Wolfgang Mulzer Institut fr Informatik Headerless Routing in

Wolfgang Mulzer Institut für Informatik Headerless Routing in Unit Disk Graphs Wolfgang Mulzer Max Willert

Routing in Graphs s 2 s t 00101010 011010 … 101001 00101001 … routing table packet header h W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs 111001010… target label t 2

Routing in Graphs s 2 s t 111101 010101 110100 … 00101010 011010 … routing table packet header h W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs 111001010… target label t 3

Routing in Graphs s s 2 s 3 s 4 s 5 s 6 t = s 7 routing path: s=s 1, s 2, s 3, s 4, s 5, s 6, s 7=t stretch: maxs, t |routing path| / |shortest path| W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs 4

Routing in Graphs s s 2 s 3 s 5 s 4 s 6 t = s 7 table size unweighted (1/ )n 2/3 header stretch labels 1/ 2+ log n [RT 15] weighted (1/ )n 2/3 log D (1/ ) log D 5+ log n [RT 15] planar (1/ )polylog n 1+ log 2 n [T 04] W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs 5

Unit Disk Graphs p q pq E |pq| 1 W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs 6

Unit Disk Graphs p q pq E |pq| 1 table size header stretch log 2 n ----- 3 log 2 n [XYD 12] -5 log 4 n log 2 n 1+ log n [KMRS 16] ----- 1+ -4 log 4 n today W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs labels 7

Additive Distance Oracle (Chan-Skrepetos) Given: unit disk graph DG(P), D = diameter of DG(P) > 1/D Query: for s, t in P, find with d(s, t) + O( D) s t W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs 8

Additive Distance Oracle (Chan-Skrepetos) Decomposition Tree: μ rooted tree of height O(log n) every node μ has sets V(μ) and π(μ) children μ 1, …, μk → V(μ) = π(μ) V(μ 1) … V(μk) μ 11 μ 2 μ 3 μ 4 μ 12 μ 13 V(μ) V(μ 1) π(μ 1) V(μ 11) π(μ) V(μ 12) V(μ 3) V(μ 13) W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs V(μ 4) P 9

Additive Distance Oracle (Chan-Skrepetos) Decomposition Tree: μ for all s, t in P: set = minμ: s, t V(μ), p π(μ) d(s, p) + d(p, t) then d(s, t) + O( D) μ 11 μ 2 μ 3 μ 4 μ 12 μ 13 V(μ) V(μ 1) π(μ) V(μ p 11) V(μ 12) V(μ 3) V(μ q 13) W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs V(μ 4) P 10

Routing Scheme with Additive Stretch Idea: turn Chan-Skrepetos Oracle into a routing scheme μ precompute shortest path trees for all points in π(μ) with respect to all V(μ) store information in labels and use for routing (Challenge: when to switch? ) μ 11 μ 2 μ 3 μ 4 μ 12 μ 13 V(μ) V(μ 1) π(μ) V(μ 11) V(μ 12) V(μ 3) V(μ 13) W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs V(μ 4) P 11

Routing Scheme with Additive Stretch Result: a headerless routing scheme with additive stretch use additional tricks to obtain multiplicative stretch μ μ 11 μ 2 μ 3 μ 4 μ 12 μ 13 V(μ) V(μ 1) π(μ) V(μ 11) V(μ 12) V(μ 3) V(μ 13) W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs V(μ 4) P 12

Conclusion First headerless routing scheme with optimum stretch and polylog labels What about general disks? The model matters! Questions? W. Mulzer and M. Willert – Headerless Routing in Unit Disk Graphs 13