Mobile Search for a Black Hole in an Anonymous Ring

Dobrev, S., Flocchini, P., Prencipe, G., & Santoro, N. (2007). 

Mobile Search for a Black Hole in an Anonymous Ring.Mengfei Peng

Network:Ring: a loop network of identical nodes, Whiteboard: Each node has a bounded

amount of storage(whiteboard), agents can write or read information from the whiteboard, O(log n) bits are sufficient.

n is known (where n is the size of the ring) Nodes are anonymous: no special marks

on any node.

Agents:computing capability; bound of storage; obey the same protocol; Asynchronous; Identical;

Result: co-located agentstwo agents are necessary and sufficient to

locate the black holeMoves: O (n log n) moves and it is optimalTime complexity: 2n-4 units of time using

n- 1 agents

If the ring is oriented, two dispersed agents are still necessary and sufficient. Moves: (θ (n log n)).

If the ring is un-oriented, three agents are necessary and sufficient; Moves: (θ (n log n)).

Result: dispersed agents

Algorithm:measure of complexity:Size: the number of agents;Cost: the number of moves;Time: the amount of time elapsed until

termination----ideal time (i.e., assuming synchronous

execution where a move can be made in one time unit)----\time" complexity is “ideal time" complexity.

Cautious Walk

Co-located:2 agents

time complexity of Algorithm Divide is also 2n log n + O(n).

n-1 agents to locate BH

Algorithm Optimal Time lets n -1 co-located agents find the black hole in 2n -4time.

Why 2n-4: if n-1 is BH, a agent must come to n-2, and come back to 0, so 2(n-2)

Dispersed agents:initially there is at most one agent at any given location

If k is known, cost in oriented rings: Ω(n log(n-k)).

If k of agents is unknown, cost in oriented rings: Ω (n log n).

Dispersed, oriental ring, k ≥ 2Three phases: pairing, elimination, and resolution.

Algorithm:

K is known When arriving at a node already visited by another agent, it proceeds to the right via the safe port. If there is no safe port, it tests how many agents are at this node; if the number of agents at the node is k- 1, the algorithm terminates.

K is unknown

A:status:alone

D:status:paired-left

C sees D’s “jion me” mark and terminates. status:paired-right

Questions:1, How (n-1) co-located agents explored the ring?

Questions:2, How k dispersed agents explored the ring while k is known?

Questions:3, How k dispersed agents explored the ring while k is unknown?