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Unique Sink Orientations of CubesMotivation and

Algorithms

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

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

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

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

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

- 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

?

?

t(2)

3

?

?

t(2) 3

?

?

?

?

- t(2) 3

t?(2) 43/20

Further small values

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

Rote 01

- WHY ???

Appearences of USOs

- Orientations of slanted geometric cubes relative

to a generic linear function, e.g. Klee-Minty

cube (acyclic)

Linear Programming the straightforward

connection

acyclic USO

Minimum

Sink

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

Smallest Enclosing Ball Problem

EXAMPLE

n points

d-space

b

a

c

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

Smallest enclosing ball

USO of cubes

b

abc

a

ab

bc

ac

a

b

c

c

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

Linear Programming the abstract connection

Polynomial algorithm for general USOs

Strongly polynomial algorithm for LP

might

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).

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

An easy first observation

- t(n) 2n-1 1

?

(?)

?

?

?

- t?(n) ¾(2n-1 1)

For arbitrary orientations

2n-1 1

is best possible

- SOMETHING BETTER
- FOR USOs?

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

?????

Are out-maps of USOs bijections

?????

- 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

- 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.

- 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

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

a

d

b

c

d

a

a

d

B b,c

A a,d

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)

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

- Something different
- for deterministic

Fibonacci Seesaw

--- Step 0

u0

Evaluate two arbitrary antipodal vertices

v0

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

- ? 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)

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)

Lower bound for tacyc (n)

Tool 1 Product of orientations

Special cases

(n-1) 1 n

1 (n-1) n

Tool 2 Local change

Important

- Tool 1 and Tool 2 preserve acyclicity

Strategy against Al

- We maintain

- Set W of requested vertices

- Partial outmap ? W ? 2A

- Subset L of labels

- Orientation f on CL

Properties

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

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

Als fourth request

CA

?

?

?

?

?

?

a

?

b

c

- Requested vertices W

- Set of labels L a,b,c

- Orientation f on CL

- 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)

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.

Forcing a recursion by Tool 2

f on CL

C

Recursion

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

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

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

Killing RandomEdge

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

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.

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?

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)

- THE END

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