Approximate Distance Oracles

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Approximate Distance Oracles
Mikkel Thorup and Uri Zwick Presented By Shiri
Chechik
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Approximate Distance Oracles

• Consider 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) .

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Approximate Distance Oracles
Reference Preproc. time Space Query time Stretch
Awerbuch-Berger-Cowen-Peleg 93 kmn1/k kn11/k kn1/k 64k
Cohen 93 kmn1/k kn11/k kn1/k 2k?
Thorup-Zwick 01 kmn1/k kn11/k k 2k-1
Constant query time!
This tradeoff isessentially optimal !
Slide from Uri Zwick
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Spanners - Formal Definition

• 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) ? kdG(u,v) .

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Spanners - Example
v
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Spanners - Equivalent Definition

• 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
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Spanners for General graphs

• Theorem
• One can efficiently find a (2k-1)-spanner with
  at most n11/k edges.

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Spanners for General graphs

• Girth Conjecture (Erdös and others)
• There are n-vertex graphs with O(n11/k) edges
  that have girth gt 2k. Known for k1,2,3,5.

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Spanners - Equivalent Definition

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

x
y
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Greedy Algorithm
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Greedy Algorithm - Properties

• H is a k-spanner of G (trivial).
• The girth of H is greater than k1.
• Assume c is a cycle of size at most k1
• Let e(u,v) be the last edge added to cycle c.
• Let w(e) be the weight of edge e.
• All other edges of cycle c are of weight less
  than w(e).
• We found a path between u and v of length less
  than kw(e).
• e was not supposed to be added to H.

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Greedy Algorithm - Example
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Lemma

• Let G(V,E) be a graph with girth 2k1.
• This is also conjectured to be the lower bound of
  the number of edges in a (2k-1)-spanner (Erdös
  65).

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Theorem

• One can efficiently find a (2k-1)-spanner with
  edges.

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Distance Oracles - Construction Preprocessing
Phase

• First build a hierarchy of centers, A0,, Ak
• A0 ?V, Ak ??
• Ai ?sample(Ai-1, n-1/k)

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Distance Oracles - Construction Preprocessing
Phase
A0
A1
A2
Ak
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A hierarchy of centers
A0?V Ak ?? Ai ?sample(Ai-1,n-1/k)
Slide from Uri Zwick
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Distance Oracles - Construction Preprocessing
Phase

• Notations
• pi(v) is the closest node to v in Ai
• For each w ? Ai\Ai1
• C(w) ? v d(v,w) lt d(v,pi1(v))

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Clusters
w
Slide from Uri Zwick
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Bunches (inverse clusters)
Slide from Uri Zwick
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Distance Oracles - Example
V
P1(V)
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Distance Oracles - Example

• 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

V3
1
2
V2
V4
1
1.5
V1
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Distance Oracles - Construction Preprocessing
Phase

• Data structures
• For 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).

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Distance Oracles

• Lemma EB(v)kn1/k
• B(v)?Ai is stochastically dominated by a
  geometric random variable with parameter pn-1/k.

Slide from Uri Zwick
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Distance Oracles

• 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 kd(u,v)

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Distance Oracles - Construction Query Phase
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Distance Oracles - Construction Query Phase
P3(v)
P2(u)
lt3?
P1(v)
2?gt
lt?
v
?
u
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Distance Oracles - Example

• What is d(v4, v1)?
• Is v1? C(v4)?
• w p1 (v1) v2
• Is v4? C(v2)?
• w p2(v4) v3
• Is v1? C(v3)? Yes
• C(v1) v1, v4, C(v4) v4
• C(v2) v2, v1
• C(v3) v1, v2, v3, v4

V3
1
2
V2
V4
1
1.5
V1
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Distance Oracles - Properties Thorup and Zwick
(2005)

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

x ? Ai1
v
w
y
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Distance Oracles - Properties Thorup and Zwick
(2005)

• v belongs to C(pi(v)) for every
• 0 i k-1.

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Distance Oracles - Construction Thorup and Zwick
(2005)

• Short Summary
• 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?

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Distance Oracles

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

v
u
w
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Distance Oracles

• The union of all these trees is a (2k-1)-spanner
  with
• O(kn11/k) edges.