Efficient Detection of Global Properties Using Weak Conjunctive Predicate Algorithm

1. Chapter 10Global Properties

2. Unstable Predicate Detection • A predicate is stable if, once it becomes true it remains true • Snapshot algorithm is not useful for detection of global properties: • Not applicable for unstable predicates • Can not compute the least global state that satisfies a given predicate • Excessive overhead if frequency of snapshots is high

3. Predicates • Any predicate B constructed from local predicates using boolean connectives can be written in a disjunctive normal form i.e. where q1, q2, …,qn are conjunctive predicates • E.g. : x = y (where x and y are boolean) can be written as

4. Weak Conjunctive Predicate (WCP) • A Weak Conjunctive Predicate (WCP) is true iff there exists a consistent global cut in which all the conjuncts are true • Disjunctive predicates are easy to detect • Given an algorithm for detecting WCP we can detect any predicate B constructed from local predicates using boolean connectives

5. WCP Algorithm outline • Non-Checker process • Maintains a vector clock • Sends vector clock to checker process when predicate becomes true

6. WCP Algorithm outline • Checker process • Maintains a separate queue for each non-checker process • Maintain a cut[1..N] (array of states of the processes) • If state cut[i] ! cut[j], then cut[i] = queuei.getNext() • Repeat above statement till all states in cut[ ] are concurrent • cut[ ] is the least CGS for which the predicate holds

7. WCP Detection - Checker Process

8. Overhead Analysis • n: number of processes involved • m: max number of messages sent or received by any process • Space : • Each local snapshot : O(n) • At most O(mn) local snapshots • O(n2m) total space • Time: n2m comparisons • O(n2m)

9. Is the time complexity optimal ? • Lemma : • Let there be n elements in a set S . Any algorithm that determines whether all elements are incomparable must make at least n(n-1)/2 comparisons.

10. Is the time complexity optimal ? • Theorem • Let S be any partially ordered finite set of size mn. We are given a decomposition of S into n sets P0 … Pn-1 such that Pi is a chain of size m . Any algorithm that determines whether there exists an anti-chain of size n must make at least mn(n-1)/2 comparisons Adversary algorithm

11. Monitor Process Node Pi Application Process A Token based algorithm for WCP • Monitor process runs on each node along with the application • Monitor processes pass the token to each other • Token stores candidate cut and information to determine if it is consistent

12. A Token based algorithm for WCP • A token is sent to a process Pi when current state from Pi happened before some other state in the candidate cut • Once the monitor process for Pi has eliminated the current state • receive a new state from the application process • check for consistency conditions again. • This process is repeated until • all states are eliminated from some process Pi or • the WCP is detected.

13. A Token based algorithm for WCP • Token consists of two vectors G and color • G represents the candidate global cut • G[i] = k indicates that state (i,k) is part of the current cut • Invariant: G[i] = k implies that any global cut C with state (i,s) 2 C and s < k cannot satisfy the WCP • color indicates which states have been eliminated • If color[i]=red, then state (i,G[i]) has been eliminated and can never satisfy the global predicate

14. A Token based algorithm for WCP

15. Applications • Distributed debugging • Detect a bad condition • E.g. There is no leader ,i.e., P1 does not have a token and P2 does not have a token and . . . Pn does not have a token