Fall 2007 CS 225 1

Recursion

Chapter 7

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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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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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Recursive Thinking

If there is one figure in the nestprocess the figure

elseprocess the outer figureprocess the nest inside the outer figure

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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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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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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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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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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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Tracing a Recursive Method

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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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Recursive Factorial Method

• Recursive definitionn! = 1 if n=0 n * (n - 1)! if n>0

• Methodpublic static int factorial( int n) { if (n==0) return 1; else return n * factorial( n-1);}

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Recursive Algorithm for xn

• Algorithmif n is 0 result is 1else result is x * xn-1

• Methodpublic static double power( double x, int n) { if (n==0) return 1; else return x * power( x, n-1);

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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

• Methodpublic 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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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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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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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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Recursive Fibonacci Method

• Recursive definitionfibn = 1 for n = 1,2

fibn-1 + fibn-2 for n>2

• Methodpublic static int fibonacci( int n) { if (n<=2) return 1; else return fibonacci(n-1) + fibonacci(n-2);

}

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Efficiency of Recursion: Exponential Fibonacci

Inefficient

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An O(n) Recursive Fibonacci Method

• To avoid excessive computation, remember the previous two values.

• Start the computationpublic 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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Efficiency of Recursion: O(n) Fibonacci

Efficient

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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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Algorithm for Recursive Linear Array Search

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Implementation of Recursive Linear Search

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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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Binary Search Algorithm

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Trace of a Binary Search

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Implementation of Binary Search

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Implementation of Binary Search

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Efficiency of Binary Search and the Comparable Interface

• At each recursive call we eliminate half the array elements from consideration– O(log2n)

• 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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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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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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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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Recursive Size Method

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Recursive toString Method

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Recursive Replace Method

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Recursive Add Method

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Recursive Remove Method

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Recursive Remove Method (continued)

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Problem Solving with Recursion

• Will look at two problems– Towers of Hanoi– Counting cells in a blob

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Towers of Hanoi

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Towers of Hanoi (continued)

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Algorithm for Towers of Hanoi

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Algorithm for Towers of Hanoi (continued)

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Algorithm for Towers of Hanoi (continued)

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Recursive Algorithm for Towers of Hanoi

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Implementation of Recursive Towers of Hanoi

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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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Classes for Counting Cells in a Blob

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Algorithm for Counting Cells in a Blob

if the cell is outside the grid result is 0if color is not abnormal result is 0else

set color to marker color result = 1 + number of cells in blob that includes a neighbor

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Implementation

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Implementation

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Counting Cells in a Blob

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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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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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Backtracking

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Recursive Algorithm for Finding Maze Path

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Maze Implementation

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Maze Implementation

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Maze Implementation

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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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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