COSC2007 Data Structures II

Chapter 5Recursion as a

Problem-Solving Technique

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Topics

8-Queen problem

Languages Algebraic expressions

Prefix & Postfix notation & conversion

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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

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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

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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

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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

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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 correctlya) If there are no columns to consider Finish – Base caseb) Place queen successfully in current column Proceed to next

column, solving the same problem on fewer columns – recursive step

c) 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

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Eight-Queens Problem Pseudocode

PlaceQueens (in currColumn: integer)// Places queens in columns numbered Column through 8if (currColumn > 8)

Succes. Reached a solution // base caseelse{ 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 columnif ( no queen possible in currColumn+1)

Remove queen from currColumn & consider next square in that column

} // end if} // end while

} // end if

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Eight-Queens Problem

Implementation: Classes:

QueenClass Data members

board:

square Indicates is the square has a queen or is empty

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Eight-Queens Problem

Implementation: Methods needed:

Public functions 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

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

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Eight-Queens Problem:

public class Queens {public static final int BOARD_SIZE = 8;

// squares per row or columnPublic static final int EMPTY =0; //use to indicate an empty squarePublic static final int QUEEN=1; //indicate square contains a queenPrivate in board [][]; //chess board public Queens() { // Creates an empty square board.

board = new int [BOARD_SIZE]{BOARD-SIZE];} public void clearBoard() { } // Sets all squares to EMPTY. public void displayBoard() { } // Displays the board.

public boolean placeQueens(int currColumn)

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Eight-Queens Problem:

// Calls: isUnderAttack, setQueen, removeQueen.{ if (currColumn > BOARD_SIZE) return true; // base case else { boolean queenPlaced = false; int row = 1; // number of square in column while ( !queenPlaced && (row <= BOARD_SIZE) ) { // if square can be attacked if (isUnderAttack(row, currColumn)) ++row; // then consider next square in currColumn else // else place queen and consider next { // column setQueen(row, currColumn); queenPlaced = placeQueens(currColumn+1); // if no queen is possible in next column, if (!queenPlaced) { // backtrack: remove queen placed earlier // and try next square in column removeQueen(row, currColumn); ++row; } // end if } // end if } // end while return queenPlaced; } // end if} // end placeQueens

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Eight-Queens Problem: Java Code

private void setQueen(int row, int column) { } // Sets the square on the board in a given row and column to QUEEN.

private void removeQueen(int row, int column) { } // Sets the square on the board in a given row and column to EMPTY.

private boolean isUnderAttack(int row, int column) // Determines whether the square on the board at a given row and column is under // attack by any queens in the columns 1 through column-1. // Precondition: Each column between 1 and column-1 has a queen placed in a // square at a specific row. None of these queens can be attacked by any other queen. // Postcondition: If the designated square is under attack, returns true; // otherwise, returns false.

private int index(int number) // Returns the array index that corresponds to // a row or column number. // Precondition: 1 <= number <= BOARD_SIZE. // Postcondition: Returns adjusted index value.} // end class Queens

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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 }

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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

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Languages

Example: Grammar for Java identifiersJava 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

Letter

Digit

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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 caseif (w is a letter)

return trueelse

return falseelse if

(the last character of w is a letter or a digit)return isId (w minus its last character)

// Point X else

return false

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Palindromes String that read the same from left-to-right as it does from

right-to-left Examples

RADAR DEED WOW ADA 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:

Empty string or one character We need a recognition algorithm

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Palindromes

Recognition algorithmisPal(in w: string) : boolean// Checks if a string w is a palindrome;// If palindrome, it returns true, false otherwise

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)// Point A

elsereturn false // Base-case failure

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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:

empty string

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An Bn Strings

Recognition algorithm

IsAnBn(in w: string): boolean

// Checks if a string w is of the form An Bn;If (w is of length 0)

return true // Base–case successelse if (w begins with ‘A’ & ends with ‘B’)

return IsAnBn( w minus first & last characters) // Point Aelse

return false // Base–case failure

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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

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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 Postfixa + b + a b a b +a + ( b * c ) + a * b c a b c * +(a + b) * c * + a b c a b + c *

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Algebraic Expressions

Converting or evaluating partially parenthesized infix expression are complicated (discussed in chapter 6-Stacks)

Postfix & prefix expressions can be evaluated easily by the computer

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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

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Prefix Expressions

How to Determine if a Given Expression is a Prefix Expression:1. Construct a recursive valued function End-Pre

(S , 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

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Prefix Expressions End-Pre Function: (Tracing: Fig 5.5, p.229)

+EndPre(in First)// Finds the end of a prefix expression, if one exists.// Precondition: The substring of strExp from index first through// the end of the string contains no blank characters.// Postcondition: Returns the index of the last character in the// prefix expression that begins at index first of strExp.// If no such prefix expression exists, endPre should return –1.

last = strExp.length –1if ((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 expressionreturn First

else if (ch is an operator) { // find the end of the first prefix expression

FirstEnd = endPre(First+1) // Point X// if the end of the first expression was found// find the end of the second prefix expresion

if (FirstEnd > -1)return EndPre(firstEnd+1) // Point Y

elsereturn -1

} // end if else return -1

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Prefix Expressions

IsPre Function Uses End-Pre() to determine whether a string is a prefix or notIsPre(): boolean// Determines whether an expression i a prefix expression.// Precondition: The class has a data member strExp that // contains a string with no blank characters.// Postcondition: Returns true if the expression is in prefix form;// otherwise return false. lastChar = endpre(0)return (LastChar >= 0) and (LastChar == strExp.length()-1)

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Prefix Expressions EvaluatePrefix Function

E is the expression beginning at IndexevaluatePrefix( in strExp: string)// Evaluates a prefix expression strExp// Precondition: strExp is a string containing a valid prefix expression // with no blanks// PostCondition: Returns the value of the prefix expression{ Ch = first character of strExp

Delete the first character from strExpif (Ch is an identifier)

return value of the identifier // base caseelse if (Ch is an operator named op){ Operand1 = EvaluatePrefix(strExp) // Point A

Operand2 = EvaluatePrefix(strRxp) // Point Breturn Operand1 op Operand2

} // end if