Basic Problems and Algorithms

1
Basic Problems and Algorithms

• One initiator
• broadcasting
• traversal
• spanning tree construction
• Multiple initiators
• wake-up
• leader election
• Setting
• no faults, bidirectional connected networks
• arbitrary graphs
• special topologies tree, clique, mesh, torus,
  hypercube, chordal rings

2
Last lecture

• Flooding algorithm for broadcasting
• works in arbitrary networks
• O(m) message complexity
• O(eccentricity(G)) bounded time
• Better algorithms for specific networks
• n-1 messages (vs cca 3n for flooding) for
  oriented mesh/torus
• n-1 messages (vs O(n log n) for flooding) for
  oriented hypercubes
• O(n) message (vs O(nk)) for unoriented compact
  k-chordal rings
• If a spanning tree is given, we can use it to
  broadcast in O(n) messages

3
Today

• Single Initiator
• Building a spanning tree
• Echo algorithm
• broadcast with an echo (convergecast)
• applications counting, naming, termination
  detection, minimum finding
• DFS Traversal
• Changs, Awerbuchs, Cidons algorithms

4
Today II

• Multiple initiators
• Wake-Up
• flooding as wake-up
• wake-up in special networks impossibility of
  beating flooding in anonymous unoriented
  hypercubes and complete graphs
• non-anonymous networks leader election
• Spanning tree construction
• impossibility in anonymous arbitrary networks
• multi-flood, motivating leader election

5
Today III

• W(n2) lower bound for wake-up in unoriented
  anonymous complete networks
• consider a computation in which all processors
  wake up simultaneously, all message delays are
  exactly 1 and port li leads to a node i steps
  ahead
• a fixed algorithm uses ports l1, l2, , lk,
  until it terminates
• consider the same algorithm in the following
  scenario
• take k nodes and connect their ports l1, l2, ,
  lk to form a clique (their remaining ports lead
  to the remaining nodes of the network)
• wake up simultaneously only these k nodes and
  set all communication delays to exactly 1
• the claim is that the algorithm cannot
  distinguish between these two scenarios
• hence, if kltn-1 then the algorithm is not
  correct (does not wake-up everybody)

6
References

• Everything is covered in the Santoros book
• Original references for traversal
• the sequential algorithm T2m-2, C2m-2
• T.Cheung, Graph traversal maximum flow problem,
  IEEE Trans. Software SE-9,(4), 1983, pp. 504-512.
  
• inform neighbours and wait for acknowledgements
  T4n-2, C4m
• B.Awerbuch, A new distributed search algorithm,
  IPL (3), 1985, pp. 147-150.
• inform neighbours and proceed immediately
  (message complexity is wrong!)
• Isreal Cidon, Yet another distributed
  depth-first-search algorithm, IPL, v.26 n.6,
  p.301-305, January 25, 1988
• all of these assume constant size messages, if
  that is not true, you can get T2n-2, C2n-2
• http//portal.acm.org/citation.cfm?id75463jmpab
  stractdlGUIDEdlACM