Brief Announcement: Distributed Broadcasting and Mapping Protocols in Directed Anonymous Networks Michael Langberg: Open University of Israel Moshe Schwartz: Ben Gurion University of the Negev Jehoshua Bruck: California Institute of Technology Presenting: Michael Elkin, Ben Gurion University of the Negev

Anonymous Networks General framework: Processors: Objectives: Do not have unique identifiers. Execute identical protocols. No knowledge of topology of network. May distinguish between incoming edges. Objectives: Broadcast. Label assignment. Graph exploration. Previously studied: Undirected graphs (or directed + strongly connected). Protocols based on message passing.

This talk Directed anonymous graphs. Not necessarily strongly connected. Has not been considered previously. Standard (message passing) protocols studied do not apply: Graph may not be strongly connected. Processors may have single chance to send message. Outline of talk: Model. Results.

Model Motivation: s Two special vertices: s Consider dynamically growing network of unknown topology. Access to network through s and t. Our model enables to perform maintenance tasks. Maintenance is triggered by s and status report obtained via t. Otherwise cannot distinguish between: s Two special vertices: Root s. Terminal t. Protocol proceeds as follows: Initial message sent from s to children. Triggers a distributed protocol. Protocol terminates when terminal vertex t is in final state – with its state as the output of protocol. Connectivity: Protocol will terminate iff all nodes lay on a path between s and t. Nodes only reachable from s or nodes that are only connected to t will not allow termination. t s Not connected t

Results Broadcast: send a message m to entire network. Trivial without termination requirement at t. Theorem: Broadcast + termination achievable with total communication O(|E|2|V|log(dout)) + |E||m|. Proof idea for termination: Messages sent will represent a certain commodity. Send unit of commodity out of s (split between children). Internal vertex: upon receiving commodity – process commodity and split among children. t terminates when over time it receives a unit of commodity. Proof is non trivial even for acyclic graphs: must split commodity carefully – otherwise exponential communication complexity. Improved complexity bounds which are “tight” for DAGs. Proof more involved for graphs with cycles.

Results II Unique label assignment: Theorem: Unique label assignment + termination achievable with total communication complexity of O(|E|2|V|log(dout)). Labels are of length O(|V|log(dout)). Proof idea: Follows commodity based protocol for broadcast. Part of the commodity is reserved for unique id. Large label length is essential (matching lower bound).

Summary Initiate study of anonymous directed networks which are not strongly connected. Present protocols for broadcast and label assignment + termination. Open questions: Close gap in communication complexity (not tight for digraphs with and without cycles). Our algorithms based on commodity preserving paradigm. Does paradigm yield optimal complexity on DAGs?