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
Result: dispersed 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)).
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).
Algorithm:
 Dispersed, oriental ring, k ≥ 2
 Three phases: pairing, elimination, and resolution.




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?