Lecture 8: Asynchronous Network Algorithms

A (1) send m1 (2) send m2 (3) receive m2 B C (4) send m2 Lecture 8: Asynchronous Network Algorithms • Completely asynchronous distributed computational model • No assumption for speed of processes. • No assumption for transmission delay of communication links. • No assumption for speed of local clocks. Nondeterministic caused by asynchronous Which one is received first, m1 or m2?

process i communication channel process i process j process k communication channel Reliable FIFO channel: any message received must have been sent at some earlier time Model of Send/Receive System • Let an n-node directed graph G=(V,E) represents an asynchronous network, where nodes are processes and directed edges are communication channels.

Problems 1 Leader Election in An asynchronous Ring • Assumption of the network • Network graph G=(V,E) is a ring, where node set V={1,2,…,n} and edge set E={(i,i+1), where i=1,2,…,n and i iscounted mod n}. • Each process has a unique distinct identifier (UID). • Each node uses only unidirectional communication and knows only its UID (it does not know the size of the ring and its neighbors). • Only the leader performs an output. • AsychLCR algorithm (informal) • At each process there is a FIFO queue of UIDs, initially containing only its own UID. • Each process removes and sends the first element of its FIFO queue. • (2) When a process receives an incoming identifier, it compares that identifier to its own. If the incoming identifier is greater than its own, it add it into its FIFO queue; if it is less than its own, it discards the incoming identifier; if it is equal to its own, the process declares itself the leader.

Theorem AsynchLCR solves the leader-election problem. • Analysis of Complexity of AsychLCR Algorithm • Communication complexity: • Time complexity:

Problem2 Leader Election in a General asynchronous Network • Assumption of the network • Undirected connected graph G=(V,E) having n node, where there is bidirectional communication on all the edges. • Processes do not know their indices, nor those of their neighbors, but refer to their neighbors by local names. • If a process i has the same process j for both incoming and outgoing neighbor, then i knows that the two processes are the same.

FloodMax algorithm (informal) • Suppose that each process has a unique distinct UID and it knows diam, the diameter of network. • Each process maintains a record of the maximum UID it has seen so far (initially its own). At each round, each process propagates it maximum on all of its outgoing edges. • After diam rounds, if the maximum value seen is the process’s own UID, the process elects itself the leader; otherwise, it is a non-leader. AsychFloodMax algorithm (informal) • Use FIFO queue for communicational channel. • Simulation the rounds of the following synchronous algorithm: • Send a round r message to tag that message with its round number r. The recipient waits to receive round r messages from all its neighbors before performing its round r transition. By simulating diam (diameter of the network) round, the algorithm can terminate correctly.

Problem 3 Spanning Tree Construction • Assumption of the network • Undirected and connected network digraph G=(V,E) having n nodes and a distinguished source note s. • Output is the structure of a spanning tree of the network graph with root s in a distributed fashion: each process other than s should have a parent component that gets set to indicate the node that is its parent in the tree. • Processes know its neighbors’ indices. They have no knowledge of the size or diameter of the network. No UIDs are needed. • AsynchBFS algorithms • Use FIFO queue for communicational channel • At any point during execution, there is some set of processes that is “marked”, initially just s. • Process s sends out a search message, to all of its outgoing neighbors. • If an unmarked process receives a search message, it marks itself and chooses one of the processes from which the search has arrived as its parent, then it sends a search message to all of its outgoing neighbors.

Application to Message Broadcast Problem AsynchBFS algorithm can be used to Message Broadcast problem: piggyback the message m on all search messages during the formation of the spanning tree.

send event time p1 p2 p3 p4 receive event space internal event Problem 4: Clock of Asynchronous Distributed Systems and Snapshot algorithms Space and Time Diagram

Causal Relation send event time p1 p2 p3 p4 receive event space internal event

Condition of Clock • Role of Clock • Determining a causal relation for events. • Local Clock • No use for determining the order of the events of different processes. • Two type of clocks: logical clock and vector clock.

Lamport’s Logical Clock

Logical clock satisfies the condition of clock send event time p1 • Clock condition does not satisfy causal relation p2 0 p3 0 p4 0 receive event space 0 internal event Property of Lamport’s logical clock 1 2 3 4 1 1 2 3 5 3 1 4 1 4 2 3 4

P1 P2 P3 $10 $20 $30 P1,2 2 $1 1 3 $2 P3,1 4 P2,4 $3 7 8 t=7.5 5 P3,6 $5 $4 9 6 P1,10 10 11 P3,8 $1 12 8 11 P1,11 $1 9 P2,12 $2 11 Application of Lamport’s Logical Clock:Bank System • CountMoney algorithm • Use a predetermined logical time t, assumed to be known to all processes. • For each process, determine the value of money after all events with logical times less than or equal to t and before all events with logical times greater than t. • For each channel, determine the amount of money in all the messages sent at logical times less than or equal to t but received at logical times strictly greater than t. $10-$1+$5=$14 $20+$1-$3+2=$20 $30-$2+$3-$5=$26

Problem 5: Global Configuration of Distribute Systems and Snapshot Algorithms • Global Configuration of Distributed Systems: • states of processes (local memory), states of communication links (message), which are used for • Dead lock detection • Termination detection • Backup at some check point (for recovery from failures) Snapshot Algorithms: Distributed algorithms for finding global configuration

Example: Snapshot at time t=7.5 P1 P2 P3 $10 $20 $30 P1,2 2 $1 1 3 $2 P3,1 4 P2,4 $3 7 8 t=7.5 5 P3,6 $5 $4 9 6 P1,10 10 11 P3,8 $1 12 8 11 P1,11 $1 9 P2,12 $2 11 • How to take a snapshot • Freeze all the processes and collect the state of each process. • ---Inefficient! • Using global clock: broadcast the global time t and collect the state of each process at time t. • --- Global clock may not exist! • Using logical clock.

p p p m1,m2 p mark m1 m1,m2,m3 m1 Start process m3 m2 mark q q q q mark mark mark Chandy and Lamport’s Snapshot Algorithm The algorithm collects the state of each process and the state of each link (the messages in communication) • One process finds its own state, then send message to each of its neighbors. • For each process p • (a) if p receives first time, it finds its own state, and then send to each of its neighbors. • (b) p continues to receive message from all its neighbors as follows: Assuming that t(0) is the time p received the first , p collect all the message from any neighbor q until p get message form q.

Exercise (1) Reconsider the banking system. Now suppose that the underlying banking system A allows deposits and withdrawals (modeled as input actions at the user interface of the system ) in addition to transfers. If we apply the same Count Money transformation as before, what can be claimed about the output of the resulting systems?

Lecture 8: Asynchronous Network Algorithms