A O (log n ) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems. Volker Turau. November 6th, 2018

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1 A O (log n ) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems Volker Turau 20th Int. Symposium on Stabilization, Safety, and Security of Distributed Systems November 6th, 2018 Institute of Telematics Hamburg University of Technology TUHH

2 Publish/Subscribe 1

3 Publish/Subscribe Publish/Subscribe Paradigm Loosely coupled distributed information dissemination middleware Subscribers declare interest in topics by subscribing to topic Highly scalable Data is distributed asynchronously and anonymously Senders are unaware of number and addresses of subscribers Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 3

4 Publish/Subscribe Publish/Subscribe Paradigm Interface subscribe(t) unsubscribe(t) publish(m, t) Publish/subscribe middleware forwards publications to subscribers Focus of our work: Low-power wireless networks with limited resources IIoT Challenge: Low memory routing structure Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 4

5 Publish/Subscribe Memory Constrained Routing Fact 1: Shortest path routing requires routing tables of size Ω(n) Path stretch of protocol P: Ratio of path length achieved by P, divided by shortest path length Fact 2: Protocols with path stretch below 3 require Ω(n) size routing tables [Gavoille et al.] Goal: Routing tables of poly-logarithmic size, e.g., O(log 2 n) Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 5

6 Publish/Subscribe Memory Constrained Routing Virtual Ring: Closed path involving each node at least once Routing Publisher forwards message around ring and subscribers read it Upon return to sender message is discarded Routing table of each node only contains id of next node May incur a linear path stretch, i.e. length of cycle To lessen stretch additional edges a.k.a. fibers are used as shortcuts at cost of increased routing tables Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 6

7 Publish/Subscribe Virtual Ring Routing with Fibers r e 1 r 0 9 e e 1 r 0 9 e e c 2 8 d c 2 8 d c a d b a 3 c 4 5 e 6 7 d b a 3 c 4 5 e 6 7 d b Communication Graph Virtual Ring Virtual Ring with Fibers Fibers are used as short cuts A node skips a ring segment if does not contain a subscriber Concurrent forwarding into disjoint segments PSVR is a pub/sub middleware that uses virtual rings Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 7

8 Publish/Subscribe New Routing Structure Observations Goal Shorter rings reduce path stretch More fibers reduce path stretch but may enlarge routing table Ring structure with smaller stretch and smaller routing tables such that PSVR can be executed without changes New concepts Not all nodes need to be on ring shorter rings Nodes on ring act as proxies for nodes outside ring (NAT) Systematic approach to select fibers Compromise between path stretch and routing table size Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 8

9 Publish/Subscribe New Routing Structure c 13 c 14 c 15 c 0 c1 c 2 c 3 c 12 c 4 c 11 c 5 c 10 c 9 c8 c 7 c 6 Nodes outside ring are homogeneously attached to proxies Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 9

10 Publish/Subscribe Optimal Fiber Structure c 15 c 0 c1 c 2 c 13 c 14 c 3 c 12 c 4 c 11 c 5 c 10 c 9 c8 c 7 c 6 Homogeneous structure of fibers inside smaller ring Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 10

11 Publish/Subscribe Contribution Efficient distributed algorithm to construct the new routing structure Time complexity O(log n) rounds Analysis of algorithm for random graphs Path stretch is w.h.p. in O(log n) Nodes outside ring attach w.h.p. homogeneously to proxies Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 11

12 Publish/Subscribe Assumptions Synchronous CON GEST model of the distributed message passing model each message contains at most O(log n) bits Unique identifiers Nodes have only limited local memory Each node knows size n of network A dedicated starting node v 0 Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 12

13 Informal Description of Algorithm A Fiber 2

14 Informal Description of Algorithm A Fiber Algorithm A Fiber A Fiber works in phases Phase 0 (O(log n) rounds) Starting in v 0 a path P of length 2 r 1 is built (2 r O(log n)) Phase 1 (O(log n) rounds) Path P is closed to a ring C of length 2 r Middle phases Concurrently edges of C are replaced by two edges Replaced edges become fibers After k middle phases, C has up to 2 k log n nodes Final phase Each node in v V \ C randomly selects a neighbor u on C Node u is the proxy for v Each middle phase and the final phase lasts O(1) rounds Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 14

15 Informal Description of Algorithm A Fiber Phases v 0 Phase 0 Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 15

16 Informal Description of Algorithm A Fiber Phases v 0 Phase 0 Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 15

17 Informal Description of Algorithm A Fiber Phases v 0 Phase 0 Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 15

18 Informal Description of Algorithm A Fiber Phases v 0 After O(log n) rounds Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 15

19 Informal Description of Algorithm A Fiber Phases v 0 Phase 1, after another O(log n) rounds Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 15

20 Informal Description of Algorithm A Fiber Phases v 0 Middle Phase Step 1 Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 15

21 Informal Description of Algorithm A Fiber Phases v 0 Middle Phase Step 2 Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 15

22 Informal Description of Algorithm A Fiber Phases v 0 After k Middle Phases: C has at most 2 k log n nodes Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 15

23 Informal Description of Algorithm A Fiber Phases c 13 c 14 c 15 c 0 c1 c 2 c 3 c 12 c 4 c 11 c 5 c 10 c 9 c8 Final Phase c 7 c 6 Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 15

24 Details of A Fiber 3

25 Details of A Fiber Middle Phases v 1 v 2... w 1 w 2 w 3 w 4 w5 {v 1 }... Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 17

26 Details of A Fiber Middle Phases {w 3 } v v 2 1 I 1 I 1 {w 3, w 4 } I 1 I 1 I 1 I 1... w 1 w 2 w 3 w 4 w5 {v 1 }... Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 17

27 Details of A Fiber Middle Phases {w 3 } v 1 v 2 {w 3, w 4 }... w 1 w 2 w 3 w 4 w5 {v 1 }... Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 17

28 Details of A Fiber Middle Phases v 1 v 2 I 2 I 2... w 1 w 2 w 3 w 4 w5 {v 1, v 2 }... Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 17

29 Details of A Fiber Middle Phases v 1 v 2... w 1 w 2 w 3 w 4 w5 {v 1, v 2 }... Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 17

30 Details of A Fiber Middle Phases v 1 v 2 I 3... w 1 w 2 w 3 w 4 w5 {v 1 }... Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 17

31 Details of A Fiber Middle Phases v 1 v 2... w 1 w 2 w 3 w 4 w5 {v 1 }... Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 17

32 Details of A Fiber Observations Individual extensions do not interfere with each other because Each node outside C sends in each middle phase at most one request to integrate and each edge of C accepts at most one integration request Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 18

33 Analysis of A Fiber 4

34 Analysis of A Fiber Analysis of A Fiber for Random Graphs We analyze algorithm A Fiber for the class of random graphs G(n, p) with p O(log n/ n) Claim: In each middle phase w.h.p. size of C is increased by constant factor Random variable Y Number of nodes that are integrated into C in a middle phase. Goal: Lower bound for Y that holds w.h.p. First we compute number of nodes outside C that send an invitation I 2? Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 20

35 Analysis of A Fiber Analysis of Middle Phases How many nodes outside C send an invitation I 2? v 1 v 2 I 2 I 2... w 1 w 2 w 3 w 4 w5 {v 1, v 2 }... Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 21

36 Analysis of A Fiber Analysis of Middle Phases A node v V \ C sends an invitation if it is connected to at least one pair of consecutive nodes on C This event has probability p 2, but these events are not independent Event π v : v V \ C forms a triangle with at least one evenly numbered edge of C π v has probability 1 (1 p 2 ) c/2 and π v s are independent Random Variable X X is the number of nodes v where π v occurs. X is a lower bound for number of nodes that send invitation I 2. E[X] = (n c)(1 (1 p 2 ) c/2 ) (1) Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 22

37 Analysis of A Fiber Analysis of Middle Phases How many nodes on C send an accept message I 3? v 1 v 2 I 3... w 1 w 2 w 3 w 4 w5 {v 1 }... Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 23

38 Analysis of A Fiber Analysis of Middle Phases Computation of Y can be reduced to bins and balls model X number of balls; c number of bins Each ball is thrown randomly in any of the c bins Probability that v C is connected to w in V \ C is independently of v and w equal to p. Y is equal to number of nonempty bins ( ( E[Y X = x] = c ) x ) c (2) Use Chernoff bound to find upper bound for Y that holds w.h.p. Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 24

39 Analysis of A Fiber Result Theorem For a random graph G(n, p) with p O(log n/ n) the following holds w.h.p. 1. After phase 1 ring C consist of O(log n) nodes 2. After O(log n) middle phases C has size O(log n n) and the fiber graph has diameter O(log n) 3. After final phase the maximum number of nodes attached to a single proxy is less than e(n/c 1) with probability 1 1/c Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 25

40 Conclusion & Extensions 5

41 Conclusion & Extensions Conclusion Proposal for a new routing structure for pub/sub systems in wireless networks O(log n) distributed algorithm to build this structure Analysis for random graphs G(n, p) with p O(log n/ n) Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 27

42 Conclusion & Extensions Extension I c 13 c 14 c 15 c 0 c1 c 2 c 3 c 12 c 4 c 11 c 5 c 10 c 9 c8 c 7 c 6 Nodes outside ring need not to be connected directly to proxies. This allows to have even shorter rings Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 28

43 Conclusion & Extensions Extension II c 13 c 14 c 15 c 0 c1 c 2 c 3 c 12 c 4 c 11 c 5 c 10 c 9 c8 c 7 c 6 The double ring structure allows PSVR to send more messages concurrently Volker Turau A O(log n) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems 29

A O (log n ) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems Volker Turau lker Turau VoSafety, and Security of Distributed Systems 20th Int.

44 A O (log n ) Distributed Algorithm to Construct Routing Structures for Pub/Sub Systems Volker Turau lker Turau VoSafety, and Security of Distributed Systems 20th Int. Symposium on Stabilization, Professor th November 6, 2018 Phone +49 / (0) turau@tuhh.de e/staff/turau Institute of Telematics Hamburg University of Technology TUHH

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