Think Global Act Local

Transcription

1 Think Global Act Local Roger Wattenhofer ETH Zurich Distributed Computing

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3 Town Planning Patrick Geddes

4 Architecture Buckminster Fuller

5 Computer Architecture Caching

6 Robot Gathering e.g., [Degener et al., 2011]

7 Natural Algorithms [Bernard Chazelle, 2009]

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9 Algorithmic Trading

10 Think Global Act Local is there a theory?

11 Complexity Theory Can a Computer Solve Problem P in Time t?

12 (Think Global Act Local) Distributed Complexity Theory Network Can a Computer Solve Problem P in Time t?

13 Distributed (Message-Passing) Algorithms Nodes are agents with unique ID s that can communicate with neighbors by sending messages. In each synchronous round, every node can send a (different) message to each neighbor

14 Distributed (Message-Passing) Algorithms Nodes are agents with unique ID s that can communicate with neighbors by sending messages. In each synchronous round, every node can send a (different) message to each neighbor Distributed (Time) Complexity: How many rounds does problem take?

15 An Example

16 How Many Nodes in Network?

17 How Many Nodes in Network?

18 How Many Nodes in Network?

19 How Many Nodes in Network?

20 How Many Nodes in Network?

21 How Many Nodes in Network?

22 How Many Nodes in Network?

23 How Many Nodes in Network? With a simple flooding/echo process, a network can find the number of nodes in time O(D), where D is the diameter (size) of the network.

24 Diameter of Network? Distance between two nodes = Number of hops of shortest path

25 Diameter of Network? Distance between two nodes = Number of hops of shortest path

26 Diameter of Network? Distance between two nodes = Number of hops of shortest path Diameter of network = Maximum distance, between any two nodes

27 Diameter of Network?

28 Diameter of Network?

29 Diameter of Network?

30 Diameter of Network?

31 Diameter of Network?

32 Diameter of Network?

33 Diameter of Network?

34 Networks Cannot Compute Their Diameter in Sublinear Time! (even if diameter is just a small constant) Pair of rows connected neither left nor right? Communication complexity: Transmit Θ(n 2 ) information over O(n) edges Ω(n) time! [Frischknecht, Holzer, W, 2012]

35 What about a local task?

36 Example: Minimum Vertex Cover (MVC) Given a network with n nodes, nodes have unique IDs. Find a Minimum Vertex Cover (MVC) a minimum set of nodes such that all edges are adjacent to node in MVC

37 Example: Minimum Vertex Cover (MVC) Given a network with n nodes, nodes have unique IDs. Find a Minimum Vertex Cover (MVC) a minimum set of nodes such that all edges are adjacent to node in MVC

38 Example: Minimum Vertex Cover (MVC) Given a network with n nodes, nodes have unique IDs. Find a Minimum Vertex Cover (MVC) a minimum set of nodes such that all edges are adjacent to node in MVC

39 On MVC Find an MVC that is close to minimum (approximation) Trade-off between time complexity and approximation ratio MVC: Various simple (non-distributed) 2-approximations exist! What about distributed algorithms?!?

40 Finding the MVC (by Distributed Algorithm) Given the following bipartite graph with S 0 = δ S 1 The MVC is just all the nodes in S 1 Distributed Algorithm S 1 S 0

41 Finding the MVC (by Distributed Algorithm) Given the following bipartite graph with S 0 = δ S 1 The MVC is just all the nodes in S 1 Distributed Algorithm S 1 S 0

42 Finding the MVC (by Distributed Algorithm) Given the following bipartite graph with S 0 = δ S 1 The MVC is just all the nodes in S 1 Distributed Algorithm S 1 S 0

43 N 2 (node in S 0 ) N 2 (node in S 1 ) S 1 S 0

44 N 2 (node in S 0 ) N 2 (node in S 1 ) Graph is symmetric, yet highly non-regular! S 1 S 0

45 Lower Bound: Results We can show that for ε > 0, in t time, the approximation ratio is at least t t Constant approximation needs at least Ω(log Δ) and Ω( log n) time. Polylog approximation Ω(log Δ/ log log Δ) and Ω( log n/ log log n). [Kuhn, Moscibroda, W, journal version in submission]

46 Lower Bound: Results We can show that for ε > 0, in t time, the approximation ratio is at least tight for MVC t t Constant approximation needs at least Ω(log Δ) and Ω( log n) time. Polylog approximation Ω(log Δ/ log log Δ) and Ω( log n/ log log n). [Kuhn, Moscibroda, W, journal version in submission]

47 Lower Bound: Reductions Many local looking problems need non-trivial t, in other words, the bounds Ω(log Δ) and Ω( log n) hold for a variety of classic problems. [Kuhn, Moscibroda, W, journal version in submission]

48 Lower Bound: Reductions Many local looking problems need non-trivial t, in other words, the bounds Ω(log Δ) and Ω( log n) hold for a variety of classic problems. cloning line graph MVC through MM line graph [Kuhn, Moscibroda, W, journal version in submission]

49 Olympics!

50 Distributed Complexity Classification 1 log n polylog n D poly n e.g., dominating set approximation in planar graphs MIS, approx. of dominating set, vertex cover,... diameter, MST, verification of e.g. spanning tree, various problems in growth-bounded graphs count, sum, spanning tree,...

51 Distributed Complexity Classification easy hard 1 log n polylog n D poly n e.g., dominating set approximation in planar graphs MIS, approx. of dominating set, vertex cover,... diameter, MST, verification of e.g. spanning tree, various problems in growth-bounded graphs count, sum, spanning tree,...

52 Locality Local Algorithms Sublinear Algorithms

53 Locality is Everywhere! Self- Assembly Applications e.g. Multi-Core Self- Stabilization Local Algorithms Sublinear Algorithms Dynamic Networks

54 Locality is Everywhere! Self- Assembly Applications e.g. Multi-Core Self- Stabilization Local Algorithms Sublinear Algorithms Dynamic Networks

55 [Afek, Alon, Barad, et al., 2011]

56 [Afek, Alon, Barad, et al., 2011]

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58 Maximal Independent Set (MIS) Given a network with n nodes, nodes have unique IDs. Find a Maximal Independent Set (MIS) a non-extendable set of pair-wise non-adjacent nodes

59 Maximal Independent Set (MIS) Given a network with n nodes, nodes have unique IDs. Find a Maximal Independent Set (MIS) a non-extendable set of pair-wise non-adjacent nodes

60 Maximal Independent Set (MIS) Given a network with n nodes, nodes have unique IDs. Find a Maximal Independent Set (MIS) a non-extendable set of pair-wise non-adjacent nodes

61 given: id, degree synchronized while (true) { p = 1 /(2*degree); if (random value between 0 and 1 < p) { transmit (degree, id) ;

62 given: id, degree synchronized while (true) { p = 1 /(2*degree); if (random value between 0 and 1 < p) { transmit (degree, id) ;?!

63 Distributed Computing Without Computing!

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65 Nano-Magnetic Computing

66 Stone Age Distributed Computing

67 nfsm: networked Finite State Machine Every node is the same finite state machine, e.g. no IDs Apart from their state, nodes cannot store anything Nodes know nothing about the network, including e.g. their degree Nodes cannot explicitly send messages to selected neighbors, i.e. nodes can only implicitly communicate by changing their state Operation is asynchronous Randomized next state okay, as long as constant number Nodes cannot compute, e.g. cannot count

68 One, Two, Many Principle Piraha Walpiri

69 One, Two, Many Principle Not okay while (k < log n) { At least half of neighbors in state s? More neighbors in state s than in state t? Okay No neighbor in state s? Some neighbor in state s? At most two neighbors in state s?

70 u 1 u 0 u 2

71 u 1 u 0 alone MIS u 2

72 u 1 u 0 alone MIS not alone d 2 u 2

73 u 1 not d 1 u 0 MIS d alone 2 alone u 2

74 u 1 not d 1 u MIS d MIS 0 2 not alone neighbor alone u 2

75 nfsm solves MIS whp in time O(log 2 n) u 1 not d 1 u MIS d MIS 0 2 not alone neighbor alone u 2 [Emek, Smula, W, in submission]

76 Overview General Graph Growth-Bounded Graph Bounded Degree Graph Diameter MVC MIS

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78 Stone Age Wireless Message Passing Overview General Graph Growth-Bounded Graph Bounded Degree Graph Diameter MVC MIS

79 Summary

80 Thank You! Questions & Comments? Thanks to my co-authors Yuval Emek Silvio Frischknecht Stephan Holzer Fabian Kuhn Thomas Moscibroda Jasmin Smula