Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees

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Improved Compact Routing Tables for Planar
Networks via Orderly Spanning Trees

• Hsueh-I Lu (???)
• Academia Sinica, Taiwan

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Outline

• The local routing table problem
• Previous work and our result
• Our tool orderly spanning tree
• Technical details
• Conclusion future work

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The problem of designing local routing tables

• Input
• a network G
• a routing tree Tr for each node r in G
• Output
• a labeling scheme for the nodes in G
• a local routing table Rr for each node r in G
• such that
• each r knows how to forward a network packet to
  the appropriate neighbor of r only by looking up
  Rr with the packets destination.

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For example,
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Add node label
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The routing table for node 2
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The routing table for node 5
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Things to minimize

• Space
• table size the number of bits required to
  encode each routing table.
• Time
• computation time the time to compute each
  routing table from its routing tree.
• query time the time to compute from the routing
  table the appropriate neighbor to pass a network
  packet.

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Explicit encoding

• table size O(n log n)
• computation time O(n)
• query time O(1)
• best possible for dense netwoks

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A natural question

• Can one reduce the table sizes for sparse
  networks without increasing the computation time
  and query time by too much?

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HIDED THIS PAGEThings to be exploited

• We have the freedom to determine the node
  labeling.
• restriction
• The labeling has to be consistent in all routing
  tables, though.

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Gavoille-Hanusse ICALP99

• The network is given with a k-page book
  embedding
• table size 2kn o(n) bits
• (a lower bound kn bits)
• computation time O(n)
• query time O(log2c n) for any cgt0

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Planar networks
Planar networks

• the focus of our paper
• important in routing under geometric metrics
• Karp 00, Narasimhan-Smid 00, Bose et al. 01, Gao
  et al. 01, Li et al. 02,

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Corollary

• Planar network
• table size 8n o(n) bits (i.e., k4)
• (a lower bound 2n bits, i.e., k2)
• computation time O(n)
• query time O(log2c n) for any cgt0

planar network
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Our result

• More compact routing tables for planar networks

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Specifically,

• Gavoille-Hanusse, ICALP99
• table size 8n o(n) bits
• computation time worst-case O(n)
• query time O(log2c n) for any cgt0
• our result
• table size 7.181n o(n) bits
• computation time expected O(n)
• query time same

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8n -gt 7.181n

• The improvement does not seem to be a big deal,
  but hopefully you will find our tool interesting.

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Orderly Spanning Tree

• Ladies and gentlemen
• introducing
• my favorite algorithmic tool for planar graphs

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An Orderly Spanning Tree
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Orderly spanning tree

• debut Chiang-Lin-Lu-SODA01
• applications for planar graphs
• SODA01 a) 2-visibility drawing
• b) encoding with query support
• GD01 floor planning
• GD02 podevs drawing
• COCOON02 local routing tables

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Why is OST versatile?

• Orderly spanning tree generalizes two powerful
  tools
• canonical ordering
• for planar graphs not required to be 3-connected
• realizer
• for planar graphs not required to be triangulated

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Technical details 2 steps
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Step 1

• Compute an orderly spanning tree T for the input
  planar network G.
• Label the nodes of G using the preordering of the
  orderly spanning tree T.

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Orderly Spanning Tree
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Step 2

• Combine the routing tree Tr and the orderly
  spanning tree T into a planar graph Gr.
• Encode Gr with support for the parent query of Tr
  in O(log n) time.

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Gr T Tr
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Table size 7.181n o(n),based upon

• Chiang-Lin-Lu-SODA01
• best know encoding for planar graphs with support
  for degree/adjacency query.
• Raman-Raman-Rao-SODA02
• best known dictionary with support for
  rank/select query.

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Computation time expected O(n)

• Bottleneck the expected O(n) time for
  constructing the dictionary of Raman-Raman-Rao-SO
  DA02

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Query time O(log2c n) for any cgt0.

• O(log1c n) parent queries suffice.
• Each parent query takes O(log n) time.

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Summary of our results

• Compact routing table for planar networks
• table size 7.181n o(n) bits
• computation time expected O(n)
• query time O(log2c n) for any cgt0
• A new application of orderly spanning tree.

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Future work

• Other applications of orderly spanning tree
• a possible candidate 1-visibility drawing
• Different approaches to attack the routing table
  problems
• a possible candidate separator-based approach.