CS222 Algorithms First Semester 2003/2004

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CS222 AlgorithmsFirst Semester 2003/2004

• Dr. Sanath Jayasena
• Dept. of Computer Science Eng.
• University of Moratuwa
• Lecture 7 (28/10/2003)
• String Matching Part 2
• Greedy Approach

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Overview

• Previous lecture String Matching Part 1
• Naïve Algorithm, Rabin-Karp Algorithm
• This lecture
• String Matching Part 2
• String Matching using Finite Automata
• Knuth-Morris-Pratt (KMP) Algorithm
• Greedy Approach to Algorithm Design

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String Matching

• PART 2

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Finite Automata

• A finite automaton M is a 5-tuple (Q, q0, A, ?,
  d), where
• Q is a finite set of states
• q0 e Q is the start state
• A ? Q is a set of accepting states
• ? is a finite input alphabet
• d is the transition function that gives the next
  state for a given current state and input

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How a Finite Automaton Works

• The finite automaton M begins in state q0
• Reads characters from ? one at a time
• If M is in state q and reads input character a, M
  moves to state d(q,a)
• If its current state q is in A, M is said to have
  accepted the string read so far
• An input string that is not accepted is said to
  be rejected

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Example

• Q 0,1, q0 0, A1, ? a, b
• d(q,a) shown in the transition table/diagram
• This accepts strings that end in an odd number of
  as e.g., abbaaa is accepted, aa is rejected

a
input
a
b
state
1
0
0
0
1
b
0
0
1
a
transition table
b
transition diagram
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String-Matching Automata

• Given the pattern P 1..m, build a finite
  automaton M
• The state set is Q0, 1, 2, , m
• The start state is 0
• The only accepting state is m
• Time to build M can be large if ? is large

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String-Matching Automata contd

• Scan the text string T 1..n to find all
  occurrences of the pattern P 1..m
• String matching is efficient T(n)
• Each character is examined exactly once
• Constant time for each character
• But time to compute d is O(m ?)
• d Has O(m ? ) entries

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Algorithm

• Input Text string T 1..n, d and m
• Result All valid shifts displayed
• FINITE-AUTOMATON-MATCHER (T, m, d)
• n ? lengthT
• q ? 0
• for i ? 1 to n
• q ? d (q, T i)
• if q m
• print pattern occurs with shift i-m

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Knuth-Morris-Pratt (KMP) Method

• Avoids computing d (transition function)
• Instead computes a prefix function p in O(m) time
• p has only m entries
• Prefix function stores info about how the pattern
  matches against shifts of itself
• Can avoid testing useless shifts

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Terminology/Notations

• String w is a prefix of string x, if xwy for
  some string y (e.g., srilan of srilanka)
• String w is a suffix of string x, if xyw for
  some string y (e.g., anka of srilanka)
• The k-character prefix of the pattern P
  1..m denoted by Pk
• E.g., P0 e, Pm P P 1..m

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Prefix Function for a Pattern

• Given that pattern prefix P 1..q matches text
  characters T (s1)..(sq), what is the least
  shift s gt s such that
• P 1..k T (s1)..(sk) where sksq?
• At the new shift s, no need to compare the first
  k characters of P with corresponding characters
  of T
• Since we know that they match

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Prefix Function Example 1
b
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Pq
Compare pattern against itself longest prefix of
P that is also a suffix of P5 is P3 so p5 3
Pk
a
b
a
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Prefix Function Example 2
i 1 2 3 4 5 6 7 8 9 10
P i a b a b a b a b c a
pi 0 0 1 2 3 4 5 6 0 1
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Knuth-Morris-Pratt (KMP) Algorithm

• Information stored in prefix function
• Can speed up both the naïve algorithm and the
  finite-automaton matcher
• KMP Algorithm on the board
• 2 parts KMP-MATCHER, PREFIX
• Running time
• PREFIX takes O(m)
• KMP-MATCHER takes O(mn)

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Greedy Approach to Algorithm Design
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Introduction

• Greedy methods typically apply to optimization
  problems in which a set of choices must be made
  to arrive at an optimal solution
• Optimization problem
• There can be many solutions
• Each solution has a value
• We wish to find a solution with the optimal
  (minimum or maximum) value

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Example Optimization Problems

• How to give a balance in minimum number of coins?
• How to allocate resources to maximize profit from
  your business?
• A thief has a knapsack of capacity c what items
  to put in it to maximize profit?
• 0-1 knapsack problem (binary choice)
• Fractional knapsack problem

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Greedy Approach

• Make each choice in a locally optimal manner
• Always makes the choice that looks best at the
  moment
• We hope that this will lead to a globally optimal
  solution
• Greedy method doesnt always give optimal
  solutions, but for many problems it does

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Example

• A cashier gives change using coins of Rs.10, 5, 2
  and 1
• Suppose the amount is Rs. 37
• Need to minimize the number of coins
• Try to use the largest coin to cover the
  remaining balance
• So, we get 10 10 10 5 2
• Does this give the optimal solution?

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Elements of Greedy Approach

• Greedy-choice property
• A globally optimal solution can be arrived at by
  making a locally optimal (greedy) choice
• Proving this may not be trivial
• Optimal substructure
• Optimal solution to the problem contains within
  it optimal solutions to subproblems

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Applications of Greedy Approach

• Graph algorithms
• Minimum spanning tree
• Shortest path
• Data compression
• Huffman coding
• Activity selection (scheduling) problems
• Fractional knapsack problem
• Not the 0-1 knapsack problem

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Announcements

• Assignment 4
• assigned today
• due next week
• Next 2 lectures
• Topic Graphs
• By Ms Sudanthi Wijewickrema