Distributed Systems Logical Clocks Paul Krzyzanowski pxk@cs.rutgers.edu Except as otherwise noted, the content of this presentation is licensed under the Creative Commons Attribution 2.5 License. Page 1 Page 1
Logical clocks Assign sequence numbers to messages – All cooperating processes can agree on order of events – vs. physical clocks : time of day Assume no central time source – Each system maintains its own local clock – No total ordering of events • No concept of happened-when Page 2
Happened-before Lamport’s “happened - before” notation a b event a happened before event b e.g.: a : message being sent, b : message receipt Transitive: if a b and b c then a c Page 3
Logical clocks & concurrency Assign “clock” value to each event – if a b then clock( a ) < clock( b ) – since time cannot run backwards If a and b occur on different processes that do not exchange messages, then neither a b nor b a are true – These events are concurrent Page 4
Event counting example • Three systems: P 0 , P 1 , P 2 • Events a , b , c , … • Local event counter on each system • Systems occasionally communicate Page 5
Event counting example e a b c d f P 1 3 4 5 1 2 6 g h i P 2 2 1 3 j k P 3 1 2 Page 6
Event counting example e a b c d f P 1 3 4 5 1 2 6 g h i P 2 2 1 3 j k P 3 1 2 Bad ordering: e h f k Page 7
Lamport’s algorithm • Each message carries a timestamp of the sender’s clock • When a message arrives: – if receiver’s clock < message timestamp set system clock to (message timestamp + 1) – else do nothing • Clock must be advanced between any two events in the same process Page 8
Lamport’s algorithm Algorithm allows us to maintain time ordering among related events – Partial ordering Page 9
Event counting example e a b c d f P 1 3 4 5 1 2 6 g h i P 2 2 1 7 6 j k P 3 1 2 7 Page 10
Summary • Algorithm needs monotonically increasing software counter • Incremented at least when events that need to be timestamped occur • Each event has a Lamport timestamp attached to it • For any two events, where a b: L(a) < L(b) Page 11
Problem: Identical timestamps e a b c d f P 1 3 4 5 1 2 6 g h i P 2 6 1 7 j k P 3 1 7 a b, b c, …: local events sequenced i c, f d , d g, … : Lamport imposes a send receive relationship Concurrent events (e.g., a & i) may have the same timestamp … or not Page 12
Unique timestamps (total ordering) We can force each timestamp to be unique – Define global logical timestamp (T i , i) • T i represents local Lamport timestamp • i represents process number (globally unique) – E.g. (host address, process ID) – Compare timestamps: (T i , i) < (T j , j) if and only if T i < T j or T i = T j and i < j Does not relate to event ordering Page 13
Unique (totally ordered) timestamps e a b c d f P 1 3.1 4.1 5.1 1.1 2.1 6.1 g h i P 2 6.2 1.2 7.2 j k P 3 1.3 7.3 Page 14
Problem: Detecting causal relations If L(e) < L(e’) – Cannot conclude that e e’ Looking at Lamport timestamps – Cannot conclude which events are causally related Solution: use a vector clock Page 15
Vector clocks Rules: 1. Vector initialized to 0 at each process V i [ j ] = 0 for i, j =1, …, N 2. Process increments its element of the vector in local vector before timestamping event: V i [i] = V i [i] +1 3. Message is sent from process P i with V i attached to it 4. When P j receives message, compares vectors element by element and sets local vector to higher of two values V j [i] = max(V i [i], V j [i]) for i =1, …, N Page 16
Comparing vector timestamps Define V = V’ iff V [i ] = V’[ i ] for i = 1 … N V V’ iff V [i ] V’[ i ] for i = 1 … N For any two events e, e’ if e e’ then V(e) < V(e’) • Just like Lamport’s algorithm if V(e) < V(e’) then e e’ Two events are concurrent if neither V(e) V(e’) nor V(e’) V(e) Page 17
Vector timestamps (0,0,0) a b P 1 (0,0,0) c d P 2 (0,0,0) e f P 3 Page 18
Vector timestamps (1,0,0) (0,0,0) a b P 1 (0,0,0) c d P 2 (0,0,0) e f P 3 Event timestamp a (1,0,0) Page 19
Vector timestamps (1,0,0) (2,0,0) (0,0,0) a b P 1 (0,0,0) c d P 2 (0,0,0) e f P 3 Event timestamp a (1,0,0) b (2,0,0) Page 20
Vector timestamps (1,0,0) (2,0,0) (0,0,0) a b P 1 (2,1,0) (0,0,0) c d P 2 (0,0,0) e f P 3 Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) Page 21
Vector timestamps (1,0,0) (2,0,0) (0,0,0) a b P 1 (2,1,0) (2,2,0) (0,0,0) c d P 2 (0,0,0) e f P 3 Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0) Page 22
Vector timestamps (1,0,0) (2,0,0) (0,0,0) a b P 1 (2,1,0) (2,2,0) (0,0,0) c d P 2 (0,0,1) (0,0,0) e f P 3 Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0) e (0,0,1) Page 23
Vector timestamps (1,0,0) (2,0,0) (0,0,0) a b P 1 (2,1,0) (2,2,0) (0,0,0) c d P 2 (0,0,1) (2,2,2) (0,0,0) e f P 3 Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) d (2,2,0) e (0,0,1) f (2,2,2) Page 24
Vector timestamps (1,0,0) (2,0,0) (0,0,0) a b P 1 (2,1,0) (2,2,0) (0,0,0) c d P 2 (0,0,1) (2,2,2) (0,0,0) e f P 3 Event timestamp a (1,0,0) b (2,0,0) concurrent c (2,1,0) events d (2,2,0) e (0,0,1) f (2,2,2) Page 25
Vector timestamps (1,0,0) (2,0,0) (0,0,0) a b P 1 (2,1,0) (2,2,0) (0,0,0) c d P 2 (0,0,1) (2,2,2) (0,0,0) e f P 3 Event timestamp a (1,0,0) b (2,0,0) concurrent c (2,1,0) events d (2,2,0) e (0,0,1) f (2,2,2) Page 26
Vector timestamps (1,0,0) (2,0,0) (0,0,0) a b P 1 (2,1,0) (2,2,0) (0,0,0) c d P 2 (0,0,1) (2,2,2) (0,0,0) e f P 3 Event timestamp a (1,0,0) b (2,0,0) concurrent c (2,1,0) events d (2,2,0) e (0,0,1) f (2,2,2) Page 27
Vector timestamps (1,0,0) (2,0,0) (0,0,0) a b P 1 (2,1,0) (2,2,0) (0,0,0) c d P 2 (0,0,1) (2,2,2) (0,0,0) e f P 3 Event timestamp a (1,0,0) b (2,0,0) c (2,1,0) concurrent d (2,2,0) events e (0,0,1) f (2,2,2) Page 28
Summary: Logical Clocks & Partial Ordering • Causality – If a -> b then event a can affect event b • Concurrency – If neither a -> b nor b -> a then one event cannot affect the other • Partial Ordering – Causal events are sequenced • Total Ordering – All events are sequenced Page 29
The end. Page 30 Page 30
Recommend
More recommend
