Tree Construction (BFS, DFS, MST) Chapter 5. Breadth-First Spanning Tree (BFS)

Size: px

Start display at page:

Download "Tree Construction (BFS, DFS, MST) Chapter 5. Breadth-First Spanning Tree (BFS)"

Delphia Harper
6 years ago
Views:

1 Tree Construction (BFS, DFS, MST) Chapter 5 Advanced Algorithms for Communication Networks and VLSI CAD - CS 6/75995 Presented By:Asrar Haque Breadth-First Spanning Tree (BFS) Definition: The breadth-first spanning tree, or the BFS tree, of G with respect to a given root r o is a spanning tree T B with the property that for every other vertex v, the path leading from the root to v is of minimum (unweighted) length possible. Basic Algorithm (Dijkstra s Algorithm) BFS developed layer by layer from the root to downwards At any stage, build next layer by adding vertices that are not yet in the tree but are adjacent to some vertex that is in the tree 1

2 Layer-Synchronized BFS Construction (DIST_DIJK) Vertex r o initiates and governs the algorithm A new layer is added in each phase After p phases the constructed tree is T p rooted at r o ; every vertex of T p knows its parent, children (if exists) and depth Layer-Synchronized BFS Construction (DIST_DIJK) 2

3 Example DIST_DIJK Time During phase p, broadcast and convergecast each take p time units Exploration stage take 2 additional time units Time (Phase p) = 2p Time(DIST_DIJK) = Time ( Phasep ) = 2p+ 2 = O( Diam ( G)) Messages Analysis (DIST_DIJK) V p = set of vertices in layer p of T E p = edges internal to V p E p, p+1 = edges connecting V p to V p+1 Message(DIST_DIJK) p = O( ndiam ( G) + E ) P Message ( Phase p) = O( n) + O( E + E, 1 ) = p p p + p p 3

Analysis (DIST_DIJK) Update Based BFS (DIST_BF) 1. Initially, the root sets L(ro) 0, and others vertices v sets L(v) 2. Root sends out message Layer(0) to all its neighbors 3.

4 Analysis (DIST_DIJK) Update Based BFS (DIST_BF) 1. Initially, the root sets L(ro) 0, and others vertices v sets L(v) 2. Root sends out message Layer(0) to all its neighbors 3. Every vertex reacts to incoming messages as follows: Upon receiving a message Layer (d) from a neighbor w do: if d+1 < L(v), then do: a) parent (v) w; b) L(v) c) Send Layer (d+1) to all neighbors except w 4

5 Synchronous Model Analysis (DIST_BF) Once a vertex, v, sets its L(v) to a value smaller than changed v sends messages on its out going edges exactly once Time (DIST_BF) = O(Diam(G) Message (DIST_DIJK) = O( E ) Asynchronous Model Lemma 5.3.2: For every d 1, after d time units from the start of execution every vertex v at distance d from the root has already received a Layer (d-1) message from some neighbor and, consequently set L(v) = d and chose a parent w with L(w) = d-1 The first value a vertex, v, sets L(v) is at most n-1 (longest possible simple path) v can change L(v) at most n-2 times and every time sends message over out going edges Time (DIST_BF) = O(Diam(G) Message (DIST_BF) = O(n E ) Complexity Of BFS Construction Algorithms On G=(V,E) With Dia D 5

6 DFS Basic Algorithm The search pattern is based on continuing further into the graph Visits new vertex as long as possible Retracts from region only after exhausting it Search starts at some vertex r o When search reaches vertex v, if v has neighbors not yet visited one them Otherwise return to the vertex from which v was visited DFS Tree r o root The parent of vertex v, is the w from which v was visited first Distributed DFS In reality sequential Control carried via a token which traverses in depth first manner Time (DIST_DFS) = Message (DIST_DFS) = ( E ) Due to need to explore every edge lower bound for message ( E ) Time complexity can be improved, Time (DIST_DFS) = O ( n ) 6

7 Minimal Spanning Tree Problem Definition: G(V, E, &) is a connected, undirected graph where V is the set of nodes and E is the set of edges and for each edge (u,v) E &(u,v) is cost of edge (u,v). We need to find an acyclic subset T E that connects all the vertices and whose total weight &(T)= ω( u, v) is minimum ( u, ν ) T MST Fragment And The Blue Rule MST Fragment Given a weighted grah G = (V, E, ), a tree T in G is called MST Fragment of G if there exists an MST T M of G such that T is a sub-tree of T M Outgoing edge An edge e = (u, ) of fragment T if only one end points belongs to T MWOE(T) The minimum weight outgoing edge of fragment T Blue Rule For a graph G = (V, E, ) and fragment T in G, if e = MWOE(T) then T # {e} is a fragment as well. Pick a fragment T and add to it the MWOE(T) 7

8 Distributed Prim s Algorithm Sequential Prim s Algorithm Designate vertex r o as the initial fragment Repeatedly apply blue rule until reaching spanning tree Distributed Algorithm (DIST_PRIM) Asynchronous CONGEST model Vertex r o initiates and governs the algorithm A new edge is added in each phase After p phases the constructed tree is T rooted at r o ; every vertex of T p knows its parent, children (if exists) and which adjacent edges are internal and which are outgoing Distributed Prim s Algorithm 8

9 Distributed Prim s Algorithm Analysis DIST_PRIM Each phase p requires up to O (p) time and message Time (DIST_PRIM) = Message (DIST_PRIM) = O(n 2 ) Synchronous GHS Algorithm Kruskal s Algorithm Unlike Prim s algorithm, multiple fragments exist in parallel Initially each vertex forms singleton fragments At each step, merge two fragments by selecting minimum weight outgoing edge from all fragments At the single fragment i.e. MST remains In sequential n-1 steps Distributed GHS Algorithm Synchronous CONGEST model At any given moment the vertices are partitioned into fragments F 1, F k Fragment F i is a rooted tree Each vertex in a F i knows its parent, children, identifier of F i 9

10 Phases of GHS Algorithm Single Phase Each fragment F find e = MWOE(F) using convergecast as in DIST_PRIM Request_to_merge message send over edge e to chosen fragment F with id of fragment and root If other fragment also sends over the same edge then merges at the same point Otherwise the merged fragment will be composed of more than two fragments Fˆ = collection of fragments Eˆ = set of arcs containing the arc( F, F ) for every fragment F whose MWOE( F) leads to F Gˆ = ( Fˆ, Eˆ) resulting graph Each connected component of Ĝ consists of a directed treewhose arc points are directed upwards towards the root, except that the root is not a single fragment, but rather a pair of fragments F 1 and F 2 poiting at each other Phases of GHS Algorithm Single Phase (Cont) Each connected component of consists of a directed treewhose arc points are directed upwards towards the root, except that the root is not a single fragment, but rather a pair of fragments F 1 and F 2 poiting at each other Choosing New root e = MWOE(F 1 ) = MWOE(F 2 ) and e =(v 1, v 2 ) If (v 1 > v 2 ) then v 1 is the new root v 1 sends NewFragment message informing all vertices its id Any vertex receiing NewFragment message updates fragment id and its parent, and neighbors Next Phase New fragment, F, finds its MWOE(F) and repeats the previous phase until one single fragment remains Ĝ 10

11 Analysis of GHS Algorithm Single Phase Each phase consist of a broadcast followed by conergecast involving not more than n vertices Time and message complexity for both broadcast followed by conergecast O (n) In fragment merge requires same comleity; additionally in final step each vertex let its neighbor know its new fragment if which costs ( E ) messages At most lg n phases Overall Complexity Time (GHS) = O(n lg n) Message (GHS) = O ( E lg n) Lower Bound For clean network Message = ( E ) For arbitrary network Message = ( n lg n ) Using Red Rule In MST For G = (V, E, ) MST solution gives weight * G spanning subgraph of G, still contains spanning tree of * C is a cycle in G and e is the heaviest edge is C G \ {e} still contains spanning tree of weight * Red Rule Pick an arbitrary cycle in the remaining graph and erase the heaviest edge in the cycle d f 11 g 11

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

GraphTheory Models & Complexity BFS Trees DFS Trees Min Spanning Trees. Tree Constructions. Martin Biely Tree Constructions Martin Biely Embedded Computing Systems Group Vienna University of Technology European Union November 29, 2006 Biely Tree Constructions 1 Outline 1 A quick Repetition on Graphs 2 Models