ECE 242 Spring 2003 Data Structures in Java

1
ECE 242 Spring 2003Data Structures in Java

• http//rio.ecs.umass.edu/ece242
• Recursion
• Prof. Lixin Gao

2
Todays Topics

• Introduction to recursion
• How recursion works
• Some examples

3
Summation

• 12345100
• Sum(100) 123100
• 12 99 100
• sum (99) 100
• sum(n) sum (n-1) n
• sum(0) 0

Recursion step
Base step
4
Summation Method

• int sum(int n)
• //base step
• if( nlt0 )
• System.out.println(Only positive integers.)
• return 1
• else if(n 0)
• return 0
• //recursive step
• int temp n sum (n-1)
• return temp

5
Factorial

• 4! 4 3 2 1
• n! n (n-1) 1
• 4! 4 3!
• n! n (n-1)!
• factorial(n) n factorial(n-1)//recursive step
• factorial(1) 1 //base step

6
Method Factorial

• // Assume n is a positive integer.
• int fact(int n)
• //base step
• if (n 1)
• return 1
• //recursive step
• int temp n fact(n-1)
• return temp

7
Iteration

• Seen in almost all the programs
• When processing a large number of related tasks
• for
• while loop
• dowhile

8
Recursion

• A close sibling of iteration
• Instead of a loop structure, a method that calls
  itself or partially defined by itself.

9
A Recursive Method Invokes Itself
sum(n) sum (n-1) n sum(0) 0
n! n (n-1)! 1! 1
10
Recursion

• Duplicate itself but with different parameters
• Reduce a complex problem into a smaller one
• Divide and conquer

11
Recursion

• Define an object in terms of itself
• Consists of two parts
• Recursive step
• Base step

f(n) f(n-1) n f(1) 1
12
Method Factorial

• int fact(int n)
• //base step
• if (n 1)
• return 1
• //recursive step
• int temp n fact(n-1)
• return temp

13
Method Factorial (Cont.)
fact(4) temp ?, n4
fact(3) temp ?, n3
fact(2) temp?, n2
fact(1) return 1
temp2 return 2
temp6 return 6
temp24 return 24 4! 24
14
Fibonacci Numbers

• 0, 1, 1, 2, 3, 5, 8, 13,
• Fib(n) Fib(n-1) Fib(n-2)
• Fib(0) 0
• Fib(1) 1

Recursive case
Base case
15
Iterative Implementation of Fibonacci Numbers

• int Fib(int n)
• int fib11,fib00, fibn, i
• if(nlt1)
• return n
• else
• for(i2iltni)
• fibnfib1fib0
• fib0fib1
• fib1fibn
• return fibn

16
Recursive Implementation of Fibonacci Numbers

• int Fib(int n)
• if(nlt1)
• return n
• else
• return (Fib(n-1)Fib(n-2))

5
Fib(5)
2
3
Fib(3)
Fib(4)
1
2
1
1
Fib(1)
Fib(2)
Fib(2)
Fib(3)
0
1
1
1
0
1
Fib(2)
Fib(1)
Fib(1)
Fib(0)
Fib(1)
Fib(0)
0
1
Fib(1)
Fib(0)
17
Print A Linked List Recursively
head
node1
node2
node3
print the rest of the list
print the first node
18
Code To Print A Linked List Recursively

• public void printListRec(Node head)
• if(isEmpty(head) )
• return
• else
• //print the first node
• System.out.println(head.value)
• //print the rest of the list
• printListRec(head.next)

19
Stack For Calling Method

• public methodA(int argA)
• int x, y
• methodB(y)
• return x
• public methodB(int argB)
• int i, j
• return i

Instruction pointer (IP)
20
Calling Stack
push in

• public methodA(int argA)
• int x, y
• methodB(y)
• public methodB(int argB)
• int i, j
• return i

Instruction pointer (IP)
x, y
Stack frame of methodA
IP
argA
stack
21
Calling Stack Of Recursive Call

• int fact(int n)
• if(nlt0)
• return 1
• else
• return (fact(n-1)n)

Calling stack
fact(n-1)
fact(n)
22
Calling Stack

• int fact(int n)
• //base step
• if (n 1)
• return 1
• //recursive step
• int temp n fact(n-1)
• return temp
• fact(4) 4! 24

Calling stack
temp1
fact(0)
n0
temp?
temp1
fact(1)
n1
temp?
temp2
fact(2)
n2
temp?
temp6
fact(3)
n3
temp24
temp?
fact(4)
n4
23
What Does Mystery Do?

• void mystery(int level)
• int i
• if(level lt3)
• for(i0iltleveli) System.out.print( )
• System.out.println(Enter mystery at level
  level)
• for(i0iltleveli) System.out.print( )
• System.out.println(I work on level level
  problem)
• mystery(level1)
• for(i0iltleveli) System.out.print( )
• System.out.println(Exit mystery at level
  level)

24
Mystery Output

• Enter mystery at level1
• I work on level1 problem
• Enter mystery at level2
• I work on level2 problem
• Enter mystery at level3
• I work on level3 problem
• Exit mystery at level 3
• Exit mystery at level 2
• Exit mystery at level 1

25
Why Recursive Solutions?

• It is natural to solve some problems recursively
• Fibonacci Numbers
• Hanoi Tower
• MergeSort

26
Tower Of Hanoi

• Goal Move all disks from A to C
• Rule
• No bigger disk is on top of smaller disk at any
  time
• Move one disk at a time

C
B
A
4 disks
27
Tower Of Hanoi

• 1 disk moves from A to C Simple!
• 2 disks move from A to C

C
B
A
3
1
2
2 disks
28
N disks from A to C?

• Move n-1 disks from A to B
• Move disk n from A to C
• Move n-1 disks from B to C

3
1
C
B
A
n-1 disks
2
disk n
29
Recursive Code For Hanoi Tower

• void move(int n, char start, char finish, char
  buf)
• if(n1)
• System.out.println(Move fromstart
  tofinish)
• else
• move(n-1,start,buf,finish)
• System.out.println(Move fromstart
  tofinish)
• move(n-1,buf,finish,start)

30
Trace Of Recursive Hanoi Tower (Example of 3
disks)

• move(3, A, C, B)
• move(2, A, B, C)
• move(1, A, C, B)
• move from A to C
• move from A to B
• move(1, C, B, A)
• move from C to B
• move from A to C
• move(2, B,C, A )
• move(1, B, A, C)
• move from B to A
• move B to C
• move(1, A, C, B)
• move from A to C
• End of 1
• End of 2
• End of 3

31
4-disk Example
C
B
A
32
4-disk Example
C
B
A
33
4-disk Example
C
B
A
34
Demo Of Hanoi

• A nice animation of Hanoi online
• http//www.mazeworks.com/hanoi/index.htm

35
Sorting

• Sorting algorithms
• Selection Sort
• Insertion Sort
• Both algorithms are of O(N2)
• Another Sorting algorithm MergeSort

36
Recursive MergeSort

• Divide and Conquer MergeSort
• Divide Split the list into two or more equal
  sub-lists
• Recursion Sort each sub-list using a recursive
  call
• Conquer Merge the sorted sub-lists into a
  solution for the original problem

37
MergeSort Split
38
MergSort Join
39
Recursive Merging

• S1, S2 are sorted sub-sequences

5
3
S1
8
2
S2
Temp buffer
5
3
8
2
40
MergeSort Needs Extra Storage

• Unlike selection sort, merge sort does not work
  in place
• A temporary collection is needed so twice as much
  memory is required.

41
Code For MSort

• public class Msort
• public static void main(String args)
• //data to be sorted.
• int data 54, 45, 32, 20, 21, 12,
  14, 8, 9, 4, 0, 3
• //print out the original data
• System.out.print (" ")
• for(int i 0 i lt data.length i)
• System.out.print(datai" ")
• System.out.println(" ")
• //call sorting method to sort the data
• sorting( data )

42
Code For MSort (Cont.)

• System.out.println("")
• //after sorting, print out the sorted array.
• System.out.print(" ")
• for(int i 0 i lt data.length i)
• System.out.print(datai" ")
• System.out.println(" ")

43
sorting Method

• static void sorting( int data)
• int start 0
• int end data.length-1
• // call mergeSort(...)
• mergeSort(data, start, end)

44
Code For mergeSort

• static void mergeSort(int data, int start,
  int end)
• // if start gt end, means the sub-array is
  already sorted.
• if( start lt end)
• //calculate the middle of the array and split it
  into two halves.
• int mid (startend)/2
• //recursively sort the first half
• mergeSort(data, start, mid)
• //recursively sort the second half
• mergeSort(data, mid1, end)
• //merge the two sorted half-array together.
• merge(data, start, mid, end)

45
Code For merge

• private static void merge(int data, int
  start, int mid, int end)
• //firstCopied number of elements copied in the
  first half arrray
• //secondCopied number of elements copied in the
  second half array.
• int firstCopied 0, secondCopied 0
• int length end-start 1, index 0
• int temp new intend-start 1
• //size of the first half array
• int firstSize mid-start1
• //size of the second half array
• int secondSize end-mid

46
Code For merge (Cont.)

• //while both half arrays still have elements
  remain
• while(firstCopied lt firstSize secondCopied lt
  secondSize)
• //compare which element is smaller, and copy
  that element to the buffer
• if (datastartfirstCopied lt
  datamid1secondCopied)
• temp index
  datastartfirstCopied
• firstCopied
• else
• tempindex
  datamid1secondCopied
• secondCopied

47
Code For merge (Cont.)

• //copy the remains of either the first array
  or the second array
• while(firstCopied lt firstSize)
• temp index datastartfirstCopie
  d
• firstCopied
• while(secondCopied lt secondSize)
• tempindex datamid1secondCopie
  d
• secondCopied
• //copy the sorted elements from the buffer
  back to the original array
• for(int i0 iltend-start1 i)
• datastartitempi

48
Complexity Of MergeSort

• Each round of Merging needs N comparisons
• There are log2N round of merging
• Average number of comparison is about O(NlogN)
  for an array of size N

49
Link To The Sorting Demo

• A set of sorting algorithm demos
• Merge Sort

50
Code For MergeSort

• MSort.java