1. COSC 2006 Data Structures I Recursion II

2. Topics
More Recursive Examples
Writing Strings backward
Binary Search

3. Recursive Functions Criteria
1. Function calls itself
2. Each call solves an simpler (smaller in size), but identical problem
3. Base case is handled differently from all other cases and enables recursive calls to stop
4. The size of the problem diminishes and ensures reaching the base case

4. What Should be Considered in Recursive Solutions?
How to define the problem in terms of smaller problems of the same type?
How will each recursive call diminish the size of the problem?
What instance will serve as the base case?
As the problem size diminishes, will it reach the base case?

5. Example: Writing Strings Backward
Given:
A string of characters
Required:
Write the string in reverse order
Recursive Algorithm:
Idea:
Divide the string of length n into two parts; one of length 1 and the other of length n-1,
Exchange both strings and perform the same operation on the n-1 length string;
Stop when the length of the n-1 string becomes either 0 or 1 (base case)

6. Example: Writing Strings Backward
A recursive solution:
WriteBackward ( S )
WriteBackward ( S minus last character )

7. Example: Writing Strings Backward
Algorithm: First Draft
WriteBackward1 (in s: string)
if (string is empty)
Do nothing - - Base case
else
{ write the last character of S
writeBackward1( S minus its last character) }

8. Writing a String Backward
void writeBackward1(string s, int size) //
// Writes a character string backward.
// Precondition: The string s contains size characters, where size>= 0
// Postcondition: s is written backward, but remains unchanged.
{ if (size > 0) // Enforcing the pre-condition
{ // write the last character
System.out.print( s.substr (size-1, 1));
// write the rest of the string backward
writeBackward1 (s, size-1); // Point A
} // end if
// size == 0 is the base case - do nothing
} // end writeBackward

9. Example: Writing Strings Backward
Algorithm box trace:
Figure 2-7a: Box trace of writeBackward("cat", 3)

10. Example: Writing Strings Backward
Algorithm box trace:
Figure 2-7b: Box trace of writeBackward("cat", 3)

11. Example: Writing Strings Backward
Algorithm box trace:
Figure 2-7c: Box trace of writeBackward("cat", 3)

12. Example: Writing Strings Backward
2rd Option: (WriteBackward2)
Attach first character to the end
WriteBackward2 (in s: string)
if (string is empty)
Do nothing - - Base case
else
{ writeBackward2( S minus its first character)
System.out.print( “About to write last character of string: “+S);
write the first character of S }
System.out.println(“Leave WriteBackward with string: “+S );

13. Writing a String Backward
Observations:
The 1-length string can be chosen either as the
first character from the n-length string
last character from the n-length string
Recursive calls to WriteBackward function use smaller values of Size
WriteBackward1 writes a character just before generating a new box
WriteBackward2 writes character after returning from recursive call

14. Example: Binary Search
Assumptions:
Array must be sorted
Size = size of the array
A[0]  A[1]  A[3]   A[Size-1]
Idea:
Divide the array into 3 parts
One half from A [First] to A [Mid - 1]
An element A [Mid]
Another half from A [Mid + 1] to A [Last]
Check if A[Mid] equals, less than, or greater than the value you are seeking

15. Example: Binary Search
Pseudocode
binarySearch(in A: ArrayType, in Value: ItemType) {
if (A is of size 1)
Determine if A’s only item = Value // Base-case
else
{ Find the midpoint of A
Determine which half of A contains Value
if (Value in first half of A)
binarySearch(first half of A, Value)
binarySearch (second half of A, Value) }

16. Binary Search Details Split around a middle value array(mid)
How do you pass half an array?
first, last parameters
How do you determine which half contains the value?
Split around a middle value array(mid)
What should the base case(s) be?
Value found at mid
Array empty
How to indicate the result, including failure?
Return index or negative number

17. Example: Binary Search
Two base cases
First > Last:
Value not found in original array
Search fails
Return either a Boolean value or a negative index
Value == A [Mid]:
Value found
Search succeeds
Return the index corresponding to Value
The array should be passed to the function by reference. It shouldn't be considered as part of the local environment.

18. Binary Search Code (abbreviated)
binarySearch (int anArray[], int first, int last, int value) {
int index;
if (first > last)
index = -1;
else {
int mid = (first + last)/2;
if (value == anArray[mid])
index = mid;
else if (value < anArray[mid])
index = binarySearch(anArray, first, mid-1, value);
else
index = binarySearch(anArray, mid+1, last, value); }
return index;

19. Example: Binary Search
Using Run-Time Stack to trace
Box contents:
Value
First
Last
Mid
The array is not considered a part of the box. It is passed by reference

20. Example: Binary Search
Searching for 9
Searching for 6
Value = 9
First = 0
Last = 7
Mid =(0+7)/2=3
Value < A[3]
Last = 2
Mid = (0+2)/2=1
Value < A[1]
First = 2
Mid = (2+2)/2=2
Value = A[2]
return 2
Value = 6
First = 0
Last = 7
Mid =(0+7)/2=3
Value < A[3]
Last = 2
Mid =(0+2)/2=2
Value < A[2]
First = 2
Mid =(2+2)/2=2
Last = 1
First > Last
return -1

21. Review
In a recursive method that writes a string of characters in reverse order, the base case is ______.
a string with a length of 0
a string whose length is a negative number
a string with a length of 3
a string that is a palindrome

22. Review
Which of the following is a precondition for a method that accepts a number n and computes the nth Fibonacci number?
n is a negative integer
n is a positive integer
n is greater than 1
n is an even integer

23. Review
The midpoint of a sorted array can be found by ______, where first is the index of the first item in the array and last is the index of the last item in the array.
first / 2 + last / 2
first / 2 – last / 2
(first + last) / 2
(first – last) / 2

24. Review
If the value being searched for by a recursive binary search algorithm is in the array, which of the following is true?
the algorithm cannot return a nonpositive number
the algorithm cannot return a nonnegative number
the algorithm cannot return a zero
the algorithm cannot return a negative number

25. Review
An array is a(n) ______.
class
method
object
variable

26. Review
For anArray = <2, 3, 5, 6, 9, 13, 16, 19>, what is the value returned by a recursive binary search algorithm if the value being searched for is 10? –1 1 10

27. Review
A recursive binary search algorithm always reduces the problem size by ______ at each recursive call. 1 2
half
one-third