1. Models and Techniques for Communication in Dynamic Networks Christian Scheideler Dept. of Computer Science Johns Hopkins University

2. 2Communication in Dynamic Networks Dynamic networks Faulty networksPeer-to-peer networksWireless networks

3. 3Communication in Dynamic Networks Overview In this talk: Basic problems Basic assumptions Faulty networks Wireless networks Adversarial networks More can be found in the paper…

4. 4Communication in Dynamic Networks Basic Problems - Connectivity

5. 5Communication in Dynamic Networks Basic Problems – Routing

6. 6Communication in Dynamic Networks Basic Problems – Admission

7. 7Communication in Dynamic Networks Basic Assumptions All packets are atomic (unsplittable, uncompressable) Time proceeds in synchronous steps Only point-to-point connections (links) Each link can forward one packet per step in each direction

8. 8Communication in Dynamic Networks Focus of Talk Communication in dynamic networks (connectivity, routing, admission control) Online, distributed algorithms online: no knowledge about future online: no knowledge about future distributed: runs locally at each node distributed: runs locally at each node Adversarial or random network changes adversary: tries to harm algorithm adversary: tries to harm algorithm Adversarial packet injections

9. 9Communication in Dynamic Networks Faulty Networks

10. 10Communication in Dynamic Networks Faulty Networks Basic models: Adversarial or random node/edge faults Central questions: What repair rate needed to ensure large connected component large connected component similar routing properties similar routing properties How to ensure uninterrupted service How to achieve a high throughput

11. 11Communication in Dynamic Networks Large Connected Components Information dispersal [Rabin 89]: no information lost Assumption: random edge faults Is there critical survival probability p* s.t. p>(1+  )p* : constant fraction CC p<(1-  )p* : no constant fraction CC

12. 12Communication in Dynamic Networks Large connected components p* = 1/(n-1) [Erdos and Renyi 60] p*=1/dp*=1/d [Ajtai, Komlos and Szemeredi 82]p*=1/2 [Kesten 80]0.337 < p* < 0.436 [Karlin, Nelson and Tamaki 94]

13. 13Communication in Dynamic Networks Large connected components r = d p*. Robustness of graph of ave. degree d: complete graph: r = 1 d-dimensional hypercube: r = 1 random graph of dn/2 edges: r = 1 d-dimensional butterfly: 1.348 < r < 1.744 2-dimensional mesh: r = 2

14. 14Communication in Dynamic Networks Structurally robust networks When can faulty network still keep original routing properties? When can faulty network still simulate original network with (amortized) constant slowdown? Constant slowdown: 1 step in original - O(1) steps in faulty network

15. 15Communication in Dynamic Networks Simulations with O(1) slowdown up to n worst-case faults [Leighton, Maggs and Sitaraman 98] 1-  up to n worst-case faults [Cole, Maggs and Sitaraman 97] 1-  up to n worst-case faults? [Leighton, Maggs and Sitaraman 97] 1-  failure with any const. p<1 OK [Hastad, Leighton and Newman 87]

16. 16Communication in Dynamic Networks Open questions Hypercube very robust! What about constant degree networks?? How to exploit knowledge about robustness of networks for the design of efficient routing protocols?

17. 17Communication in Dynamic Networks Fault-tolerant circuit switching How can uninterrupted service be ensured? Use more than one path. Allocate more bandwidth than needed. A B

18. 18Communication in Dynamic Networks Fault-tolerant circuit switching k-edge disjoint paths problem (k-EDP): Given a graph G and a set T of pairs of nodes, find a maximum set of pairs s.t. each pair is connected by k disjoint paths and all paths are disjoint. k-disjoint flows problem (k-DFP): Given a network G with edge capacities and a set T of pairs of nodes with demands d, find a set of pairs of maximum total demand s.t. pair i in set is connected by k disjoint paths with flow d /k and all capacity constraints are kept. i i

19. 19Communication in Dynamic Networks Fault-tolerant circuit switching Bounded greedy algorithm (BGA) For each pair: if k disjoint paths available that use at most L edges: take them L= O(k  log n): BGA is O(k  log n) - competitive 3 3  : maximum degree,  : expansion, n: number of nodes [Bagchi & Chaughari & Kolman & Sch. 02]

20. 20Communication in Dynamic Networks Wireless ad hoc networks How to ensure connectivity?How to select and maintain routes?How to send packets along selected routes?

21. 21Communication in Dynamic Networks Connectivity [Yao 82, Ruppert & Seidel 91,Lukovszki 99]: partition space into sectors connect to nearest neighbor in each sector

22. 22Communication in Dynamic Networks Route selection Strategy: Always go to neighbor that lies in sector of destination (if possible…)

23. 23Communication in Dynamic Networks Wireless ad hoc networks Problem with sector-based approach: Nodes have to know location (GPS) Solution when nodes move at random: Just connect to k nearest neighbors What if nodes have limited range??

24. 24Communication in Dynamic Networks Adversarial networks

25. 25Communication in Dynamic Networks Adversarial networks Situation Network and packet injections under adversarial control Problems How to select paths? How to schedule packet movements? How to perform admission control?

26. 26Communication in Dynamic Networks Adversarial networks “Friendly” adversary All injected packets can be delivered Buffer size for this at most B Only gathering scenario All packets have the same destination Is there online algorithm s.t. number of packets in system finite?

27. 27Communication in Dynamic Networks Adversarial gathering Balancing algorithm: For every time step t and every edge (v,w) #packets(v) - #packets(w) > T: send packet receive all incoming and injected packets vw [Awerbuch et al 89, Aiello et al 98,…]

28. 28Communication in Dynamic Networks Adversarial gathering [Awerbuch & Berenbrink & Brinkmann & Sch. 01] For T>2 , number of packets in node at most (B+T)(n-1)+B where  is maximum degree of node. Bound tight up to constant factor. How about more complex communication modes? (unicasting, anycasting, multicasting,…)

29. 29Communication in Dynamic Networks Communication modes Unicasting : one destination per packetAnycasting : optional set of destinationsMulticasting : multiple destinations

30. 30Communication in Dynamic Networks Anycasting and multicasting Approach: Adversary controls topology, injections, now unfriendly Compare number of packet deliveries (throughput) with optimal algorithm Algorithm is (c,s)-competitive: Reaches c-fraction of optimal throughput with s times more buffer space

31. 31Communication in Dynamic Networks Anycasting Originial model: each node has buffer for each anycast set each step adversary offers edges Option set model: buffer: virtual node edge: set of virtual edges between corr. buffers combine all virt. dest. nodes in single dest. Anycasting can be reduced to gathering! vw

32. 32Communication in Dynamic Networks Gathering in option set model Balancing algorithm: For every time step t and every option set S select edge (v,w) with largest #pack(v)-#pack(w) #pack(v) - #pack(w) > T: send packet #pack(v) - #pack(w) > T: send packet receive all incoming and injected packets (if not possible: delete packet) [Awerbuch, Brinkmann and Sch. 02] balancing is (1- ,O(L/  )) – competitive (L: ave. path length used by packets in OPT)

33. 33Communication in Dynamic Networks Multicasting Major problem: Multicast packets can split Approach: Option set model for anycasting can be generalized to multicasting Balancing algorithm can be extended accordingly Balancing (1- , O(f(  )))-competitive

34. 34Communication in Dynamic Networks Adversarial networks Major problem: Solutions show feasibility but FAR from being practical Possible solutions: Use clustering strategies Combine balancing with other methods

35. 35Communication in Dynamic Networks Conclusion Routing in dynamic networks is interesting (also systems people are searching in dark) Techniques and models are available Many open problems left: Robust, fault-tolerant networks of constant degree Robust, fault-tolerant networks of constant degree Routing in fault-tolerant and wireless networks Routing in fault-tolerant and wireless networks