Distributed Maximal Matching: Greedy is Optimal

Transcription

1 Distributed Maximal Matching: Greedy is Optimal Jukka Suomela Helsinki Institute for Information Technology HIIT University of Helsinki Joint work with Juho Hirvonen Weimann Institute of Science, 7 November 0

2 Background

3 Maximal Matchings Input Output

4 Distributed Algorithms Graph G = input = communication network node = computer edge = communication link

5 Distributed Algorithms Initially, each node only knows its incident edges i.e., each node knows its radius-0 neighbourhood 5

6 Distributed Algorithms Nodes can exchange messages to learn more about graph G communication round: discover radius- neighbourhood 6

7 Distributed Algorithms Nodes can exchange messages to learn more about graph G communication rounds: discover radius- neighbourhood 7

8 Distributed Algorithms Nodes can exchange messages to learn more about graph G communication rounds: discover radius- neighbourhood all nodes can do this in parallel 8

9 Distributed Algorithms After T rounds, all nodes know their radius-t neighbourhoods in G 9

10 Distributed Algorithms After T rounds, all nodes know their radius-t neighbourhoods in G local view 0

11 Distributed Algorithms Mapping: local view local output Each node decides whether it is matched and with whom

12 Distributed Algorithms Time = number of communication rounds Equivalent: Distributed algorithm that runs in time T All nodes run the same algorithms; after T synchronous communication rounds all nodes announce their local outputs Mapping from radius-t neighbourhoods to local outputs

13 Distributed Algorithms Time = number of communication rounds How fast can we find a maximal matching? O(n)? O(log n)? O()?

14 Distributed Algorithms Time = number of communication rounds How fast can we find a maximal matching? O(n)? O(log n)? O()? Maybe we should first make sure that we can find a maximal matching at all

15 Symmetry Breaking Some kind of symmetry-breaking is needed! identical local views, identical local outputs 5

16 Symmetry Breaking Unique identifiers Port numbering Node colouring Edge colouring Randomness Geometry 6

17 Symmetry Breaking Unique identifiers Port numbering Node colouring Edge colouring Randomness Geometry } another world 7

18 Symmetry Breaking Unique identifiers Port numbering Node colouring } not enough! Edge colouring Randomness Geometry } another world 8

19 Symmetry Breaking Unique identifiers n nodes: identifiers subset of {,, poly(n)} I will refer to this when discussing related work Edge colouring proper k-colouring of edges enough for our purposes this what we use today 9

20 Greedy Algorithm Given: k-edge coloured graph Input 0

21 Greedy Algorithm Greedily add edges of colour, Input Greedy algorithm

22 Greedy Algorithm Greedily add edges of colour,, Input Greedy algorithm

23 Greedy Algorithm Greedily add edges of colour,,, Input Greedy algorithm

24 Greedy Algorithm Greedily add edges of colour,,, k Input Greedy algorithm

25 Greedy Algorithm That s it maximal matching in time O(k) Input Greedy algorithm Output 5

26 Greedy Algorithm Running time is exactly k initially each node knows the colours of incident edges 6

27 Greedy Algorithm Running time is exactly k Analysis is tight Example for case k = Identical radius- neighbourhoods, different outputs: 7

28 Greedy Algorithm Running time is exactly k Analysis is tight But could we design a faster algorithm? turns out that this is connected to some fundamental open questions of the field 8

29 Related Work 9

30 n and Running time as a function of what? Two parameters commonly used: n = number of nodes = maximum degree We often assume that n and are known or some upper bounds of them 0

31 Problem maximal matching ( +)-vertex colouring ( -)-edge colouring maximal edge packing vertex cover -approx. Upper bound + log* n + log* n + log* n Lower bound polylog( ) + log* n polylog( ) + log* n polylog( ) + log* n polylog( ) polylog( ) Positive: Panconesi Rii (00), Barenboim Elkin (009), Kuhn (009), Åstrand Suomela (00), Negative: Linial (99), Kuhn et al. (00, 006)

32 n and Time complexity is well-understood as a function of n asymptotically tight upper and lower bounds But we do not really understand time complexity as a function of exponential gap

33 k and What about edge-coloured graphs? k = number of colours, = maximum degree Maximal matching: upper bound: O( + log* k) lower bound: Ω(polylog( ) + log* k) Again, an exponential gap for

34 k and What about edge-coloured graphs? k = number of colours, = maximum degree Maximal matching: upper bound: O( + log* k) lower bound: Ω(polylog( ) + log* k) Ω( + log* k) Again, an exponential gap for

35 Contributions Time complexity of finding maximal matchings in k-edge-coloured graphs General graphs: k matching upper bound: k (greedy) Bounded-degree graphs: Ω( + log* k) matching upper bound: O( + log* k) (an adaptation of Panconesi Rii 00) 5

36 Lower Bound 6

37 Plan Focus: d-regular k-edge-coloured graphs If d = k: trivial to find a maximal matching in constant time (pick a colour class) If d = k : as difficult as the general case! we show that we need at least d rounds 7

38 Plan Given k, define d = k, assume: algorithm A finds a maximal matching in any d-regular k-edge-coloured graph We construct a pair of infinite trees T, T: root nodes have identical (k )-neighbourhoods output of A: root of T matched, root of T unmatched running time of A is at least k 8

39 Tools Now we need tools for constructing and manipulating infinite edge-coloured trees Warning: the manuscript uses a very different formalism (with some group-theoretic constructions) in this talk I ll try to keep everything lightweight, and just present the key ideas with illustrations 9

40 Node Colours Node colour = the unique missing colour 0

41 Templates Degree < d

42 Templates Degree < d: add loops

43 Templates Degree < d: add loops, unfold loops

44 Templates Unfolding preserves traversals

45 Templates 5 Natural homomorphism

46 Templates Compact representations of trees = 6

47 Templates = = 7

48 Templates = = 8

49 Templates origin 9

50 Templates same origin same local view same output 50

51 Templates What is the output of A here? 5

52 Templates What is the output of A here? 5

53 Templates 5 What is the output of A here? = What is the output of A here? Definition!

54 Templates What is the output of A here? From now on we can study the output of algorithm A on templates 5

55 Templates unmatched unmatched non-maximal Template of degree < d: all nodes are matched 55

56 Templates 56 Output x: matched along the edge of colour x

57 Templates 57 Output x: matched along the edge of colour x

58 Base Case Apply algorithms A to templates of degree ero Defines a mapping template output from node colours to outputs? 58

59 Base Case h: {,,,k} {,,,k} no fixed points template output h 59

60 Base Case h: {,,,k} {,,,k} no fixed points there are distinct x, y, with h(x) = y h() y template x y output 60

61 Base Case h: {,,,k} {,,,k} no fixed points there are distinct x, y, with h(x) = y h() y template x y output 6

62 Base Case x y edge of colour y exists, in matching x y edge of colour y exists, but not in matching 6

63 Base Case x y x y x K y y L x x x 6

64 Base Case x y x y x K x y X x x y x y x L output in X cannot be copied from K & L something must change! 6

65 Base Case x y x x y x y x x y x degree templates, same radius-0 view, different output 65

66 Base Case x y x x y x y x x y x degree templates, same radius-0 view, different output 66

67 Inductive Step x S Given: degree i templates, same radius-(i ) view, different output T Construct: degree i+ templates, same radius-i view, different output (here i = ) 67

68 Inductive Step S x Choose one loop per node Prefer loops that are matched in T T Then unfold these loops 68

69 Inductive Step K x x x L 69

70 Inductive Step K x x x X x L again, something must change in the output! 70

71 Inductive Step K x x x X x L 7

72 Inductive Step same radius-0 view x x x x 7

73 Inductive Step same radius-0 view same radius-0 view x x x x 7

74 Inductive Step same radius- view x x x x 7

75 Inductive Step same radius- view x degree templates, same radius- view, different output 75

76 Inductive Step x Given: degree i templates, same radius-(i ) view, different output x Construct: degree i+ templates, same radius-i view, different output (here i = ) 76

77 Inductive Step K L x x 77

78 Inductive Step K X L x x x something must change 78

79 Inductive Step K X L x x x 79

80 Inductive Step K X same radius- view x 80

81 Inductive Step K X L x x x 8

82 Inductive Step same radius- view L X x x x 8

83 Conclusions x x x x x By induction, we can construct: two degree-d trees same radius-(d ) view different output 8

84 Conclusions x x x x x Algorithm A requires at least k rounds in a k-edge-coloured graph Algorithm A cannot be faster than the greedy algorithm 8

85 Conclusions Greedy is optimal 85

86 Conclusions Maximal matching in k-edge-coloured graphs requires: k communication rounds in general Θ( + log* k) rounds in graphs of degree Still open: what if we have unique identifiers? or both edge colouring and node colouring? 86