ShortestNontrivialCycles inDirectedSurfaceGraphs JeffErickson - PowerPoint PPT Presentation

Shortest
Nontrivial
Cycles in
Directed
Surface
Graphs Jeff
Erickson SOCG
2011

Drink! 2

The
problem Given
an
edge‐weighted
 directed 
graph
 G 
embedded
on
a
surface, find
the
shortest
non‐contractible
or
non‐separating
cycle
in
 G . 3

The
problem Given
an
edge‐weighted
 directed 
graph
 G 
embedded
on
a
surface, find
the
shortest
non‐contractible
or
non‐separating
cycle
in
 G . non‐separating separating separating non‐contractible non‐contractible contractible 3

Why? Finding
short
nontrivial
cycles
is
a
critical
 subproblem
of
several
more
complex
 surface‐graph
algorithms. •Topological
noise/feature
removal •Surface
parametrization •Shortest
homotopic
paths •Approximate
TSP
tours
and
Steiner
trees •Graph
isomorphism •Drawing
in
the
plane
with
few
crossings •Probabilistic
embedding
into
planar
graphs •... 4

Time
bounds
(undirected) Non‐contractible Non‐separating O( n 3 ) O( n 3 ) Thomassen
90 O( n 2 
log
 n ) O( n 2 
log
 n ) Erickson
Har‐Peled
03,
04 g O( g ) 
 n 3/2 O( g 3/2
 n 3/2 
log
 n ) Cabello
Mohar
05,
07 g O( g ) 
 n 4/3 
log
 n — Cabello
06 g O( g ) 
 n 
 log
 n g O( g ) 
 n 
log
 n Kutz
06 O( g 3 
 n 
log
 n ) O( g 3 
 n 
log
 n ) Cabello
Chambers
07 O( g 2 
 n 
log
 n ) O( g 2 
 n 
log
 n ) Cabello
Chambers
Erickson In
 some 
of
these
bounds,
O(log
 n )
factors
can
be
reduced
to
O(log
log
 n )
 [Italiano
Nussbaum
Sankowski
Wulff‐Nielsen
STOC
11] 5

Time
bounds
(directed) Non‐contractible Non‐separating O( n 2 
log
 n ) O( n 2 
log
 n ) Cabello,
Colin
de
Verdière,
 Lazarus
10 O( g 1/2
 n 3/2 
log
 n ) O( g 1/2
 n 3/2 
log
 n ) — 2 O( g ) 
 n 
log
 n Erickson
Nayyeri
11 6

Time
bounds
(directed) Non‐contractible Non‐separating O( n 2 
log
 n ) O( n 2 
log
 n ) Cabello,
Colin
de
Verdière,
 Lazarus
10 O( g 1/2
 n 3/2 
log
 n ) O( g 1/2
 n 3/2 
log
 n ) — 2 O( g ) 
 n 
log
 n Erickson
Nayyeri
11 g O( g ) 
 n 
log
 n O( g 2 
 n 
log
 n ) This
paper Same
as
Kutz
for
 Same
as
Cabello
et
al.
for
 undirected
graphs undirected
graphs 6

Two
main
ideas ‣ 3‐path
condition: 
The
shortest
 ‣ 1‐crossing
condition: 
The
 nontrivial
cycle
consists
of
two
 shortest
nontrivial
cycle
 shortest
paths
between
 any 
 crosses
any
shortest
path
at
 pair
of
 antipodal 
points.

 most
once.






























 [Thomassen
90,
Erickson
Har‐Peled
04] [Cabello
Mohar
05,
all
later
papers] 7

Why
directed
graphs
are
harder ‣ Undirected
shortest
paths
cross
at
most
once. 8

Why
directed
graphs
are
harder ‣ Undirected
shortest
paths
cross
at
most
once. ‣ Directed
shortest
paths
can
cross
arbitrarily
many
times! 8

Time
bounds
(directed) modified Non‐contractible Non‐separating 3‐path
 condition O( n 2 
log
 n ) O( n 2 
log
 n ) Cabello,
Colin
de
Verdière,
 Lazarus
10 O( g 1/2
 n 3/2 
log
 n ) O( g 1/2
 n 3/2 
log
 n ) — 2 O( g ) 
 n 
log
 n Erickson
Nayyeri
11 g O( g ) 
 n 
log
 n O( g 2 
 n 
log
 n ) This
paper Same
as
Kutz
for
 Same
as
Cabello
et
al.
for
 modified undirected
graphs undirected
graphs 1–crossing
 condition 9

Key
insight ‣ Let
 γ 
be
the
shortest
nontrivial
 directed 
cycle. ‣ Let
 σ 
be
any
 directed 
shortest
path
that
crosses
γ. 10

Key
insight ‣ Let
 γ 
be
the
shortest
nontrivial
 directed 
cycle. ‣ Let
 σ 
be
any
 directed 
shortest
path
that
crosses
γ. ‣ Contracting
σ
to
a
point
transforms
γ
into
a
set
of
loops. 10

Key
insight ‣ Let
 γ 
be
the
shortest
nontrivial
 directed 
cycle. ‣ Let
 σ 
be
any
 directed 
shortest
path
that
crosses
γ. ‣ Contracting
σ
to
a
point
transforms
γ
into
a
set
of
loops. ‣ At
most
one
of
these
loops
is
nontrivial. 10

Shortest
non‐separating
cycles 11

Greedy
homology
basis ‣ Greedy
 tree‐cotree
decomposition
 ( T , 
L , 
C ): ▹ T 
=
single‐source
shortest‐path
tree
in
 G ▹ C 
=
dual
of
any
spanning
tree
of
( G \ T )* ▹ L 
=
 G \ ( T ∪ C )
=
set
of
2 g 
edges ‣ For
each
edge
 e i 
in
 L ,
let
 λ i 
=
unique
 undirected 
cycle
in
 T ∪ e i [Eppstein
03,
Erickson
Whittlesey
05,
Cabello
Mohar
05] 12

Greedy
homology
basis Undirected
cycles
λ 1 ,
λ 2 ,
...,
λ 2 g 
such
that: ‣ Homology
basis:
 Every
nonseparating
cycle
crosses
some
cycle
λ i 
 an
odd
number
of
times. ‣ Greedy:
 Every
cycle
λ i 
consists
of
two
shortest
paths
σ i 
and
τ i 
and
 an
edge. [Eppstein
03,
Erickson
Whittlesey
05,
Cabello
Mohar
05] 13

Greedy
homology
basis Undirected
cycles
λ 1 ,
λ 2 ,
...,
λ 2 g 
such
that: ‣ Homology
basis:
 Every
nonseparating
cycle
crosses
some
cycle
λ i 
 an
odd
number
of
times. ‣ Greedy:
 Every
cycle
λ i 
consists
of
two
shortest
paths
σ i 
and
τ i 
and
 an
edge. e i σ i τ i [Eppstein
03,
Erickson
Whittlesey
05,
Cabello
Mohar
05] 13

Cyclic
double‐cover
ΣN λ ‣ Fix
a
non‐separating
cycle
λ. λ 
Σ 14

Cyclic
double‐cover
ΣN λ ‣ Fix
a
non‐separating
cycle
λ. λ 
Σ ‣ The
surface
Σ ʹ 
:=
Σ
 ✂ 
λ
has
two
 boundary
cycles
λ + 
and
λ – . λ + 
Σ ʹ λ – 14

Cyclic
double‐cover
ΣN λ ‣ Fix
a
non‐separating
cycle
λ. λ 
Σ ‣ The
surface
Σ ʹ 
:=
Σ
 ✂ 
λ
has
two
 boundary
cycles
λ + 
and
λ – . (λ + ,0) 
(Σ ʹ ,0) ‣ Make
2
copies
(Σ ʹ ,0)
and
(Σ ʹ ,1) (λ – ,0) (λ + ,1) 
(Σ ʹ ,1) (λ – ,1) 14

Cyclic
double‐cover
ΣN λ ‣ Fix
a
non‐separating
cycle
λ. λ 
Σ ‣ The
surface
Σ ʹ 
:=
Σ
 ✂ 
λ
has
two
 boundary
cycles
λ + 
and
λ – . λ + 
Σ ʹ ‣ Make
2
copies
(Σ ʹ ,0)
and
(Σ ʹ ,1) λ – ‣ Identify
(λ + ,1)
=
(λ – ,0)
and






 (λ + ,0)
=
(λ – ,1) 
(Σ ʹ ,0) 
(Σ ʹ ,1) 
Σq 14

Cyclic
double‐cover
ΣN λ Any
cycle
γ
in
Σ
crossing
λ
an
odd
number
of
times is
the
projection
of
a path
γ̂
in
Σq λ 
from
( s ,0)
to
( s ,1),
for
 any 
point
 s 
in
γ 15

Cyclic
double‐cover
ΣN λ Any
cycle
γ
in
Σ
crossing
λ
an
odd
number
of
times is
the
projection
of
a path
γ̂
in
Σq λ 
from
( s ,0)
to
( s ,1),
for
 any 
point
 s 
in
γ 15

Cyclic
double‐cover
ΣN λ The
 shortest 
cycle
γ
in
Σ
crossing
λ
an
odd
number
of
times is
the
projection
of
a shortest 
path
γ̂
in
Σq λ 
from
( s ,0)
to
( s ,1),
for
 any 
point
 s 
in
γ 16

At
most
one
nontrivial
crossing ‣ Fix
a
non‐separating
cycle
λ
covered
by
two
shortest
paths
σ
and
τ. ‣ Let
γ
=
shortest
cycle
crossing
λ
an
odd
number
of
times. ‣ Suppose
γ
intersects
σ. σ τ 17

At
most
one
nontrivial
crossing ‣ Fix
a
non‐separating
cycle
λ
covered
by
two
shortest
paths
σ
and
τ. ‣ Let
γ
=
shortest
cycle
crossing
λ
an
odd
number
of
times. ‣ Suppose
γ
intersects
σ. σ 17

At
most
one
nontrivial
crossing ‣ Suppose
γ
intersects
σ
=
 s 0 → s 1 → ⋅⋅⋅ → s k . (σ,0) (σ,1) 18

At
most
one
nontrivial
crossing ‣ Suppose
γ
intersects
σ
=
 s 0 → s 1 → ⋅⋅⋅ → s k . ‣ Let
 s i 
=
 minimum‐index 
vertex
of
σ
that
lies
on
γ. (σ,0) ( s i ,0) (σ,1) ( s i ,1) 18

At
most
one
nontrivial
crossing ‣ Suppose
γ
intersects
σ
=
 s 0 → s 1 → ⋅⋅⋅ → s k . ‣ Let
 s i 
=
 minimum‐index 
vertex
of
σ
that
lies
on
γ. ‣ Let
γ̂
=
shortest
path
in
Σq λ
 from
( s i ,0)
to
( s i ,1). (σ,0) ( s i ,0) (σ,1) ( s i ,1) 18

At
most
one
nontrivial
crossing ‣ Suppose
γ
intersects
σ
=
 s 0 → s 1 → ⋅⋅⋅ → s k . ‣ Let
 s i 
=
 minimum‐index 
vertex
of
σ
that
lies
on
γ. ‣ Let
γ̂
=
shortest
path
in
Σq λ
 from
( s i ,0)
to
( s i ,1). (σ,0) ( s i ,0) (σ,1) ( s i ,1) 18