An Efficient and Fault-Tolerant Solution for Distributed Mutual Exclusion by D. Agrawal, A.E. Abbadi Presentation by Peter Tsui for COEN 317, F/03

Goals Fault-Tolerant - sites, links, and network partitioning Low Communication Cost Distributed Solution

Background – Other Solutions Primary Site Distributed (Lamport) Majority Maekawa Token-Based (Raymond)

2 Ideas Coterie, C, is a set of sets or quorums. –Intersection Property – If g and h are quorums in C, then g and h has non-empty intersection. –Minimality Property – There are no two quorums g and h in C such that g is a superset of h. Use Logical Structure – Organize sites into a logical (complete) binary tree (easy extension to n-ary tree)

Tree Quorum TQ(T) Construction If AVAIL(T->Node) return – T->Node U TQ(T->Left-C) or – T->Node U TQ(T->Right-C) Else return –TQ(T->Left-C) U TQ(T->Right-C) –If either of above is empty, error

Example No fail : {1,2,4}, {1,2,5}, {1,3,6}, {1,3,7} {1} fail : {2,4,3,6}, {2,4,3,7}, {2,5,3,6}, {2,5,3,7} {2 or 3} fail : {1,4,5}, {1,6,7} {1, 2} fail : {4,5,3,6}, {4,5,3,7} {1,2,3} fail : {4,5,6,7} level complete binary tree

Correctness The set of Tree Quorums satisfy intersection and minimality properties of coteries. Proof by inducation of level of tree. Induction Step: Consider binary tree of level k+1. Any tree quorum is from one of following 3 classes:- {s1} U {members from quorum set of left subtree} {s1} U {members from quorum set of right subtree} {members from quorum set of left subtree} U {members from quorum set of right subtree}

Discussion Best case tree quorum size is (ceiling function on) log n (base 2). Worst case tree quorum size is (ceiling function on ) (n+1)/2. If {1,2,4} fail, no tree quorum possible. When {3,5,6,7} fail, a majority, tree quorum still possible. When less than log n nodes fail, tree quorum always possible.

Message Cost f = fraction of tree quorum that include ROOT of tree Recurrence relation of cost of tree quorum at level (l+1) in terms of l.

Availability Availability of Tree Quorum indistinguishable with Majority Quorum when p, probability of site availability, > Availability is only inferior when p is between [0.5, 0.75].

Some Applications Mutual Exclusion – send request to a tree quorum, get all replies to enter CS, send relinquish when done. Use Inquire/Yield to break deadlocks. Replication – use tree quorums for both read and write quorums. Get data with highest version number.

Conclusions Tree Quorum provide log n messaging cost in best case and (n+1)/2 in worst case. Tree Quorum provides fault tolerance up to about log n down sites in worst case. Tree Quorum availability is comparable with majority alogirthm when p > 0.75.