Approximate Distance Oracles

1
Approximate Distance Oracles

• Mikkel Thorup ATT Research
• Uri Zwick Tel Aviv University

2
Finite metric space
Approximate distance oracle
O(n2) time
n by ndistancematrix
Compact datastructure
O(1) query timeExact answers?(n2) space
O(k) query timestretch 2k-1O(kn11/k) space
3
APSPalgorithm
n by ndistancematrix
Graph
kmn1/k timekn11/k space
Compact datastructure
4
Approximate Distance Oracles
Constant query time!
This tradeoff isessentially optimal !
5
Approximate Distance Oracles
6
Stretch/space tradeoff
Let G(V,E) be a graph with Vn and
girth(G)2k2. Any subgraph G(V,E) of G must
have a distinct data structure! If (u,v)?E,
then ?G(u,v)1. Otherwise ?G(u,v) 2k1. As
there are 2E different subgraphs of G, some
subgraphs must have data structures of at least
E bits. Conjecture (Erdös 65) For every k1,
there are infinitely many n-vertex graphs with
?(n11/k) edges that have girth2k2.
7
Spanners
Let G be a weighted undirected graph. A subgraph
H of G is a t-spanner of G iff ?u,v?G, ?H(u,v)
? t ?G(u,v) .
Awerbuch 85 Peleg-Schäffer 89
8
Trivial example
9
Theorem
For every k1, every weighted undirected graph on
n vertices has a (2k-1)-spanner with at most
m2k1(n) n11/k edges.
Tight for k1,2,3,5. Conjectured to be tight for
any k
Maximal number of edges in an n-vertex graph with
girth 2k1
10
Proof/Algorithm
Consider the edges in non-decreasing order of
weight. Add each edge to the spanner if it does
not close a cycle of size at most 2k. The
resulting graph is a (2k-1)-spanner and it does
not contain a cycle of size at most 2k. Hence the
number of edges is at most m2k1(n) n11/k.
Althöfer, Das, Dobkin, Joseph, Soares 93
11
If cycle?2k, then red edge can be removed.
12
A hierarchy of centers
A0?V Ak ?? Ai ?sample(Ai-1,n-1/k)
13
Clusters
w
14
Bunches (inverse clusters)
15
Bunches
p2(v)
v
p1(v)
16
The data structure

• For every vertex v?V
• The centers p1(v), p2(v),, pk-1(v)
• A hash table holding B(v)
• For every w?V, we can check, in constant time,
  whether w?B(v), and if so, what is ?(v,w).

17
Lemma EB(v)kn1/k
Proof B(v)?Ai is stochastically dominated by a
geometric random variable with parameter pn-1/k.
18
Query answering algorithm
Algorithm distk(u,v) w?u , i?0 while
w?B(v) i ?i1 (u,v) ?(v,u) w
?pi(u) return ?(w,u) ?(w,v)
19
Query answering algorithm
w3p3(v)?A3
w2p2(u)?A2
w1p1(v)?A1
u
v
20
Analysis
wipi(u)?Ai
wi-1pi-1(v)?Ai-1
i?
(i1)?
i?
(i-1)?
v
u
?
21
Spanners / Tree covers
In each cluster, construct a tree of shortest
paths
The union of all these trees in a (2k-1)-spanner
with kn11/k edges.
w
Constructed in O(kmn1/k) time!
22
Tree Cover
Each vertex contained in at most n1/k log n
trees. For every u,v, there is a tree with a path
of stretch at most 2k-1 between them.
23
Applications

• Routing
• Distance labels
• Sub-quadratic approximation algorithms for metric
  space problems

24
Open Problems

• Deterministic construction of(2k-1,n11/k,k)-dist
  ance oracles in o(mn) time?
• Constructing a (3,n3/2,1)-distance oracle in
  n2o(1) time?
• Distance oracles withadditive errors?