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 agents two agents are necessary and sufficient to locate the black hole Moves: O (n log n) moves and it is optimal Time 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 ≥ 2 Three 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?