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Some Results in Interval Routing

Francis C.M. LauHKU and ITCS Tsinghua

December 2, 2007

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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.

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Example: XY-routing in Grid

• “X first, and then Y”.

• What is in a node’srouting table?

• Size = O(log n) bits,n is the number of nodes.

• Length of path is the shortest.

“My coordinates”

0,02,0

5,2

X

Y

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Interval Routing

• A well-known technique to do compact routing.

• To label the nodesfrom 0 to n-1.

• To label eachoutgoing link byan interval <p,q>,p and q arenode numbers.

edge label node label

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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

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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].

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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

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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 (n1/2) for planar graphs [Tse-Lau 99 & 04].

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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]:

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• 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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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.

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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.

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• How to emulateXY-routing with IR?

• The original XY-routingwon’t work if somenode goes down.

• With IR, you can adjustthe labels.

• Problems:– IR (CR) for “injured” graphs– Deadlock-free IR– IR for mobile nodes (localized changes to topology)

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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.

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Thank you!