GIAN Course on Distributed Network Algorithms. Spanning Tree Constructions
|
|
|
- Ella Floyd
- 6 years ago
- Views:
Transcription
1 GIAN Course on Distributed Network Algorithms Spanning Tree Constructions Stefan T-Labs, 2011
2 Spanning Trees Attactive infrastructure : sparse subgraph ( loop-free backbone ) connecting all nodes. E.g., cheap flooding. Spanning Tree Cycle-free subgraph spanning all nodes. 2
3 Spanning Trees Is this a spanning tree?
4 Spanning Trees No, not cycle-free! 4
5 Spanning Trees Is this a spanning tree?
6 Spanning Trees No, not spanning this node! 6
7 Spanning Trees Is this a spanning tree?
8 Spanning Trees No, disconnected: spanning forest, not spanning tree! 8
9 Spanning Trees Is this a spanning tree?
10 Spanning Trees Yes, a spanning tree 10
11 Applications Efficient Broadcast and Aggregation Algebraic Gossip Disseminating multiple messages in large communication network Used in Ethernet network to avoid Layer-2 forwarding loops: Spanning Tree Protocol In ad-hoc networks: efficient backbone: broadcast and aggregate data using a linear number of transmissions Random communication pattern with neighbors Gossip: based on local interactions 11
12 Applications ConvergeCast: a fundamental network service. Efficient with spanning trees! Efficient Broadcast and Aggregation Algebraic Gossip Disseminating multiple messages in large communication network Used in Ethernet network to avoid Layer-2 forwarding loops: Spanning Tree Protocol In ad-hoc networks: efficient backbone: broadcast and aggregate data using a linear number of transmissions Random communication pattern with neighbors Gossip: based on local interactions 12
13 Types of Spanning Trees BFS MST a.k.a. shortest distance spanning tree (may also be weighted) Spanning tree includes shortest paths from a given root to all nodes Interesting e.g. for fast broadcast Minimum link cost spanning tree Interesting, e.g., for least routing cost or energy cost How to compute a spanning tree in the LOCAL model? 13
14 Benefits from Spanning Tree: Fundamental ConvergeCast 14
15 Benefits from Spanning Tree: Fundamental ConvergeCast Combination of broadcast from root to leaves and information aggregation from leaves to root! 15
16 Benefits from Spanning Tree: Fundamental ConvergeCast Want to know average temperature! 16
17 Benefits from Spanning Tree: Fundamental ConvergeCast Want to know average temperature! Broadcast along spanning tree: O(n) rather than O(m) messages! Broadcast Round 1 17
18 Benefits from Spanning Tree: Fundamental ConvergeCast Want to know average temperature! Broadcast along spanning tree: O(n) rather than O(m) messages! Broadcast Round 2 18
19 Benefits from Spanning Tree: Fundamental ConvergeCast Want to know average temperature! Broadcast along spanning tree: O(n) rather than O(m) messages! Broadcast Round 3 19
20 Benefits from Spanning Tree: Fundamental ConvergeCast Want to know average temperature! Broadcast along spanning tree: O(n) rather than O(m) messages! O(n) messages for broadcast! 20
21 Benefits from Spanning Tree: Fundamental ConvergeCast But how to aggregate information now in O(n) messages?! 21
22 Benefits from Spanning Tree: Fundamental ConvergeCast But how to aggregate information now in O(n) messages?! Aggregate from leaves toward root, with in-network processing! 22
23 Benefits from Spanning Tree: Fundamental ConvergeCast Temp! Temp! Temp! Temp! Temp! Aggregate Round 1 23
24 Benefits from Spanning Tree: Fundamental ConvergeCast Wait for Wait for children children Temp! Wait for children Wait for children Temp! Temp! Temp! Aggregate Round 1 Wait for children 24
25 Benefits from Spanning Tree: Fundamental ConvergeCast Wait for Wait for children children Temp! Temp! Temp! Wait for children Inner nodes should not send yet: avoid multiple messages over same spanning tree link! Wait for children Temp! Aggregate Round 1 Wait for children 25
26 Benefits from Spanning Tree: Fundamental ConvergeCast agg Aggregate agg Round 2 26
27 Benefits from Spanning Tree: Fundamental ConvergeCast agg Aggregate Round 3 27
28 Benefits from Spanning Tree: Fundamental ConvergeCast agg Aggregate Round 4 28
29 Benefits from Spanning Tree: Fundamental ConvergeCast agg Finished! 29
30 Benefits from Spanning Tree: Fundamental ConvergeCast agg How good is this algorithm? Can we do ConvergeCast with less messages?? Finished! 30
31 Benefits from Spanning Tree: Fundamental ConvergeCast agg How good is this algorithm? Can we do ConvergeCast with less messages? Finished! Let s talk about lower bounds! 31
32 Recall: Local Algorithm Send receive compute. 32
33 Let us introduce some definitions Distance, Radius, Diameter Distance between two nodes is # hops. Radius of a node is max distance to any other node. Radius of graph is minimum radius of any node. Diameter of graph is max distance between any two nodes. Relationship between R and D? 33
34 Let us introduce In general: some R definitions D 2R. Upper bound: max distance cannot be longer than going via any given node (2x radius). Lower bound: contradict definition of radius. Distance, Radius, Diameter Distance between two nodes is # hops. Radius of a node is max distance to any other node. Radius of graph is minimum radius of any node. Diameter of graph is max distance between any two nodes. In the complete graph, for all nodes: R=D. On the line, for broder nodes: 2R=D. 34
35 Let us introduce In general: some R definitions D 2R. Upper bound: max distance cannot be longer than going via any given node (2x radius). Lower bound: contradict definition of radius. Distance, Radius, Diameter Distance between two nodes is # hops. Radius of a node is max distance to any other node. Radius of graph is minimum radius of any node. Diameter of graph is max distance between any two nodes. In the complete graph, for all nodes: R=D. On the line, for broder nodes: 2R=D. Therefore, in the following, we will express all asymptotic results in D (not R). 35
36 Relevance: Radius People enjoy identifying nodes of small radius in a graph! E.g., Erdös number, Kevin Bacon number, joint Erdös-Bacon number, etc. 36
37 Lower Bounds for Broadcast Message complexity? Time complexity? 37
38 Lower Bounds for Broadcast Message complexity? Each node must receive message: so at least n-1. Time complexity? The radius of the source: each node needs to receive message. 38
39 Lower Bounds for Broadcast Message complexity? Each node must receive message: so at least n-1. Time complexity? The radius of the source: each node needs to receive message. How to achieve this? 39
40 Lower Bounds for Broadcast Message complexity? Each node must receive message: so at least n-1. Time complexity? The radius of the source: each node needs to receive message. How to achieve this? Compute a breadth first spanning tree! But how? Next! 40
41 Idea: Compute BFS using Flooding in LOCAL Model! 41
42 Idea: Compute BFS using Flooding in LOCAL Model! Send to all neighbors! Round 1 42
43 Idea: Compute BFS Choose using Flooding parent in LOCAL Model! for spanning tree! Choose parent for spanning tree! Choose parent for spanning tree! Round 1 Invariant: parent has shorter distance to root: loop-free! 43
44 Idea: Compute Send BFS using to all Flooding in LOCAL Model! neighbors! Round 2 44
45 Idea: Compute BFS using Flooding in LOCAL Model! Choose a parent: if multiple arrive at same time, take arbitrary! arbitrary parent with shorter distance! Round 2 Invariant: parent has shorter distance to root: loop-free! 45
46 Idea: Compute BFS using Flooding in LOCAL Model! Round 3 46
47 Idea: Compute BFS using Flooding in LOCAL Model! Round 3 Invariant: parent has shorter distance to root: loop-free! 47
48 Idea: Compute BFS using Flooding in LOCAL Model! Round 4 48
49 Idea: Compute BFS using Flooding in LOCAL Model! BFS! Invariant: parent has shorter distance to root: loop-free! 49
50 Idea: Compute BFS using Flooding in LOCAL Model! BFS! But careful! We assumed that messages propagate in synchronous manner! What if not? 50
51 Bad example Careful: in asynchronous environment, should not make first successful sender my parent! 51
52 Bad example How to overcome? Two options: Dijkstra and Bellman-Ford Careful: in asynchronous environment, should not make first successful sender my parent! 52
53 Distributed BFS: Dijkstra Flavor Essentially a synchronizer for the LOCAL model! Idea: overcome asynchronous problem by proceeding in phases! 53
54 Distributed BFS: Dijkstra Flavor Phase 1 Explore 1-neighborhood only: set Round-Trip-Time to 1. (Round 1) Idea: overcome asynchronous problem by proceeding in phases! 54
55 Distributed BFS: Dijkstra Flavor Phase 1 (Round 2) Idea: overcome asynchronous problem by proceeding in phases! 55
56 Distributed Start Phase BFS: 2! (Propagate Dijkstra Flavor along existing spanning tree!) Phase 2 (Round 1) Idea: overcome asynchronous problem by proceeding in phases! 56
57 Distributed BFS: Dijkstra Flavor Start Phase 2! I am at distance 1 from root! Phase 2 (Round 2) Idea: overcome asynchronous problem by proceeding in phases! 57
58 Distributed BFS: Dijkstra Flavor join! Phase 2 (Round 3) no! no! join! join! Idea: overcome asynchronous problem by proceeding in phases! Choose parent with smaller distance! 58
59 Distributed BFS: Dijkstra Flavor Start Phase 3! Phase 3 Start Phase 3! Start Phase 3! I am at distance 2 from root! Idea: overcome asynchronous problem by proceeding in phases! 59
60 General Scheme In each phase: expand spanning tree by one hop: explore neighborhood! Phase i Phase i+1 60
61 General Scheme Start i Start i+1 Phase i Phase i+1 61
62 General Scheme For efficiency: can propagate start i messages along pre-established spanning tree! Start i Start i+1 At edge we need to try all. Phase i Phase i+1 62
63 Distributed BFS: Dijkstra Flavor join i join i Same for responses! (Aggregated along existing BFS) Phase i Phase i+1 63
64 Distributed BFS: Dijkstra Flavor Time Complexity? Phase i Phase i+1 64
65 Distributed BFS: Dijkstra Flavor Time Complexity? O(D) phases, take time O(D): O(D 2 ) where D is the radius from the root. Phase i Phase i+1 65
66 Distributed BFS: Dijkstra Flavor Message Complexity? Phase i Phase i+1 66
67 Distributed BFS: Dijkstra Flavor Message Complexity? Plus: test each edge once: join, ACK/NAK at edge: total O(m). start and join propagation inside spanning tree: O(n) per phase: O(nD) in total. Phase i Phase i+1 67
68 Distributed BFS: Dijkstra Flavor Message Complexity? Plus: test each edge once: join, ACK/NAK at edge: total O(m). start and join propagation inside spanning tree: O(n) per phase: O(nD) in total. O(nD+m) Phase i Phase i+1 68
69 Distributed BFS: Dijkstra Flavor Dijkstra: find next closest node ( on border ) to the root Dijkstra Style Divide execution into phases. In phase p, nodes with distance p to the root are detected. Let T p be the tree of phase p. T 1 neighbors. Repeat (until no new nodes discovered): is the root plus all direct 1. Root starts phase p by broadcasting start p within T p 2. A leaf u of T p (= node discovered only in last phase) sends join p+1 to all quiet neighbors v (u has not talked to v yet) 3. Node v hearing join for first time sends back ACK : it becomes leave of tree T p+1 ; otherwise v replied NACK (needed since async!) 4. The leaves of T p collect all answers and start Echo Algorithm to the root 5. Root initates next phase 69
70 Distributed BFS: Bellman-Ford Flavor 70
71 Distributed BFS: Bellman-Ford Flavor Idea: Don t go through these time-consuming phases but blast out messages, but with distance information! 71
72 Distributed BFS: Bellman-Ford Flavor 0 Idea: Don t go through these time-consuming phases but blast out messages, but with distance information! init to Initialize: root distance 0, other nodes 72
73 Distributed BFS: Bellman-Ford Flavor distance 1 0 Idea: Don t go through these time-consuming phases but blast out messages, but with distance information! Start: root sends distance 1 packet to neighbors 73
74 Distributed BFS: Bellman-Ford Flavor 0 1 Idea: Don t go through these time-consuming phases but blast out messages, but with distance information! 1 1 Repeat: whenever receive new packet: check whether new minimal distance (if so change parent), and propagate!
75 Distributed BFS: Bellman-Ford Flavor Idea: Don t go through these time-consuming phases but blast out messages, but with distance information! Packet may race through network: asynchronous! Repeat: whenever receive new packet: check whether new minimal distance (if so change parent), and propagate! 75
76 Distributed BFS: Bellman-Ford Flavor Idea: Don t go through these time-consuming phases but blast out messages, but with distance information! Packet may race through network: asynchronous! 1 But sooner or later, node will learn shorter distance! Repeat: whenever receive new packet: check whether new minimal distance (if so change parent), and propagate! 76
77 Distributed BFS: Bellman-Ford Flavor Bellman-Ford: compute shortest distances by flooding an all paths; best predecessor = parent in tree Bellman-Ford Style Each node u stores d u, the distance from u to the root. Initially, d root =0 and all other distances are 1. Root starts algo by sending 1 to all neighbors. 1. If a node u receives message y with y<d u d u := y send y+1 to all other neighbors 77
78 Analysis How is this defined?! Assuming a unit upper bound on per link delay! Time Complexity? Message Complexity? 78
79 Analysis Time Complexity? Worst propagation time is simply the diameter. O(D) where D is diameter of graph. By induction: By time d, node at distance d got d. Clearly true for d=0 and d=1. A node at distance d has neighbor at distance d-1 that got d-1 on time by induction hypothesis. It will send d in next time slot... Message Complexity? O(mn) where m is number of edges, n is number of nodes. Because: A node can reduce its distance at most n- 1 times (recall: asynchronous!). Each of these times it sends an upate message to all its neighbors 79
80 Bellman-Ford with Many Messages 1 d=1 2 root d=2 3 d=5 d=3 5 4 d=4 80
81 Bellman-Ford with Many Messages d=1 root 1 2 d=1 d=4 d=2 4 3 d=3 Everyone has a new best distance and informs neighbors! 81
82 Bellman-Ford with Many Messages d=1 root d=1 1 d=3 d=1 3 2 d=2 Everyone has a new best distance and informs neighbors! 82
83 Discussion Which algorithm is better? Dijkstra has better message complexity, Bellman-Ford better time complexity. Can we do better? Yes, but not in this course... Remark: Asynchronous algorithms can be made synchronous... (e.g., by central controller or better: local synchronizers) 83
84 How to compute an MST? MST Tree with edges of minimal total weight Stefan T-Labs, 2011
85 Idea: Exploit Basic Fact of MST: Blue Edges Blue Edge Let T be an MST and T a subgraph of T. Edge e=(u,v) is outgoing edge if u ϵ T and v T. The outgoing edge of minimal weight is called blue edge. Lemma If T is the MST and T a subgraph of T, then the blue edge of T is also part of T. It holds: exactly one tree edge across and cut and the lightest edge across a cut must be part of the MST! By contradiction: otherwise get a cheaper MST by swapping the two cut edges! 1 3 5
86 Gallager-Humblet-Spira 86
87 Gallager-Humblet-Spira Basic idea: Grow components in parallel and merge them at the blue edge! Using Covergecast. 87
88 Gallager-Humblet-Spira Assume some components have already emerged: Basic idea: Grow components in parallel and merge them at the blue edge! Using Covergecast. T 1 blue for T 1 T blue for T 2 and T 3 T 3 Each component has only one blue edge (cheapest outgoing): loops impossible, can take them in parallel! 88
89 Gallager-Humblet-Spira Basic idea: Grow components and in parallel merge them and merge at the them blue edge, at the using blue edge! Using Covergecast! Covergecast. leader of T 1! 8 T 1 blue for T 1 T leader of T 2! blue for T 2 and T 3 leader of T 3! T 3 Idea: a leader in each cluster responsible of finding blue edge with convergecast! Inside component: maintain spanning tree directed to leader! 89
90 Gallager-Humblet-Spira: High-level View Phase 3 Phase 1 Idea: components grow in parallel and merge in a loop-free manner! Phase 2 90
91 Gallager-Humblet-Spira: High-level View After round i, minimal component has size at least 2 i : doubles in each round! Phase 3 Phase 1 Idea: components grow in parallel and merge in a loop-free manner! Phase 2 So at most log(n) phases in total! 91
92 Gallager-Humblet-Spira: High-level View But how to determine blue edge quickly and re-elect new leader in merged larger component? Phase 3 Phase 1 Minimal fragment size in round i? 2 i Keep spanning tree in each component! Can do efficient covergecast there. Phase 2 Total number of phases? 92
93 Example: Agree on a New Root How to merge T and T across (u,v)? root T 1 T root v root u T 93
94 Example: Agree on a New Root How to merge T and T across (u,v)? root T blue edge of T and T T root v root Invariant: directed and rooted spanning tree in each component! Links point to root. blue edge of T u T 94
95 Example: Agree on a New Root How to merge T and T across (u,v)? root T 1 T root v root u T Step 1: invert path from root1 to u and root2 to v. 95
96 Example: Agree on a New Root How to merge T and T across (u,v)? root T 1 T root v root u T Step 1: invert path from root1 to u and root2 to v. Step 2: send merge request across blue edge (u,v). Here (u,v) is only the blue edge for T so one message! Step 3: consequence of link reversal: v becomes new root overall! 96
97 Example: Agree on a New Root How to merge T and T across (u,v)? root T Rooted again! T root v root u T Step 1: invert path from root1 to u and root2 to v. Step 2: send merge request across blue edge (u,v). Here (u,v) is only the blue edge for T so one message! Step 3: consequence of link reversal: v becomes new root overall! 97
98 Example: Agree on a New Root How to merge T and T across (u,v)? Rooted again! root T T root v What if (u,v) was blue link for both T and T? Just make tie-breaking who becomes root, u or v, e.g., ID based! 7 root u T Step 1: invert path from root1 to u and root2 to v. Step 2: send merge request across blue edge (u,v). Here (u,v) is only the blue edge for T so one message! Step 3: consequence of link reversal: v becomes new root overall! 98
99 Distributed Kruskal Idea: Grow components by learning blue edge! But do many fragments in parallel! Gallager-Humblet-Spira Initially, each node is root of its own fragment. Repeat (until all nodes in same fragment) 1. nodes learn fragment IDs of neighbors 2. root of fragment finds blue edge (u,v) by convergecast 3. root sends message to u (inverting parent-child) 4. if v also sent a merge request over (u,v), u or v becomes new root depending on smaller ID (make trees directed) 5. new root informs fragment about new root (convergecast on MST of fragment): new fragment ID 99
100 Analysis Time Complexity? Message Complexity? Each phase mainly consists of two convergecasts, so O(D) time and O(n) messages per phase? 100
101 Analysis Time Complexity? O(n log n) where n is graph size. Message Complexity? Log n phases with O(n) time convergecast: spanning tree is not BFS! The size of the smallest fragment at least doubles in each phase, so it s logarithmic. But converge cast may take n hops. Log n phases but in each phase need to learn leader ID of neighboring fragments, for all neighbors! O(m log n) where m is number of edges: at most O(1) messages on each edge in a phase. Really needed? Each phase mainly consists of two convergecasts, so O(n) time and O(n) messages. In order to learn fragment IDs of neighbors, O(m) messages are needed (again and again: ID changes in each phase). Yes, we can do better. Stefan T-Labs,
102 Analysis Time Complexity? O(n log n) where n is graph size. Message Complexity? Log n phases with O(n) time convergecast: spanning tree is not BFS! The size of the smallest fragment at least doubles in each phase, so it s logarithmic. But converge cast may take n hops. Log n phases but in each phase need to learn leader ID of neighboring fragments, for all neighbors! O(m log n) where m is number of edges: at most O(1) messages on each edge in a phase. Really needed? Each phase mainly consists of two convergecasts, so O(n) time and O(n) messages. In order to learn fragment IDs of neighbors, O(m) messages Note: are needed this (again algorithm and again: ID can changes solve in each leader phase). election! Leader = last surviving root! Yes, we can do better. Stefan T-Labs,
103 Homework: License to Match 103
104 Literature for further reading: - Peleg s book End of lecture
105 Exercise 2: License to Match Only a subset participates in the mission! NOT OK! Link used twice. Only know whether I myself participate initially Need to build pairs Communication along tree: use link only once!
106 Exercise 2: License to Match Idea: Convergecast! Pair up from leaves upward At most one unmatched guy in subtree: either match with myself or propagate upward (links used at most once!) If total number of active nodes even, there is a solution
107 Exercise 2: License to Match
108 Exercise 2: License to Match Idea: Convergecast! Pair up from leaves upward 3. At most one unmatched guy in subtree: either match with myself or propagate upward (links used at most once!) 2. If total number of active nodes even, there is a solution 1.
109 Exercise 2: License to Match Analysis Runtime in the order of tree depth 3. Number of messages linear in tree size 2. 1.
GIAN Course on Distributed Network Algorithms. Spanning Tree Constructions
GIAN Course on Distributed Network Algorithms Spanning Tree Constructions Stefan Schmid @ T-Labs, 2011 Spanning Trees Attactive infrastructure : sparse subgraph ( loop-free backbone ) connecting all nodes.
Clearly there is a close relation between the radius R and the diameter D of a graph, such as R D 2R.
Chapter 2 Tree Algorithms In this chapter we learn a few basic algorithms on trees, and how to construct trees in the first place so that we can run these (and other) algorithms. The good news is that
A synchronizer generates sequences of clock pulses at each node of the network satisfying the condition given by the following definition.
Chapter 8 Synchronizers So far, we have mainly studied synchronous algorithms because generally, asynchronous algorithms are often more di cult to obtain and it is substantially harder to reason about
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
Distributed Computing
Principles of Distributed Computing Roger Wattenhofer wattenhofer@tik.ee.ethz.ch Assistant: Christoph Lenzen (lenzen@tik.ee.ethz.ch) Spring 2010 2 Introduction What is Distributed Computing? In the last
Broadcast: Befo re 1
Broadcast: Before 1 After 2 Spanning Tree ffl assume fixed spanning tree ffl asynchronous model 3 Processor State parent terminated children 4 Broadcast: Step One parent terminated children 5 Broadcast:Step
Distributed Algorithms 6.046J, Spring, Nancy Lynch
Distributed Algorithms 6.046J, Spring, 205 Nancy Lynch What are Distributed Algorithms? Algorithms that run on networked processors, or on multiprocessors that share memory. They solve many kinds of problems:
All-to-All Communication
Network Algorithms All-to-All Communication Thanks to Thomas Locher from ABB Research for basis of slides! Stefan Schmid @ T-Labs, 2011 Repetition Tree: connected graph without cycles Spanning subgraph:
MITOCW watch?v=w_-sx4vr53m
MITOCW watch?v=w_-sx4vr53m The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To
THE FIRST APPROXIMATED DISTRIBUTED ALGORITHM FOR THE MINIMUM DEGREE SPANNING TREE PROBLEM ON GENERAL GRAPHS. and
International Journal of Foundations of Computer Science c World Scientific Publishing Company THE FIRST APPROXIMATED DISTRIBUTED ALGORITHM FOR THE MINIMUM DEGREE SPANNING TREE PROBLEM ON GENERAL GRAPHS
Distributed Computing over Communication Networks: Leader Election
Distributed Computing over Communication Networks: Leader Election Motivation Reasons for electing a leader? Reasons for not electing a leader? Motivation Reasons for electing a leader? Once elected, coordination
6.852 Lecture 10. Minimum spanning tree. Reading: Chapter 15.5, Gallager-Humblet-Spira paper. Gallager-Humblet-Spira algorithm
6.852 Lecture 10 Gallager-Humblet-Spira algorithm Reading: Chapter 15.5, Gallager-Humblet-Spira paper Assume undirected graph (i.e., bidirectional communication) distinct edge weights size and diameter
Communication Networks I December 4, 2001 Agenda Graph theory notation Trees Shortest path algorithms Distributed, asynchronous algorithms Page 1
Communication Networks I December, Agenda Graph theory notation Trees Shortest path algorithms Distributed, asynchronous algorithms Page Communication Networks I December, Notation G = (V,E) denotes a
Distributed Algorithms 6.046J, Spring, 2015 Part 2. Nancy Lynch
Distributed Algorithms 6.046J, Spring, 2015 Part 2 Nancy Lynch 1 This Week Synchronous distributed algorithms: Leader Election Maximal Independent Set Breadth-First Spanning Trees Shortest Paths Trees
GIAN Course on Distributed Network Algorithms. Distributed Objects
GIAN Course on Distributed Network Algorithms Distributed Objects Stefan Schmid @ T-Labs, 2011 GIAN Course on Distributed Network Algorithms Distributed Objects How to access and manipulate data in an
Distributed minimum spanning tree problem
Distributed minimum spanning tree problem Juho-Kustaa Kangas 24th November 2012 Abstract Given a connected weighted undirected graph, the minimum spanning tree problem asks for a spanning subtree with
managing an evolving set of connected components implementing a Union-Find data structure implementing Kruskal s algorithm
Spanning Trees 1 Spanning Trees the minimum spanning tree problem three greedy algorithms analysis of the algorithms 2 The Union-Find Data Structure managing an evolving set of connected components implementing
COMP 182: Algorithmic Thinking Prim and Dijkstra: Efficiency and Correctness
Prim and Dijkstra: Efficiency and Correctness Luay Nakhleh 1 Prim s Algorithm In class we saw Prim s algorithm for computing a minimum spanning tree (MST) of a weighted, undirected graph g. The pseudo-code
Tree Construction (BFS, DFS, MST) Chapter 5. Breadth-First Spanning Tree (BFS)
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
The Power of Locality
GIAN Course on Distributed Network Algorithms The Power of Locality Case Study: Graph Coloring Stefan Schmid @ T-Labs, 2011 Case Study: Graph Coloring Case Study: Graph Coloring Assign colors to nodes.
Algorithm Design (8) Graph Algorithms 1/2
Graph Algorithm Design (8) Graph Algorithms / Graph:, : A finite set of vertices (or nodes) : A finite set of edges (or arcs or branches) each of which connect two vertices Takashi Chikayama School of
Dijkstra s algorithm for shortest paths when no edges have negative weight.
Lecture 14 Graph Algorithms II 14.1 Overview In this lecture we begin with one more algorithm for the shortest path problem, Dijkstra s algorithm. We then will see how the basic approach of this algorithm
CS 161 Lecture 11 BFS, Dijkstra s algorithm Jessica Su (some parts copied from CLRS) 1 Review
1 Review 1 Something I did not emphasize enough last time is that during the execution of depth-firstsearch, we construct depth-first-search trees. One graph may have multiple depth-firstsearch trees,
A family of optimal termination detection algorithms
Distrib. Comput. (2007) 20:141 162 DOI 10.1007/s00446-007-0031-3 A family of optimal termination detection algorithms Neeraj Mittal S. Venkatesan Sathya Peri Received: 28 January 2005 / Accepted: 20 April
2. True or false: even though BFS and DFS have the same space complexity, they do not always have the same worst case asymptotic time complexity.
1. T F: Consider a directed graph G = (V, E) and a vertex s V. Suppose that for all v V, there exists a directed path in G from s to v. Suppose that a DFS is run on G, starting from s. Then, true or false:
Master Thesis: Animations and Analysis of Distributed Algorithms. Boris Koldehofe
Master Thesis: Animations and Analysis of Distributed Algorithms Boris Koldehofe 12th April 1999 Contents 1 Introduction 1 1.1 Distributed System and Models... 2 1.2 Complexity Measures and Causality..................
Network Layer: Routing
Network Layer: Routing The Problem A B R 1 R 2 R 4 R 3 Goal: for each destination, compute next hop 1 Lecture 9 2 Basic Assumptions Trivial solution: Flooding Dynamic environment: links and routers unreliable:
Binary heaps (chapters ) Leftist heaps
Binary heaps (chapters 20.3 20.5) Leftist heaps Binary heaps are arrays! A binary heap is really implemented using an array! 8 18 29 20 28 39 66 Possible because of completeness property 37 26 76 32 74
22 Elementary Graph Algorithms. There are two standard ways to represent a
VI Graph Algorithms Elementary Graph Algorithms Minimum Spanning Trees Single-Source Shortest Paths All-Pairs Shortest Paths 22 Elementary Graph Algorithms There are two standard ways to represent a graph
Redes de Computadores. Shortest Paths in Networks
Redes de Computadores Shortest Paths in Networks Manuel P. Ricardo Faculdade de Engenharia da Universidade do Porto » What is a graph?» What is a spanning tree?» What is a shortest path tree?» How are
CS521 \ Notes for the Final Exam
CS521 \ Notes for final exam 1 Ariel Stolerman Asymptotic Notations: CS521 \ Notes for the Final Exam Notation Definition Limit Big-O ( ) Small-o ( ) Big- ( ) Small- ( ) Big- ( ) Notes: ( ) ( ) ( ) ( )
Distributed Systems. Basic Algorithms. Rik Sarkar. University of Edinburgh 2015/2016
Distributed Systems Basic Algorithms Rik Sarkar University of Edinburgh 2015/2016 Distributed ComputaEon Ref: NL How to send messages to all nodes efficiently How to compute sums of values at all nodes
Distributed Systems. Basic Algorithms. Rik Sarkar. University of Edinburgh 2016/2017
Distributed Systems Basic Algorithms Rik Sarkar University of Edinburgh 2016/2017 Distributed ComputaEon Ref: NL How to send messages to all nodes efficiently How to compute sums of values at all nodes
Distributed Minimum Spanning Tree
Distributed Minimum Spanning Tree Outline I. Introduction II. GHS algorithm (1983) III. Awerbuch algorithm (1987) IV. Other results Introduction Problem and Model Undirected G(N,E) asynchronous algorithm
Routing. Information Networks p.1/35
Routing Routing is done by the network layer protocol to guide packets through the communication subnet to their destinations The time when routing decisions are made depends on whether we are using virtual
ADHOC SYSTEMSS ABSTRACT. the system. The diagnosis strategy. is an initiator of. can in the hierarchy. paper include. this. than.
Journal of Theoretical and Applied Information Technology 2005-2007 JATIT. All rights reserved. www.jatit.org A HIERARCHICAL APPROACH TO FAULT DIAGNOSIS IN LARGE S CALE SELF-DIAGNOSABLE WIRELESS ADHOC
Lecture 19. Broadcast routing
Lecture 9 Broadcast routing Slide Broadcast Routing Route a packet from a source to all nodes in the network Possible solutions: Flooding: Each node sends packet on all outgoing links Discard packets received
Tutorial. Question There are no forward edges. 4. For each back edge u, v, we have 0 d[v] d[u].
![Tutorial. Question There are no forward edges. 4. For each back edge u, v, we have 0 d[v] d[u].](/thumbs/88/114871846.jpg "Tutorial. Question There are no forward edges. 4. For each back edge u, v, we have 0 d[v] d[u].")
Tutorial Question 1 A depth-first forest classifies the edges of a graph into tree, back, forward, and cross edges. A breadth-first tree can also be used to classify the edges reachable from the source
Chapter 4. Greedy Algorithms. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 4 Greedy Algorithms Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 4.5 Minimum Spanning Tree Minimum Spanning Tree Minimum spanning tree. Given a connected
CSE 100 Minimum Spanning Trees Prim s and Kruskal
CSE 100 Minimum Spanning Trees Prim s and Kruskal Your Turn The array of vertices, which include dist, prev, and done fields (initialize dist to INFINITY and done to false ): V0: dist= prev= done= adj:
Name: Lirong TAN 1. (15 pts) (a) Define what is a shortest s-t path in a weighted, connected graph G.
1. (15 pts) (a) Define what is a shortest s-t path in a weighted, connected graph G. A shortest s-t path is a path from vertex to vertex, whose sum of edge weights is minimized. (b) Give the pseudocode
CS 341: Algorithms. Douglas R. Stinson. David R. Cheriton School of Computer Science University of Waterloo. February 26, 2019
CS 341: Algorithms Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo February 26, 2019 D.R. Stinson (SCS) CS 341 February 26, 2019 1 / 296 1 Course Information 2 Introduction
Chapter 8 DOMINATING SETS
Chapter 8 DOMINATING SETS Distributed Computing Group Mobile Computing Summer 2004 Overview Motivation Dominating Set Connected Dominating Set The Greedy Algorithm The Tree Growing Algorithm The Marking
Chapter 8 DOMINATING SETS
Distributed Computing Group Chapter 8 DOMINATING SETS Mobile Computing Summer 2004 Overview Motivation Dominating Set Connected Dominating Set The Greedy Algorithm The Tree Growing Algorithm The Marking
CSE 521: Design and Analysis of Algorithms I
CSE 521: Design and Analysis of Algorithms I Greedy Algorithms Paul Beame 1 Greedy Algorithms Hard to define exactly but can give general properties Solution is built in small steps Decisions on how to
Lecture Summary CSC 263H. August 5, 2016
Lecture Summary CSC 263H August 5, 2016 This document is a very brief overview of what we did in each lecture, it is by no means a replacement for attending lecture or doing the readings. 1. Week 1 2.
A Time- and Message-Optimal Distributed Algorithm for Minimum Spanning Trees
A Time- and Message-Optimal Distributed Algorithm for Minimum Spanning Trees Gopal Pandurangan University of Houston Houston, TX, USA gopalpandurangan@gmail.com ABSTRACT This paper presents a randomized
Lecture 3: Graphs and flows
Chapter 3 Lecture 3: Graphs and flows Graphs: a useful combinatorial structure. Definitions: graph, directed and undirected graph, edge as ordered pair, path, cycle, connected graph, strongly connected
How Routing Algorithms Work
How Routing Algorithms Work A router is used to manage network traffic and find the best route for sending packets. But have you ever thought about how routers do this? Routers need to have some information
Graph Algorithms. Many problems in networks can be modeled as graph problems.
Graph Algorithms Graph Algorithms Many problems in networks can be modeled as graph problems. - The topology of a distributed system is a graph. - Routing table computation uses the shortest path algorithm
Ad hoc and Sensor Networks Topology control
Ad hoc and Sensor Networks Topology control Goals of this chapter Networks can be too dense too many nodes in close (radio) vicinity This chapter looks at methods to deal with such networks by Reducing/controlling
11.9 Connectivity Connected Components. mcs 2015/5/18 1:43 page 419 #427
mcs 2015/5/18 1:43 page 419 #427 11.9 Connectivity Definition 11.9.1. Two vertices are connected in a graph when there is a path that begins at one and ends at the other. By convention, every vertex is
CS261: Problem Set #1
CS261: Problem Set #1 Due by 11:59 PM on Tuesday, April 21, 2015 Instructions: (1) Form a group of 1-3 students. You should turn in only one write-up for your entire group. (2) Turn in your solutions by
Λέων-Χαράλαμπος Σταματάρης
Λέων-Χαράλαμπος Σταματάρης INTRODUCTION Two classical problems of information dissemination in computer networks: The broadcasting problem: Distributing a particular message from a distinguished source
CSE 331: Introduction to Algorithm Analysis and Design Graphs
CSE 331: Introduction to Algorithm Analysis and Design Graphs 1 Graph Definitions Graph: A graph consists of a set of verticies V and a set of edges E such that: G = (V, E) V = {v 0, v 1,..., v n 1 } E
CSc Leader Election
CSc72010 Leader Election Reading Skim 3.5 Read 3.6, General synchronous networks Read 4.1 Leader election algorithms Lelann/Chang/Roberts: O(n) time complexity O(n 2 ) message complexity Hirschberg/Sinclair
ICS 351: Today's plan. distance-vector routing game link-state routing OSPF
ICS 351: Today's plan distance-vector routing game link-state routing OSPF distance-vector routing game 1. prepare a list of all neighbors and the links to them, and the metric for each link 2. create
Copyright 2000, Kevin Wayne 1
Linear Time: O(n) CS 580: Algorithm Design and Analysis 2.4 A Survey of Common Running Times Merge. Combine two sorted lists A = a 1,a 2,,a n with B = b 1,b 2,,b n into sorted whole. Jeremiah Blocki Purdue
Matching Algorithms. Proof. If a bipartite graph has a perfect matching, then it is easy to see that the right hand side is a necessary condition.
18.433 Combinatorial Optimization Matching Algorithms September 9,14,16 Lecturer: Santosh Vempala Given a graph G = (V, E), a matching M is a set of edges with the property that no two of the edges have
A DISTRIBUTED SYNCHRONOUS ALGORITHM FOR MINIMUM-WEIGHT SPANNING TREES
ISSN: 2778-5795 A DISTRIBUTED SYNCHRONOUS ALGORITHM FOR MINIMUM-WEIGHT SPANNING TREES Md. Mohsin Ali 1, Mst. Shakila Khan Rumi 2 1 Department of Computer Science, The University of Western Ontario, Canada
UNIT 2 ROUTING ALGORITHMS
UNIT ROUTING ALGORITHMS Routing Algorithms Structure Page Nos..0 Introduction 3. Objectives 3. Flooding 3.3 Shortest Path Routing Algorithm 5.4 Distance Vector Routing 6.4. Comparison.4. The Count-to-Infinity
Computer Networks. Routing
Computer Networks Routing Topics Link State Routing (Continued) Hierarchical Routing Broadcast Routing Sending distinct packets Flooding Multi-destination routing Using spanning tree Reverse path forwarding
Review of Graph Theory. Gregory Provan
Review of Graph Theory Gregory Provan Overview Need for graphical models of computation Cloud computing Data centres Telecommunications networks Graph theory Graph Models for Cloud Computing Integration
Lecture 6 Basic Graph Algorithms
CS 491 CAP Intro to Competitive Algorithmic Programming Lecture 6 Basic Graph Algorithms Uttam Thakore University of Illinois at Urbana-Champaign September 30, 2015 Updates ICPC Regionals teams will be
Advanced Algorithms Class Notes for Monday, October 23, 2012 Min Ye, Mingfu Shao, and Bernard Moret
Advanced Algorithms Class Notes for Monday, October 23, 2012 Min Ye, Mingfu Shao, and Bernard Moret Greedy Algorithms (continued) The best known application where the greedy algorithm is optimal is surely
Trees. Arash Rafiey. 20 October, 2015
20 October, 2015 Definition Let G = (V, E) be a loop-free undirected graph. G is called a tree if G is connected and contains no cycle. Definition Let G = (V, E) be a loop-free undirected graph. G is called
Parallel Breadth First Search
CSE341T/CSE549T 11/03/2014 Lecture 18 Parallel Breadth First Search Today, we will look at a basic graph algorithm, breadth first search (BFS). BFS can be applied to solve a variety of problems including:
21. Distributed Algorithms
21. Distributed Algorithms We dene a distributed system as a collection of individual computing devices that can communicate with each other [2]. This denition is very broad, it includes anything, from
Single Source Shortest Path
Single Source Shortest Path A directed graph G = (V, E) and a pair of nodes s, d is given. The edges have a real-valued weight W i. This time we are looking for the weight and the shortest path from s
Topics. Trees Vojislav Kecman. Which graphs are trees? Terminology. Terminology Trees as Models Some Tree Theorems Applications of Trees CMSC 302
Topics VCU, Department of Computer Science CMSC 302 Trees Vojislav Kecman Terminology Trees as Models Some Tree Theorems Applications of Trees Binary Search Tree Decision Tree Tree Traversal Spanning Trees
CSE 431/531: Analysis of Algorithms. Greedy Algorithms. Lecturer: Shi Li. Department of Computer Science and Engineering University at Buffalo
CSE 431/531: Analysis of Algorithms Greedy Algorithms Lecturer: Shi Li Department of Computer Science and Engineering University at Buffalo Main Goal of Algorithm Design Design fast algorithms to solve
CSE 100: GRAPH ALGORITHMS
CSE 100: GRAPH ALGORITHMS Dijkstra s Algorithm: Questions Initialize the graph: Give all vertices a dist of INFINITY, set all done flags to false Start at s; give s dist = 0 and set prev field to -1 Enqueue
Minimum-Spanning-Tree problem. Minimum Spanning Trees (Forests) Minimum-Spanning-Tree problem
Minimum Spanning Trees (Forests) Given an undirected graph G=(V,E) with each edge e having a weight w(e) : Find a subgraph T of G of minimum total weight s.t. every pair of vertices connected in G are
What is a minimal spanning tree (MST) and how to find one
What is a minimal spanning tree (MST) and how to find one A tree contains a root, the top node. Each node has: One parent Any number of children A spanning tree of a graph is a subgraph that contains all
Outline. Routing. Introduction to Wide Area Routing. Classification of Routing Algorithms. Introduction. Broadcasting and Multicasting
Outline Routing Fundamentals of Computer Networks Guevara Noubir Introduction Broadcasting and Multicasting Shortest Path Unicast Routing Link Weights and Stability F2003, CSG150 Fundamentals of Computer
Shortest Paths. Nishant Mehta Lectures 10 and 11
Shortest Paths Nishant Mehta Lectures 0 and Communication Speeds in a Computer Network Find fastest way to route a data packet between two computers 6 Kbps 4 0 Mbps 6 6 Kbps 6 Kbps Gbps 00 Mbps 8 6 Kbps
Distributed Leader Election Algorithms in Synchronous Networks
Distributed Leader Election Algorithms in Synchronous Networks Mitsou Valia National Technical University of Athens School of Applied Mathematics and Physics 1/66 Distributed Computing Distributed computing
CS4450. Computer Networks: Architecture and Protocols. Lecture 11 Rou+ng: Deep Dive. Spring 2018 Rachit Agarwal
CS4450 Computer Networks: Architecture and Protocols Lecture 11 Rou+ng: Deep Dive Spring 2018 Rachit Agarwal 2 Goals for Today s Lecture Learning about Routing Protocols Link State (Global view) Distance
Algorithm Design and Analysis
Algorithm Design and Analysis LECTURE 4 Graphs Definitions Traversals Adam Smith 9/8/10 Exercise How can you simulate an array with two unbounded stacks and a small amount of memory? (Hint: think of a
Parallel Pointers: Graphs
Parallel Pointers: Graphs Breadth first Search (BFS) 101 Given a vertex S, what is the distance of all the other vertices from S? Why would this be useful? To find the fastest route through a network.
T Seminar on Theoretical Computer Science Spring 2007 Distributed Computation. Constructing a spanning tree Toni Kylmälä
T-79.4001 Seminar on Theoretical Computer Science Spring 2007 Distributed Computation Constructing a spanning tree Toni Kylmälä toni.kylmala@tkk.fi 1 Spanning tree To construct a spanning tree we must
From Shared Memory to Message Passing
Message Passing From Shared Memory to Message Passing but pass messages along networks: classic social networks Stefan Schmid T-Labs & TU Berlin Some parts of the lecture (e.g., Skript and exercises) are
ECE 333: Introduction to Communication Networks Fall 2001
ECE : Introduction to Communication Networks Fall 00 Lecture : Routing and Addressing I Introduction to Routing/Addressing Lectures 9- described the main components of point-to-point networks, i.e. multiplexed
Lecture 10 Graph algorithms: testing graph properties
Lecture 10 Graph algorithms: testing graph properties COMP 523: Advanced Algorithmic Techniques Lecturer: Dariusz Kowalski Lecture 10: Testing Graph Properties 1 Overview Previous lectures: Representation
Algorithms for Minimum Spanning Trees
Algorithms & Models of Computation CS/ECE, Fall Algorithms for Minimum Spanning Trees Lecture Thursday, November, Part I Algorithms for Minimum Spanning Tree Sariel Har-Peled (UIUC) CS Fall / 6 Sariel
Shortest Path Algorithms
Shortest Path Algorithms Andreas Klappenecker [based on slides by Prof. Welch] 1 Single Source Shortest Path Given: a directed or undirected graph G = (V,E) a source node s in V a weight function w: E
Multicast EECS 122: Lecture 16
Multicast EECS 1: Lecture 16 Department of Electrical Engineering and Computer Sciences University of California Berkeley Broadcasting to Groups Many applications are not one-one Broadcast Group collaboration
Lecture 11: Analysis of Algorithms (CS ) 1
Lecture 11: Analysis of Algorithms (CS583-002) 1 Amarda Shehu November 12, 2014 1 Some material adapted from Kevin Wayne s Algorithm Class @ Princeton 1 2 Dynamic Programming Approach Floyd-Warshall Shortest
Leader Election in Rings
Leader Election Arvind Krishnamurthy Fall 2003 Leader Election in Rings Under different models: Synchronous vs. asynchronous Anonymous vs. non-anonymous (knowledge of unique id) Knowledge of n (non-uniform)
Depth First Search A B C D E F G A B C 5 D E F 3 2 G 2 3
Depth First Search A B C D E F G A 4 3 2 B 4 5 4 3 C 5 D 3 4 2 E 2 2 3 F 3 2 G 2 3 Minimum (Weight) Spanning Trees Let G be a graph with weights on the edges. We define the weight of any subgraph of G
Throughout this course, we use the terms vertex and node interchangeably.
Chapter Vertex Coloring. Introduction Vertex coloring is an infamous graph theory problem. It is also a useful toy example to see the style of this course already in the first lecture. Vertex coloring
Kruskal s MST Algorithm
Kruskal s MST Algorithm CLRS Chapter 23, DPV Chapter 5 Version of November 5, 2014 Main Topics of This Lecture Kruskal s algorithm Another, but different, greedy MST algorithm Introduction to UNION-FIND
Discrete mathematics
Discrete mathematics Petr Kovář petr.kovar@vsb.cz VŠB Technical University of Ostrava DiM 470-2301/02, Winter term 2018/2019 About this file This file is meant to be a guideline for the lecturer. Many
Optimization I : Brute force and Greedy strategy
Chapter 3 Optimization I : Brute force and Greedy strategy A generic definition of an optimization problem involves a set of constraints that defines a subset in some underlying space (like the Euclidean
cs/ee 143 Communication Networks
cs/ee 143 Communication Networks Chapter 5 Routing Text: Walrand & Parakh, 2010 Steven Low CMS, EE, Caltech Warning These notes are not self-contained, probably not understandable, unless you also were
Routing Algorithms for Safety Critical Wireless Sensor Networks
Routing Algorithms for Safety Critical Wireless Sensor Networks Tunc Ikikardes Master Thesis, SS-2006 Advisors: Dr. Markus Hofbauer, Siemens Dr. August Kälin, Siemens Dr. Martin May Professor: Prof. Bernhard
Graph Algorithms. Many problems in networks can be modeled as graph problems.
Graph Algorithms Many problems in networks can be modeled as graph problems. - The topology of a distributed system is a graph. - Routing table computation uses the shortest path algorithm - Efficient
Lecture 5 Using Data Structures to Improve Dijkstra s Algorithm. (AM&O Sections and Appendix A)
Lecture Using Data Structures to Improve Dijkstra s Algorithm (AM&O Sections 4.6 4.8 and Appendix A) Manipulating the Data in Dijkstra s Algorithm The bottleneck operation in Dijkstra s Algorithm is that
Shortest Path Routing Communications networks as graphs Graph terminology Breadth-first search in a graph Properties of breadth-first search
Shortest Path Routing Communications networks as graphs Graph terminology Breadth-first search in a graph Properties of breadth-first search 6.082 Fall 2006 Shortest Path Routing, Slide 1 Routing in an
Algorithms and Data Structures
Algorithms and Data Structures Priority Queues Ulf Leser Special Scenarios for Searching Up to now, we assumed that all elements of a list are equally important and that any of them could be searched next