Unique Sink Orientations of Cubes Motivation and Algorithms

1
Unique Sink Orientations of CubesMotivation and
Algorithms

• Tibor Szabó
• ETH Zürich
• (Bernd Gärtner, Ingo Schurr, Emo Welzl)

2
Outline of Talk

• The Problem
• Definition
• Examples
• Connections to algorithmic problems in Geometry
• Linear Programming
• Smallest Enclosing Ball Problem
• Unique Sink Orientation of Cubes
• Properties
• Algorithms
• Lower bound
• (Lots of) open questions

3
Cubes --- Notation
1,2,3,4
A 1,2,,n n
2,3,4
CA is an edge labeled graph
1,3,4
1,2,4
1,2,3
3,4
2,3
2,4
1,4
1,3

• V(CA)2A

1,2

• E(CA)u,v u?v 1

4
3
2
1
3
4
1
2

• ?(u,v) a, where a u?v

4
Unique Sink Orientation

• A sink is a vertex without outgoing edges
• A unique sink orientation (USO) of a cube is an
  orientation where
• every face has a unique sink

5
The Problem

• Find the sink in a USO
• The USO is given implicitly it can be accessed
  through vertex evaluations
• Vertex evaluation returns orientations of all
  incident edges

6

• How many vertices have to be evaluated in a USO
  of an n-cube, until we have evaluated the sink?
• t(n) worst-case deterministic
• t?(n) worst-case expected for randomized

t?(1)
t(1)
2
3/2
?
½
½
?
7
?
t(2)
3

?
?
8
t(2) 3
?
?
?
?
9

• t(2) 3

t?(2) 43/20
10
Further small values

• t(3) 5 t(3) 4074633/1369468
• t(4) 7 t(4) ?
• t(5) ?

Rote 01
11

• WHY ???

12
Appearences of USOs

• Orientations of slanted geometric cubes relative
  to a generic linear function, e.g. Klee-Minty
  cube (acyclic)

13
Linear Programming the straightforward
connection
acyclic USO
Minimum
Sink
14
Appearences of USOs

• Orientations of slanted geometric cubes relative
  to a generic linear function, e.g. Klee-Minty
  cube (acyclic)
• Model for smallest enclosing ball problem (can
  have cycles as well) Gärtner et al., 01

15
Smallest Enclosing Ball Problem
EXAMPLE
n points
d-space
b
a
c
16
Smallest enclosing ball

• of a set P of d1 affinely independent points
  in d-space.
• Define orientation on the cube CP For x?Q ? P,
• Q ? QUx iff x?ß(Q),
• where ß(Q) is the smallest ball with Q on
  boundary.
• This is a USO
• S is the sink iff ß(S) is the smallest ball

enclosing P
17
Smallest enclosing ball
USO of cubes
b
abc
a
ab
bc
ac
a
b
c
c
18
Appearences of USOs

• Orientations of slanted geometric cubes relative
  to a generic linear function, e.g. Klee-Minty
  cube (acyclic)
• Model for smallest enclosing ball problem (can
  have cycles as well) Gärtner et al., 01
• Model for linear complementarity problems
  corresponding to P-matrices (can have cycles)
  Stickney-Watson 78
• Model for general LP Gärtner-Schurr, 03

19
Linear Programming the abstract connection
Polynomial algorithm for general USOs

Strongly polynomial algorithm for LP

might
20
USOs Related work

• t(n) 2n (???) t?(n) (3/2)n
• s?(n) (v2)n(1o(1)) Aldous 84
• s?(n) is the expected number of evaluations
  the fastest randomized algorithm takes while
  finding the sink in a cube with an orientation
  given by a function on the vertices (? acyclic),
  which has a unique global sink (no restriction on
  the faces).

21
USOs Specific algorithms

• RandomEdge can take ((n-1)/2)! steps Morris
  00
• Analysis of RandomEdge for Klee-Minty cube
  Gärtner, Henk, Ziegler 98
• RandomEdge on decomposable orientations is O(n2)
  Williamson Hoke 88
• RandomFacet t?acyc(n) e2vn Gärtner 95

22
An easy first observation

• t(n) 2n-1 1

?
(?)
?
?
?

• t?(n) ¾(2n-1 1)

23
For arbitrary orientations
2n-1 1
is best possible
24

• SOMETHING BETTER
• FOR USOs?

25
Out-map
1,2,3

• Out-map sf of an orientation f of Cn is defined
  by
• v ? set of labels of outgoing edges incident
  with v
• v is a sink iff sf(v) ?

3
2
1,3
1
1,2
1
2
3
2,3
26
?????
Are out-maps of USOs bijections
?????
27

• Lemma Reversing the orientation of the edges of
  a fixed label does not affect the unique sink
  property.

Proof
a
a
a
a
a
a
a
28

• Lemma Out-maps of USOs are bijections between
  the vertex set of Cn and 2n.
• Proof
• Suppose u and v have the same out-map value.
• Reverse all edges of label from s(u)s(v).
• This makes both u and v a sink, a
  contradiction.

29

• Lemma Out-maps of USOs are bijections between
  the vertex set of Cn and 2n.
• Inherited orientation fB
• of USO f relative to B ? A.
• fB defined on CA , where A A \ B

30
Inherited Orientation

of f on CA relative to B
A.
UI
The out-map on the sinks of B-labeled faces of
CA gives a mapping s 2A ? 2A, A A \ B.

• s is well-defined
• s is a USO

31
a
d
b
c
d
a
a
d
B b,c
A a,d
32
Product Algorithm

• Choose B with Bk
• Perform a search in the (n-k)-cube with the
  inherited orientation fB takes t(n-k)
  hyper-vertex evaluation
• Each hyper-vertex evaluation is a search for
  the sink of the corresponding k-cube takes t(k)
  vertex evaluations
• t(n) t(k) t(n-k)
• t?(n) t?(k) t?(n-k)

33
Consequences

• t(2) 3 ? t(n) (v3)n 1.73n
• t(3) 5 ? t(n) (v5)n 1.71n
• t?(2) 43/20 ? t?(n) 1.47n
• t?(3) 4074633/1369468 ? t?(n) 1.44n

3
Rote 01
34

• Something different
• for deterministic

35
Fibonacci Seesaw
--- Step 0
u0
Evaluate two arbitrary antipodal vertices
v0
36
Fibonacci Seesaw
--- Step i
dim F i-1
F
ui-1
ui
dim F i
b
F
G
?
t(i-1) evaluations
b
dim G i-1
G
vi-1
dim G i
37

• ? 0 ? 1 ? i ? (i1) ? (n-1)
• t(n) 2 t(0) t(1) t(n-2)
• Thus t(n) Fn2 O(1.62n)
• where Fn is the Fibonacci sequence
• F0 0, F1 1, Fk2 F k1 Fk.
• t(4) 8 lt 9 23 1 3 3 t(2) t(2)

2
t(0)
t(1)
t(i)
t(n-2)
38
Is the Fibonacci Seesaw optimal?

• NO!!
• The SevenStepsToHeaven Algorithm solves the
  4-dimensional problem using at most 7
  evaluations.
• That is optimal t(4)7
• Combining it with the Fibonacci Seesaw gives

t(n)O(1.61n)o(Fn)
39

Lower bound for tacyc (n)
40
Tool 1 Product of orientations
41
Special cases
(n-1) 1 n
1 (n-1) n
42
Tool 2 Local change
43
Important

• Tool 1 and Tool 2 preserve acyclicity

44
Strategy against Al

• We maintain

• Set W of requested vertices

• Partial outmap ? W ? 2A

• Subset L of labels

• Orientation f on CL

45
Properties

• L W
• w n L ? w n L for ? w,w?W
• ?(w) f(w) U (n L) for ? w?W

46
What Al sees at before its fourth request
CA
a
b
c

• Requested vertices W

• Set of labels L a,b,c

• Orientation f on CL

47
Als fourth request
CA
?
?
?
?
?
?
a
?
b
c

• Requested vertices W

• Set of labels L a,b,c

• Orientation f on CL

48

• New request w is above at most one old request
  v in CL

• Pick l ? w?v
• (otherwise any l? L)

w n L
L L?l

• Update

v n L
(Tool 1)
49
Second phase
when L W n-log n

• Choose a translate C of CL which is empty, i.e.
  there are no requests in it so far (2n-L
  n !!!)
• Tell Al the sink is in C and force it solve an
• (n-log n)-dimensional problem on C.

50
Forcing a recursion by Tool 2
f on CL
C
51
Recursion

• tacyc (n) n-logn tacyc(n-logn) i.e.
• tacyc(n) n2 / 2logn

52
Currently known

• Deterministic n2-o(1) tacyc(n) t(n)
  O(1.61n)
• Randomized t?(n) O(1.44n)
• Randomized acyclic t?acyc(n) e2vn
• Log ( of USOs) T(2nlog n)

Gärtner 95
Matousek 01
53
Killing RandomFacet

• F(n) ½ F(n-1)
• ½ F(n-1)
• ½n F(0)½n F(1)...
• . ½n F(n-1)

Special case of Tool 1
A n-1
rand A n-1
F(n) evn
54
Killing RandomEdge

• Is harder. One needs
• Repeated application of Tool 1 and 2
• Lots of copies of the Klee-Minty cube

55
A large class

• Let ? be the uniform orientation
• M be a matching of the hypercube
• Flip the edges of the matching
• By Tool 2 this is a USO
• There are a lot of them
• For this class there is an algorithm finding the
  sink in FIVE(!!!) steps.

56
Outlook for Acyclic USOs

• Exploit acyclicity in order to improve the
  existing best deterministic algorithm
• Analyze RandomEdge on AUSO it could take ec³?n
  time Matousek-Szabo, 04
  (it is even worse for general USOs Morris) No
  upper bound is known for AUSO.
• Is the class of orientation constructed by Tool
  1 and 2 realizable as an LP?

57
Outlook for General USOs

• Provide a faster randomized algorithm, which is
  conceptually different from the Product Algorithm
• Prove any lower bound for the expected running
  time of a randomized algorithm
• Behaviour of RandomFacet is unresolved (on
  acyclic USOs the expected running time is eT(vn)
  Matousek, Gärtner)

58

• THE END