java programming: guided learning with early objects
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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
Java Programming: Guided Learning with Early Objects 3
Objectives (continued)
• Learn about recursive methods• Become familiar with direct and indirect
recursion• Learn how to use recursive methods to
implement recursive algorithms
Java Programming: Guided Learning with Early Objects 4
Recursive Definitions
• Recursion: reducing a problem to successively smaller versions of itself– Powerful way to solve problems for which the
solution is otherwise complicated
Java Programming: Guided Learning with Early Objects 5
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
Java Programming: Guided Learning with Early Objects 6
Recursive Definitions (continued)
• Recursion definition: defined in terms of a smaller version of itself
• Every recursive definition must have at least one base case
Java Programming: Guided Learning with Early Objects 7
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
Java Programming: Guided Learning with Early Objects 8
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
Java Programming: Guided Learning with Early Objects 9
Recursive Definitions (continued)
• Factorial definition:
public static int fact(int num) {
if (num == 0)
return 1;
else
return num * fact(num -1);
}
Java Programming: Guided Learning with Early Objects 10
Figure 11-1 Execution of fact(4)
Java Programming: Guided Learning with Early Objects 11
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
Java Programming: Guided Learning with Early Objects 12
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
Java Programming: Guided Learning with Early Objects 13
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
Java Programming: Guided Learning with Early Objects 14
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
Java Programming: Guided Learning with Early Objects 15
Infinite Recursion (continued)
• Computer memory is finite• Infinitely recursive method continues until
system runs out of memory
Java Programming: Guided Learning with Early Objects 16
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
Java Programming: Guided Learning with Early Objects 17
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]))
Java Programming: Guided Learning with Early Objects 18
Figure 11-2 List with six elements
Java Programming: Guided Learning with Early Objects 19
Figure 11-3 List with four elements
Java Programming: Guided Learning with Early Objects 20
Figure 11-4 Execution of largest(list, 0, 3)
Java Programming: Guided Learning with Early Objects 21
Fibonacci Numbers
• Recall Chapter 5 designed a program to determine a Fibonacci number– Each Fibonacci number is the sum of the
previous two
Java Programming: Guided Learning with Early Objects 22
Fibonacci Numbers (continued)
2
2
1
)2,,()1,,(
),,(
n
n
n
nbaFibnbaFib
b
a
nbaFib
Java Programming: Guided Learning with Early Objects 23
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)}
Java Programming: Guided Learning with Early Objects 24
Figure 11-5 Execution of rFibNum(2,3,5)
Java Programming: Guided Learning with Early Objects 25
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
Java Programming: Guided Learning with Early Objects 26
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
Java Programming: Guided Learning with Early Objects 27
Figure 11-5 Towers of Hanoi with three disks
Java Programming: Guided Learning with Early Objects 28
Towers of Hanoi (continued)
• One disk: – Base case
– Move disk from needle one to needle three
Java Programming: Guided Learning with Early Objects 29
Towers of Hanoi (continued)
• Two disks:– First disk moves to second needle
– Second disk moves to third needle
– First disk moves to third needle
Java Programming: Guided Learning with Early Objects 30
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
Java Programming: Guided Learning with Early Objects 31
Figure 11-6 Solution to Towers of Hanoi with three disks
Java Programming: Guided Learning with Early Objects 32
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);
}}
Java Programming: Guided Learning with Early Objects 33
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
Java Programming: Guided Learning with Early Objects 34
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
Java Programming: Guided Learning with Early Objects 35
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
Java Programming: Guided Learning with Early Objects 36
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;
Java Programming: Guided Learning with Early Objects 37
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
Java Programming: Guided Learning with Early Objects 38
Figure 11-8 A sorted list
Java Programming: Guided Learning with Early Objects 39
Figure 11-9 Tracing the recursive binary search algorithm
Java Programming: Guided Learning with Early Objects 40
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
Java Programming: Guided Learning with Early Objects 41
Recursion or Iteration? (continued)
• No general answer to which is better• Guidelines:
– Nature of the solution
– Efficiency of solution
Java Programming: Guided Learning with Early Objects 42
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
Java Programming: Guided Learning with Early Objects 43
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
Java Programming: Guided Learning with Early Objects 44
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
Java Programming: Guided Learning with Early Objects 45
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
Java Programming: Guided Learning with Early Objects 46
Summary (continued)
• General solution breaks problem into smaller versions of itself
• General case eventually reduced to a base case
• Base case stops the recursion
Java Programming: Guided Learning with Early Objects 47
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
Java Programming: Guided Learning with Early Objects 48
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
Java Programming: Guided Learning with Early Objects 49
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
Java Programming: Guided Learning with Early Objects 50
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
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