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CS222 Algorithms First Semester 2003/2004

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Title: CS222 Algorithms Lecture 7 String Matching 2 + Greedy Approach Author: Sanath Jayasena Last modified by: Sanath Jayasena Created Date: 9/7/2003 3:36:19 PM – PowerPoint PPT presentation

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Title: CS222 Algorithms First Semester 2003/2004


1
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

2
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

3
String Matching
  • PART 2

4
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

5
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

6
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
7
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

8
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

9
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

10
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

11
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

12
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

13
Prefix Function Example 1
b
a
c
b
a
b
a
b
a
a
b
c
b
a
T
s
a
b
a
b
a
c
a
P
q
b
a
c
b
a
b
a
b
a
a
b
c
b
a
T
s
a
b
a
b
a
c
a
P
k
a
b
a
b
a
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
14
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
15
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)

16
Greedy Approach to Algorithm Design
17
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

18
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

19
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

20
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?

21
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

22
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

23
Announcements
  • Assignment 4
  • assigned today
  • due next week
  • Next 2 lectures
  • Topic Graphs
  • By Ms Sudanthi Wijewickrema
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