Approximate Distance Approximate Distance OraclesOracles

Mikkel Thorup and Uri Zwick Presented ByShiri Chechik

Approximate Distance Approximate Distance OraclesOraclesConsider a graph G=(V,E).An approximate distance oracle with a

stretch k for the graph G is a data-structure that can answer an approximate distance query for any two vertices with a stretch of at most k.

For every u,v in V the data structure returns in “short time” an approximate distance d’ such that:

dG(u,v) d’ k· dG(u,v) .

Stretch

Query

timeSpace

Preproc.

timeReference

64k kn1/k

kn1+1/k kmn1/k

Awerbuch-Berger-Cowen-

Peleg ‘93

2k+ kn1/k Cohen ‘93

2k-1 k Thorup-Zwick ‘01

Constant query time!This tradeoff is

essentially optimal !

Approximate Distance Approximate Distance OraclesOracles

Slide from Uri Zwick

SpannersSpanners - Formal - Formal DefinitionDefinition

Consider a graph G=(V,E) with positive edge weights.

A subgraph H is a k-spanner of G if for every u,v in V:

dH(u,v) k·dG(u,v) .

SpannersSpanners - Example - Example

v

SpannersSpanners - Equivalent - Equivalent DefinitionDefinition

A subgraph H is a k-spanner of G if for every edge (u,v) in E:

dH(u,v) k· w(u,v) .

x

y

Spanners for General Spanners for General graphsgraphsTheoremOne can efficiently find a (2k-1)-spanner with at most n1+1/k edges.

Spanners for General Spanners for General graphsgraphsGirth Conjecture (Erdös and

others):There are n-vertex graphs with Ω(n1+1/k) edges that have girth > 2k.

Known for k=1,2,3,5.

Distance Oracles Distance Oracles -- Construction Construction Preprocessing PhasePreprocessing Phase

First build a hierarchy of centers, A0,…, Ak

A0 V, Ak

Ai sample(Ai-1, n-1/k)

A0 =

A1 =

A2 =

Ak =

Distance Oracles Distance Oracles -- Construction Construction Preprocessing PhasePreprocessing Phase

A hierarchy of centersA hierarchy of centers

A0V ; Ak ; Ai sample(Ai-1,n-1/k) ;

Slide from Uri Zwick

Notationspi(v) is the closest node to v in Ai

For each w Ai\Ai+1

◦C(w) {v| δ(v,w) < δ(v,pi+1(v))}

Distance Oracles Distance Oracles -- Construction Construction Preprocessing PhasePreprocessing Phase

ClustersClustersA0=A1=A2=

1 1( ) { | ( , ) ( , )} ,i i iC w v V w v A v w A A

w

Slide from Uri Zwick

Bunches (inverse Bunches (inverse clusters)clusters)

1

1

1

1

( ) { | ( , ) ( , )} ,

( ) { | ( , ) (

( ) (

, }

)

)i i

i

ii

i i

C w v V w v A v

if w

w

B v w A A w v A

B

A A

v v C w

v

Slide from Uri Zwick

Distance OraclesDistance Oracles - - ExampleExample

V

P1(V)

A0 = {v1, v2, v3, v4} A1 = {v2, v3} A2 = {v3} A3 = {}

C(v1)= {v1, v4}, C(v4)= {v4} C(v2)= {v2, v1} C(v3)= {v1, v2, v3, v4}

Distance OraclesDistance Oracles - - ExampleExample

1 2

11.5

V2

V1

V3

V4

Data structuresFor every v V

◦p1(v),…,pk-1(v) and the distance from v to pi(v).

◦C(v) (hash table) and the distance from v to every u in C(v).

Distance Oracles Distance Oracles -- Construction Construction Preprocessing PhasePreprocessing Phase

Lemma: E[|B(v)|]≤kn1/k

|B(v)Ai| is stochastically dominated by a geometric random variable with parameter p=n-1/k.

Distance OraclesDistance Oracles

Slide from Uri Zwick

Lemma For every two nodes u and v, there exists a node w such that◦u,v C(w)◦ The distance from u to w and

from v to w is at most k·d(u,v)

Distance OraclesDistance Oracles

Distance Oracles Distance Oracles -- Construction Construction Query PhaseQuery Phase

u v

P1(v)

P2(u)P3(v)

<2>

<3

Distance Oracles Distance Oracles -- Construction Construction Query PhaseQuery Phase

Distance OraclesDistance Oracles - - ExampleExample

1 2

11.5

V2

V1

V3

V4

Consider a node v C(w), every node on a shortest path from v to w also belongs to C(w).

Distance Oracles Distance Oracles -- Properties Properties Thorup and Zwick (2005)Thorup and Zwick (2005)

v w

x Ai+1

y

v belongs to C(pi(v)) for every

0≤ i ≤k-1.

Distance Oracles Distance Oracles -- Properties Properties Thorup and Zwick (2005)Thorup and Zwick (2005)

From each cluster, construct a tree T(w) containing shortest path.

Distance OraclesDistance Oracles

wu

v

The union of all these trees is a (2k-1)-spanner with O(kn1+1/k) edges.

Distance OraclesDistance Oracles