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Set 15: Broadcast CSCE 668 DISTRIBUTED ALGORITHMS AND SYSTEMS CSCE 668 Fall 2011 Prof. Jennifer Welch 1 Broadcast Specifications 2 Recall the specification of a broadcast service given in the last set of slides: Inputs: bc-sendi(m) an input to the broadcast service pi wants to use the broadcast service to send m to all the procs Outputs: bc-recvi(m,j) an output of the broadcast service broadcast service is delivering msg m, sent by pj, to pi Set 15: Broadcast CSCE 668 Broadcast Specifications 3 A sequence of inputs and outputs (bc-sends and bcrecvs) is allowable iff there exists a mapping from each bc-recvi(m,j) event to an earlier bc-sendj(m) event s.t. is well-defined: every msg bc-recv'ed was previously bcsent (Integrity) restricted to bc-recvi events, for each i, is one-to-one: no msg is bc-recv'ed more than once at any single proc. (No Duplicates) restricted to bc-recvi events, for each i, is onto: every msg bc-sent is received at every proc. (Liveness) Set 15: Broadcast CSCE 668 Ordering Properties 4 Sometimes we might want a broadcast service that also provides some kind of guarantee on the order in which messages are delivered. We can add additional constraints on the mapping : single-source FIFO or totally ordered or causally ordered Set 15: Broadcast CSCE 668 Single-Source FIFO Ordering 5 For all messages m1 and m2 and all pi and pj, if pi sends m1 before it sends m2, and if pj receives m1 and m2, then pj receives m1 before it receives m2. Phrased carefully to avoid requiring that both messages are received. that is the responsibility of a liveness property Set 15: Broadcast CSCE 668 Totally Ordered 6 For all messages m1 and m2 and all pi and pj, if both pi and pj receive both messages, then they receive them in the same order. Phrased carefully to avoid requiring that both messages are received by both procs. that is the responsibility of a liveness property Set 15: Broadcast CSCE 668 Happens Before for Broadcast Messages 7 Earlier we defined "happens before" relation for events. Now extend this definition to broadcast messages. Assume all communication is through broadcast sends and receives. Msg m1 happens before msg m2 if some bc-recv event for m1 happens before (in the old sense) the bc-send event for m2, or m1 and m2 are bc-sent by the same proc. and m1 is bc-sent before m2 is bc-sent. Set 15: Broadcast CSCE 668 Example of Happens Before for Broadcast Messages 8 m1 m3 m2 m4 m1 happens before m3 and m4 m2 happens before m4 m3 happens before m4 Set 15: Broadcast CSCE 668 Causally Ordered 9 For all messages m1 and m2 and all pi, if m1 happens before m2, and if pi receives both m1 and m2, then pi receives m1 before it receives m2. Phrased carefully to avoid requiring that both messages are received. that is the responsibility of a liveness property Set 15: Broadcast CSCE 668 Example 10 a b single-source FIFO? totally ordered? causally ordered? Set 15: Broadcast CSCE 668 Example 11 a b single-source FIFO? totally ordered? causally ordered? Set 15: Broadcast CSCE 668 Example 12 a b single-source FIFO? totally ordered? causally ordered? Set 15: Broadcast CSCE 668 Algorithm BB to Simulate Basic Broadcast on Top of Point-to-Point 13 When bc-sendi(m) occurs: pi sends a separate copy of m to every processor (including itself) using the underlying point-to-point message passing communication system When can pi perform bc-recvi(m)? when it receives m from the underlying point-topoint message passing communication system Set 15: Broadcast CSCE 668 Basic Broadcast Simulation 14 bc-sendi bc-recvi … Alg BB BB0 sendi bc-sendj recvi bc-recvj BBn-1 basic broadcastsendj recvj asynch pt-to-pt message passing Set 15: Broadcast CSCE 668 Correctness of Basic Broadcast Algorithm 15 Assume the underlying point-to-point message passing system is correct (i.e., conforms to the spec given in previous set of slides). Check that the simulated broadcast service satisfies: Integrity No Duplicates Liveness Set 15: Broadcast CSCE 668 Single-Source FIFO Algorithm 16 Assume the underlying communication system is basic broadcast. when ssf-bc-sendi(m) occurs: pi uses the underlying basic broadcast service to bcast m together with a sequence number pi increments sequence number by 1 each time it initiates a bcast when can pi perform ssf-bc-recvi(m)? when pi has bc-recv'ed m with sequence number T and has ssf-bc-recv'ed messages from pj (the ssf-bc-sender of m) with all smaller sequence numbers Set 15: Broadcast CSCE 668 Single-Source FIFO Algorithm 17 user of SSF bcast ssf-bc-send ssf-bc-recv ssf bcast SSF alg (timestamps) bc-send bc-recv basic bcast alg (n copies) send basic bcast recv point-to-point message passing Set 15: Broadcast CSCE 668 Asymmetric Algorithm for Totally Ordered Broadcast 18 Assume underlying communication service is basic broadcast. There is a distinguished proc. pc when to-bcasti(m) occurs: pi sends m to pc (either assume the basic broadcast service also has a point-to-point mechanism, or have recipients other than pc ignore the msg) when pc receives m from pi from the basic broadcast service: append a sequence number to m and bc-send it Set 15: Broadcast CSCE 668 Asymmetric Algorithm for Totally Ordered Broadcast 19 when can pi perform to-bc-recv(m)? when pi has bc-recv'ed m with sequence number T and has to-bc-recv'ed messages with all smaller sequence numbers Set 15: Broadcast CSCE 668 Asymmetric Algorithm Discussion 20 Simple Only requires basic broadcast But pc is a bottleneck Alternative approach next… Set 15: Broadcast CSCE 668 Symmetric Algorithm for Totally Ordered Broadcast 21 Assume the underlying communication service is single-source FIFO broadcast. Each proc. tags each msg it sends with a timestamp (increasing). Break ties using proc. ids. Each proc. keeps a vector of estimates of the other proc's timestamps: If pi 's estimate for pj is k, then pi will not receive any later msg from pj with timestamp k. Estimates are updated based on msgs received and "timestamp update" msgs Set 15: Broadcast CSCE 668 Symmetric Algorithm for Totally Ordered Broadcast 22 Each proc. keeps its timestamp to be ≥ all its estimates: when pi has to increase its timestamp because of the receipt of a message, it sends a timestamp update msg A proc. can deliver a msg with timestamp T once every entry in the proc's vector of estimates is at least T. Set 15: Broadcast CSCE 668 Symmetric Algorithm 23 when to-bc-sendi(m) occurs: ts[i]++ add (m,ts[i],i) to pending invoke ssf-bc-sendi((m,ts[i])) when ssf-bc-recvi((m,T)) from pj occurs: ts[j] := T add (m,T,j) to pending if T > ts[i] then ts[i] := T invoke ssf-bc-sendi("ts-up",T) invoke to-bc-recvi(m,j) when: (m,T,j) is entry in pending with smallest (T,j) T ≤ ts[k] for all k result: remove (m,T,j) from pending when ssf-bc-recvi("ts-up",T) from pj occurs: ts[j] := T Set 15: Broadcast CSCE 668 user of TO bcast 24 to-bc-send to-bc-recv TO bcast symmetric TO alg ssf-bc-send ssf-bc-recv SSF alg (timestamps) bc-send bc-recv basic bcast alg (n copies) send ssf bcast basic bcast recv point-to-point message passing Set 15: Broadcast CSCE 668 Correctness of Symmetric Algorithm 25 Lemma (8.2): Timestamps assigned to msgs form a total order (break ties with id of sender). Theorem (8.3): Symmetric algorithm simulates totally ordered broadcast service. Proof: Must show top-level outputs of symmetric algorithm satisfy 4 properties, in every admissible execution (relies on underlying ssf-bcast service being correct). Set 15: Broadcast CSCE 668 Correctness of Symmetric Alg. 26 Integrity: follows from same property for ssf-bcast. No Duplicates: follows from same property for ssf-bcast. Liveness: Suppose in contradiction some pi has some entry (m,T,j) stuck in its pending set forever, where (T,j) is the smallest timestamp of all stuck entries. Eventually (m,T,j) has the smallest timestamp of all entries in pi's pending set. Why is (m,T,j) stuck at pi? Because pi's estimate of some pk's timestamp is stuck at some value T' < T. But that would mean either pk never receives (m,T,j) or pk's timestamp-update msg resulting from pk receiving (m,T,j) is never received at pi, contradicting correctness of the SSF broadcast. Set 15: Broadcast CSCE 668 Correctness of Symmetric Alg. 27 Total Ordering: Suppose pi invokes to-bc-recv for msg m with timestamp (T,j), and later it invokes to-bc-recv for msg m' with timestamp (T',j'). Show (T,j) < (T',j'). By the code, if (m',T',j') is in pi's pending set when pi invokes the to-bc-recv for m, then (T,j) < (T',j'). Suppose (m',T',j') is not yet in pi's pending set at that time. When pi invokes the to-bc-recv for m, precondition ensures that T ≤ ts[j']. So pi has received a msg from pj' with timestamp ≥ T. By the SSF property, every subsequent msg pi receives from pj' will have timestamp > T, so T' must be > T. Set 15: Broadcast CSCE 668 Causal Ordering Algorithms 28 The symmetric total ordering algorithm ensures causal ordering: timestamp order extends the happens-before order on messages. Causal ordering can also be attained without the overhead of total ordering, by using an algorithm based on vector clocks… Set 15: Broadcast CSCE 668 Causal Order Algorithm 29 Code for pi : when co-bc-sendi(m) occurs: vt[i]++ invoke co-bc-recvi(m) invoke bc-sendi((m,vt)) when bc-recvi((m,w)) from pj occurs: add (m,w,j) to pending invoke co-bc-recvi(m,j) when: (m,w,j) is in pending w[j] = vt[j] + 1 w[k] ≤ vt[k] for all k ≠ j result: remove (m,w,j) from pending vt[j]++ Note: vt[j] records how many msgs from pj have been co-bc-recv'ed by pi Set 15: Broadcast CSCE 668 Causal Order Algorithm Discussion 30 Vector clocks are implemented slightly differently than in the point-to-point case. In point-to-point case, we exploited indirect (transitive) information about messages received by other procs. In the broadcast case, we don't need to do that, since very proc will eventually receive every message directly. Set 15: Broadcast CSCE 668 Causal Order Algorithm Example 31 Algorithm delays the delivery of the C.O. msgs until causal order property won't be violated. (1,3,0) (0,1,0) (0,2,0) (0,3,0) Set 15: Broadcast CSCE 668 Correctness of Causal Order Algorithm (Sketch) 32 Lemma (8.6): The local array variables vt serve as vector clocks. Theorem (8.7): The algorithm simulates causally ordered broadcast, if the underlying communication system satisfies (basic) broadcast. Proof: Integrity and No Duplicates follow from the same properties of the basic broadcast. Liveness requires some arguing. Causal Ordering follows from the lemma. Set 15: Broadcast CSCE 668 Reliable Broadcast 33 What do we require of a broadcast service when some of the procs can be faulty? Specifications differ from those of the corresponding non-fault-tolerant specs in two ways: 1. 2. proc indices are partitioned into "faulty" and "nonfaulty" Liveness property is modified… Set 15: Broadcast CSCE 668 Reliable Broadcast Specification 34 Nonfaulty Liveness: Every msg bc-sent by a nonfaulty proc is eventually bc-recv'ed by all nonfaulty procs. Faulty Liveness: Every msg bc-sent by a faulty proc is bc-recv'ed by either all the nonfaulty procs or none of them. Set 15: Broadcast CSCE 668 Discussion of Reliable Bcast Spec 35 Specification is independent of any particular fault model. We will only consider implementations for crash faults. No guarantee is given concerning which messages are received by faulty procs. Can extend this spec to the various ordering variants: msgs that are received by nonfaulty procs must conform to the relevant ordering property. Set 15: Broadcast CSCE 668 Spec of Failure-Prone Point-to-Point Message Passing System 36 Before we can design an algorithm to implement reliable (i.e., fault-tolerant) broadcast, we need to know what we can rely on from the lower layer communication system. Modify the previous point-to-point spec from the no-fault case in two ways: 1. 2. partition proc indices into "faulty" and "nonfaulty" Liveness property is modified… Set 15: Broadcast CSCE 668 Spec of Failure-Prone Point-to-Point Message Passing System 37 Nonfaulty Liveness: every msg sent by a nonfaulty proc to any nonfaulty proc is eventually received. Note that this places no constraints on the eventual delivery of messages to faulty procs. Set 15: Broadcast CSCE 668 Reliable Broadcast Algorithm 38 when rel-bc-sendi(m) occurs: invoke sendi(m) to all procs when recvi(m) from pj occurs: if m has not already been recv'ed then invoke sendi(m) to all procs invoke rel-bc-recvi(m) Set 15: Broadcast CSCE 668 Correctness of Reliable Bcast Alg 39 Integrity: follows from Integrity property of underlying point-to-point msg system. No Duplicates: follows from No Duplicates property of underlying point-to-point msg system and the check that this msg was not already received. Nonfaulty Liveness: follows from Nonfaulty Liveness property of underlying point-to-point msg system. Faulty Liveness: follows from relaying and underlying Nonfaulty Liveness. Set 15: Broadcast CSCE 668