Transcript Koffman_Ch07

Recursion
Chapter 7
Chapter Objectives
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To understand how to think recursively
To learn how to trace a recursive method
Show how recursion is used in math formulas
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
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Chapter Objectives (continued)
• To understand how to use recursion to process twodimensional images
• To learn how to apply backtracking to solve search
problems such as finding a path through a maze
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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 splits a problem into one or more simpler
versions of itself
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Recursive Thinking
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Recursive Algorithm
• A given problem is a candidate for recursive solution if
you can define a base case and a recursive case
• Base case: There is at least one case, for a small
value of n, that can be solved directly
• Recursive case: A problem of a given size n can be
split into one or more smaller versions of the same
problem
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Steps to Design 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
• Smaller versions must progress toward base case
• Then to solve the original problem…
• …combine the solutions of the smaller (or smallest)
problems in such a way as to solve the problem
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Recursive Algorithm to Search Array
• Given: Array of n elements, sorted in increasing order
• Replace problem of searching n elements by problem of
searching n/2 elements
• To find target element…
• Look at element in the middle
• If middle element equals target, then done
• Else if target is less than middle element
• Repeat search on first half
• Else (target is greater than middle element)
• Repeat search on second half
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Recursive Algorithm to Search Array
• Suppose we search for “7” in this array of 25 elements:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89
,97
• Middle element is 41
• Since 7 is less than 41, repeat search using 1st half:
2,3,5,7,11,13,17,19,23,29,31,37
• Middle element is 13
• Since 7 is less then 13, repeat using 1st half:
2,3,5,7,11
• Middle element is 5
• Since 7 is greater than 5, repeat using 2nd half
7,11
• And so on…
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Recursive Algorithm to Search Array
• If array is empty
• Return -1
• Else if middle element matches target
• Return subscript of element
• Else if target is less than middle element
• Recursively search first half of array
• Else
• Recursively search second half of array
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General Recursive algorithm
• If the problem can be solved for current value of n
• Solve it (base case)
• Else
• Recursively apply the algorithm to one or more
problems with smaller value(s) of n (recursive case)
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Steps to Design Recursive Algorithm
• In general…
• A base case (small n) that can be solved directly
• Problem of size n can be split into smaller version(s)
of the same problem (recursive case)
• To design a recursive algorithm, we must
• Recognize base case, and provide solution to it
• Devise a strategy to split problem into smaller
versions of itself
• Combine solutions to smaller problems so that large
problem is solved correctly
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Recursive String Length Algorithm
• How to find length of a string?
• Lots of ways to do this
• How about a recursive solution?
• Length of empty string is 0
• Length of non-empty string is length of first character
plus length of the “rest of the string”
• That is, 1 + length of rest of the string
• For example, length of “abcd” is 1 + length of “bcd”
• And so on…
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Recursive String Length Algorithm
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Recursive String Length Algorithm
• In Java
public static int length(String str) {
if (str == null || str.equals(""))
return 0;
else
return 1 + length(str.substring(1));
}
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Run-Time Stack and Activation Frames
• “Activation frame” pushed onto run-time stack
• Run-time stack is like “scratch paper”
• OS can save information (keep state)
• For later use
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Tracing Recursive String Length Method
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Proving that a Recursive Method is Correct
• 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
• Proof of recursion is similar to induction
• Verify that the base case is solved correctly
• Verify that each recursive case progresses 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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Prove Recursive String Length is Correct
• Base case?
• Empty string is of length 0
• Recursive case makes progress towards base case?
• String gets smaller by 1 each time
• Show that if smaller problem solved correctly, then
original problem is solved correctly
• Smaller problem: length(str.substring(1))
• If small problem correct, then length of original string
is correct: 1 + length(str.substring(1))
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Recursive Math Formulas
• Mathematics has many naturally recursive formulas
• Examples include:
• Factorial
• Powers
• Greatest common divisor
• Note that if recursive problem is too big, a stack overflow
error will occur
• Space available for run-time stack is limited
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Factorial
• How do you pronounce “n!” ?
• As you know, n! = n  (n-1)  (n-2)  …  1
• And 0! = 1
• Recursive definition?
• We have: n! = n  (n-1)!
• Can easily write a recursive method for factorial…
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Recursive Factorial Method
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Exponentiation
• If n is a non-negative integer, xn is x times itself, n times
• And x0 = 1
• For example, 27 = 2  2  2  2  2  2  2 = 128
• Recursive definition?
• We have: xn = x  xn-1
• Can easily write recursive method…
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Recursive Exponentiation Method
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Recursion Versus Iteration
• What is iteration?
• A loop repetition condition determines whether to
repeat the loop body or exit from the loop
• In recursion, the condition tests for a base case
• There are similarities between recursion and iteration
• You can always write an iterative solution to a
problem that is solvable by recursion
• You are probably more familiar with iteration
• So, why bother with recursion?
• Recursive code often simpler than an iterative
algorithm and thus easier to write, read, and debug
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Tail Recursion
• “Tail recursion” or “last-line recursion”
• Single recursive call, and it is last line of the method
• All of the examples we have considered so far are
examples of tail recursion
• Easy to convert tail recursive method to iterative method
• Example on the next slide…
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Iterative Factorial Method
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Efficiency of Recursion
• Is recursion more or less efficient than iteration?
• Recursive methods often slower than iteration: Why?
• The overhead for loop repetition is smaller than the
overhead for a method call and return
• Recall, run-time stack
• 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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Fibonacci Numbers
• Fibonacci numbers developed to model rabbit population
• Defined as
fib1 = 1, fib2 = 1, fibn = fibn-1 + fibn-2
• The first several Fibonacci numbers are
1,1,2,3,5,8,13,21,34,55,89,144,233,377,…
• Easy to write a recursive method for Fibonacci numbers
• See next slide…
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Inefficient Recursive Fibonacci Method
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Inefficient Recursive Fibonacci Method
• Why is the obvious Fibonacci recursion inefficient?
• Consider fibonacci(7)
• Compute fibonacci(6) and fibonacci(5)
• Then fibonacci(5), fibonacci(4), fibonacci(4)
fibonacci(3)
• Then fibonacci(4), fibonacci(3), fibonacci(3),
fibonacci(2), fibonacci(3), fibonacci(2), fibonacci(2),
fibonacci(1)
• And so on…
• Why is this inefficient?
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Inefficient Recursive Fibonacci
• How inefficient is this?
• Exponential time!!!
• If n = 100, need about 2100 activation frames
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Efficient Recursive Fibonacci Method
• An O(n) Fibonacci algorithm…
• If we know current and previous Fibonacci numbers
• Then next Fibonacci number is current + previous
• Method fibo
• 1st argument is current Fibonacci (fibCurrent)
• 2nd argument is previous Fibonacci (fibPrevious)
• 3rd argument is n
• When n = 1, base case, so return fibCurrent
• Otherwise, return
fibo(fibCurrent + fibPrevious, fibCurrent, n - 1)
• Start by computing: fibo(1, 0, n)
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Efficient Recursive Fibonacci Method
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Efficient Recursive Fibonacci Method
How efficient?
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Dynamic Programming
• We want to find “best” stagecoach route from A to J
• Where ”best” == shortest distance
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Dynamic Programming
• Let F(X) be shortest distance from A to X
• Note that, for example,
F(J) = min{F(H) + 3, F(I) + 4}
• This provides a recursive method to find F(J)…
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Dynamic Programming
• F(A) = 0
0
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Dynamic Programming
• F(B) = 2
• F(C) = 4
• F(D) = 3
2
4
0
3
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Dynamic Programming
• F(E) = min{F(B) + 7, F(C) + 3, F(D) + 4} = min{9,7,7} = 7
• F(F) = min{F(B) + 4, F(C) + 2, F(D) + 1} = min{6,6,4} = 4
• F(G) = min{F(B) + 6, F(C) + 4, F(D) + 5} = min{8,8,8} = 8
2
4
0
3
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Dynamic Programming
• F(H) = min{F(E) + 1, F(F) + 6, F(G) + 3} = min{8,10,11} = 8
• F(I) = min{F(E) + 4, F(F) + 3, F(G) + 3} = min{11,7,11} = 7
2
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8
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0
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7
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Dynamic Programming
• F(J) = min{F(H) + 3, F(I) + 4} = min{11,11} = 11
2
7
8
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0
11
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7
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Dynamic Programming
• The shortest path(s) from A to J have distance 11
• In this example, the shortest path is not unique
• How to find best path?
• As opposed to shortest distance
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11
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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 a Binary Search Algorithm
• Binary search can be performed only on an array that
has been sorted
• We assume array is in increasing order
• Stop cases
• The array is empty
• Element being examined matches the target
• Check the middle element for a match with the target
• Throw away the half of the array
• Throw away the half that the target cannot be in
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Binary Search Algorithm
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Binary Search Example
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Implementation of Binary Search
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Implementation of Binary Search
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Efficiency of Binary Search
• At each recursive call we eliminate half the array
elements from consideration
• O(log2n)
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Comparable Interface
• Classes that implement the Comparable interface must
define a compareTo method that enables its objects to
be compared in a standard way
• CompareTo allows you to define ordering of elements
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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 useful for processing recursive data
structures
• The first language developed for artificial intelligence
research was a recursive language called LISP
• “LISt Processing” language
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Recursive Definition of a Linked List
• A non-empty linked list is a collection of nodes where
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
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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
• Goal: Move stack of disks to one of the empty pegs
• Rules:
• Only the top disk on a peg can be moved
• A larger disk cannot be placed on top of a smaller disk
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Towers of Hanoi
• Suppose we can get top 2 disks onto middle peg
• Then we have smaller version of original problem
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Towers of Hanoi
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Algorithm for Towers of Hanoi
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Algorithm for Towers of Hanoi
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Algorithm for Towers of Hanoi
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Recursive Algorithm for Towers of Hanoi
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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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Counting Cells in a Blob (continued)
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Counting Cells in a Blob (continued)
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Implementation
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Implementation (continued)
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Counting Cells in a Blob (continued)
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Backtracking
• Backtracking is an approach to implementing systematic
trial and error in a search for a solution
• An example is finding a path through a maze
• If you are attempting to walk through a maze, you will
probably walk 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
• Backtracking is a systematic approach to trying
alternative paths and eliminating them if they don’t work
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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 Path Implementation
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Maze Path Implementation
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Maze Path Implementation
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Recursion Java Code
• http://www.cs.sjsu.edu/~stamp/CS46B/other/recursion/
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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
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