Distributed Algorithms

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Distributed Algorithms

• Broadcast and spanning tree

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Model

• Node Information
• Adjacent links
• Neighboring nodes.
• Unique identity.
• Graph model undirected graph.
• Nodes V
• Edges E
• Diameter D

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Model Definition

• Synchronous Model
• There is a global clock.
• Packet can be sent only at clock ticks.
• Packet sent at time t is received by t1.
• Asynchronous model
• Packet sent is eventually received.

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Complexity Measures

• Message complexity
• Number of messages sent (worst case).
• Usually assume small messages.
• Time complexity
• Synchronous Model number of clock ticks
• Asynchronous Model
• Assign delay to each message in (0,1.
• Worse case over possible delays

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Problems

• Broadcast - Single initiator
• The initiator has an information.
• At the end of the protocol all nodes receive the
  information
• Example Topology update.
• Spanning Tree
• Single/Many initiators.
• On termination there is a spanning tree.
• BFS tree
• A minimum distance tree with respect to the
  originator.

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Basic Flooding Algorithm

• Initiator
• Initially send a packet p(info) to all neighbors.
• Every node
• When receiving a packet p(info)
• Sends a packet p(info) to all neighbors.

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Basic Flooding Example
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Basic Flooding Correctness

• Correctness Proof
• By induction on the distance d from initiator.
• Basis d0 trivial.
• Induction step
• Consider a node v at distance d.
• Its neighbor u of distance d-1 received the info.
• Therefore u sends a packet p(info) to all
  neighbors (including v).
• Eventually v receives p(info).

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Basic Flooding Complexity

• Time Complexity
• Synchronous (and asynchronous)
• Node at distance d receives the info by time d.
• Message complexity
• There is no termination!
• Message complexity is unbounded!
• How can we correct this?

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Basic Flooding Improved

• Each node v has a bit sentv, initially
  sentvFALSE.
• Initiator
• Initially Send packet p(info) to all neighbors
  and set sentinitiatorTRUE.
• Every node v
• When receiving a packet p(info) from u
• IF sentvFALSE THEN
• Send packet p(info) to all neighbors.

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Basic Flooding Improved

• Message Complexity
• Each node sends once on each edge!
• Number of messages equals to 2E.
• Time complexity
• Diameter D.
• Drawbacks
• Termination detection

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Spanning tree

• Goal
• At the end of the protocol have a spanning tree
• Motivation for building spanning tree
• Construction costs O(E).
• Broadcast on tree only V-1 messages.

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Spanning tree Algorithm

• Data Structure
• Add to each node v a variable fatherv.
• Initially fatherv null.
• Each node v
• When receiving packet p(info) from u and
  fatherv null
• THEN Set father vu.

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Spanning Tree Example
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Spanning tree Algorithm Correctness

• Correctness Proof
• Every node v (except the initiator) sets
  fatherv.
• There are no cycles.
• Is it a shortest path tree?
• Synchronous
• Asynchronous

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Flood with Termination

• Motivation Inform Initiator of termination.
• Initiator
• Initially Send packet p(info) to all neighbors
  and set setinitiatorTrue
• Non-Initiator node w
• When receiving a packet p(info) from u
• IF sentwFALSE
• THEN fatherwu send packet p(info) to all
  neighbors (except u).
• ELSE Send Ack to u.
• After receiving Ack from all neighbors except
  fatherw
• Send Ack to fatherw.

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Flood with Termination Correctness

• Tree Construction Proof as before.
• Termination
• By induction on the distance from the leaves of
  the tree.
• Basis d0 (v is a leaf).
• Receives immediately Ack on all packets (except
  fatherv).
• Sends Ack packet to fatherv.
• Induction step
• All nodes at distance d-1 received and sent Ack.
• Nodes at distance d will eventually receive Ack,
• and eventually will send Ack.

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DFS Tree

• Data Structure in node v
• Edge-statusv,u for each neighbor u.
• Possible values are NEW,SENT, Done.
• Initially statusv,uNEW.
• Node-statusv.
• Possible Values New, Visited, Terminated
• Initially Node-StatusvNEW (except the
  Initiator which have Node-StatusinitiatorVisite
  d).
• Parentv Initially ParentvNull
• Packets type
• Token traversing edges.

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DFS Algorithm

• Token arrives at a node v from u
• Node_StatusvVisited Edge_Statusv,uNEW
• Edge_Statusv,uDone Send token to u.
• Node_StatusvVisited Edge_Statusv,uSent
• Edge_Statusv,uDone.
• () IF for some w Edge_Statusv,wNew THEN
• Edge_Statusv,wSent Send token to w.
• ELSE Node_StatusTerminated send token to
  parentv.
• If v is the initiator then DFS terminates
• Node_StatusvNew
• Parentvu Node_StatusvVisited
  Edge_Statusu,vSent
• Same as Node_StatusvVisited from ()

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DFS Example
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DFS Tree Performance

• Correctness
• There is only one packet in transit at any time.
• On each edge one packet is sent in each
  direction.
• The parent relationship creates a DFS tree.
• Complexity
• Messages 2E
• Time 2E
• How can we improve the time to O(V).

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Spanning Tree BFS

• Synchronous model have seen.
• Asynchronous model
• Distributed Bellman-Ford
• Distributed Dijkstra

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BFS Bellman-Ford

• Data structure
• Each node v has a variable levelv.
• Initially levelinitiator0 at the initiator and
  levelv? at any other node v.
• Variable parent, initialized at every node v to
  parentv null.
• Packet types
• layer(d) there is a path of length at most d to
  the initiator.

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BFS Bellman-Ford Algorithm

• Initiator
• Send packet layer(0) to all neighbors.
• Every node v
• When receiving packet layer(d) from w
• IF d1 lt levelv THEN
• Parentvw
• Levelvd1
• Send layer(d1) to all neighbors (except w).

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BFS Bellman-Ford Correctness

• Correctness Proof
• Termination Each node u updates levelu at most
  V times.
• Relationship parent creates a tree
• After termination if parentuw then
  levelulevelw1 BSF Tree
• Complexity
• Time O(Diameter)
• Messages O(DiameterE)

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BFS Bellman-Ford Example
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Bellman Ford Multiple initiators

• Set distance (id-initiator,d).
• Use Lexicographic ordering.
• Each node can change level at most
• Number of initiators V
• At most V2

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BSF Dijkstra Illustrated
k1
1
k
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BFS Dijkstra

• Asynchronous model
• Advance one level at a time.
• Go back to root after adding each level.
• Time O( D2 )
• Messages O( E D V )
• DDiameter

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BFS Dijkstra

• Data Structures
• Fatherv the identity of the father of v.
• Childv,u u is a child of v in the tree.
• Upv,u u is closer to the initiator than v.
• Packets
• Pulse
• forward
• Father
• Ack.

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BFS Dijkstra

• In phase k
• Initiator sends a packet Pulse to its children.
• Every node of the tree that receives a packet
  Pulse sends a packet Pulse to its children.
• Every leaf v of the tree
• Sends a packet forward to all neighbors w s.t.
  upv,wFALSE.
• When receiving packet fatheru from u set
  childv,uTRUE.
• When receiving either Ack or fatherw from all
  neighbors w with upv,wFALSE send Ack to
  fatherv.
• Every internal node v that receives Ack from all
  its children send Ack to its father fatherv.

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BFS Dijkstra

• A node u that receives a forward packet (from v)
• First forward packet
• Sets fatheruv
• Sends fatheru to node v.
• second (or more) forward packet
• Send Ack to v.

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BFS Dijkstra

• End of phase k
• Initiator receives Ack from all its children.
• Initiator start phase k1.
• Termination
• If no new node is reached in last phase.

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BFS Dijkstra

• Correctness
• Induction on the phases
• Asynchronous model.
• Time O( D2 )
• Messages O( E D V )
• DDiameter

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Conclusion

• Broadcast
• Flooding
• Flooding with termination
• Spanning tree
• Unstructured spanning tree (using flooding)
• DFS Tree
• BFS Tree