1. Approximate Distance Oracles
Mikkel Thorup AT&T Research
Uri Zwick Tel Aviv University

2. Approximate distance oracle Compact data structure
Finite metric space
Approximate distance oracle
O(n2) time
n by n distance matrix
Compact data structure
O(1) query time Exact answers (n2) space
O(k) query time stretch 2k-1 O(kn1+1/k) space

3. Compact data structure
APSP algorithm
n by n distance matrix
Graph
kmn1/k time kn1+1/k space
Compact data structure

4. Approximate Distance Oracles
Reference
Preproc. time
Space
Query time
Stretch
Awerbuch-Berger-Cowen-Peleg ‘93
kmn1/k
kn1+1/k
kn1/k
64k
Cohen ‘93
2k+
Thorup- Zwick ‘01 k
2k-1
Constant query time!
This tradeoff is essentially optimal !

5. Approximate Distance Oracles
Preproc. time
Space
Query time
Stretch
Ω(n2)
2.99
mn1/2
O(n3/2) 1 3
Ω(n3/2)
4.99
mn1/3
O(n4/3) 5 …
m log n
n log n
log n

6. Stretch/space tradeoff
Let G=(V,E) be a graph with |V|=n and girth(G)≥2k+2.
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) ≥2k+1.
As there are 2|E| different subgraphs of G, some subgraphs must have data structures of at least |E| bits.
Conjecture: (Erdös ’65) For every k≥1, there are infinitely many n-vertex graphs with Ω(n1+1/k) edges that have girth≥2k+2.

7. Spanners H(u,v)  t G(u,v) . 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 k≥1, every weighted undirected graph on n vertices has a (2k-1)-spanner with at most m2k+1(n) ≤ n1+1/k edges.
Tight for k=1,2,3,5. Conjectured to be tight for any k
Maximal number of edges in an n-vertex graph with girth ≥ 2k+1

10. [Althöfer, Das, Dobkin, Joseph, Soares ‘93]
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 m2k+1(n) ≤ n1+1/k.
[Althöfer, Das, Dobkin, Joseph, Soares ‘93]

11. If |cycle|2k, then red edge can be removed.

12. A0V ; Ak  ; Ai sample(Ai-1,n-1/k) ;
A hierarchy of centers
A0V ; Ak  ; Ai sample(Ai-1,n-1/k) ;

13. A0= A1= A2=
Clusters w

14. Bunches (inverse clusters)

15. Bunches
A0= A1= A2=
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: E[|B(v)|]≤kn1/k
Proof: |B(v)Ai| is stochastically dominated by a geometric random variable with parameter p=n-1/k.

18. Query answering algorithm
Algorithm distk(u,v)
wu , i0
while wB(v)
{ i i+1
(u,v) (v,u)
w pi(u) }
return (w,u)+ (w,v)

19. Query answering algorithm
w3=p3(v)A3
w2=p2(u)A2
w1=p1(v)A1 u v

20. Analysis
wi=pi(u)Ai
wi-1=pi-1(v)Ai-1 i
(i+1) i
(i-1) u v 

21. Spanners / Tree covers w
In each cluster, construct a tree of shortest paths
The union of all these trees in a (2k-1)-spanner with kn1+1/k edges. w
Constructed in O(kmn1/k) time!

22. Each vertex contained in at most n1/k log n trees.
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. Subsequent work
Approximate distance oracles for planar networks (Thorup ’01)
Roundtrip spanners for directed networks (Roditty-Thorup-Z ’02)

25. Open Problems
Deterministic construction of (2k-1,n1+1/k,k)-distance oracles in o(mn) time?
Constructing a (3,n3/2,1)-distance oracle in n2+o(1) time?
Distance oracles with additive errors?