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**Lecture 7: Synchronous Network Algorithms**

Basic technology of distributed algorithms (1) Correctness of algorithms (2) Efficiency of algorithms (3) Fault-tolerance

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**Complexity of distributed algorithms**

Communication complexity: (1) the number of messages transmitted in whole system, (2) the number of message bits transmitted in whole system Memory complexity: (1) the amount of memory used in whole system, (2) the amount of memory used in one process. Computation complexity: (1) processing time in processes, (2) upper bound of transmission delay in communication links.

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**Problems 1 Leader Election in A synchronous 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 its UID (it does not know the size of the ring, its own index and anything of its neighbors). Only the leader performs an output. LCR algorithm (informal) (1) Each process sends its identifier around the ring. (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 keeps passing the identifier; 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.

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**LCR algorithm (formal)**

Suppose that each node i has A UID u, initially i’s UID, send, a UID or null, initially i’s UID, A status with value in {unknown, leader}, initially unknown. Algorithm Send the current value of send to clockwise neighbor send := null if the incoming message is v, a UID, than case v>u: send :=v v=u: status := leader v<u: do nothing endcase Theorem LCR solves the leader-election problem in a synchronous ring. Proof: exact one process outputs the value leader.

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**Analysis of the Complexity of LCR Algorithm**

Time complexity: n rounds until a leader is announced. Communication complexity: Remark 1: If it is required that all the nodes halt, the time complexity is 2n and the communication complexity is still Remark 2: If it is required that the leader and nonleaders all provide output and all processes halt, the extra cost of obtaining the extra outputs and the halting is only n rounds and n messages.

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**Improved Leader-Election Algorithm**

In the following algorithm, the communication is supposed to be bidirectional. HS algorithm (informal) Each process i operates in phases 0,1,2,…. In each phase l, process i send out “tokens” consists of UID in both directions. These are intended to travel distance then return to their original i. If both tokens make it back safely, process i continues with the following phase. However, the tokens might not make it back safely. While a token is proceeding in the outbound direction, each other process j on path compares with its own UID If then j simply discards the token, whereas if , then j relays If , then it means that process j has received its own UID before the token has turned around, so process j elects itself as the leader. All processes always relay all tokens in the inbound direction.

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**Analysis of the Complexity of HS Algorithm **

find leader relay Analysis of the Complexity of HS Algorithm Time complexity: O(n) rounds (the time for each phase is ). Communication complexity: O(n log n) (at most processes altogether initiate tokens at phase l. Therefore, the total messages sent out at phase l is bounded by ).

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**Problem2 Leader Election in a General Network**

Assumption of the network An strongly connected network digraph G=(V,E) having n nodes. 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 its 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.

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**Theorem FloodMax algorithm solves the leader-election problem in a synchronous general network.**

Proof: exact one process outputs the value leader. Analysis of the Complexity of FloodMax Algorithm Time complexity: (The time until the leader is elected and all other processes know that they are not the leader is diam rounds.) Communication complexity: (The number of messages is , where |E| is the number of directed edges in the digraph, because a message is sent on every directed edge for each of the first diam rounds.)

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**Problem 3 Breadth-First Search**

Bread-First Search Given a graph G=(V,E) and a distinguished source vertex i, breadth-first search explores the edges of G to discover every vertex that is reachable from s. It provides the distance from i to all such reachable vertices. It also provide a breadth-first tree with root i that contains all such reachable vertices. s Bread-First Search in an undirected graph s Bread-First Search in a directed graph Spanning Tree A spanning tree of graph G=(V,E) is a rooted tree which contains all vertices of V and the path from the root to any vertex. A breadth-firs tree is a spanning tree.

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**Assumption of the network**

An strongly connected network digraph G=(V,E) having n nodes and a distinguished source note s. Output is the structure of a breadth-first search 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 their neighbors’ indices. They have no knowledge of the size or diameter of the network. UIDs are not needed. SynchBFS algorithms At any point during execution, there is some set of processes that is “marked”, initially just s. Process s sends out a search message at round 1, to all of its outgoing neighbors. At any round, 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.

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**Theorem SynchBFS algorithm solves the Breadth-First Search problem.**

Analysis of the complexity of SynchBFS algorithm Time complexity: At most diam rounds Communication complexity: |E| Application to Message Broadcast Problem SynchBFS algorithm can be used to Message Broadcast problem: piggyback the message m in the Breadth-First Search.

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**Problem 4 Shortest Path Problem**

Shortest Path Problem Consider a strong directed graph G=(V,E), where ach edge e=(i,j) in E has a nonnegative real-value weight w(i,j). The problem is to find a shortest path form a distinguished node s to each other node in G. s 2 1 4 3 Shortest paths from s s 2 1 4 3

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**BellmanFord algorithm**

Assumption of network Each process initially knows its neighbors’ indices and the weight of all its incident edges. Each process knows the number n of nodes in the network. Output: each process knows its parent in the shortest path tree, and its distance from s. BellmanFord algorithm Each process i keeps track of parent and dist, the shortest distance from s to i it knows so far. Initially, dist(s)=0, dist(i)= for and the parent components are not defined. At each round process i sends its dist(i) to all its outgoing neighbors. Then each process i updates its dist(i) to be If dist(i) is updated, the parent is also updated accordingly. After n-1 rounds, dist contains the shortest distance, and parent contains the parent in the shortest tree.

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**Analysis of the complexity of BellFord algorithm Time complexity: n-1 **

2 1 4 3 s 2 1 4 3 s 2 1 4 3 s 2 1 4 3 s 2 1 4 3 s 2 1 4 3 Analysis of the complexity of BellFord algorithm Time complexity: n-1 Communication complexity: (n-1)|E|

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Assignment Improve algorithm FloodMax to reduce the communication complexity. Describe an algorithm that extends SynchBFS to allow the source process s to broadcast a message to all other processes and obtain an acknowledge that all processes have received it. Your algorithm should use O(|E|) messages and O(diam) time. You may assume that the network graph is undireted.

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