DISTRIBUTED SYSTEMS II A POLYNOMIAL LOCAL SOLUTION TO MUTUAL EXCLUSION Prof Philippas Tsigas Distributed Computing and Systems Research Group

Doorways  Doorway is a separation mechanism of two areas.  Processes which pass a doorway at time T prevent neighbor processes entering the same area at a greater time than T until exiting the doorway.

Asynchronous Doorway Enry CodeExit Code m1 is the “I am going in” message

Behavior

Problem with the Asynchronous Doorway

Synchronous Doorway  A process which desires to enter a synchronous doorway is required to wait for a situation in which all its neighbors are outside the doorway. This is implemented by checking for a process the states of its neighbors before it enters the doorway.

Synchronous Doorway Entry code:Exit code:

What Can Go Wrong?

What if We Combine?

LINK REVERSAL ALGORITHMS Based on slides from Jennifer Welch

12 What is Link Reversal?  Distributed algorithm design technique  Used in solutions for a variety of problems –routing, leader election, mutual exclusion, scheduling, resource allocation,…  Model problem as a directed graph and reverse the direction of links appropriately  Use local knowledge to decide which links to reverse

Routing [Gafni & Bertsekas 1981]  Undirected connected graph represents communication topology of a system  Unique destination node  Assign virtual directions to the graph edges (links) s.t. –if nodes forward messages over the links, they reach the destination  Directed version of the graph (orientation) must –be acyclic –have destination as only sink Thus every node has path to destination. 13

Routing Example 14 D

Mending Routes  What happens if some edges go away? –Might need to change the virtual directions on some remaining edges (reverse some links)  More generally, starting with an arbitrary directed graph, each node should decide independently which of its incident links to reverse 15

Mending Routes Example 16 D

Sinks  A vertex with no outgoing links is a sink.  The property of being a sink can be detected locally.  A sink can then reverse some incident links  Basis of several algorithms… 17

Full Reversal Routing Algorithm 18  Input: directed graph G with destination vertex D  Let S(G) be set of sinks in G other than D  while S(G) is nonempty do –reverse every link incident on a vertex in S(G) –G now refers to resulting directed graph

19 Full Reversal (FR) Routing Example D D D D D D

Why Does FR Terminate?  Suppose it does not.  Let W be vertices that take infinitely many steps.  Let X be vertices that take finitely many steps; includes D.  Consider neighboring nodes w in W, x in X.  Consider first step by w after last step by x: link is w  x and stays that way forever.  Then w cannot take any more steps, contradiction. 20

Why is FR Correct?  Assume input graph is acyclic.  Acyclicity is preserved at each iteration: –Any new cycle introduced must include a vertex that just took a step, but such a vertex is now a source (has no incoming links)  When FR terminates, no vertex, except possibly D, is a sink.  A DAG must have at least one sink: –if no sink, then a cycle can be constructed  Thus output graph is acyclic and D is the unique sink. 21

References Resource Allocation and Link Reversal  Dijkstra, E. W. (1971, June). Hierarchical ordering of sequential processes. Acta Informatica 1(2):  Chandy, K.M.; Misra, J. (1984). The Drinking Philosophers Problem. ACM Transactions on Programming Languages and Systems.  Nancy A. Lynch. Upper bounds for static resource allocation in a distributed system Journal for Computer and System Sciences, 23: , 1981  Manhoi Choy and Ambuj K. Singh. Efficient fault-tolerant algorithms for distributed resource allocation. ACM Transactions on Programming Languages and Systems, 17(3): , May  Gafni and Bertsekas, “Distributed Algorithms for Generating Loop-Free Routes in Networks with Frequently Changing Topology,” IEEE Trans. Comm  Welch and Walter, “Link Reversal Algorithms,” Synthesis Lectures on Distributed Computing Theory #8, Morgan & Claypool Publishers,