GraphTheory Models & Complexity BFS Trees DFS Trees Min Spanning Trees. Tree Constructions. Martin Biely

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1 Tree Constructions Martin Biely Embedded Computing Systems Group Vienna University of Technology European Union November 29, 2006 Biely Tree Constructions 1

2 Outline 1 A quick Repetition on Graphs 2 Models & Complexity 3 Breadth First Search Trees BFS Tree through Flooding? Asynchronous BFS Overview 4 Depth First Search Trees 5 Minimumweight Spanning Trees non matroid The end Biely Tree Constructions 2

3 Outline 1 A quick Repetition on Graphs 2 Models & Complexity 3 Breadth First Search Trees 4 Depth First Search Trees 5 Minimumweight Spanning Trees Biely Tree Constructions 3

4 Graphs v 1 v 2 v 7 v 3 v 6 v 5 v 4 G = (V, E) V = {v 1,..., v n } E = {e 1,..., e n } e = (u, v) paths(u, v) = {(u, v 1 )(v 1, v 2 )... e x 1 (v x 1, v),... } = {paths from u to v} dist(u, v) = min p paths(u,v) p diam(g) = max u,v V 2(dist(u, v)) Biely Tree Constructions 4

5 Trees v 1 v 2 v 7 v 3 v 6 v 5 v 4 Definition 1 (Spanning Tree) A spanning tree of a graph (V, E) is an acyclic connected sub-graph (V, E ), with E E. Biely Tree Constructions 5

6 Rooted Trees v 1 v 2 v 7 v 3 v 6 v 5 v 4 Definition 2 (Rooted Spanning Tree) A rooted spanning tree of a spanning Tree with one dedicated root node. Biely Tree Constructions 6

7 Rooted Trees v 1 v 2 v 7 v 3 Definition 3 v 6 v 5 v 4 Leaves (v 2, v 7 and v 3 ) are vertices which have no children. The root (v 6 ) has no parent. Biely Tree Constructions 7

8 Outline 1 A quick Repetition on Graphs 2 Models & Complexity 3 Breadth First Search Trees 4 Depth First Search Trees 5 Minimumweight Spanning Trees Biely Tree Constructions 8

9 Computational Models Asynchronous Synchronous (Lockstep) Biely Tree Constructions 9

10 Complexity Measures Definition 4 For any Algorithm A we will denote by, Time(A) the time complexity and by Message(A) the message complexity of the algorithm. Definition 5 The message complexity is the maximum (over all executions) of the total number of messages sent. Definition 6 In synchronous systems, the time complexity is the number of rounds until termination. In asynchronous systems, the time complexity is the number of consecutive message delays until termination. Biely Tree Constructions 10

11 Outline 1 A quick Repetition on Graphs 2 Models & Complexity 3 Breadth First Search Trees BFS Tree through Flooding? Asynchronous BFS Overview 4 Depth First Search Trees 5 Minimumweight Spanning Trees Biely Tree Constructions 11

12 BFS Tree through Flooding? when starting tree construction at vertex root do send ConstructTree to all when receiving ConstructTree for the first time do parent the sender of that message send ConstructTree to all (except v ) when receiving ConstructTree again do do nothing Biely Tree Constructions 12

13 Synchronous Execution of Flooding: Round 1 v 1 v 2 v 7 v 3 v 6 v 5 v 4 Biely Tree Constructions 13

14 Synchronous Execution of Flooding: Round 2 v 1 v 2 v 7 v 3 v 6 v 5 v 4 Biely Tree Constructions 14

15 Asynchronous Execution of Flooding: End v 1 v 2 v 7 v 3 v 6 v 5 v 4 Biely Tree Constructions 15

16 when receiving Pulse do if children = (i.e., we are a leaf) then send Layer to all (except parent) else send Pulse to all children Biely Tree Constructions 16 Layer Synchronized Construction (Algorithm LS Part I) (Distributed version of Dijkstra s single source shortest path algorithm) Variables: children new = 0 when starting tree construction at vertex root do while new 0 do send Pulse to self // start new phase wait

17 Layer Synchronized Construction (Algorithm LS Part II) when receiving Layer from v (for the first Layer message) do parent v send (Ack, 1) when receiving Layer from v (for later Layer messages) do send (Ack, 0) when received (Ack, x c ) from all children c do new Σ c childrenx c if v = root then continue else send (Ack, new) to parent Biely Tree Constructions 17

18 Complexity (Lower Bound) Theorem 7 For any n-vertex graph G = (V, E), distributed BFS Tree construction requires Ω( E ) messages and Ω(Diam(G)) time. Biely Tree Constructions 18

19 Complexity (Layer Synchronized Algorithm LS) Definition 8 Let E p denote the edges within level p, and E p,p+1 the edges between levels p and p + 1 of the constructed tree. Time(phase p of LS) = 2p + 2 Time(LS) = Σ phases Time(one phase) = O(Diam(G) 2 ) Message(phase p of LS) = O( E p + E p,p+1 ) + O(n) Message(LS) = Σ phases Message(one phase) = = O(nDiam(G) + E ) Biely Tree Constructions 19

20 Update-Based BFS Construction (Algorithm BF) (Distributed version of the Bellman-Ford single source shortest path algorithm) Variables: L when starting tree construction at vertex root do L 0 send (Layer, L) to all when receiving (Layer, L ) from v do if L + 1 < L then parent v L L + 1 send (Layer, L) to all except v Biely Tree Constructions 20

21 Complexity (Algorithm BF) Time(BF) = O(Diam(G)) Message(BF) = O(n E ) Biely Tree Constructions 21

22 BFS Complexities Overview Message Time Lower Bound E Diam Best sync. E Diam LS ndiam + E Diam 2 BF n E Diam Best async. E + n log 3 n Diam log 3 n Biely Tree Constructions 22

23 Outline 1 A quick Repetition on Graphs 2 Models & Complexity 3 Breadth First Search Trees 4 Depth First Search Trees 5 Minimumweight Spanning Trees Biely Tree Constructions 23

24 DFS Algorithm DF Variables: visited function do dfs() while n N \ visited do send Token to n wait for Token from n when starting tree construction at vertex root do do dfs() when receiving Token from v do parent v do dfs() send Token to parent Biely Tree Constructions 24

25 DFS Algorithm DF Variables: visited function do dfs() while n N \ visited do send Token to n wait for Token from n when starting tree construction at vertex root do do dfs() when receiving Token from v do parent v send Visited to all (except visited) wait for Ack from all (except visited) do dfs() send Token to parent when receiving Visited from v do visited visited {v } send Ack to v Biely Tree Constructions 24

26 Complexity of DFS Message(DF) = Θ( E ) Time(DF) = O(n) Biely Tree Constructions 25

27 Outline 1 A quick Repetition on Graphs 2 Models & Complexity 3 Breadth First Search Trees 4 Depth First Search Trees 5 Minimumweight Spanning Trees non matroid The end Biely Tree Constructions 26

28 Weight? Definition 9 v v 7 v 2 v 6 v 5 v In a weighted Graph G = (V, E, ω), where ω(e) is the weight function returning the weight of the edge e, we define the weight of a path p as ω(p) = Σ e p ω(e) and the weight of a (sub)graph G = (V, E, ω) of G as ω(g ) = Σ e E ω(e) v 3 Biely Tree Constructions 27

29 Weight of a tree v v v v v 6 v 5 v 4 ω(tree) = = 51 Biely Tree Constructions 28

30 Weight of the MST v v v v v 6 v 5 v 4 ω(tree) = = 23 Biely Tree Constructions 29

31 Lemma 10 If the edge weights are all distinct, the MST of a graph is unique.... so we assume unique weights from now on. Biely Tree Constructions 30

32 MST fragment Definition 11 (MST fragment) In a weighted graph G = (V, E, ω), a tree T in G is called an MST fragment of G, iff there exists an MST of G such that T is a subgraph of that MST. Definition 12 An edge e is an outgoing edge of a MST fragment T, iff exactly one of its endpoints belongs to T. The minimum weight outgoing edge is denoted MWOE(T ). Biely Tree Constructions 31

33 Lemma 13 Consider a MST fragment T of a graph G = (V, E, ω). Let e = MWOE(T ). Then T e is a MST fragment as well. Proof. Let T M be an MST containing T. If T M contains T we are done. Otherwise, let e be an edge that connects T to the rest of T M. Clearly, e is an outgoing edge of T and ω(e ) ω(e). Adding e to T M, creates a graph C with a cycle through e and e. Discarding e from C yields a new T M with ω(t M ) ω(t M). Biely Tree Constructions 32

34 The blue rule Pick a MST fragment T and add to it the minimum weight outgoing edge is denoted MWOE(T ). Biely Tree Constructions 33

35 Distributed Prim s algorithm DP Phase p+1 if v = root then broadcast (Pulse, MWOE(T p )) on fragment T p when receiving (Pulse, MWOE (T p )) do if MWOE(T p ) is a local edge then tell that neighbour to join the tree else update types of outgoing edges convergecastthe minimum the outgoing edge e if v = root then MWOE(T p+1 ) = result of the convergecast Time(DP) = Message(DP) = O(#phases) = O( V 2 ) Biely Tree Constructions 34

36 Distributed Kruskal s algorithm GHS ( Gallager, Humblet, and Spira ) Phase each fragment F determines its MWOE(F ) = e as in DP a RequestToMergeis sent over e to the fragment F (now all the Fragments form an fragment forest... Biely Tree Constructions 35

37 Fragment Forest Definition 14 (Fragment Forest) Each connected component of a fragment forest consists of a directed tree of fragments (each identified by the id of its root) whose arcs point towards the root, which is a pair of fragments F 1, F 2 pointing at each other. F 1 is the fragment with the smaller id. Biely Tree Constructions 36

38 Distributed Kruskal s algorithm GHS Phase p+1 each fragment F determines its MWOE(F ) = e as in DP a RequestToMergeis sent over e to the fragment F (now all the Fragments form an fragment forest Each F 1 root broadcasts (NewFragment, id(f 1 )) #phases = log n Time(GHS) = O(n log n) Message(GHS) = O( E log n) Biely Tree Constructions 37

39 that s it for now see you on Friday: Vertex coloring Maximal independent sets Matroid-based MST construction Biely Tree Constructions 38