A superglue for string comparison

A superglue for string comparison
Alexander Tiskin
Department of Computer Science, University of Warwick
http://go.warwick.ac.uk/alextiskin
Alexander Tiskin (Warwick) Semi-local LCS 1 / 164

1 Introduction
2 Matrix distance multiplication
3 Semi-local string comparison
4 The seaweed method
5 Periodic string comparison
6 Sparse string comparison
7 Compressed string comparison
8 Parallel string comparison
9 The transposition network
method
10 Beyond semi-locality

1 Introduction
method

Introduction
String matching: ﬁnding an exact pattern in a string
String comparison: ﬁnding similar patterns in two strings

Introduction
global: whole string a vs whole string b
semi-local: whole string a vs substrings in b (approximate matching);
preﬁxes in a vs suﬃxes in b
local: substrings in a vs substrings in b

Introduction
global: whole string a vs whole string b
semi-local: whole string a vs substrings in b (approximate matching);
preﬁxes in a vs suﬃxes in b
local: substrings in a vs substrings in b
We focus on semi-local comparison:
fundamentally important for both global and local comparison
exciting mathematical properties

Introduction
Overview
Standard approach to string comparison: dynamic programming
Our approach: the algebra of seaweed matrices/braids
Can be used either for dynamic programming, or for divide-and-conquer
Divide-and-conquer is more eﬃcient for:
comparing dynamic strings (truncation, concatenation)
comparing compressed strings (e.g. LZ-compression)
comparing strings in parallel
The “conquer” step involves a magic “superglue” (eﬃcient multiplication
of seaweed matrices/braids)

Introduction
Terminology and notation
The “squared paper” notation:
Integers {. . . − 2, −1, 0, 1, 2, . . .} x− = x − 1
2 x+ = x + 1
2
Half-integers . . . − 3
2, −1
2, 1
2, 3
2, 5
2, . . . = . . . (−2)+, (−1)+, 0+, 1+, 2+
Planar dominance:
(i, j) (i , j ) iﬀ i < i and j < j (the “above-left” relation)
(i, j) (i , j ) iﬀ i > i and j < j (the “below-left” relation)
where “above/below” follow matrix convention (not the Cartesian one!)

Introduction
A permutation matrix is a 0/1 matrix with exactly one nonzero per row
and per column


0 1 0
1 0 0
0 0 1



Introduction
Given matrix D, its distribution matrix is made up of -dominance sums:
DΣ(i, j) = ˆı>i,ˆ<j D(i, j)

Introduction
Given matrix E, its density matrix is made up of quadrangle diﬀerences:
E (ˆı, ˆ) = E(ˆı−, ˆ+) − E(ˆı−, ˆ−) − E(ˆı+, ˆ+) + E(ˆı+, ˆ−)
where DΣ, E over integers; D, E over half-integers

Introduction


0 1 0
1 0 0
0 0 1


Σ
=




0 1 2 3
0 1 1 2
0 0 0 1
0 0 0 0








0 1 2 3
0 1 1 2
0 0 0 1
0 0 0 0



 =


0 1 0
1 0 0
0 0 1



Introduction


0 1 0
1 0 0
0 0 1


Σ
=




0 1 2 3
0 1 1 2
0 0 0 1
0 0 0 0








0 1 2 3
0 1 1 2
0 0 0 1
0 0 0 0



 =


0 1 0
1 0 0
0 0 1


(DΣ) = D for all D
Matrix E is simple, if (E )Σ = E; equivalently, if it has all zeros in the left
column and bottom row

Introduction
Matrix E is Monge, if E is nonnegative
Intuition: boundary-to-boundary distances in a (weighted) planar graph
G. Monge (1746–1818)
G (planar)
i
j
0
1 2 3
4
0
1 2 3
4

Introduction
Matrix E is Monge, if E is nonnegative
Intuition: boundary-to-boundary distances in a (weighted) planar graph
G. Monge (1746–1818)
i
j
0
1 2 3
4
0
1 2 3
4
blue + blue ≤ red + red

Introduction
Matrix E is unit-Monge, if E is a permutation matrix
Intuition: boundary-to-boundary distances in a grid-like graph (in
particular, an alignment graph for a pair of strings)

Introduction
Matrix E is unit-Monge, if E is a permutation matrix
Intuition: boundary-to-boundary distances in a grid-like graph (in
particular, an alignment graph for a pair of strings)
Simple unit-Monge matrix (sometimes called “rank function”): PΣ, where
P is a permutation matrix
Seaweed matrix: P used as an implicit representation of PΣ


0 1 0
1 0 0
0 0 1


Σ
=




0 1 2 3
0 1 1 2
0 0 0 1
0 0 0 0





Introduction
Implicit unit-Monge matrices
Eﬃcient PΣ queries: range tree on nonzeros of P [Bentley: 1980]
binary search tree by i-coordinate
under every node, binary search tree by j-coordinate
•
•
•
•
−→ •
•
•
•
−→ •
•
•
•
↓
•
•
•
•
−→ •
•
•
•
−→ •
•
•
•
↓
•
•
•
•
−→ •
•
•
•
−→ •
•
•
•

Introduction
Eﬃcient PΣ queries: (contd.)
Every node of the range tree represents a canonical range (rectangular
region), and stores its nonzero count
Overall, ≤ n log n canonical ranges are non-empty
A PΣ query is equivalent to -dominance counting: how many nonzeros
are -dominated by query point?
Answer: sum up nonzero counts in ≤ log2
n disjoint canonical ranges
Total size O(n log n), query time O(log2
n)

Introduction
Efficient PΣ queries: (contd.)
Every node of the range tree represents a canonical range (rectangular
region), and stores its nonzero count
Overall, ≤ n log n canonical ranges are non-empty
A PΣ query is equivalent to -dominance counting: how many nonzeros
are -dominated by query point?
Answer: sum up nonzero counts in ≤ log2
n disjoint canonical ranges
Total size O(n log n), query time O(log2
n)
There are asymptotically more efficient (but less practical) data structures
Total size O(n), query time O log n
log log n [JáJá+: 2004]
[Chan, Pˇatra¸scu: 2010]

1 Introduction
method

Matrix distance multiplication
Seaweed braids
Distance semiring (or (min, +)-semiring, special case of tropical algebra):
addition ⊕ given by min, multiplication given by +
Matrix -multiplication
A B = C C(i, k) = j A(i, j) B(j, k) = minj A(i, j) + B(j, k)
Intuition: gluing distances in a composition of planar graphs
i
G
j
G
k

Seaweed braids
Matrix classes closed under -multiplication (for given n):
general (integer, real) matrices ∼ general weighted graphs
Monge matrices ∼ planar weighted graphs
simple unit-Monge matrices ∼ grid-like graphs
Implicit -multiplication: deﬁne P Q = R as PΣ QΣ = RΣ
The seaweed monoid Tn:
simple unit-Monge matrices under
equivalently, permutation (seaweed) matrices under
Isomorphic to the 0-Hecke monoid of the symmetric group H0(Sn)

Seaweed braids
P Q = R can be seen as combing of seaweed braids
P Q
=
R

Seaweed braids
P Q
=
R
P
Q
=

Seaweed braids
P Q
=
R
P
Q
= =

Seaweed braids
P Q
=
R
P
Q
= = R

Seaweed braids
The seaweed monoid Tn:
n! elements (permutations of size n)
n − 1 generators g1, g2, . . . , gn−1 (elementary crossings)
Idempotence:
g2
i = gi for all i =
Far commutativity:
gi gj = gj gi j − i > 1 · · · = · · ·
Braid relations:
gi gj gi = gj gi gj j − i = 1 =

Seaweed braids
Special elements of the seaweed monoid Tn
Identity: 1 x = x
1 = =




• · · ·
· • · ·
· · • ·
· · · •




Zero: 0 x = 0
0 = =




· · · •
· · • ·
· • · ·
• · · ·





Seaweed braids
Seaweed monoid: g2
i = gi ; far comm; braid
Related structures:
classical braid group: gi g−1
i = 1; far comm; braid
Sn (Coxeter presentation): g2
positive braid monoid: far comm; braid
locally free idempotent monoid: g2
i = gi ; far comm
nil-Hecke monoid: g2
Generalisations:
0-Hecke monoid of a Coxeter group
Hecke–Kiselman monoid
J -trivial monoid

Seaweed braids
Computation in the seaweed monoid: a conﬂuent rewriting system can be
obtained by software (Semigroupe, GAP, Sage)

Seaweed braids
T3: 1, a = g1, b = g2; ab, ba; aba = 0
aa → a bb → b bab → 0 aba → 0

Seaweed braids
T3: 1, a = g1, b = g2; ab, ba; aba = 0
T4: 1, a = g1, b = g2, c = g3; ab, ac, ba, bc, cb, aba, abc, acb, bac,
bcb, cba, abac, abcb, acba, bacb, bcba, abacb, abcba, bacba; abacba = 0
aa → a
bb → b
ca → ac
cc → c
bab → aba
cbc → bcb
cbac → bcba
abacba → 0

Seaweed braids
T3: 1, a = g1, b = g2; ab, ba; aba = 0
T4: 1, a = g1, b = g2, c = g3; ab, ac, ba, bc, cb, aba, abc, acb, bac,
bcb, cba, abac, abcb, acba, bacb, bcba, abacb, abcba, bacba; abacba = 0
aa → a
bb → b
ca → ac
cc → c
bab → aba
cbc → bcb
cbac → bcba
abacba → 0
Easy to use as a “black box”, but not eﬃcient

Seaweed matrix multiplication
Seaweed matrix -multiplication
Given permutation matrices P, Q, compute R, such that PΣ QΣ = RΣ
(equivalently, P Q = R)

Seaweed matrix -multiplication
Given permutation matrices P, Q, compute R, such that PΣ QΣ = RΣ
(equivalently, P Q = R)
Seaweed matrix -multiplication: running time
type time
general O(n3) standard
O n3(log log n)3
log2
n
[Chan: 2007]
Monge O(n2) via [Aggarwal+: 1987]
implicit unit-Monge O(n1.5) [T: 2006]
O(n log n) [T: 2010]

P
Q
R
?

Plo, Phi
Qlo, Qhi

Plo, Phi
Qlo, Qhi
Rlo + Rhi

Plo, Phi
Qlo, Qhi
R

P
Q
R

Seaweed matrix -multiplication: Steady Ant algorithm
RΣ(i, k) = minj PΣ(i, j) + QΣ(j, k)
Divide-and-conquer on the range of j: two subproblems of size n/2
PΣ
lo QΣ
lo = RΣ
lo PΣ
hi QΣ
hi = RΣ
hi
Conquer: tracing a balanced trail from bottom-left to top-right R
Trail invariant: equal number of -dominated nonzeros in PC,hi and
-dominating nonzeros in PC,lo
All such nonzeros are “errors” in the output, must be removed
Must compensate by placing nonzeros on the path, time O(n)
Overall time O(n log n)

Bruhat order
Bruhat order
Permutation A is lower (“more sorted”) than permutation B in the Bruhat
order (A B), if B A by successive pairwise sorting (equivalently,
A B by anti-sorting) of arbitrary pairs
Permutation matrices: P Q, if Q P by successive 2 × 2 submatrix
sorting: ( 0 1
1 0 ) → ( 1 0
0 1 )

Bruhat order
Bruhat order
Permutation A is lower (“more sorted”) than permutation B in the Bruhat
order (A B), if B A by successive pairwise sorting (equivalently,
A B by anti-sorting) of arbitrary pairs
Permutation matrices: P Q, if Q P by successive 2 × 2 submatrix
sorting: ( 0 1
1 0 ) → ( 1 0
0 1 )
Plays an important role in group theory and algebraic geometry (inclusion
order of Schubert varieties)
Describes pivoting order in Gaussian elimination (matrix Bruhat
decomposition)

Bruhat order
Bruhat comparability: running time
O(n2) [Ehresmann: 1934; Proctor: 1982; Grigoriev: 1982]
O(n log n) [T: 2013]
O n log n
log log n [Gawrychowski: NEW]

Bruhat order
Ehresmann’s criterion (dot criterion, related to tableau criterion)
P Q iﬀ PΣ ≤ QΣ elementwise


1 0 0
0 0 1
0 1 0


Σ
=




0 1 2 3
0 0 1 2
0 0 1 1
0 0 0 0



 ≤




0 1 2 3
0 1 2 2
0 0 1 1
0 0 0 0



 =


0 0 1
1 0 0
0 1 0


Σ


1 0 0
0 0 1
0 1 0


Σ
=




0 1 2 3
0 0 1 2
0 0 1 1
0 0 0 0



 ≤




0 1 2 3
0 1 1 2
0 0 0 1
0 0 0 0



 =


0 1 0
1 0 0
0 0 1


Σ
Time O(n2)

Bruhat order
Seaweed criterion
P Q iﬀ PR Q = IdR
, where PR = counterclockwise rotation of P
Intuition: permutations represented by seaweed braids
P Q, iﬀ
no pair of seaweeds is crossed in P, while the “corresponding” pair is
uncrossed in Q
equivalently, no pair is uncrossed in PR, while the “corresponding”
pair is uncrossed in Q
equivalently, PR Q = IdR
Time O(n log n) by seaweed matrix multiplication

Bruhat order


1 0 0
0 0 1
0 1 0




0 0 1
1 0 0
0 1 0




1 0 0
0 0 1
0 1 0


R 

0 0 1
1 0 0
0 1 0

 =


0 1 0
0 0 1
1 0 0




0 0 1
1 0 0
0 1 0

 =


0 0 1
0 1 0
1 0 0

 = IdR
P

Bruhat order


1 0 0
0 0 1
0 1 0




0 0 1
1 0 0
0 1 0




1 0 0
0 0 1
0 1 0


R 

0 0 1
1 0 0
0 1 0

 =


0 1 0
0 0 1
1 0 0




0 0 1
1 0 0
0 1 0

 =


0 0 1
0 1 0
1 0 0

 = IdR
P Q

Bruhat order


1 0 0
0 0 1
0 1 0




0 0 1
1 0 0
0 1 0




1 0 0
0 0 1
0 1 0


R 

0 0 1
1 0 0
0 1 0

 =


0 1 0
0 0 1
1 0 0




0 0 1
1 0 0
0 1 0

 =


0 0 1
0 1 0
1 0 0

 = IdR
P Q
PR
Q
=

Bruhat order


1 0 0
0 0 1
0 1 0




0 0 1
1 0 0
0 1 0




1 0 0
0 0 1
0 1 0


R 

0 0 1
1 0 0
0 1 0

 =


0 1 0
0 0 1
1 0 0




0 0 1
1 0 0
0 1 0

 =


0 0 1
0 1 0
1 0 0

 = IdR
P Q
PR
Q
= IdR

Bruhat order


1 0 0
0 0 1
0 1 0




0 1 0
1 0 0
0 0 1




1 0 0
0 0 1
0 1 0


R 

0 1 0
1 0 0
0 0 1

 =


0 1 0
0 0 1
1 0 0




0 1 0
1 0 0
0 0 1

 =


0 1 0
0 0 1
1 0 0

 = IdR
P

Bruhat order


1 0 0
0 0 1
0 1 0




0 1 0
1 0 0
0 0 1




1 0 0
0 0 1
0 1 0


R 

0 1 0
1 0 0
0 0 1

 =


0 1 0
0 0 1
1 0 0




0 1 0
1 0 0
0 0 1

 =


0 1 0
0 0 1
1 0 0

 = IdR
P Q

Bruhat order


1 0 0
0 0 1
0 1 0




0 1 0
1 0 0
0 0 1




1 0 0
0 0 1
0 1 0


R 

0 1 0
1 0 0
0 0 1

 =


0 1 0
0 0 1
1 0 0




0 1 0
1 0 0
0 0 1

 =


0 1 0
0 0 1
1 0 0

 = IdR
P Q
PR
Q
=

Bruhat order


1 0 0
0 0 1
0 1 0




0 1 0
1 0 0
0 0 1




1 0 0
0 0 1
0 1 0


R 

0 1 0
1 0 0
0 0 1

 =


0 1 0
0 0 1
1 0 0




0 1 0
1 0 0
0 0 1

 =


0 1 0
0 0 1
1 0 0

 = IdR
P Q
PR
Q
= = IdR

Bruhat order
Alternative solution: clever implementation of Ehresmann’s criterion
[Gawrychowski: 2013]
The online partial sums problem: maintain array X[1 : n], subject to
update(k, ∆): X[k] ← X[k] + ∆
preﬁxsum(k): return 1≤i≤k X[i]
Query time:
Θ(log n) in semigroup or group model
Θ log n
log log n in RAM model on integers [Pˇatra¸scu, Demaine: 2004]
Gives Bruhat comparability in time O n log n
log log n in RAM model
Open problem: seaweed multiplication in time O n log n
log log n ?

1 Introduction
method

Semi-local string comparison
Semi-local LCS and edit distance
Consider strings (= sequences) over an alphabet of size σ
Contiguous substrings vs not necessarily contiguous subsequences
Special cases of substring: preﬁx, suﬃx
Notation: strings a, b of length m, n respectively
Assume where necessary: m ≤ n; m, n reasonably close

Consider strings (= sequences) over an alphabet of size σ
Contiguous substrings vs not necessarily contiguous subsequences
Special cases of substring: prefix, suffix
Notation: strings a, b of length m, n respectively
Assume where necessary: m ≤ n; m, n reasonably close
The longest common subsequence (LCS) score:
length of longest string that is a subsequence of both a and b
equivalently, alignment score, where score(match) = 1 and
score(mismatch) = 0
In biological terms, “loss-free alignment” (unlike efficient but “lossy”
BLAST)

The LCS problem
Give the LCS score for a vs b

The LCS problem
Give the LCS score for a vs b
LCS: running time
O(mn) [Wagner, Fischer: 1974]
O mn
log2
n
σ = O(1) [Masek, Paterson: 1980]
[Crochemore+: 2003]
O mn(log log n)2
log2
n
[Paterson, Danˇc´ık: 1994]
[Bille, Farach-Colton: 2008]
Running time varies depending on the RAM model version
We assume word-RAM with word size log n (where it matters)

LCS computation by dynamic programming
lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
max(lcs(aα, b), lcs(a, bβ)) if α = β
lcs(a, b) + 1 if α = β

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0
E 0
S 0
I 0
G 0
N 0

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1
E 0
S 0
I 0
G 0
N 0

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1
E 0
S 0
I 0
G 0
N 0

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1 1 1 1 1
E 0
S 0
I 0
G 0
N 0

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1 1 1 1 1
E 0 1
S 0
I 0
G 0
N 0

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1 1 1 1 1
E 0 1 2
S 0
I 0
G 0
N 0

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1 1 1 1 1
E 0 1 2 2 2 2 2
S 0
I 0
G 0
N 0

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1 1 1 1 1
E 0 1 2 2 2 2 2
S 0 1 2 2 2 2 2
I 0
G 0
N 0

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1 1 1 1 1
E 0 1 2 2 2 2 2
S 0 1 2 2 2 2 2
I 0 1 2 2 3 3 3
G 0
N 0

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1 1 1 1 1
E 0 1 2 2 2 2 2
S 0 1 2 2 2 2 2
I 0 1 2 2 3 3 3
G 0 1 2 2 3 3 3
N 0 1 2 2 3 4 4

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1 1 1 1 1
E 0 1 2 2 2 2 2
S 0 1 2 2 2 2 2
I 0 1 2 2 3 3 3
G 0 1 2 2 3 3 3
N 0 1 2 2 3 4 4
lcs(“DEFINE”, “DESIGN”) = 4

lcs(a, ∅) = 0
lcs(∅, b) = 0
lcs(aα, bβ) =
∗ D E F I N E
∗ 0 0 0 0 0 0 0
D 0 1 1 1 1 1 1
E 0 1 2 2 2 2 2
S 0 1 2 2 2 2 2
I 0 1 2 2 3 3 3
G 0 1 2 2 3 3 3
N 0 1 2 2 3 4 4
lcs(“DEFINE”, “DESIGN”) = 4
LCS(a, b) can be traced back through
the dynamic programming table at no
extra asymptotic time cost

LCS on the alignment graph: directed grid of match and mismatch cells
B
A
A
B
C
B
C
A
B A A B C A B C A B A C A blue = 0
red = 1
score(“BAABCBCA”, “BAABCABCABACA”) = len(“BAABCBCA”) = 8
LCS = highest-score path from top-left to bottom-right

LCS: dynamic programming [WF: 1974]
Sweep cells in any -compatible order
Cell update: time O(1)
Overall time O(mn)

LCS: micro-block dynamic programming [MP: 1980; BF: 2008]
Sweep cells in micro-blocks, in any -compatible order
Micro-block size:
t = O(log n) when σ = O(1)
t = O log n
log log n otherwise
Micro-block interface:
O(t) characters, each O(log σ) bits, can be reduced to O(log t) bits
O(t) small integers, each O(1) bits
Micro-block update: time O(1), by precomputing all possible interfaces
Overall time O mn
log2
n
when σ = O(1), O mn(log log n)2
log2
n
otherwise

‘Begin at the beginning,’ the King said gravely, ‘and
go on till you come to the end: then stop.’
L. Carroll, Alice in Wonderland

‘Begin at the beginning,’ the King said gravely, ‘and
go on till you come to the end: then stop.’
L. Carroll, Alice in Wonderland
Dynamic programming: begins at empty strings,
proceeds by appending characters, then stops
What about:
prepending/deleting characters (dynamic LCS)
concatenating strings (LCS on compressed
strings; parallel LCS)
taking substrings (= local alignment)

Dynamic programming from both ends:
better by ×2, but still not good enough
Is dynamic programming strictly necessary to
solve sequence alignment problems?
Eppstein+, Eﬃcient algorithms for sequence
analysis, 1991

The semi-local LCS problem
Give the (implicit) matrix of O (m + n)2 LCS scores:
string-substring LCS: string a vs every substring of b
prefix-suffix LCS: every prefix of a vs every suffix of b
suffix-prefix LCS: every suffix of a vs every prefix of b
substring-string LCS: every substring of a vs string b

Semi-local LCS on the alignment graph
B
A
A
B
C
B
C
A
B A A B C A B C A B A C AC A B C A B A blue = 0
red = 1
score(“BAABCBCA”, “CABCABA”) = len(“ABCBA”) = 5
String-substring LCS: all highest-score top-to-bottom paths
Semi-local LCS: all highest-score boundary-to-boundary paths

Score matrices and seaweed matrices
The score matrix H
0 1 2 3 4 5 6 6 7 8 8 8 8 8
-1 0 1 2 3 4 5 5 6 7 7 7 7 7
-2 -1 0 1 2 3 4 4 5 6 6 6 6 7
-3 -2 -1 0 1 2 3 3 4 5 5 6 6 7
-4 -3 -2 -1 0 1 2 2 3 4 4 5 5 6
-5 -4 -3 -2 -1 0 1 2 3 4 4 5 5 6
-6 -5 -4 -3 -2 -1 0 1 2 3 3 4 4 5
-7 -6 -5 -4 -3 -2 -1 0 1 2 2 3 3 4
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 3 4
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
5
a = “BAABCBCA”
b = “BAABCABCABACA”
H(i, j) = score(a, b i : j )
H(4, 11) = 5
H(i, j) = j − i if i > j

Semi-local LCS: output representation and running time
size query time
O(n2) O(1) trivial
O(m1/2n) O(log n) string-substring [Alves+: 2003]
O(n) O(n) string-substring [Alves+: 2005]
O(n log n) O(log2
n) [T: 2006]
. . . or any 2D orthogonal range counting data structure
running time
O(mn2) naive
O(mn) string-substring [Schmidt: 1998; Alves+: 2005]
O(mn) [T: 2006]
O mn
log0.5
n
[T: 2006]
O mn(log log n)2
log2
n
[T: 2007]

The score matrix H and the seaweed matrix P
H(i, j): the number of matched characters for a vs substring b i : j
j − i − H(i, j): the number of unmatched characters
Properties of matrix j − i − H(i, j):
simple unit-Monge
therefore, = PΣ, where P = −H is a permutation matrix
P is the seaweed matrix, giving an implicit representation of H
Range tree for P: memory O(n log n), query time O(log2
n)

0 1 2 3 4 5 6 6 7 8 8 8 8 8
-1 0 1 2 3 4 5 5 6 7 7 7 7 7
-2 -1 0 1 2 3 4 4 5 6 6 6 6 7
-3 -2 -1 0 1 2 3 3 4 5 5 6 6 7
-4 -3 -2 -1 0 1 2 2 3 4 4 5 5 6
-5 -4 -3 -2 -1 0 1 2 3 4 4 5 5 6
-6 -5 -4 -3 -2 -1 0 1 2 3 3 4 4 5
-7 -6 -5 -4 -3 -2 -1 0 1 2 2 3 3 4
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 3 4
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
5
a = “BAABCBCA”
H(4, 11) = 5
H(i, j) = j − i if i > j

0 1 2 3 4 5 6 6 7 8 8 8 8 8
-1 0 1 2 3 4 5 5 6 7 7 7 7 7
-2 -1 0 1 2 3 4 4 5 6 6 6 6 7
-3 -2 -1 0 1 2 3 3 4 5 5 6 6 7
-4 -3 -2 -1 0 1 2 2 3 4 4 5 5 6
-5 -4 -3 -2 -1 0 1 2 3 4 4 5 5 6
-6 -5 -4 -3 -2 -1 0 1 2 3 3 4 4 5
-7 -6 -5 -4 -3 -2 -1 0 1 2 2 3 3 4
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 3 4
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
5
a = “BAABCBCA”
H(4, 11) = 5
H(i, j) = j − i if i > j
blue: diﬀerence in H is 0
red: diﬀerence in H is 1

0 1 2 3 4 5 6 6 7 8 8 8 8 8
-1 0 1 2 3 4 5 5 6 7 7 7 7 7
-2 -1 0 1 2 3 4 4 5 6 6 6 6 7
-3 -2 -1 0 1 2 3 3 4 5 5 6 6 7
-4 -3 -2 -1 0 1 2 2 3 4 4 5 5 6
-5 -4 -3 -2 -1 0 1 2 3 4 4 5 5 6
-6 -5 -4 -3 -2 -1 0 1 2 3 3 4 4 5
-7 -6 -5 -4 -3 -2 -1 0 1 2 2 3 3 4
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 3 4
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
5
a = “BAABCBCA”
H(4, 11) = 5
H(i, j) = j − i if i > j
blue: diﬀerence in H is 0
red: diﬀerence in H is 1
green: P(i, j) = 1
H(i, j) = j − i − PΣ(i, j)

a = “BAABCBCA”
H(4, 11) =
11 − 4 − PΣ(i, j) =
11 − 4 − 2 = 5

The (combed) seaweed braid in the alignment graph
B
A
A
B
C
B
C
A
B A A B C A B C A B A C AC A B C A B A a = “BAABCBCA”
H(4, 11) =
11 − 4 − PΣ(i, j) =
11 − 4 − 2 = 5
P(i, j) = 1 corresponds to seaweed top i bottom j

The (combed) seaweed braid in the alignment graph
B
A
A
B
C
B
C
A
B A A B C A B C A B A C AC A B C A B A a = “BAABCBCA”
H(4, 11) =
11 − 4 − PΣ(i, j) =
11 − 4 − 2 = 5
P(i, j) = 1 corresponds to seaweed top i bottom j
Also deﬁne seaweeds top right, left right, left bottom
Represent implicitly semi-local LCS for each preﬁx of a vs b

Seaweed braid: a highly symmetric object (element of H0(Sn))
Can be built by assembling subbraids: divide-and-conquer
Flexible approach to local alignment, compressed approximate matching,
parallel computation. . .

Weighted alignment
The LCS problem is a special case of the weighted alignment problem
Scoring scheme: match score wM , mismatch score wX , gap score wG
LCS score: wM = 1, wX = wG = 0
Levenshtein score: wM = 2, wX = 1, wG = 0

Weighted alignment
Scoring scheme is rational, if wM , wX , wG are rational numbers

Weighted alignment
Scoring scheme is rational, if wM , wX , wG are rational numbers
Edit distance: minimum cost to transform a into b by weighted character
edits (insertion, deletion, substitution)
Corresponds to a scoring scheme wM = 0:
insertion/deletion cost −wG
substitution cost −wX

Weighted alignment
Weighted alignment graph for a, b
B
A
A
B
C
B
C
A
B A A B C A B C A B A C AC A B C A B A blue = 0
red (solid) = 2
red (dotted) = 1
Levenshtein(“BAABCBCA”, “CABCABA”) = 11

Weighted alignment
Reduction: ordinary alignment graph for blown-up a, b
$B
$A
$A
$B
$C
$B
$C
$A
$B $A $A $B $C $A $B $C $A $B $A $C $A$C $A$B $C $A$B$A blue = 0
red = 1 or 2
Levenshtein(“BAABCBCA”, “CABCABA”) =
lcs(“$B$A$A$B$C$B$C$A”, “$C$A$B$C$A$B$A”) = 11

Weighted alignment
Reduction: rational-weighted semi-local alignment to semi-local LCS
$B
$A
$A
$B
$C
$B
$C
$A
$B $A $A $B $C $A $B $C $A $B $A $C $A$C $A$B $C $A$B$A
Let wM = 1, wX = µ
ν , wG = 0
Increase × ν2 in complexity (can be reduced to ν)

1 Introduction
method

The seaweed method
Seaweed combing
B
A
A
B
C
B
C
A
B A A B C A B C A B A C A

The seaweed method
Seaweed combing
B
A
A
B
C
B
C
A

The seaweed method
Seaweed combing
Semi-local LCS: seaweed combing [T: 2006]
Initialise uncombed seaweed braid: mismatch cell = crossing
match cell: skip (keep uncrossed)
mismatch cell: comb (uncross) iﬀ the same seaweed pair already
crossed before
Overall time O(mn)
Correctness: by seaweed monoid relations

The seaweed method
Micro-block seaweed combing
B
A
A
B
C
B
C
A

The seaweed method
B
A
A
B
C
B
C
A

The seaweed method
Semi-local LCS: micro-block seaweed combing [T: 2007]
Sweep cells in micro-blocks, in any -compatible order
Micro-block size: t = O log n
log log n
Micro-block interface:
O(t) characters, each O(log σ) bits, can be reduced to O(log t) bits
O(t) integers, each O(log n) bits, can be reduced to O(log t) bits
Micro-block update: time O(1), by precomputing all possible interfaces
Overall time O mn(log log n)2
log2
n

The seaweed method
Cyclic LCS
The cyclic LCS problem
Give the maximum LCS score for a vs all cyclic rotations of b

The seaweed method
Cyclic LCS
The cyclic LCS problem
Give the maximum LCS score for a vs all cyclic rotations of b
Cyclic LCS: running time
O mn2
log n naive
O(mn log m) [Maes: 1990]
O(mn) [Bunke, B¨uhler: 1993; Landau+: 1998; Schmidt: 1998]
O mn(log log n)2
log2
n
[T: 2007]

The seaweed method
Cyclic LCS
Cyclic LCS: the algorithm
Micro-block seaweed combing on a vs bb, time O mn(log log n)2
log2
n
Make n string-substring LCS queries, time negligible

The seaweed method
Longest repeating subsequence
The longest repeating subsequence problem
Find the longest subsequence of a that is a square (a repetition of two
identical strings)

The seaweed method
The longest repeating subsequence problem
Find the longest subsequence of a that is a square (a repetition of two
identical strings)
Longest repeating subsequence: running time
O(m3) naive
O(m2) [Kosowski: 2004]
O m2(log log m)2
log2
m
[T: 2007]

The seaweed method
Longest repeating subsequence: the algorithm
Micro-block seaweed combing on a vs a, time O m2(log log m)2
log2
m
Make m − 1 preﬁx-suﬃx LCS queries, time negligible
Open question: generalise to longest subsequence that is a cube, etc.

The seaweed method
Approximate matching
The approximate pattern matching problem
Give the substring closest to a by alignment score, starting at each
position in b
Assume rational scoring scheme
Approximate pattern matching: running time
O(mn) [Sellers: 1980]
O mn
log n σ = O(1) via [Masek, Paterson: 1980]
O mn(log log n)2
log2
n
via [Bille, Farach-Colton: 2008]

The seaweed method
Approximate matching
Approximate pattern matching: the algorithm
Micro-block seaweed combing on a vs b (with blow-up), time
O mn(log log n)2
log2
n
The implicit semi-local edit score matrix:
an anti-Monge matrix
approximate pattern matching ∼ row minima
Row minima in O(n) element queries [Aggarwal+: 1987]
Each query in time O(log2
n) using the range tree representation,
combined query time negligible
Overall running time O mn(log log n)2
log2
n
, same as [Bille, Farach-Colton: 2008]

1 Introduction
method

Periodic string comparison
Wraparound seaweed combing
The periodic string-substring LCS problem
Give (implicit) LCS scores for a vs each substring of b = . . . uuu . . . = u±∞
Let u be of length p
May assume that every character of a occurs in u (otherwise delete it)
Only substrings of b of length ≤ mp (otherwise LCS score = m)

B
A
A
B
C
B
C
A

Periodic string-substring LCS: Wraparound seaweed combing
Sweep cells row-by-row: each row starts at match cell, wraps at boundary
match cell: skip (keep uncrossed)
mismatch cell: comb (uncross) iﬀ the same seaweed pair already
crossed before, possibly in a diﬀerent period
Overall time O(mn)
String-substring LCS score: count seaweeds with multiplicities

The tandem LCS problem
Give LCS score for a vs b = uk
We have n = kp; may assume k ≤ m
Tandem LCS: running time
O(mkp) naive
O m(k + p) [Landau, Ziv-Ukelson: 2001]
O(mp) [T: 2009]
Direct application of wraparound seaweed combing

The tandem alignment problem
Give the substring closest to a in b = u±∞ by alignment score among
global: substrings uk of length kp across all k
cyclic: substrings of length kp across all k
local: substrings of any length
Tandem alignment: running time
O(m2p) all naive
O(mp) global [Myers, Miller: 1989]
O(mp log p) cyclic [Benson: 2005]
O(mp) cyclic [T: 2009]
O(mp) local [Myers, Miller: 1989]

Cyclic tandem alignment: the algorithm
Periodic seaweed combing for a vs b (with blow-up), time O(mp)
For each k ∈ [1 : m]:
solve tandem LCS (under given alignment score) for a vs uk
obtain scores for a vs p successive substrings of b of length kp by LCS
batch query: time O(1) per substring
Running time O(mp)

1 Introduction
method

Sparse string comparison
Semi-local LCS between permutations
The LCS problem on permutation strings (LCSP)
Give LCS score for a vs b; in each of a, b all characters distinct
Equivalent to
longest increasing subsequence (LIS) in a string
maximum clique in a permutation graph
maximum planar matching in an embedded bipartite graph
LCSP: running time
O(n log n) implicit in [Erd¨os, Szekeres: 1935]
[Robinson: 1938; Knuth: 1970; Dijkstra: 1980]
O(n log log n) unit-RAM [Chang, Wang: 1992]
[Bespamyatnikh, Segal: 2000]

Semi-local LCSP
Give semi-local LCS scores for a vs b; in each of a, b all characters distinct
Equivalent to
longest increasing subsequence (LIS) in every substring of a string
Semi-local LCSP: running time
O(n2 log n) naive
O(n2) restricted [Albert+: 2003; Chen+: 2005]
O(n1.5 log n) randomised, restricted [Albert+: 2007]
O(n1.5) [T: 2006]
O(n log2
n) [T: 2010]

C
F
A
E
D
H
G
B
D E H C B A F G

Semi-local LCSP: the algorithm
Divide-and-conquer on the alignment graph
Divide graph (say) horizontally; two subproblems of eﬀective size n/2
Conquer: seaweed matrix -multiplication, time O(n log n)
Overall time O(n log2
n)

Longest piecewise monotone subsequences
A k-increasing sequence: a concatenation of k increasing sequences
A (k − 1)-modal sequence: a concatenation of k alternating increasing and
decreasing sequences
The longest k-increasing (or (k − 1)-modal) subsequence problem
Give the longest k-increasing ((k − 1)-modal) subsequence of string b
Longest k-increasing (or (k − 1)-modal) subsequence: running time
O(nk log n) (k − 1)-modal [Demange+: 2007]
O(nk log n) via [Hunt, Szymanski: 1977]
O(n log2
n) [T: 2010]
Main idea: lcs(idk
, b) (respectively, lcs((idid)k/2, b))

Longest k-increasing subsequence: algorithm A
Sparse LCS for idk
vs b: time O(nk log n)

Longest k-increasing subsequence: algorithm A
Sparse LCS for idk
vs b: time O(nk log n)
Longest k-increasing subsequence: algorithm B
Semi-local LCSP for id vs b: time O(n log2
n)
Extract string-substring LCSP for id vs b
String-substring LCS for idk
vs b by at most 2 log k instances of
-square/multiply: time log k · O(n log n) = O(n log2
n)
Query the LCS score for idk
vs b: time negligible
n)
Algorithm B faster than Algorithm A for k ≥ log n

Maximum clique in a circle graph
The maximum clique problem in a circle graph
Given a circle with n chords, ﬁnd the maximum-size subset of pairwise
intersecting chords

The maximum clique problem in a circle graph
Given a circle with n chords, ﬁnd the maximum-size subset of pairwise
intersecting chords
Standard reduction to an interval model: cut the circle and lay it out on
the line; chords become intervals (here drawn as square diagonals)
Chords intersect iﬀ intervals overlap, i.e. intersect without containment

Maximum clique in a circle graph: running time
exp(n) naive
O(n3) [Gavril: 1973]
O(n2) [Rotem, Urrutia: 1981; Hsu: 1985]
[Masuda+: 1990; Apostolico+: 1992]
O(n1.5) [T: 2006]
O(n log2
n) [T: 2010]

Maximum clique in a circle graph: the algorithm
Helly property: if any set of intervals intersect pairwise, then they all
intersect at a common point
Compute seaweed matrix, build range tree: time O(n log2
n)
Run through all 2n + 1 possible common intersection points
For each point, find a maximum subset of covering overlapping segments
by a prefix-suffix LCS query: time (2n + 1) · O(log2
n) = O(n log2
n)
n) + O(n log2
n) = O(n log2
n)

Parameterised maximum clique in a circle graph
The maximum clique problem in a circle graph, sensitive e.g. to
number e of edges; e ≤ n2
size l of maximum clique; l ≤ n
cutwidth d of interval model (max number of intervals covering a
point); l ≤ d ≤ n
Parameterised maximum clique in a circle graph: running time
O(n log n + e) [Apostolico+: 1992]
O(n log n + nl log(n/l)) [Apostolico+: 1992]
O(n log n + n log2
d) NEW

Parameterised maximum clique in a circle graph: the algorithm
For each diagonal block of size d, compute seaweed matrix, build range
tree: time n/d · O(d log2
d) = O(n log2
d)
Extend each diagonal block to a quadrant: time O(n log2
d)
Run through all 2n + 1 possible common intersection points
For each point, find a maximum subset of covering overlapping segments
by a prefix-suffix LCS query: time O(n log2
d)
d)

1 Introduction
method

Compressed string comparison
Grammar compression
Notation: pattern p of length m; text t of length n
A GC-string (grammar-compressed string) t is a straight-line program
(context-free grammar) generating t = t¯n by ¯n assignments of the form
tk = α, where α is an alphabet character
tk = uv, where each of u, v is an alphabet character, or ti for i < k
In general, n = O(2¯n)
Example: Fibonacci string “ABAABABAABAAB”
t1 = A t2 = t1B t3 = t2t1 t4 = t3t2 t5 = t4t3 t6 = t5t4

Grammar compression
Grammar-compression covers various compression types, e.g. LZ78, LZW
(not LZ77 directly)
Simplifying assumption: arithmetic up to n runs in O(1)
This assumption can be removed by careful index remapping

Extended substring-string LCS on GC-strings
LCS: running time (r = m + n, ¯r = ¯m + ¯n)
p t
plain plain O(mn) [Wagner, Fischer: 1974]
O mn
log2
m
[Masek, Paterson: 1980]
[Crochemore+: 2003]
plain GC O(m3 ¯n + . . .) gen. CFG [Myers: 1995]
O(m1.5 ¯n) ext subs-s [T: 2008]
O(m log m · ¯n) ext subs-s [T: 2010]
GC GC NP-hard [Lifshits: 2005]
O(r1.2¯r1.4) R weights [Hermelin+: 2009]
O(r log r · ¯r) [T: 2010]
O r log(r/¯r) · ¯r [Hermelin+: 2010]
O r log1/2
(r/¯r) · ¯r [Gawrychowski: 2012]

Extended substring-string LCS on GC-strings
Substring-string LCS (plain pattern, GC text): the algorithm
For every k, compute by recursion the appropriate part of semi-local LCS
for p vs tk, using seaweed matrix -multiplication: time O(m log m · ¯n)
Overall time O(m log m · ¯n)

Subsequence recognition on GC-strings
The global subsequence recognition problem
Does pattern p appear in text t as a subsequence?
Global subsequence recognition: running time
p t
plain plain O(n) greedy
plain GC O(m¯n) greedy

The local subsequence recognition problem
Find all minimally matching substrings of t with respect to p
Substring of t is matching, if p is a subsequence of t
Matching substring of t is minimally matching, if none of its proper
substrings are matching

Local subsequence recognition: running time ( + output)
p t
plain plain O(mn) [Mannila+: 1995]
O mn
log m [Das+: 1997]
O(cm + n) [Boasson+: 2001]
O(m + nσ) [Troniˇcek: 2001]
plain GC O(m2 log m¯n) [C´egielski+: 2006]
O(m1.5 ¯n) [T: 2008]
O(m log m · ¯n) [T: 2010]
O(m · ¯n) [Yamamoto+: 2011]

ˆı0+ ˆı1+ ˆı2+
ˆ0+ ˆ1+ ˆ2+
0 n n
b i : j matching iﬀ box [i : j] not pierced left-to-right
Determined by -chain of -maximal seaweeds
b i : j minimally matching iﬀ (i, j) is in the interleaved skeleton -chain

ˆ0+ ˆ1+ ˆ2+n n m+n
−m
0
n
ˆı0+
ˆı1+
ˆı2+

Local subsequence recognition (plain pattern, GC text): the algorithm

Given an assignment t = t t , ﬁnd by recursion
minimally matching substrings in t

Given an assignment t = t t , ﬁnd by recursion
Then, ﬁnd -chain of -maximal seaweeds in time ¯n · O(m) = O(m¯n)
Its skeleton -chain: minimally matching substrings in t overlapping t , t
Overall time O(m log m · ¯n) + O(m¯n) = O(m log m · ¯n)

Threshold approximate matching
The threshold approximate matching problem
Find all matching substrings of t with respect to p, according to a
threshold k
Substring of t is matching, if the edit distance for p vs t is at most k

Threshold approximate matching: running time ( + output)
p t
plain plain O(mn) [Sellers: 1980]
O(mk) [Landau, Vishkin: 1989]
O(m + n + nk4
m ) [Cole, Hariharan: 2002]
plain GC O(m¯nk2) [K¨arkk¨ainen+: 2003]
O(m¯nk + ¯n log n) [LV: 1989] via [Bille+: 2010]
O(m¯n + ¯nk4 + ¯n log n) [CH: 2002] via [Bille+: 2010]
O(m log m · ¯n) [T: NEW]
(Also many specialised variants for LZ compression)

ˆı0+ ˆı1+ ˆı2+ ˆı3+ ˆı4+
ˆ0+ ˆ1+ ˆ2+ ˆ3+ ˆ4+
0 n n
Blow up: weighted alignment on strings p, t of size m, n equivalent to
LCS on strings p, t of size m = νm, n = νn

ˆ0+ ˆ1+ ˆ2+ ˆ3+ ˆ4+n n m+n
ˆı0+
ˆı1+
ˆı2+
ˆı3+
ˆı4+
−m
0
n

Threshold approx matching (plain pattern, GC text): the algorithm
Algorithm structure similar to local subsequence recognition
-chains replaced by m × m submatrices
Extra ingredients:
the blow-up technique: reduction of edit distances to LCS scores
implicit matrix searching, replaces -chain interleaving
Monge row minima: “SMAWK” O(m) [Aggarwal+: 1987]
Implicit unit-Monge row minima:
O(m log log m) [T: 2012]
O(m) [Gawrychowski: 2012]
Overall time O(m log m · ¯n) + O(m¯n) = O(m log m · ¯n)

1 Introduction
method

Parallel string comparison
Parallel LCS
Bulk-Synchronous Parallel (BSP) computer [Valiant: 1990]
Simple, realistic general-purpose parallel
model
COMM. ENV . (g, l)
0
P
M
1
P
M
p − 1
P
M· · ·

Parallel LCS
Bulk-Synchronous Parallel (BSP) computer [Valiant: 1990]
Simple, realistic general-purpose parallel
model
COMM. ENV . (g, l)
0
P
M
1
P
M
p − 1
P
M· · ·
Contains
p processors, each with local memory (1 time unit/operation)
communication environment, including a network and an external
memory (g time units/data unit communicated)
barrier synchronisation mechanism (l time units/synchronisation)

Parallel LCS
BSP computation: sequence of parallel supersteps
0
1
p − 1

Parallel LCS
0
1
p − 1 · · ·
· · ·
· · ·
· · ·

Parallel LCS
0
1
p − 1 · · ·
· · ·
· · ·
· · ·
Asynchronous computation/communication within supersteps (includes
data exchange with external memory)
Synchronisation before/after each superstep

Parallel LCS
0
1
p − 1 · · ·
· · ·
· · ·
· · ·
Asynchronous computation/communication within supersteps (includes
data exchange with external memory)
Synchronisation before/after each superstep
Cf. CSP: parallel collection of sequential processes

Parallel LCS
Compositional cost model
For individual processor proc in superstep sstep:
comp(sstep, proc): the amount of local computation and local
memory operations by processor proc in superstep sstep
comm(sstep, proc): the amount of data sent and received by
processor proc in superstep sstep

Parallel LCS
Compositional cost model
For individual processor proc in superstep sstep:
comp(sstep, proc): the amount of local computation and local
memory operations by processor proc in superstep sstep
comm(sstep, proc): the amount of data sent and received by
processor proc in superstep sstep
For the whole BSP computer in one superstep sstep:
comp(sstep) = max0≤proc<p comp(sstep, proc)
comm(sstep) = max0≤proc<p comm(sstep, proc)
cost(sstep) = comp(sstep) + comm(sstep) · g + l

Parallel LCS
For the whole BSP computation with sync supersteps:
comp = 0≤sstep<sync comp(sstep)
comm = 0≤sstep<sync comm(sstep)
cost = 0≤sstep<sync cost(sstep) = comp + comm · g + sync · l

Parallel LCS
The input/output data are stored in the external memory; the cost of
input/output is included in comm

Parallel LCS
The input/output data are stored in the external memory; the cost of
input/output is included in comm
E.g. for a particular linear system solver with an n × n matrix:
comp = O(n3/p) comm = O(n2/p1/2) sync = O(p1/2)

Parallel LCS
BSP software: industrial projects
Google’s Pregel [2010]
Apache Spark (hama.apache.org) [2010]
Apache Giraph (giraph.apache.org) [2011]
BSP software: research projects
Oxford BSP (www.bsp-worldwide.org/implmnts/oxtool) [1998]
Paderborn PUB (www2.cs.uni-paderborn.de/~pub) [1998]
BSML (traclifo.univ-orleans.fr/BSML) [1998]
BSPonMPI (bsponmpi.sourceforge.net) [2006]
Multicore BSP (www.multicorebsp.com) [2011]
Epiphany BSP (www.codu.in/ebsp) [2015]
Petuum (petuum.org) [2015]

Parallel LCS
The ordered 2D grid dag
grid2(n)
nodes arranged in an n × n grid
edges directed top-to-bottom, left-to-right
≤ 2n inputs (to left/top borders)
≤ 2n outputs (from right/bottom borders)
size n2 depth 2n − 1

Parallel LCS
The ordered 2D grid dag
grid2(n)
nodes arranged in an n × n grid
edges directed top-to-bottom, left-to-right
≤ 2n inputs (to left/top borders)
≤ 2n outputs (from right/bottom borders)
size n2 depth 2n − 1
Applications: triangular linear system; discretised PDE via Gauss–Seidel
iteration (single step); 1D cellular automata; dynamic programming
Sequential work O(n2)

Parallel LCS
Parallel ordered 2D grid computation
grid2(n)
Consists of a p × p grid of blocks, each
isomorphic to grid2(n/p)
The blocks can be arranged into 2p − 1
anti-diagonal layers, with ≤ p independent
blocks in each layer

Parallel LCS
Parallel ordered 2D grid computation (contd.)
The computation proceeds in 2p − 1 stages, each computing a layer of
blocks. In a stage:
every block assigned to a diﬀerent processor (some processors idle)
the processor reads the 2n/p block inputs, computes the block, and
writes back the 2n/p block outputs

Parallel LCS
blocks. In a stage:
comp: (2p − 1) · O (n/p)2 = O(p · n2/p2) = O(n2/p)
comm: (2p − 1) · O(n/p) = O(n)

Parallel LCS
blocks. In a stage:
comp: (2p − 1) · O (n/p)2 = O(p · n2/p2) = O(n2/p)
comm: (2p − 1) · O(n/p) = O(n)
n ≥ p
comp O(n2/p) comm O(n) sync O(p)

Parallel LCS
Parallel LCS computation
The 2D grid algorithm solves the LCS problem (and many others) by
dynamic programming
comp = O(n2/p) comm = O(n) sync = O(p)

Parallel LCS
The 2D grid algorithm solves the LCS problem (and many others) by
dynamic programming
comp = O(n2/p) comm = O(n) sync = O(p)
comm is not scalable (i.e. does not decrease with increasing p) :-(
Can scalable comm be achieved for the LCS problem?

Parallel LCS
Solve the more general semi-local LCS problem:
each string vs all substrings of the other string
all preﬁxes of each string against all suﬃxes of the other string
Divide-and-conquer on substrings of a, b: log p recursion levels
Conquer phase: assembles substring semi-local LCS from smaller ones by
parallel seaweed multiplication
Base level: p semi-local LCS subproblems on n
p1/2 -strings
Sequential time still O(n2)

Parallel LCS
Parallel LCS computation (cont.)
Parallel semi-local LCS: running time, comm, sync
comp comm sync
O(n2/p) O(n) O(p) 2D grid
O(n2/p) O n
p1/2 O(log2
p) [Krusche, T: 2007]
O n log p
p1/2 O(log p) [Krusche, T: 2010]
O n
p1/2 O(log p) [T: NEW]
Based on parallel Steady Ant algorithm

Parallel LCS
Seaweed matrix -multiplication: parallel Steady Ant
Divide-and-conquer on the range of j: two subproblems on n/2-matrices
PΣ
lo QΣ
lo = RΣ
lo PΣ
hi QΣ
hi = RΣ
hi
Conquer: tracing a balanced trail from bottom-left to top-right R
Partition n × n index space into a regular p × p grid of blocks
Sample R in block corners: total p2 samples
Steady Ant’s trail passes through ≤ 2p blocks
Partition blocks across processors: comp, comm = O(n/p)
comp = O n log n
p comm = O n log p
p sync = O(log p)

Parallel LCS
Seaweed matrix -multiplication: generalised parallel Steady Ant
Divide-and-conquer on the range of j: p subproblems on n
p -matrices
Conquer: tracing p balanced trails from bottom-left to top-right R
Partition n × n index space into a regular p × p grid of blocks
Sample R in block corners: total p2 samples
Steady Ant’s trails pass through ≤ 2p1+ blocks
Partition blocks across processors: comp, comm = O n
p1−
comp = O n log n
p + n
p1− comm = O n
p1− sync = O(1/ )

Parallel LCS
Divide-and-conquer on substrings of a, b: log p recursion levels
In every level, semi-local LCS subproblems on input substrings
Conquer phase: generalised parallel Steady Ant algorithm, = 1/3
base level: p = p1/2 × p1/2 subproblems on n
p1/2 -substrings; one
proc/subproblem; comm = O n
p1/2
middle level: p1/2 = p1/4 × p1/4 subproblems on n
p1/4 -substrings; p1/2
procs/subproblem; comm = O n/p1/4
(p1/2)1−1/3 = O n
p7/12
top level: one subproblem on n-strings; p procs;
comm = O n
p1−1/3 = O n
p2/3
sync = O 1
1/3 = O(1) in every level

Parallel LCS
comm dominated by base level:
O n
p1/2 + . . . + O n
p7/12 + . . . + O n
p2/3 = O n
p1/2
comp = O n2
p comm = O n
p1/2 sync = O(log p)

Parallel LCS
Parallel LCS on permutation strings (LCSP = LIS)
Sequential time O(n log n), no good way known to parallelise directly
Solution via the more general semi-local LCSP is no longer eﬃcient:
sequential time O(n log2
n)
Divide-and-conquer on substrings/subsequences of a, b
Parallelising normal O(n log n) seaweed multiplication: [Krusche, T: 2010]
comp O n log2
n
p comm O n log p
p sync O(log2
p)
Open problem: can we achieve
comp O n log n
p for LCSP?
comp O n log2
n
p , comm O(n
p ), sync O(log2
p) for semi-local LCSP?

1 Introduction
method

The transposition network method
Transposition networks
Comparison network: a circuit of comparators
A comparator sorts two inputs and outputs them in prescribed order
Comparison networks traditionally used for non-branching merging/sorting
Classical comparison networks
# comparators
merging O(n log n) [Batcher: 1968]
sorting O(n log2
n) [Batcher: 1968]
O(n log n) [Ajtai+: 1983]

Comparison network: a circuit of comparators
A comparator sorts two inputs and outputs them in prescribed order
Comparison networks traditionally used for non-branching merging/sorting
Classical comparison networks
# comparators
merging O(n log n) [Batcher: 1968]
sorting O(n log2
n) [Batcher: 1968]
O(n log n) [Ajtai+: 1983]
Comparison networks are visualised by wire diagrams
Transposition network: all comparisons are between adjacent wires

Seaweed combing as a transposition network
A
C
B
C
A B C A
+7
+1
+5
+5
+3
+7
+1
−5
−1
−3
−3
+3
−5
−1
−7
−7
Character mismatches correspond to comparators
Inputs anti-sorted (sorted in reverse); each value traces a seaweed

Global LCS: transposition network with binary input
A
C
B
C
A B C A
1
1
1
1
1
1
1
0
0
0
0
1
0
0
0
0
Inputs still anti-sorted, but may not be distinct
Comparison between equal values is indeterminate

Parameterised string comparison
String comparison sensitive e.g. to
low similarity: small λ = LCS(a, b)
high similarity: small κ = distLCS (a, b) = m + n − 2λ
Can also use weighted alignment score or edit distance
Assume m = n, therefore κ = 2(n − λ)

Low-similarity comparison: small λ
sparse set of matches, may need to look at them all
preprocess matches for fast searching, time O(n log σ)
High-similarity comparison: small κ
set of matches may be dense, but only need to look at small subset
no need to preprocess, linear search is OK
Flexible comparison: sensitive to both high and low similarity, e.g. by both
comparison types running alongside each other

Parameterised string comparison: running time
Low-similarity, after preprocessing in O(n log σ)
O(nλ) [Hirschberg: 1977]
[Apostolico, Guerra: 1985]
[Apostolico+: 1992]
High-similarity, no preprocessing
O(n · κ) [Ukkonen: 1985]
[Myers: 1986]
Flexible
O(λ · κ · log n) no preproc [Myers: 1986; Wu+: 1990]
O(λ · κ) after preproc [Rick: 1995]

Parameterised string comparison: the waterfall algorithm
Low-similarity: O(n · λ)
0 0 0 0 0 0 0 0
1
1
1
1
1
1
1
1
0 0 1 1 0 0 1 0
0
1
0
1
1
0
1
1

0 0 0 0 0 0 0 0
1
1
1
1
1
1
1
1
0 0 1 1 0 0 1 0
0
1
0
1
1
0
1
1
High-similarity: O(n · κ)
0 0 0 0 0 0 0 0
1
1
1
1
1
1
1
1
1 1 1 0 1 1 0 0
0
0
0
0
1
1
1
0

0 0 0 0 0 0 0 0
1
1
1
1
1
1
1
1
0 0 1 1 0 0 1 0
0
1
0
1
1
0
1
1
High-similarity: O(n · κ)
0 0 0 0 0 0 0 0
1
1
1
1
1
1
1
1
1 1 1 0 1 1 0 0
0
0
0
0
1
1
1
0
Trace 0s through network in contiguous blocks and gaps

Dynamic string comparison
The dynamic LCS problem
Maintain current LCS score under updates to one or both input strings
Both input strings are streams, updated on-line:
prepending/appending characters at left or right
deleting characters at left or right
Assume for simplicity m ≈ n, i.e. m = Θ(n)
Goal: linear time per update
O(n) per update of a (n = |b|)
O(m) per update of b (m = |a|)

Dynamic string comparison
Dynamic LCS in linear time: update models
left right
– app+del standard DP [Wagner, Fischer: 1974]
prep app a ﬁxed [Landau+: 1998], [Kim, Park: 2004]
prep app [Ishida+: 2005]
prep+del app+del [T: NEW]
Dynamic semi-local LCS in linear time: update models
left right
prep+del app+del amortised [T: NEW]

Bit-parallel string comparison
String comparison using standard instructions on words of size w
Bit-parallel string comparison: running time
O(mn/w) [Allison, Dix: 1986; Myers: 1999; Crochemore+: 2001]

Bit-parallel string comparison: binary transposition network
In every cell: input bits s, c; output bits s , c ; match/mismatch ﬂag µ
¬µ
s
c
s
c
s 0 1 0 1 0 1 0 1
c 0 0 1 1 0 0 1 1
µ 0 0 0 0 1 1 1 1
s 0 1 1 1 0 0 1 1
c 0 0 0 1 0 1 0 1
+
∧
µ
s
c
s
c
s 0 1 0 1 0 1 0 1
c 0 0 1 1 0 0 1 1
µ 0 0 0 0 1 1 1 1
s 0 1 1 0 0 0 1 1
c 0 0 0 1 0 1 0 1

Bit-parallel string comparison: binary transposition network
In every cell: input bits s, c; output bits s , c ; match/mismatch ﬂag µ
¬µ
s
c
s
c
s 0 1 0 1 0 1 0 1
c 0 0 1 1 0 0 1 1
µ 0 0 0 0 1 1 1 1
s 0 1 1 1 0 0 1 1
c 0 0 0 1 0 1 0 1
+
∧
µ
s
c
s
c
s 0 1 0 1 0 1 0 1
c 0 0 1 1 0 0 1 1
µ 0 0 0 0 1 1 1 1
s 0 1 1 0 0 0 1 1
c 0 0 0 1 0 1 0 1
2c + s ← (s + (s ∧ µ) + c) ∨ (s ∧ ¬µ)
S ← (S + (S ∧ M)) ∨ (S ∧ ¬M), where S, M are words of bits s, µ

High-similarity bit-parallel string comparison
κ = distLCS (a, b) Assume κ odd, m = n
Waterfall algorithm within diagonal band of width κ + 1: time O(nκ/w)
Band waterfall supported from below by separator matches

High-similarity bit-parallel multi-string comparison: a vs b0, . . . , br−1
κi = distLCS (a, bi ) ≤ κ 0 ≤ i < r
Waterfalls within r diagonal bands of width κ + 1: time O(nrκ/w)
Each band’s waterfall supported from below by separator matches

1 Introduction
method

Beyond semi-locality
Fixed-length substring LCS
Given window length w, let w-window be any substring of length w
The window-substring LCS problem
Give LCS score for every w-window of a vs every substring of b

Given window length w, let w-window be any substring of length w
The window-substring LCS problem
Give LCS score for every w-window of a vs every substring of b
Window-substring LCS: running time
O(mn3w) naive
O(mnw) multiple seaweed
O(mn) [Krusche, T: 2010]

Window-substring LCS: the algorithm
Compute seaweed matrices for canonical substrings of a vs b recursively
Compute seaweed matrices for w-windows of a vs b, using the
decomposition of each w-window into canonical substrings
Both stages use matrix -multiplication, overall time O(mn)

The window-window LCS problem
Give LCS score for every w-window of a vs every w-window of b
Provides an LCS-scored alignment plot (by analogy with Hamming-scored
dot plot), a useful tool for studying weak conservation in the genome

The window-window LCS problem
Give LCS score for every w-window of a vs every w-window of b
Provides an LCS-scored alignment plot (by analogy with Hamming-scored
dot plot), a useful tool for studying weak conservation in the genome
Window-window LCS: running time
O(mnw2) naive
O(mnw) repeated seaweed combing
O(mn) [Krusche, T: 2010]

An implementation of alignment plots [Krusche: 2010]
Based on O(mnw) repeated seaweed combing approach, using some ideas
from the optimal O(mn) algorithm: resulting time O(mnw0.5)
Weighted alignment (match 1, mismatch 0, gap −0.5) via alignment dag
blow-up: ×4 slowdown
C++, Intel assembly (x86, x86 64, MMX/SSE2 data parallelism)
SMP parallelism (currently two processors)

Krusche’s implementation used at Warwick Systems Biology Centre to
study conservation in plant non-coding DNA
More sensitive than BLAST and other heuristic (seed-and-extend) methods
Found hundreds of conserved non-coding regions between papaya (Carica
papaya), poplar (Populus trichocarpa), thale cress (Arabidopsis thaliana)
and grape (Vitis vinifera)
Single processor:
×10 speedup over heavily optimised, bit-parallel naive algorithm
×7 speedup over biologists’ heuristics
Two processors: extra ×2 speedup, near-perfect parallelism

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by Katy Edgington
06 December 2012
We can study
selected
conserved
regions
experimentally
to understand
Researchers from the University of Warwick have found that hundreds of regions
of non-coding DNA are conserved among various plant species that have been
evolving separately for around 100 million years. The findings, which have been
detailed in the journal The Plant Cell, have led the team to believe that these
sequences, although they are not genes, could provide vital information about
plant development and functioning.
Computational analysis of the genomes of papaya (Carica papaya), poplar
(Populus trichocarpa), thale cress (Arabidopsis thaliana) and grape (Vitis vinifera)
showed that all four species contained conserved non-coding sequences (CNSs)
Conserved DNA could prove crop science shortcut MOST READ COMMENTS
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Conserved DNA could prove crop science shortcut - Science Omega http://www.scienceomega.com/article/732/conserved-dna-could-prove-...

Publications:
[Picot+: 2010] Evolutionary Analysis of Regulatory Sequences (EARS) in
Plants. Plant Journal.
[Baxter+: 2012] Conserved Noncoding Sequences Highlight Shared
Components of Regulatory Networks in Dicotyledonous Plants. Plant Cell.
Web service: http://wsbc.warwick.ac.uk/ears
Warwick press release (with video): http://www2.warwick.ac.uk/
newsandevents/pressreleases/discovery_of_100
Future work: genomic repeats; genome-scale comparison

Quasi-local LCS and spliced alignment
Given m (possibly overlapping) prescribed substrings in a
The quasi-local LCS problem
Give LCS score for every prescribed substring of a vs every subsring of b

Given m (possibly overlapping) prescribed substrings in a
The quasi-local LCS problem
Give LCS score for every prescribed substring of a vs every subsring of b
Quasi-local LCS: running time
O(m2n) repeated seaweed combing
O(mn log2
m) [T: unpublished]

Quasi-local LCS: the algorithm
Compute seaweed matrices for canonical substrings of a vs b recursively
Compute seaweed matrices for prescribed substrings of a vs b, using the
decomposition of each prescribed substring into canonical substrings
Both stages use matrix -multiplication, time O(mn log2
m)

The spliced alignment problem
Give the chain of non-overlapping prescribed substrings in a, closest to b
by alignment score
Describes gene assembly from candidate exons, given a known similar gene

The spliced alignment problem
Give the chain of non-overlapping prescribed substrings in a, closest to b
by alignment score
Describes gene assembly from candidate exons, given a known similar gene
Assume m = n; let s = total size of prescribed substrings
Spliced alignment: running time
O(ns) = O(n3) [Gelfand+: 1996]
O(n2.5) [Kent+: 2006]
O(n2 log2
n) [T: unpublished]
O(n2 log n) [Sakai: 2009]

Local alignment
Assume a fixed set of character alignment scores
The Smith–Waterman local alignment problem
For every prefix of a and every prefix of b, give a pair of respective suffixes
of these prefixes that have the highest alignment score against each other
Smith–Waterman local alignment: running time
O(m2n2) [naive]
O(mn) [Smith, Waterman: 1981]

Local alignment
Smith–Waterman local alignment [SW: 1981]
Overall time O(mn)
Smith–Waterman local alignment maximises alignment score irrespective
of length
May not always be the most relevant alignment, e.g. may be
too short (would prefer a longer alignment even with a lower score)
too long (would prefer a signiﬁcantly shorter alignment with slightly
lower score)

Local alignment
The bounded substring alignment problem
For every prefix of string a and every prefix of string b, give a pair of
respective suffixes of these prefixes of length at least/at most t that have
the highest alignment score against each other
Solved efficiently using window-substring alignment as a subroutine, e.g.
for alignments of length at least t:
solve the window-substring alignment problem, obtaining implicit
alignment score for every t-window in a against every substring in b;
obtain explicit optimal alignment score for every t-window in a
against all substrings in b terminating in a given position;
extend locally optimal window-substring alignment scores to optimal
substring-substring scores as in Smith–Waterman algorithm

1 Introduction
method

Conclusions and open problems (1)
Implicit unit-Monge matrices:
equivalent to seaweed braids
distance multiplication in time O(n log n)
Semi-local LCS problem:
representation by implicit unit-Monge matrices
generalisation to rational alignment scores
Seaweed combing and micro-block speedup:
a simple algorithm for semi-local LCS
semi-local LCS in time O(mn), and even slightly o(mn)
Sparse string comparison:
fast LCS on permutation strings
fast max-clique in a circle graph

Compressed string comparison:
LCS and approximate matching on plain pattern against GC-text
Parallel string comparison:
comm- and sync- eﬃcient parallel LCS
Transposition networks:
simple interpretation of parameterised and bit-parallel LCS
dynamic LCS
Beyond semi-locality:
window alignment and sparse spliced alignment

Open problems (theory):
real-weighted alignment?
max-clique in circle graph: O(n log2
n) O(n log n)?
compressed LCS: other compression models?
parallel LCS: comm per processor O n log p
p1/2 O n
p1/2 ?
lower bounds?
Open problems (biological applications):
real-weighted alignment?
good pre-ﬁltering (e.g. q-grams)?
application for multiple alignment? (NB: NP-hard)

Thank you

A superglue for string comparison

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