Java Programming: Guided Learning with Early Objects

Chapter 11Recursion

Java Programming: Guided Learning with Early Objects 2

Objectives

• Learn about recursive definitions• Determine the base case and general case of a

recursive definition• Learn about recursive algorithms

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Objectives (continued)

• Learn about recursive methods• Become familiar with direct and indirect

recursion• Learn how to use recursive methods to

implement recursive algorithms

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Recursive Definitions

• Recursion: reducing a problem to successively smaller versions of itself– Powerful way to solve problems for which the

solution is otherwise complicated

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Recursive Definitions (continued)

• Factorial:– 0! = 1 equation 11-1

– n! = n ( n – 1)! if n > 0 equation 11-2

• Equation 11-1 is the base case• Equation 11-2 is the general (recursive) case

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Recursive Definitions (continued)

• Recursion definition: defined in terms of a smaller version of itself

• Every recursive definition must have at least one base case

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Recursive Definitions (continued)

• General case eventually must be reduced to a base case

• Base case stops the recursion• Recursive algorithm: finds solution by

reducing problems to smaller versions of itself

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Recursive Definitions (continued)

• Recursive method: method that calls itself– Body contains a statement that calls same

method before completing the current call

– Must have one or more base cases

– General solution eventually must reduce to base case

• Recursive algorithms implemented with recursive methods

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Recursive Definitions (continued)

• Factorial definition:

public static int fact(int num) {

if (num == 0)

return 1;

else

return num * fact(num -1);

}

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Figure 11-1 Execution of fact(4)

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Recursive Definitions (continued)

• Think of a recursive method as having unlimited copies of itself

• Every recursive call has its own code, parameters, and local variables

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Recursive Definitions (continued)

• After completing a recursive call, control goes back to previous call

• Current call must execute completely • Execution in previous call begins from point

immediately following recursive call

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Direct and Indirect Recursion

• Directly recursive method calls itself• Indirectly recursive method calls another

method– Eventually original method is called

– Involves several methods

– Can be elusive; take extra care in design

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Infinite Recursion

• If every recursive call results in another recursive call, method is infinitely recursive– Base case never executes

• Every recursive call allocates memory– System saves information to transfer control

back to caller

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Infinite Recursion (continued)

• Computer memory is finite• Infinitely recursive method continues until

system runs out of memory

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Designing Recursive Algorithms and Methods

• Determine limiting conditions• Identify base cases

– Provide direct solution to each base case

• Identify general cases– Provide solution to each general case in terms

of smaller version of itself

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Problem Solving Using Recursion

• Largest element in an array– list is name of array containing list elements

– If list has length 1, single element is the largest

– Find largest element by:max(list[a],largest(list[a+1]…list[b]))

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Figure 11-2 List with six elements

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Figure 11-3 List with four elements

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Figure 11-4 Execution of largest(list, 0, 3)

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Fibonacci Numbers

• Recall Chapter 5 designed a program to determine a Fibonacci number– Each Fibonacci number is the sum of the

previous two

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Fibonacci Numbers (continued)

2

2

1

)2,,()1,,(

),,(

n

n

n

nbaFibnbaFib

b

a

nbaFib

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Fibonacci Numbers (continued)

public static int Fib(int a, int b, int n){if (n==1)

return a;else if (n == 2)

return belse

return Fib(a,b,n-1) + Fib(a,b,n-2)}

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Figure 11-5 Execution of rFibNum(2,3,5)

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Towers of Hanoi

• At creation of universe, priests in the temple of Brahma given three diamond needles

• One needle contained 64 golden disks• Each disk slightly smaller than disks below it• Task: move all 64 disks from first needle to third

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Towers of Hanoi (continued)

• Rules:– Only one disk moved at a time

– Removed disk must be placed on one of the other two needles

– Larger disk cannot be placed on smaller disk

• Once all disks moved from first needle to third, universe comes to an end

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Figure 11-5 Towers of Hanoi with three disks

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Towers of Hanoi (continued)

• One disk: – Base case

– Move disk from needle one to needle three

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Towers of Hanoi (continued)

• Two disks:– First disk moves to second needle

– Second disk moves to third needle

– First disk moves to third needle

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Towers of Hanoi (continued)

• Three disks:– Two problems of moving two disks

• 64 disks:– Two problems of moving 63 disks

• n disks:– Two problems of moving n-1 disks

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Figure 11-6 Solution to Towers of Hanoi with three disks

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Towers of Hanoi (continued)

public static void moveDisks(int count, int needle1, int needle3, int needle2)

{if (count > 0) {

moveDisks (count-1, needle1, needle2,needle3);

moveDisks (count-1, needle2, needle3, needle1);

}}

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Towers of Hanoi: Analysis

• Needle 1 contains 64 disks– Number of moves to needle 3: 264-1 ≈ 1.6 x 1019

• Number of seconds in one year: 3.2 x 107

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Towers of Hanoi: Analysis (continued)

• Priests move one disk per second without resting: 5 x 1011 years

• Estimated age of universe: 1.5 x 1010 years• Computer: 1 billion moves per second, finishes

in 500 years

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Recursive Binary Search

• Recall binary search from Chapter 9• Find middle element• Compare sought element with middle• Repeat on half of list

– Use method call

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Recursive Binary Search (continued)

public static int rBin(int[] list, int first,int last,int srchItm ) {

int mid;int location = 0;if (first <= last) {

mid = (first + last)/2;if (list[mid] == srchItm)

location = mid;

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Recursive Binary Search (continued)

else if (list[mid] > srchItm) location = rBin(list, first,

mid – 1, srchItm);

else location = rBin(list, mid + 1,

last, srchItm);}// end if first <= lastif (first > location || last < location)

location = -1;return location;

}//end rBin

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Figure 11-8 A sorted list

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Figure 11-9 Tracing the recursive binary search algorithm

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Recursion or Iteration?

• Often two ways to solve a problem:– Recursion

– Iteration

• Iterative algorithm often seems simpler• Iterative control structure: uses a looping

structure to repeat a set of statements

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Recursion or Iteration? (continued)

• No general answer to which is better• Guidelines:

– Nature of the solution

– Efficiency of solution

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Recursion or Iteration? (continued)

• Every recursive call has its own parameters and local variables– Requires system to allocate space when method

is called

– Memory deallocated when method terminates

• Recursive calls have overhead in memory and execution time

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Recursion or Iteration? (continued)

• Efficiency of programmer’s time also important consideration– Balance with execution efficiency

• Choice may be a matter of personal preference• Any program that can be written recursively can

be written iteratively• If iterative solution is at least as obvious and

easy as recursive solution, choose iterative

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Summary

• Recursion: solving a problem by reducing it to smaller versions of itself

• Recursive definition defines problem in terms of smaller versions of itself

• Every recursive definition has one or more base cases

• Recursive algorithm solves a problem by reducing it to smaller versions of itself

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Summary (continued)

• Solution to a problem in a base case obtained directly

• Recursive method calls itself• Recursive algorithms implemented as recursive

methods• Recursive method must have one or more base

cases

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Summary (continued)

• General solution breaks problem into smaller versions of itself

• General case eventually reduced to a base case

• Base case stops the recursion

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Summary (continued)

• Tracing a recursive method:– Think of recursive method as having unlimited

copies of itself

– Every call to recursive method executes the code with its own set of parameters and variables

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Summary (continued)

• Tracing a recursive method (continued):– After completing recursive call, control goes

back to calling environment

– Current call executes completely before control returns

– Execution in previous call continues from point following recursive call

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Summary (continued)

• Method is directly recursive if it calls itself• Method is indirectly recursive if it:

– Calls another method

– Eventually results in call to itself

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Summary (continued)

• Design a recursive method:– Understand problem requirements

– Determine limiting conditions

– Identify base cases• Provide direct solution to base cases

– Identify general cases• Provide recursive solution to each general case