1. COSC2007 Data Structures II
April 23, 2017
COSC Data Structures II
Chapter 6 Recursion as a
Problem-Solving Technique
Recursion as a Problem-Solving Technique

2. Topics 8-Queen problem Languages Algebraic expressions
April 23, 2017
Topics
8-Queen problem
Languages
Algebraic expressions
Prefix & Postfix notation & conversion
Recursion as a Problem-Solving Technique

3. Introduction Backtracking:
April 23, 2017
Introduction
Backtracking:
A problem-solving technique that involves guesses at a solution, backing up, in reverse order, when a dead-end is reached, and trying a new sequence of steps
Applications
8-Queen problems
Tic-Tac-Toe
Recursion as a Problem-Solving Technique

4. Eight-Queens Problem Given: Required:
April 23, 2017
Eight-Queens Problem
Given:
64 square that form 8 x 8 - rows x columns
A queen can attack any other piece within its row, column, or diagonal
Required:
Place eight queens on the chessboard so that no queen can attack any other queen
Recursion as a Problem-Solving Technique

5. Eight-Queens Problem Any idea? Observations:
April 23, 2017
Eight-Queens Problem
Observations:
Each row or column contain exactly one queen
There are 4,426,165,368 ways to arrange 8 queens on the chessboard
Not all arrangements lead to correct solution
Wrong arrangements could be eliminated to reduce the number of solutions to the problem
8! = 40,320 arrangements
Any idea?
Recursion as a Problem-Solving Technique

6. Eight-Queens Problem Solution Idea:
April 23, 2017
Eight-Queens Problem
Solution Idea:
Start by placing a queen in the first square of column 1 and eliminate all the squares that this queen can attack
At column 5, all possibilities will be exhausted and you have to backtrack to search for another layout
When you consider the next column, you solve the same problem with one column less (Recursive step)
The solution combines recursion with backtracking
Recursion as a Problem-Solving Technique

7. April 23, 2017
Eight-Queens Problem
How can we use recursion for solving this problem?
For placing a queen in a column, assuming that previous queens were placed correctly
If there are no columns to consider  Finish – Base case
Place queen successfully in current column  Proceed to next column, solving the same problem on fewer columns – recursive step
No queen can be placed in current column  Backtrack
Base case:
Either we reach a solution to the problem with no columns left
 Success & finish
Or all previous queens will be under attack
 Failure & Backtrack
Recursion as a Problem-Solving Technique

8. Eight-Queens Problem Pseudocode PlaceQueens (in currColumn: integer)
April 23, 2017
Eight-Queens Problem
Pseudocode
PlaceQueens (in currColumn: integer)
// Places queens in columns numbered Column through 8
if (currColumn > 8)
Success. Reached a solution // base case
else
{ while (unconsidered squares exist in the currColumn & problem is unsolved)
{determine the next square in column “currColumn” that is not under attack in earlier column
if (such a square exists)
{ Place a queen in the square
PlaceQuenns(currColumn + 1) // Try next column
if ( no queen possible in currColumn+1)
Remove queen from currColumn &
consider next square in that column
} // end if
} // end while
Recursion as a Problem-Solving Technique

9. Eight-Queens Problem Implementation: Classes: Data members QueenClass
April 23, 2017
Eight-Queens Problem
Implementation:
Classes:
QueenClass
Data members
board:
How?
square
Indicates that the square has a queen or is empty
Recursion as a Problem-Solving Technique

10. Eight-Queens Problem Textbook provides partial implementation
April 23, 2017
Eight-Queens Problem
Implementation:
Methods needed:
Public functions (methods)
ClearBoard: Sets all squares to empty
DisplayBoard: Displays the board
PlaceQueens: Places the queens in board’s columns & returns either success or failure
Private functions (methods)
SetQueen: Sets a given square to QUEEN
RemoveQueen: Sets a given square to EMPTY
IsUnderAttack: Checks if a given square is under attack
Index: Returns the array index corresponding to a row or column
Textbook provides partial implementation
Recursion as a Problem-Solving Technique

11. Languages Language: A set of strings of symbols
April 23, 2017
Languages
Language:
A set of strings of symbols
Can be described using syntax rules
Examples:
Java program
program =
{ w : w is a syntactically correct java program}
Algebraic expressions
Algebraic Expression = { w : w is an algebraic expression }
Recursion as a Problem-Solving Technique

12. Languages Language recognizer: Recognition algorithm:
April 23, 2017
Languages
Language recognizer:
A program (or machine) that accepts a sentence and determines whether the sentence belongs to the language
Recognition algorithm:
An algorithm that determines whether a given string belongs to a specific language, given the language grammar
If the grammar is recursive, we can write straightforward recursive recognition algorithms
Recursion as a Problem-Solving Technique

13. Languages Example: Grammar for Java identifiers Grammar Syntax graph
April 23, 2017
Languages
Example: Grammar for Java identifiers
Java Ids = {w:w is a legal Java identifier}
Grammar
< identifier > = < letter > | < identifier > < letter >
| < identifier > < digit >
< letter > = a | b | | z | A | B | | Z | _
< digit > = 0 | 1 | | 9
Syntax graph
Observations
The definition is recursive
Base case:
A letter ( length = 1)
We need to construct a recognition algorithm
Letter
Digit
Recursion as a Problem-Solving Technique

14. Languages Java Identifiers’ Recognition algorithm
April 23, 2017
Languages
Java Identifiers’ Recognition algorithm
isId (in w: string): boolean
// Returns true if w is a legal Java identifier;
// otherwise returns false.
if (w is of length 1 ) // base case
if (w is a letter)
return true
else
return false
else if
(the last character of w is a letter or a digit)
return isId (w minus its last character)
// Point X
Recursion as a Problem-Solving Technique

15. April 23, 2017
Palindromes
String that read the same from left-to-right as it does from right-to-left
Examples
RADAR
Madam I’m Adam (without the blanks and quotes)
Language:
Palindromes = { w : w reads the same from left to right
as from right to left }
Grammar:
<pal > = empty string | < ch > | a <pal > a | | z <pal> z |
A < pal > A | | Z < pal > Z
< ch > = a | b | | z | A | B | | Z
The definition is recursive
Base cases: ?
We need a recognition algorithm
Recursion as a Problem-Solving Technique

16. Palindromes Recognition algorithm isPal(in w: string) : boolean
April 23, 2017
Palindromes
Recognition algorithm
isPal(in w: string) : boolean
If (w is of length 0 or 1)
return true // Base-case success
else
if (first and last characters are the same)
return IsPal( w minus first and last characters)
return false // Base-case failure
Recursion as a Problem-Solving Technique

17. An Bn Strings Language: Grammar: The definition is recursive
April 23, 2017
An Bn Strings
Language:
L = { w : w is of the form An Bn for some n  0 }
Grammar:
< legal word > = empty string | A < legal word > B
The definition is recursive
Base case: ?
Recursion as a Problem-Solving Technique

18. An Bn Strings Recognition algorithm IsAnBn(in w: string): boolean
April 23, 2017
An Bn Strings
Recognition algorithm
IsAnBn(in w: string): boolean
If (w is of length 0)
return true // Base–case success
else if (w begins with ‘A’ & ends with ‘B’)
return IsAnBn( w minus first & last characters) // Point A
else
return false // Base–case failure
Recursion as a Problem-Solving Technique

19. Algebraic Expressions
April 23, 2017
Algebraic Expressions
Algebraic expression
A fully or partially parenthesized infix arithmetic expression
Examples
A + ( B – C ) * D { partially parenthesized}
( A + ( ( B – C ) * D )) {Fully parenthesized}
Solution
Find a way to get rid of the parentheses completely
Recursion as a Problem-Solving Technique

20. Algebraic Expressions
April 23, 2017
Algebraic Expressions
Infix expression:
Binary operators appear between their operands
Prefix (Polish) expression:
Binary operators appear before their operands
Postfix (Reverse Polish) expression:
Binary operators appear after their operands
Examples:
Infix Prefix Postfix
a + b + a b a b +
a + ( b * c ) a * b c a b c * +
(a + b) * c * + a b c a b + c *
Recursion as a Problem-Solving Technique

21. Prefix Expressions Language: Grammar: Definition is recursive
April 23, 2017
Prefix Expressions
Language:
L = { w : w is a prefix expression }
Grammar:
< prefix > = <identifier>
| < operator > < prefix1> < prefix2 >
< operator > = + | - | * | /
< identifier > = A | B | | Z
Definition is recursive
Size of prefix1 & prefix2 is smaller than prefix, since they are subexpressions
Base case:
When an identifier is reached
Recursion as a Problem-Solving Technique

22. April 23, 2017
Prefix Expressions
How to Determine if a Given Expression is a Prefix Expression?
Recursion as a Problem-Solving Technique

23. April 23, 2017
Prefix Expressions
How to Determine if a Given Expression is a Prefix Expression:
1. Construct a recursive valued function End-Pre (First, Last ) that returns either the index of the end of the prefix expression beginning at S(First), or -1 , if no such expression exists in the given string
2. Use the previous function to determine whether a character string S is a prefix expression
Recursion as a Problem-Solving Technique

24. Prefix Expressions End-Pre Function: (Tracing: Fig 6.5, p.309)
April 23, 2017
Prefix Expressions
End-Pre Function: (Tracing: Fig 6.5, p.309)
endPre(first, last)
// Finds the end of a prefix expression, if one exists.
// If no such prefix expression exists, endPre should return –1.
if ((First < 0) or (First > Last)) // string is empty
return –1
ch = character at position first of strExp
if (ch is an identifier)
// index of last character in simple prefix expression
return First
else if (ch is an operator)
{ // find the end of the first prefix expression
firstEnd = endPre(First+1, last) // Point X
// if the end of the first expression was found
// find the end of the second prefix expression
if (firstEnd > -1)
return endPre(firstEnd+1, last) // Point Y
else
return -1
} // end if
Recursion as a Problem-Solving Technique

25. Prefix Expressions IsPre Function IsPre(): boolean
April 23, 2017
Prefix Expressions
IsPre Function
Uses endPre() to determine whether a string is a prefix or not
IsPre(): boolean
size= length of expression strExp
lastChar = endpre(0, size-1)
if (lastChar >= 0 and lastChar == strExp.length()-1)
return true
else
return false
Recursion as a Problem-Solving Technique

26. Prefix Expressions EvaluatePrefix Function
April 23, 2017
Prefix Expressions
EvaluatePrefix Function
E is the expression beginning at Index
evaluatePrefix( strExp: string)
// Evaluates a prefix expression strExp {
Ch = first character of strExp
Delete the first character from strExp
if (Ch is an identifier)
return value of the identifier // base case
else if (Ch is an operator named op)
{ Operand1 = EvaluatePrefix(strExp) // Point A
Operand2 = EvaluatePrefix(strRxp) // Point B
return Operand1 op Operand2
} // end if }
Recursion as a Problem-Solving Technique

27. Review
______ is a problem-solving technique that involves guesses at a solution.
Recursion
Backtracking
Iteration
Induction

28. Review
In the recursive solution to the Eight Queens problem, the problem size decreases by ______ at each recursive step.
one square
two squares
one column
two columns

29. Review A language is a set of strings of ______. numbers letters
Alphabets
symbols

30. Review
The statement:
JavaPrograms = {strings w : w is a syntactically correct Java program} signifies that ______.
all strings are syntactically correct Java programs
all syntactically correct Java programs are strings
the JavaPrograms language consists of all the possible strings
a string is a language

31. Review
The Java ______ determines whether a given string is a syntactically correct Java program.
applet
editor
compiler
programmer

32. Review
If the string w is a palindrome, which of the following is true?
w minus its first character is a palindrome
w minus its last character is a palindrome
w minus its first and last characters is a palindrome
the first half of w is a palindrome
the second half of w is a palindrome

33. Review Which of the following strings is a palindrome? “bottle” “beep”
“coco” “r

34. Review
The symbol AnBn is standard notation for the string that consists of ______.
an A, followed by an n, followed by a B, followed by an n
an equal number of A’s and B’s, arranged in a random order
n consecutive A’s, followed by n consecutive B’s
a pair of an A and a B, followed another pair of an A and a B

35. Review Which of the following is an infix expression? / a + b c

36. Review What is the value of the prefix expression: – * 3 8 + 6 5? 1123 13 21