Java Programming: Guided Learning with Early Objects

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


Objectives (continued)

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

recursion• Learn how to use recursive methods to

implement recursive algorithms


Recursive Definitions

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

solution is otherwise complicated


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



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

• Every recursive definition must have at least one base case



• 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



• 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



• Factorial definition:

public static int fact(int num) {

if (num == 0)

return 1;

else

return num * fact(num -1);

}


Figure 11-1 Execution of fact(4)



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

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



• 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


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


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


Infinite Recursion (continued)

• Computer memory is finite• Infinitely recursive method continues until

system runs out of memory


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


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]))


Figure 11-2 List with six elements


Figure 11-3 List with four elements


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


Fibonacci Numbers

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

previous two


Fibonacci Numbers (continued)

2

2

1

)2,,()1,,(

),,(

n

n

n

nbaFibnbaFib

b

a

nbaFib


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)}


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


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


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


Figure 11-5 Towers of Hanoi with three disks



• One disk: – Base case

– Move disk from needle one to needle three



• Two disks:– First disk moves to second needle

– Second disk moves to third needle

– First disk moves to third needle



• 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


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



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);

}}


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


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


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


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;


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


Figure 11-8 A sorted list


Figure 11-9 Tracing the recursive binary search algorithm


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


Recursion or Iteration? (continued)

• No general answer to which is better• Guidelines:

– Nature of the solution

– Efficiency of solution



• 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



• 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


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


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


Summary (continued)

• General solution breaks problem into smaller versions of itself

• General case eventually reduced to a base case

• Base case stops the recursion


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


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


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


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

Java Programming: Guided Learning with Early Objects

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