2007CS 22548 Counting Cells in a Blob Consider how we might process an image that is presented as a two-dimensional array of color values Information in the image may come from –X-Ray –MRI –Satellite imagery –Etc. Goal is to determine the size of any area in the image that is considered abnormal because of its color values

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Fall 2007CS 22549 Classes for Counting Cells in a Blob

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Fall 2007CS 22550 Algorithm for Counting Cells in a Blob if the cell is outside the grid result is 0 if color is not abnormal result is 0 else set color to marker color result = 1 + number of cells in blob that includes a neighbor

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Fall 2007CS 22551 Implementation

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Fall 2007CS 22552 Implementation

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Fall 2007CS 22553 Counting Cells in a Blob

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Fall 2007CS 22554 Backtracking Backtracking is an approach to using trial and error in a search for a solution –Backtracking is a systematic approach to trying alternative paths and eliminating them if they don’t work For example: finding a path through a maze –Start by walking down a path as far as you can go Eventually, you will reach your destination or you won’t be able to go any farther If you can’t go any farther, you will need to retrace your steps You need a way to keep track of what you've already tried

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Fall 2007CS 22555 Backtracking Never try the exact same path more than once, and you will eventually find a solution path if one exists Problems that are solved by backtracking can be described as a set of choices made by some method Recursion allows us to implement backtracking in a relatively straightforward manner –Each activation frame is used to remember the choice that was made at that particular decision point A program that plays chess may involve some kind of backtracking algorithm

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Fall 2007CS 22556 Backtracking

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Fall 2007CS 22557 Recursive Algorithm for Finding Maze Path

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Fall 2007CS 22558 Maze Implementation

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Fall 2007CS 22559 Maze Implementation

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Fall 2007CS 22560 Maze Implementation

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Fall 2007CS 22561 Chapter Review A recursive method has a standard form To prove that a recursive algorithm is correct, you must –Verify that the base case is recognized and solved correctly –Verify that each recursive case makes progress toward the base case –Verify that if all smaller problems are solved correctly, then the original problem must also be solved correctly The run-time stack uses activation frames to keep track of argument values and return points during recursive method calls

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Fall 2007CS 22562 Chapter Review Mathematical Sequences and formulas that are defined recursively can be implemented naturally as recursive methods Recursive data structures are data structures that have a component that is the same data structure Towers of Hanoi and counting cells in a blob can both be solved with recursion Backtracking is a technique that enables you to write programs that can be used to explore different alternative paths in a search for a solution

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