1. Compact Routing Schemes
Mikkel Thorup Uri Zwick
AT&T Labs – Research
Tel Aviv University

2. Routing 3 2 v u 1
Packet:
label(v)
information

3. The same header is used for all messages sent from u to v
Handshaking u v
header(u,v)
Packet:
header(u,v)
information
The same header is used for all messages sent from u to v

4. Routing in Trees Each vertex is assigned a (1+o(1))log2n – bit label.
Given label(u) and label(v), it is possible to find, in constant time, the right edge to take from u.
Similar result by Fraigniaud and Gavoille [ICALP’01] u v

5. Routing in General Graphs
Handshaking?
Table Size
Stretch no
n1/2 3
yes
n1/3 5 7
n1/k
2k-1
4k-5

6. Previous Results n2/3 n1/2 n1/k 3 5 O(k2) Authors Table Size Stretch
Cowen ‘99
n2/3 3
Eilam, Gavoille Peleg ‘98
n1/2 5
Awerbuch Peleg ‘92
n1/k
O(k2)

7. Our Results Are Essentially Optimal!
Labels must be at least log2n – bit long.
In graphs, for stretch<3, the total size of the routing tables must be (n2). For stretch<5, the total size must be (n3/2).
Conjecture: For stretch<2k+1, the total size of the tables must be (n1+1/k). (Equivalent to a well known girth conjecture of Erdös.)

8. Tree Routing – A Practical Scheme
O(log2n)-bit labels. Arbitrary port numbers. 1 10 7 2 10 12
DFS numbering:
For every vertex u, let fu be the largest descendant of u. Then v is a descendant of u iff 3 11 13 14 4 5 6 7
A trivial solution with O(deg(v)) memory. 8 9

9. Tree Routing – A Practical Scheme (Cont.)
Let s(v) be the number of descendants of v. 14 8 2 1 7 4 3
Let pv be the parent of v. Then, vertex v is heavy if s(v)s(pv)/2, and light otherwise.

10. Tree Routing – A Practical Scheme (End)
The light-level lv of a vertex v is the number of light vertices on the path to it from the root.
Claim: lv<log2n
label(v)=(v,port(e1),port(e2),…)
At v we store: v, fv, hv, lv, port(v,pv) and port(v,hv). e1 1 e2 2 2 e3 3 3 e4 4 v

11. Routing in Graphs

12. centA(v) = a center closest to v
Choose a Set of Centers
centA(v) = a center closest to v

13. Construct Clusters
cluster
clusterA(v) = vertices that are closer to v than to all centers.

14. Keep Routing Info from v to AclusterA(v)

15. If vclusterA(u), Route Directlyw
For any w on the shortest path we have vclusterA(w).

16. If vclusterA(u), Route through centA(v)
Label(v)=
(v,centA(v),port(centA(v),v)) v

17. How do we choose centers?
We want A such that
|A|=O(n1/2)
clusterA(v)=O(n1/2), for every v
[Cowen does this with O(n2/3)]

18. W{wV | clusterA(w)>4n1/2 }; Return A;
Algorithm center(G)
A; WV;
While W {
AA  choose(W,n1/2);
W{wV | clusterA(w)>4n1/2 }; }
Return A;
The expected size of A is O(n1/2log n).

19. Smaller Tables, Larger stretch
Use a hierarchy of centers.
Construct a tree cover of the graph.
Identify an appropriate tree from the cover and route on it.

20. Each vertex contained in at most n1/k trees.
Tree Cover
Each vertex contained in at most n1/k trees.
For every u,v, there is a tree with a path of stretch at most 2k-1 between them.

21. Is there a routing scheme with:
Table size = O(n1/k)
Label size = O(log n)
No handshaking
???