(les09)-stringsearch

7/25/2019 (les09)-stringsearch

1/32

Algoritmen & Datastructuren

2012 2013Substring Search

(Slides by Sedgewick)

Philip Dutr & Benedict Brown

Dept. of Computer Science, K.U.Leuven


2/32

Copyright Ph.Dutr, Spring 20132

Goal: Find pattern of length M in a text of length N

Usually M


3/32

Applications


Parsers

Spam filters

Digital libraries

Word processors

Web search engines

Natural language processing

Computational molecular biology

Feature detection in digitized images


4/32

Application: spam filtering


Identify patterns indicative

of spam.

PROFITS L0SE WE1GHT

herbal V1AGRA

There is no catch L0W M0RTGAGE RATES

This is a one-time mailing.

This message is sent in

compliance with spam

regulations


5/32

Application: electronic surveillance


Look for certain phrases in email traffic

Detonate Bomb

Attack Belgians Ik zal een taart in het gezicht van Leonard werpen

Huge amount of data, incoming streams, no backup


6/32

Application: page scraping


Goal. Extract relevant data from web page.

E.g. Find string delimited by and after first

occurrence of pattern Last Trade


7/32

Brute force


Check all possible starting positions,

check entire length of pattern


8/32

Brute force



9/32

Brute force: worst case


Brute-force algorithm can be slow if text and patternare repetitive

Worst case: ~ M.N character comparisons M.(N-M+1) = M.N M.M + M ~ M.N if M


10/32

Challenges


In many applications, backup in data to be avoided

Solution 1: maintain buffer (memory!)

Solution 2: avoid backup

Challenge: we want linear time complexity!


11/32

Brute force: alternate implementation


Same sequence of character compares

i points to end of sequence of already-matched chars in text

j stores number of already-matched chars(end of sequence in pattern)


12/32

Knuth-Morris-Pratt (KMP)


Intuition: Suppose we are searching in text for pattern

BAAAAAAAAA (alphabet = {A,B})

Suppose we match 5 chars, with mismatch on 6th char We know previous 6 characters in text are BAAAAB

Don't need to back up text pointer!


13/32

Deterministic finite state automaton (DFA)


DFA is abstract string-searching machine.

Finite number of states (including start and halt)

Exactly one transition for each char in alphabet Accept if sequence of transitions leads to halt state


14/32

Interpretation of Knuth-Morris-Pratt DFA


Interpretation of DFA state after reading in txt[i]?

State-number = number of characters in pattern that have

been matched. length of longest prefix of pat[] that is at end of txt[0..i]

E.g. DFA is in state 3 after reading in character txt[6]


15/32

KMP trace



16/32

KMP Implementation


Differences from brute-force implementation:

Text pointer i never decrements (no backup needed)

Need to precompute dfa[][] from pattern (some work)

Running time Simulate DFA on text: at most N character accesses

Build DFA: how to do efficiently?


17/32

How to build DFA from pattern?


Match transition

If in state j and next char c == pat.charAt(j),then go to state j+1.

first j characters of pattern

have already been matched next char matches

now first j+1 characters of

pattern have been matched


18/32

How to build DFA from pattern?


Mismatch transition

If in state j and next char c != pat.charAt(j), then the last j characters

of input are pat[1..j-1], followed by c

To compute dfa[c][j]:

Simulate pat[1..j-1] on DFA (so far ) and take transition c

Running time: Seems to require j steps

Simulation does not start from scratch: remember state X


19/32



20/32



21/32

KMP Analysis


KMP accesses no more than M + N chars

M for constructing DFA

N for actual search

KMP constructs dfa[][] in time and space

proportional to R.M R = size of alphabet


22/32

The Art of Computer Programming


"If you think you're a really good programmer . . . read

(Knuth's) Art of Computer Programming . . . You shoulddefinitely send me a rsum if you can read the whole

thing. (Bill Gates)

Donald

Knuth


23/32

Boyer-Moore


Intuition:

Scan characters in pattern from right to left

Can skip M text chars when finding one not in the pattern


24/32

Boyer-Moore


How much to skip?

right[c] = rightmost occurrence of character c in pattern

-1 for characters not in pattern


25/32

Boyer-Moore



26/32

Boyer-Moore



27/32

Boyer-Moore Analysis


~ N/M character compares

(characters in pattern < characters in alphabet)

Worst-case: ~ M.N

KMP-like rules to avoid repetition linear ~ N


28/32

Rabin-Karp


Basic idea = modular hashing

Compute a hash of pattern characters 0 to M-1

For each i, compute a hash of text characters i to M+i-1 If pattern hash = text substring hash, check for a match


29/32

Rabin-Karp


Intuition: M-digit, base-R integer, modulo Q

ti = txt.charAt(i)

Horner's method

Linear-time method to evaluate degree-M polynomial


30/32

Rabin-Karp


Can update hash function in constant time!


31/32

Rabin-Karp



32/32

Rabin-Karp


Las Vegas variant

If hash is equal, then check pattern

100% correct Extremely likely to run in linear time (but worst case

M.N)

Monte Carlo variant

If hash is equal, we assume pattern is equal as well

Always runs in linear time

Extremely likely to return correct answer

Probability 1/Q choose large Q

Documents

(les09)-stringsearch