1. Distance and Routing Labeling Schemes in Graphs
Feodor F. Dragan
Computer Science Department Kent State University

2. Talk is based on Known results for trees Kannan&Naor&Rudich’88,
Peleg’99 and ’00,
Thorup&Zwick’01,
Fraigniaud&Gavoille’01

3. Traditional graph representation
Store adjacency information in global data structure
Requires much storage
Node labels contain no information, serve only as “pointers” to global data structure

4. Local graph representation
IDEA: Store local pieces of information, i.e.,
Associate label Label(v) with each node v, s.t. labels allow inferring adjacency information locally, without using additional memory and information
Question: Can this be done compactly & efficiently?

5. “Ancient” Example: Hamming adjacency labeling
Goal: Label each node v with m-bit label Label(v) s.t. Label(v), Label (u) allow deciding if u and v are adjacent:
[Breuer&Folkman’67]: For any n-node graph there exists a Hamming distance adjacency labeling with m=2nD, T=4D -4, where D = max node degree.
Question: Can this be done with short labels?

6. Interest Revived: Adjacency Labeling
[Kannan&Naor&Rudich’88] What’s in label?
Adjacency-labeling (for graph family F ):
Function Label labeling nodes of any graph in F with short distinct labels (compactness)
Efficient algorithm for deciding adjacency of two nodes given their labels (efficiency) (polylogarithmic labels  polylogarithmic time)
Note: - Algorithm knows nothing beyond the labels (even, it does not know which graph they come from)

7. Adjacency Labeling on Trees
[Kannan&Naor&Rudich’88]
Labeling:
1. Arbitrarily root the tree and prelabel each node v with distinct integer I(v) from [1..n].
2. Label root r by Label(r)=(I(r)).
3. Label each non-root node v with parent w by Label(v)=(I(v),I(w)).
Labels contain bits

8. Adjacency Labeling on Trees
[Kannan&Naor&Rudich’88]
Labeling: (with bits)
1. Arbitrarily root the tree and prelabel each node v with distinct integer I(v) from [1..n].
2. Label root r by Label(r)=(I(r)).
3. Label each non-root node v with parent w by Label(v)=(I(v),I(w)).
Adjacency Decoder for u,v: Check if 1st entry in Label(v) = 2nd entry in Label(u) or vice versa.
Constant time!

9. Extensions
Labels contain bits
Bounded arboricity graphs
Bounded degree
Bounded-genus graphs (e.a., planar graphs)
Intersection-based graphs (e.a., interval graphs)
c-decomposable graphs
Question (compactness): Is this doable for all graphs with bit labels?

10. Negative results
Note: Graph class F is adjacency-labelable using bit labels
 each G from F is fully defined by bits
 # of represented graphs
Therefore: Graph family F containing n-node graphs is not adjacency-labelable using bit labels.
Negative examples: Bipartite / chordal graphs

11. Question: Can this be done for other. types of information
Question: Can this be done for other types of information beyond adjacency?
In general graphs
Distance
Edge / node connectivity
Routing
Flow …
In trees
Ancestry
Lowest common ancestor (LCA or NCA)
Depth of NCA (separation level)
Distance

12. Question: Can this be done for other. types of information
Question: Can this be done for other types of information beyond adjacency?
In general graphs
Distance
Edge / node connectivity
Routing
Flow …
In trees
Ancestry
Lowest common ancestor (LCA or NCA)
Depth of NCA (separation level)
Distance

13. Distance Labeling Scheme
Goal: Short labels that encode distances and distance decoder, an algorithm for inferring the distance between two nodes only from their labels (in time polynomial in the label length)
…a method for storing and extracting distances
in a “distributed” fashion…
Labeling: v  Label(v)
Distance decoder: D(Label(v), Label(u))  dist(u,v)
Distance

14. Simple examples
Enjoy bit distance labeling schemes with constant-time decoder
Distance

15. [Gavoille,&Peleg&Perennes&Raz’01]
General graphs
Easy labeling:
Label each node in n-node graph G on
with all distances
Hence, for the class of all n-node graphs
Distance
Currently:
[Gavoille,&Peleg&Perennes&Raz’01]
Similar bounds apply for other large graph families (bipartite / chordal / …)

16. Distance labeling on trees [Peleg’99]
Preprocessing:
Arbitrarily label
each node v of T
with distinct
integers I(v)
from [1..n]

17. Recursive partitioning and decomposition tree
Tree separator: node v in n-node tree T, whose removal breaks T into subtrees of size at most n/2.
Known fact: Every tree has a separator (can be found in linear time) H T

18. Labeling H T
Construct for H Depth of NCA labeling scheme ( bit labels [Peleg’00] and constant (can be made) decision time)
Then,
v  A(v)
Hence, bit labels

19. Constant Time Distance DecoderH T
(A(u),2,2,0))
(A(v),6,2,1,0))
Given Label(u) and Label(v)
Function distance_decoder_trees(Label(u),Label(v))
Extract A(u), A(v)
Use A(u), A(v) to find the depth k of NCA(u,v) in H  1
Output , by extracting appropriate
entries from labels  2+2=4

20. Lower bounds and Planar graphs
[Gavoille,&Peleg&Perennes&Raz’01]

21. Routing Labeling Scheme
Goal: Short labels that encode the routing information and routing protocol, an algorithm for inferring port number of the first edge on a shortest path from source to destination,
giving only labels and nothing else
Labeling: v  Label(v)
Distance decoder: R(Label(v), Label(u))  port(v,u)
Distance

22. General graphs Easy labeling (full routing tables):
Label each node in n-node graph G on
with full routing information
Hence, for the class of all n-node graphs one needs nlogn -bit labels
Distance
Currently:
Nothing better can be done for general graphs if we insist on routing via shortest paths

23. Labeling trees with -bit labels
Construct for H Depth of NCA labeling scheme ( bit labels
Then,
v  A(v)
Hence, bit labels

24. Constant Time Routing DecisionH
Given Label(v) and Label(u)
Function routing_decision_trees(Label(u),Label(v))
Extract A(v), A(u)
Use A(v), A(u) to find the depth k of NCA(v,u) in H  1
If k<h(v) output from Label(v)  port(4,12)=1
else (i.e., k=h(v)) output from Label(u)

25. Best routing labeling scheme for trees and Planar graphs
[Thorup&Zwick’01] For family of n-node trees there is a routing labeling scheme with labels of size O(logn)-bits per node and constant time routing decision.
[Fraigniaud&Gavoille’01] For family of n-node trees there is a routing labeling scheme with labels of size bits per node and constant time routing decision.
[Gavoille&Hanusse’99] For family of n-node planar graphs there is a routing labeling scheme with labels of size (8n +o(n))-bits per node.