Lecture 9: 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 A C B (1) send m1 (2) send m2 (3) receive m2 (4) send m2 Which one is received first, m1 or m2?

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. process i process i process j communication channel process k communication channel Reliable FIFO channel: any message received must have been sent at some earlier time

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 is counted 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.

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. 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. FloodMax algorithm (informal)

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.

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

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

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. Condition of Clock

Lamport’s Logical Clock

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

Application of Lamport’s Logical Clock: Bank System $10 2 3 $20 $30 $1 P1,2 4 8 1 5 P2,4 $3 P3,1 $2 9 $4 P1,10 6 $5 P3,6 7 10 11 12 P3,8 P1,11 P2,12 P1 P2 P3 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. t=7.5 $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

Freeze all the processes and collect the state of each process. $10 2 3 $20 $30 $1 P1,2 4 8 1 5 P2,4 $3 P3,1 $2 9 $4 P1,10 6 $5 P3,6 7 10 11 12 P3,8 P1,11 P2,12 P1 P2 P3 t=7.5 Example: Snapshot at time t=7.5 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.

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 <mark> to each of its neighbors. For each process p (a) if p receives <mark> first time, it finds its own state, and then send <mark> to each of its neighbors. (b) p continues to receive message <mark> from all its neighbors as follows: Assuming that t(0) is the time p received the first <mark>, p collect all the message from any neighbor q until p get message <mark> form q. p q m1,m2 m3 mark p q mark m1 m2 p q p q m1,m2,m3 mark Start process mark m1

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?