Morris-Pratt algorithm

1
Morris-Pratt algorithm
A linear pattern-matching algorithm, Technical
Report 40, University of California, Berkeley,
1970.
Morris (Jr) J. H., Pratt V. R.

• Advisor Prof. R. C. T. Lee
• Reporter C. S. Ou

2
Morris-Pratt algorithm
We are given a text T and a pattern P to find all
occurrences of P in T and perform the comparisons
from left to right.
n the length of T m the length of P
Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
t A A A A A A T C A C A T T A G C A A A A
p A T C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
3
Rule 1 The Partial Window Rule
This rule means that instead of a complete window
whose is equal to the size of the pattern, we may
use a prefix of a complete window to match the
prefix of a prefix of the complete pattern.
A complete window
T
P
How do we get the partial window?
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The basic principle of MP Algorithm is still step
by step comparison. Initially, the length of the
partial window is 1. Initially, we compare T(1)
with P(1). If T(1) ? P(1), we move The pattern
one step towards the right.
Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A A A A A A T C A C A T T A G C A A A A
P C T C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
P C T C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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If T(1)P(1), we extend the partial window until
a mismatching is found. Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A T C A C A G C A C A T T A G C A A A A
P A T C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Suppose the following condition occurs, should we
move pattern P only one step towards the
right? The answer is no in this case as we may
use Rule 2, the suffix of T to prefix of P rule.
j
ij-1
jm-1
1
n
T
b
i
m
1
P
a
Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
t A A A A A A T C A C A T T A G C A A A A
p A T C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Rule 2 The Suffix of T to Prefix of P Rule

• For a window to have any chance to match a
  pattern, in some way, there must be a suffix of
  the window which is equal to a prefix of the
  pattern.

T
P
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The Implication of Rule 2
Find the longest suffix v of the window which is
equal to some prefix of P. Skip the pattern as
follows
T
v
P
v
P
v
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Now, we know that a prefix U of T is equal to a
prefix U of P. Thus, instead of finding the
longest suffix of T equal to a prefix of P, We
may simply find the longest suffix of U of P
which is equal to a prefix of P.
T
U
b
P
U
a
v
Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A A A A A C A C A C A T T A G C A A A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
t A A A A A C A C A C A T T A G C A A A A
p C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
In this case, we can see the longest suffix of U
which is equal to a prefix of P is CA. Thus, we
may apply Rule 2 to move P as follows
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
t A A A A A C A C A C A T T A G C A A A A
p C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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The MP Algorithm

• Assume that we have already found the largest
  prefix of T which is equal to a prefix of P.

t
U
b
p
U
a
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The MP Algorithm

• Skip the pattern by using Rule 1 and Rule 2.

T
v
b
P
v
a
v
c
T
v
b
P
c
v
Given a prefix U of T which is equal to a prefix
of P, how do we know the longest Suffix of U
which is equal to some prefix of U? We do this by
pre-processing.
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Preprocessing phase
for x gt 1 and
The prefix function
Let
f(j), 2 j m, for P( j) can be written as
follows
MP algorithm uses j g(j) 1 to decide the
distance that pattern P aligns in text T.
Example

p A T C A C A T C A T C A
1 2 3 4 5 6 7 8 9 10 11 12 13
prefix function 1 2 3 4 5 6 7 8 9 10
11 12 13 0 0 0 1 0 1 2 3 4 2 3 4
-1 0 0 0 1 0 1 2 3 4 2 3 4 1
1 2 3 3 5 5 5 5 5 8 8 8
j
f(j)
g(j)
j - g(j)
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Example

p A T C A C A T C A T C A
1 2 3 4 5 6 7 8 9 10 11 12 13
prefix function 1 2 3 4 5 6 7 8 9 10
11 12 13 0 0 0 1
j
f(j)
j 1 ?f(1) 0 j 2 ?P2 T? Pf
1(2-1)1P1A ?f(2)0 j 3 ? P3 C? Pf
1(3-1)1P1A ?f(3)0 j 4 ?P4 A Pf
1(4-1)1P1A ?f(4)011
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Example

p A T C A C A T C A T C A
1 2 3 4 5 6 7 8 9 10 11 12 13
prefix function 1 2 3 4 5 6 7 8 9 10
11 12 13 0 0 0 1 0 1 2 3 4
j
f(j)
j 5 ?P5 C? Pf 1(5-1)1P11T ?f(5)0 j
6 ? P6 A Pf 1(6-1)1P1A ?f(6)011
j 7 ? P7 T Pf 1(7-1)1P11T
?f(7)112 j 8 ? P8 C Pf
1(8-1)1P21C ?f(8)213 j 9 ? P9 A
Pf 1(9-1)1P31A ?f(9)314
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Example
p A T C A C A T C A T C A
1 2 3 4 5 6 7 8 9 10 11 12 13
prefix function 1 2 3 4 5 6 7 8 9 10
11 12 13 0 0 0 1 0 1 2 3 4 2 3 4
j
f(j)
We have found that f(9) 4. We now check whether
P(10)P(5) . The answer is no. Does this mean
that we should set f(9) to be 0? No.
j 10 ?P10 T? Pf 2(10-1)1Pf
(4)1P11P2T ?f(10)112 j 11 ? P11
C Pf 1(11-1)1P21C ?f(11)213 j 12
? P12 A Pf 1(12-1)1P31T ?f(12)314
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Then, after a shift, the comparisons can resume
between characters c P(f(i )) and T( i j) b
without missing any occurrence of P in T, and
avoiding a backtrack on the text.
ij-1
jm-1
1
n
T
u
b
i
m
1
P
u
a
v
P
Example
a
v
c
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A A A A A C A C A C A T T A G C A A A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 13 1 1 2
2 2 2 2 7 8 9 10 10 10
prefix function
Shift by 1
j
j - g(j)-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 13 1 1 2
2 2 2 2 7 8 9 10 10 10
prefix function
Shift by 2
j
j - g(j)-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 13 1 1 2
2 2 2 2 7 8 9 10 10 10
prefix function
Shift by 1
j
j - g(j)-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 13 1
1 2 2 2 2 2 7 8 9 10 10 10
prefix function
Shift by 1
j
j - g(j)-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 13 1
1 2 2 2 2 2 7 8 9 10 10 10
prefix function
Shift by 1
j
j - g(j)-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Example
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 13 1
1 2 2 2 2 2 7 8 9 10 10 10
prefix function
Shift by 1
j
j - g(j)-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Example
MATCH
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 13 1
1 2 2 2 2 2 7 8 9 10 10 10
Shift by 10
prefix function
j
j - g(j)-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T A C A C G T A C A C A C A G T A T C A A
P C A C A C A G T A T C A
1 2 3 4 5 6 7 8 9 10 11 12
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Time Complexity
preprocessing phase in O(m) space and time
complexity searching phase in O(nm) time
complexity
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References
AHO, A.V., HOPCROFT, J.E., ULLMAN, J.D., 1974,
The design and analysis of computer algorithms,
2nd Edition, Chapter 9, pp. 317--361,
Addison-Wesley Publishing Company. BEAUQUIER,
D., BERSTEL, J., CHRÉTIENNE, P., 1992, Éléments
d'algorithmique, Chapter 10, pp 337-377, Masson,
Paris. CROCHEMORE, M., 1997. Off-line serial
exact string searching, in Pattern Matching
Algorithms, ed. A. Apostolico and Z. Galil,
Chapter 1, pp 1-53, Oxford University Press.
HANCART, C., 1992, Une analyse en moyenne de
l'algorithme de Morris et Pratt et de ses
raffinements, in Théorie des Automates et
Applications, Actes des 2e Journées
Franco-Belges, D. Krob ed., Rouen, France, 1991,
PUR 176, Rouen, France, 99-110. HANCART, C.,
1993. Analyse exacte et en moyenne d'algorithmes
de recherche d'un motif dans un texte, Ph. D.
Thesis, University Paris 7, France. MORRIS (Jr)
J.H., PRATT V.R., 1970, A linear pattern-matching
algorithm, Technical Report 40, University of
California, Berkeley.
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Thanks for your attention.