Teacher: Chun-Yuan Lin

Introduction to Distributed AlgorithmAppendix A: Pseudocode ConventionsAppendix B: Graphs and Networks Teacher: Chun-Yuan Lin Appendices A and B

Appendix A: Pseudocode Conventions Appendices A and B

Pascal-like pseudocode • The pseudocode used in this book very much resembles the language Pascal. • Process names can be used as array indices, and set variables are used. • The variableNeighpis the set of processes to which p is connected (the neighbors of p). • Assignment ("a :== expression"). • Conditional statement (“ifcondition thenstatemenf”, with an optional “elsestatemenf” part). • loop (“whilecondition dostatemenf”). • The forall statement has the form “forallx X dostatemenf”, where x is a formal parameter and X a set. Ex. forall qNeighp doRp[q]:=0. • begin and end. Appendices A and B

Message passing(1) • The operations "send" and "receive" are used to describe message passing in the pseudocode. • Assume that each process must send to its neighbors its identity (process name) and its degree (number of neighbors). • This message is denoted as (info, a, b). • The send operation must be given a message (its type and the value of its data fields) and a destination. To send its identity and degree to process q, process p may execute: send <info, idp, |Neighp|> to q. • The shorthand "shout" is used to send a message to all neighbors. shout <info, idp, |Neighp|> as forall q  Neighp do send <info, idp, |Neighp|> to q . Appendices A and B

Message passing(2) • To receive a message, its type as well as formal parameters for the data fields of the message and its sender must be specified. To receive the information regarding the identity and degree of a neighboring node, process p may execute: receive <info, a, b> from q . • A process may need to receive a message from a specific process, receive <info, a, b> from this q0 . • The communication between processes can be either asynchronousor synchronous. Appendices A and B

Control-oriented versus event-driven notation • The control-oriented notation of an algorithm consists of a declaration of its variables and a semicolon-separated list of statements enclosed between begin and end. • Event-driven notation consists of a declaration of variables together with the initialization of these variables, followed by a list of actions. Appendices A and B

Appendices A and B

Appendix B: Graphs and Networks Appendices A and B

Definitions and Terminology(1) • As the topology of a distributed system is usually modeled by a graph, some knowledge of graph theory and its terminology is useful in the study of distributed algorithms. • A graph can be thought of as a collection of points (called the nodesof the graph) some of which are connected by lines (the edges). Appendices A and B

Undirected Graphs(1) • An undirected graph G is a pair (V, E), where Vis a set called the node set of G and Eis a collection of unordered pairs from V. • An element of E is a pair of {u, v} with u, vV. To shorten the notation, we write uvE instead of {u, v}E. (uv E is equivalent to vu E) • The edge uvis called an incident edge of u (and of v). • If uvE, the nodes u and v are said to be adjacent, or neighbors. • The degreeof a node is the number of edges incident to it, or, equivalently, its number of neighbors. • The graph is called regular if all nodes have the same degree. Appendices A and B

Undirected Graphs(2) • A pathof length k between v0and vkis a sequence P = < v0, ... , vk > of nodes such that for each i < k, vivi+1E. • A cycle is a path of which the begin node equals the end node. • A path is called simpleif the nodes v0through vkare all different. A cycle is called simpleif the nodes v0 through vkare all different. • The distance between u and v, denoted d(u, v), is the length of a shortest path between u and v. • The diameter of G is the largest distance between any two nodes. • An undirected graph is connectedif there exists a path between each pair of nodes. • An undirected graph is called acyclicif it contains no simple cycle of length three or more. Appendices A and B

Undirected Graphs(3) • A graph G’= (V’, E’)is called a subgraph of G if V’V and E’  E. • G’is called a spanning subgraphif V’=V . • G’is called an induced subgraphif E’= {uvE| uV’^v  V’}. • A connected component of G is a maximal connected induced subgraph G’ of G; that is, G’is a connected induced subgraph. • A graph is called a planar graphif it is possible to draw the graph in the plane without crossing edges. • A graph is called outerplanar if it is possible to draw the graph in the plane without crossing the edges and with all nodes on the border of the picture. Appendices A and B

Undirected Graphs(4) • The following properties of planar and outerplanar graphs are of interest. Appendices A and B

Directed Graphs(1) • A directed graph G is a pair (V, E), where Vis the node set, and Eis a collection of ordered pairs from V. • uv E is not equivalent to vu E. • The edge uv is called an outgoing edge of u and an incoming edge of v. • If uv E, v is called an out-neighbor of u, and u is called an in-neighbor of v. • The in-degreeof a node is the number of its incoming edges, and the out-degreeis the number of outgoing edges. The degree is the sum of the in-degree and the out-degree. • With same definitions for path, cycle, simple, distance by undirected graph. Appendices A and B

Directed Graphs(2) • A directed graph is strongly connectedif there exists a path from each node to each other node. • A directed graph is called acyclicif it contains no simple cycle of length two or more. Appendices A and B

Weighted Graphs • A (directed or undirected) graph is weightedif for each pair u, v with uvE, a numerical value Wuvis defined. • The weight assignment is called symmetricif, for each pair, Wuv= Wvu. • In a weighted graph, the weight of a path is defined as the sum of the edge weights over the edges in the path. • If no weight assignment is assumed for a graph, it is called unweighted. Appendices A and B

Frequently Used Graphs • Some classes of graph that occur frequently in the study of distributed algorithms, namely, rings, trees, forests, grids, tori, cliques, and hypercubes. • In the sequel, G = (V, E)be a graph, Nthe number of its nodes, and Dthe diameter. Appendices A and B

Rings • The ring topology is a circular arrangement of nodes, and is often used as a control topology in distributed computations. • Definition B.lA ring is an undirected, connected, regular graph of degree two. • Theorem B.2The following are equivalent for an undirected graph G. • A spanning ring can sometimes be defined in a graph G = (V, E); that is, a set E’  E is selected such that (V, E’)is a ring. • It is NP-complete to decide whether a given graph has spanning ring. (not every graph contains one) (virtual ring) Appendices A and B

Trees, Forests, and Stars (1) • Trees • A tree is a graph that contains a minimal number of edges connecting its nodes, and as a result computations on tree-shaped networks may have a very low communication complexity. • Definition B.3A tree is an undirected, connected, acyclic graph. • Theorem B.4The following are equivalent for an undirected graph G. Appendices A and B

Trees, Forests, and Stars (2) • A tree T = (V, E)is rootedif there is a unique designated node r called the root. • u is a node on the (unique) simple path between v and r, u is called an ancestorof v, and v is called adescendantof u. • u and v are neighbors, v is called a son(or child) of u, and u is called the father(or parent) of v. • The subgraph induced by the descendants of u is a rooted tree (with root u), called the subtree ofu. • The depth of the tree is the maximal length of any simple path from the root to a node. • Every connected graph G = (V, E)contains a spanning tree; that is, a set E’  E can be chosen, such that (V, E’)is a tree. Appendices A and B

Trees, Forests, and Stars (3) • Low-diameter spanning tree • If the spanning tree must be chosen to minimize the total time necessary for a computation on it, it is desirable that the diameter is as small as possible. • Minimal-weight spanning tree • If the total communication cost for a computation in a spanning tree must be low, the subtree must be chosen so as to minimize the weight of the tree. (If all edge weights are different, the minimal-weight spanning tree is unique) • Restricted-degree spanning tree • If the computation overhead per node must be low, a spanning tree must be selected that has a low degree for each node. • Depth-first search tree • A spanning tree is a depth-first search spanning tree if each frond edge connects a node and a descendant of that node. Appendices A and B

Trees, Forests, and Stars (4) • A graph consisting of a number of isolated trees is called a forest. • Definition B.5A forest is an undirected acyclic graph. • Theorem B.6The following are equivalent for an undirected graph G. • A forest is rooted if a root node is designated in every tree of the forest. • Every graph has a spanning forest, consisting of a spanning tree of each connected component of the graph. Appendices A and B

Trees, Forests, and Stars (5) • A star is a graph with one special node, the center, and all other nodes are connected only to this center. • Definition B.7A star is a rooted tree of depth one. The root of the star is called the center. • Theorem B.8The following are equivalent for an undirected graph G. • Stars are not usual as a physical connection topology of a distributed systems; rather, stars are the virtual topology used in computations that are controlled by a central process. Appendices A and B

Cliques • In a clique, sometimes also called a complete graph, each pair of nodes is directly connected by an edge. • Definition B.9A clique is a graph with diameter one. • Theorem B.l0The following are equivalent for an undirected graph G. • As is the case for stars, cliques are rarely used as a physical connection topology. (higher layer) Appendices A and B

Grids and Tori (1) • In an n ×n grid there are N = n2nodes, arranged in n rows of n nodes each. Each node is connected to the nodes above it, right and left of it, and under it. • The n ×ntorus is similar, but in addition the leftmost and rightmost node of each row are adjacent, and the uppermost and lowermost node of each column are adjacent. • The n ×n grid has 2n(n -1) edges, contains nodes of degree two, three, and four, and its diameter is 2(n - 1). The n ×n torus has 2n2edges, is regular of degree four, and its diameter is Appendices A and B

Grids and Tori (2) Appendices A and B

Hypercubes(1) • Like grids and tori, hypercubes are often used in the design of multiprocessor computers. (Both diameter and degree equal logN, where N is the number of nodes) • Definition B.l2The n-dimensional hypercubeis a graph on N = 2n nodes where each node can be assigned a unique element from the set oflabels {(b0, ... , bn-1): bi= 0, 1} in such a way thatE = {uv : the labels of u and v differ in one bit}. • Theorem B.13The following are equivalent for an undirected graph G. Appendices A and B

Hypercubes(2) • The hypercube is regular of degree n, and has diameter n. • An (n + 1 )-dimensional hypercube can be constructed by taking two ndimensional hypercubes and connecting corresponding nodes. Appendices A and B

Hypercubes(3) Appendices A and B

Sense of Direction (1) • It was observed that the communication complexity of distributed computations is influenced by following factors • Topological awareness In order to exploit the advantages of a specific topology, it is sometimes necessary that the processes "know" that they are connected in a topology of this class. • Sense of direction The routing of information through a network can be done more efficiently if the edges incident to each node are labelled with the "direction" to wl1ich they lead in the network. Appendices A and B

Sense of Direction (2) Appendices A and B

Sense of Direction-Ring • As the ring is regular of degree two, there are only two directions in the ring; we call them “Pred” (previous) and “Next”. • Definition B.14A labeling of the ring is an assignment at every node of different labels from the set {Prev, Next} to the edges incident to that node. A labeling S of the ring constitutes a sense of direction if for all adjacent nodes u and v, Su(v) =Prev  Sv(u) = Next. Appendices A and B

Sense of Direction-Cliques • The clique of N nodes has degree N - 1; its directions are numbered from 1 through N – 1. • Definition B.15A labeling of the clique is an assignment at every node of different labels from the set {1, ... , N - 1} to the edges incident to that node.A labeling S of the clique constitutes a sense of direction if the nodes of the clique can be numbered from 0 through N - 1, in such a way that for every i and j, Si(j)= (j - i) mod N. Appendices A and B

Sense of Direction-Tori • The four directions of the torus are called up, down, right, and left, abbreviated U, D, R, L. Appendices A and B

Sense of Direction-Hypercubes • The n dimensions of the n-dimensional hypercube are numbered from 0 through n - 1. There is a sense of direction if all parallel edges are labeled with the same number. Appendices A and B

Sense of Direction (3) Appendices A and B

Teacher: Chun-Yuan Lin