Multiinitiator computation in trees

1
Multi-initiator computation in trees

• Wake-up Broadcasting
• flooding is optimal
• Traversal
• DF-traversal is optimal
• Election, minimum, center,
• saturation technique
• rootes the non-rooted tree

2
Saturation technique

• 1.Activation/wake-up
• using flooding, at most 2(n-1) messages
• 2. Saturation
• wait until messages from all but one neighbour
  are received
• process these messages and send the result to
  the remaining neighbour, which becomes your
  parent
• at the end, there will be two nodes who are
  father of each other
• these two roots initiate the resolution phase
• n-1 messages
• 3. Resolution
• problem dependent

3
Applications of Saturation

• leader election
• the two roots elect the leader among them, e.g.
  the one with the smaller ID
• the leader may broadcast its ID to all other
  processors
• minimum finding
• remember the minimum value seen so far and send
  it to your parent
• the roots broadcast this value (no unique IDs
  needed)
• generalization distributed function evaluation
• the function must be associative and
  commutative, defined on all subsets of inputs
• examples min, max, sum, product, logical and,
  or
• cardinal statistics average, standard deviation

4
Saturation Graph Algorithms

• eccentricities for all nodes
• eccentricity of a node is the maximal distance
  from this node to another node in the graph
• finding centers
• the node(s) with minimal eccentricities
• there can be more then one centers
• finding median
• the node(s) for which average distance to all
  other nodes is minimized
• identifying median paths

5
Computing eccentricities for all nodes

• eccentricity of a node is the maximal distance
  from this node to another node in the graph
• trivial approach flood echo from every node
• each node needs to compute the depths of the
  trees rooted at it
• from the saturation technique it can compute the
  depths of its sons
• the only missing piece is the depth of the tree
  leading to its father
• that can be computed in the resolution phase
• the roots know the depths of all their sons
• send to son i the maximal depth di among the
  remaining sons
• from this the son i can compute the depth of the
  subtree containing its father it is di1

6
Finding centers

• trivial approach
• compute eccentricities
• find and broadcast minimum
• unnecessarily many messages
• refined approach
• start by first two phases of saturation with
  computing the depths of the subtrees
• go to the deepest subtree until d1-d21
• if d1d2 then there is single center, otherwise
  there are two of them
• di i-th deepest subtree