1. 1 Some Results in Interval Routing Francis C.M. Lau HKU and ITCS Tsinghua December 2, 2007

2. 2 Compact Routing Input: a network G with weighted edges. Output: a routing scheme for G. –Every node has a mechanism to route a message through its next hop(s). –Every message eventually reaches its destination. Goals: –To minimize the sizes of the routing tables. –To minimize the lengths of the routing paths. –Or a tradeoff between the two.

3. 3 Example: XY-routing in Grid “X first, and then Y”. What is in a node’s routing table? Size = O(log n) bits, n is the number of nodes. Length of path is the shortest. “My coordinates” 0,0 2,0 5,2 X Y

4. 4 Interval Routing A well-known technique to do compact routing. To label the nodes from 0 to n-1. To label each outgoing link by an interval, p and q are node numbers. edge label node label

5. 5 IR is Real The transputer (1988-): a technical wonder but a business failure. One of the best examples of software-hardware co-design: –The Occam Language (modeled after Hoare’s CSP) –“Processes” and IPC at the hardware level! The C104 router (c. 1990) –A 32-way switch (100 Mbps per link) –Implemented IR and wormhole routing in hardware A 2D mesh of 42 transputers

6. 6 Interval Routing Scheme (IRS) At most O(d log n) space is needed at a node, where d is the node’s (out) degree. In general, d < n, and so this is compact. Question: how to label the nodes and edges so that all the routing paths are shortest paths (an optimum IRS)? Optimum IRSs exist for some specific graphs (including the grid), but not for arbitrary graphs [Gavoille 00].

7. 7 IRS for Arbitrary Graphs Ruzicka gave a “globe graph” which admits no optimum IRS, and a lower bound of 1.5D-1 on the longest path, D is the diameter of the network [Ruzicka 91]. We improved that to 1.75D-1, and subsequently to 2D-3 and 2D-o(D) [Tse-Lau 97a & 97b]. The well-known upper bound using a BFS spanning tree is 2D [Santoro-Khatib 85]. This is for a family of graphs whose # nodes is large

8. 8 More Results If more space is allowed, an “M-IRS” attaches up to M labels to an edge. Question: what is the smallest M to achieve optimality in the length of the longest path (i.e., = D)? We showed that at least  (n/D) labels per edge would be necessary, and  (n 1/2 ) for planar graphs [Tse-Lau 99 & 04].

9. 9 More Results (2) An all-shortest-path IRS gives exactly all the shortest paths between any pair of nodes (some labels overlap). We proved that the following problems are NP-complete [Wang-Lau-Liu 07b] :

10. 10 The proof of the first problem is based on a transformation from the following problem, which is proved to be NP-complete in [Wang-Lau 07a]. –Starting with any instance of k-C1BS (M), we construct a graph G such that there is a row permutation on M leading to each column having not more than k consecutive 1’s blocks if and only if G supports an all-shortest-path (k+1)-IRS. The proof of the second problem is based on a transformation from the well-known Hamiltonian path problem.

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12. 12 Compact Routing Recently There’s been a flurry of activities in compact routing lately [Gavoille 07, Cowen 07]. They are more interested in other methods that give good space-stretch tradeoff than in interval routing. –Stretch = ratio between length of a route and the corresponding (shortest) distance. XY-routing in grid has optimal stretch of 1. Two variants: –Name independent: designer has no control over the labeling of nodes (e.g., assigned by some adversary). –Labeled: designer assigns the labels.

13. 13 Compact Routing for Wireless, Ad- Hoc, and Sensor Networks Small devices favor compact routing schemes. –Space and time (energy) Challenges: –Changing topology [Chen-Ganesh 06]. –Unreliable links and nodes [Gavoille-Nehéz 05]. –Susceptible to attacks (WSN). Research in this genre is just beginning.

14. 14 How to emulate XY-routing with IR? The original XY-routing won’t work if some node goes down. With IR, you can adjust the labels. Problems: –IR (CR) for “injured” graphs –Deadlock-free IR –IR for mobile nodes (localized changes to topology) 0123 4567 891011 12131415

15. References [Chen-Ganesh 06] M. Chen and G. Ganesh, “A compact routing protocol for ad-hoc networks,” Second Workshop on Spatial Stochastic Modeling of Wireless Networks, 2006. [Cowen 07] L. Cowen, Compact Routing in Theory and Practice (talk), 2007. [http://www.cs.tufts.edu/~cowen/CompactRouting2007.ppt] [Gavoille-Nehéz 05] C. Gavoille and M. Nehéz. “Interval routing in reliability networks.” Theoretical Computer Science, 333(3):415-432, 2005. [Gavoille 07] C. Gavoille. “An overview on compact routing.” Workshop on Peer-to-Peer, Routing in Complex Graphs, and Network Coding, March 2007. [http://dept-info.labri.fr/~gavoille/article/iGav07] [Gavoille 00] C. Gavoille. “A survey on interval routing.” Theoretical Computer Science, 245(2):217-253, 2000. [Ruzicka 91] Ruzicka, P. “A note on the efficiency of an interval routing algorithm.” The Computer Journal, 34:475–476, 1991. [Santoro-Khatib 85] N. Santoro and R. Khatib, “Labelling and implicit routing in networks,” The Computer Journal, 28:5-8, 1985. [Tse-Lau 04] S.S.H. Tse and F.C.M. Lau, “New Bounds for Multi-label Interval Routing”, Theoretical Computer Science, 310(1-3):61-77, 2004. [Tse-Lau 99] S.S.H. Tse and F.C.M. Lau, “On the Space Requirement of Interval Routing,” IEEE Transactions on Computers, 48(7):752-757, 1999. [Tse-Lau 97a] S.S.H. Tse and F.C.M. Lau, “A Lower Bound for Interval Routing in General Networks,” Networks, 29(1):49-53, 1997. [Tse-Lau 97b] S.S.H. Tse and F.C.M. Lau, “An Optimal Lower Bound for Interval Routing in General Networks,” Proc. of 4th International Colloquium on Structural Information and Communication Complexity (SIROCCO'97), Ascona, Switzerland, July 1997, 112-124. [Wang-Lau 07a] R. Wang, F.C.M. Lau, and Y.C. Zhao, “Hamiltonicity of Regular Graphs and Blocks of Consecutive Ones in Symmetric Matrices”, Discrete Applied Mathematics, 155(17):2312-2320, 2007. [Wang-Lau-Liu 07b] R. Wang, F.C.M. Lau, and Y.Y. Liu, “On the Hardness of Minimizing Space for All-Shortest- Path Interval Routing Schemes,” Theoretical Computer Science, 389:250–264, 2007.

16. 16 Thank you!