1
play

1 Graphs Graphs a a c c Graph algorithms Depth-first search, - PDF document

Feb 11, 2024 •254 likes •317 views

PhD course Bulk-Synchronous Parallel (BSP) Computing Big Data Analytics Bulk-Synchronous Parallel (BSP) Model [Valiant90] A Bridging Model between abstract PRAM model and real-world parallel computers, to support algorithm development

  1. PhD course Bulk-Synchronous Parallel (BSP) Computing Big Data Analytics  Bulk-Synchronous Parallel (BSP) Model [Valiant’90]  A Bridging Model between abstract PRAM model and real-world parallel computers, to support algorithm development Programming Models for Big-Graph Analytics BSP computation: sequence of supersteps time Global Global Concep- Concep- Bulk-Synchronous Parallel Computing, BSP: communi tual cation barrier Pregel, Giraph, Hama Local computation … Parallelism Christoph Kessler IDA, Linköping University superstep BSP machine model: March 2017 Distributed memory Christoph Kessler, IDA, Linköpings universitet. 2 C. Kessler, IDA, Linköpings universitet. BSP Example: BSP-Model Global Maximum (NB: non-optimal algorithm) 3 4 C. Kessler, IDA, Linköpings universitet. C. Kessler, IDA, Linköpings universitet. BSP Remarks Graphs a c  Local variables of BSP processes persist to next superstep  G=(V,E)  In superstep s , contents of messages sent (and received) in  V: set of nodes b step s -1 are accessible  E: set of edges (directed or undirected) d  Two-sided and/or one-sided message passing possible  Storage: – Adjacency matrix – random access but too inefficient for sparse graphs  BSP implementations in the 1990s include  BSP implementations in the 1990s include – Adjacency list – each node holds a list of its neighbors  BSPlib – library for C atop MPI [Hill et al. ’98]  Graph examples  One-sided communication (put, get)  WWW  PUB – library for C atop MPI [Bonorden et al.’03]  Social network graphs  One-sided communication  Transportation networks  NestStep [K. 2000, 2004]  Similarity of documents  Partitioned global address space (PGAS) language  Citation relationships extension of C  … 5 6 C. Kessler, IDA, Linköpings universitet. C. Kessler, IDA, Linköpings universitet. 1

  2. Graphs Graphs a a c c  Graph algorithms  Depth-first search, Breadth-first search b b  Single-source shortest paths  Today, many graphs are very large d d  All-pairs shortest paths  Clustering  Page rank  How to distribute a graph?  How to distribute a graph?  Minimum cut / Maximum flow Minimum cut / Maximum flow  Usually, partition and distribute the array of vertices  Connected components (each vertex with a local value  Strongly connected components and its adjacency list of outgoing edges)  Hamiltonian, Euler Tour, Traveling Salesman Problem  If locality matters in partitioning?  …  Typical properties  Replace default partitioning with a user-defined partitioning  Poor locality of memory accesses  Graph partitioner  Little work per vertex e.g. METIS (used in HPC e.g. for FEM meshes)  Changing degree of parallelism 7 8 C. Kessler, IDA, Linköpings universitet. C. Kessler, IDA, Linköpings universitet. Example Example Server 0 Server 1 0 0 4 4 1 1 5 5 2 2 6 6 3 3 Hash vertices over servers. Ex.: S servers, N vertices, hashfn = id , B = ceil( N / S ): • owner (i) = i div B • local index (i) = i mod B 9 10 C. Kessler, IDA, Linköpings universitet. C. Kessler, IDA, Linköpings universitet. Graph Traversal Graph Analytics a c  Global analysis of a graph  Exploration starts at some vertex  Consider each vertex and each edge  For each vertex, b consider all its neighbors  2 flavors: d  Visit each vertex once and traverse each edge once  Compute one value over all vertices / edges  Sequential algorithms:  E.g. sum-up/maximize/… attribute values over all vertices Depth first search, Breadth first search Depth first search, Breadth first search  Compute one value for each vertex / each edge  Compute one value for each vertex / each edge Special case: Tree traversals  E.g. page rank for each vertex (web page)  Much (dynamic) parallelism  Often, iterative (unless graph has very special structure)  multiple sweeps over the graph  Little/no data locality (unless graph has very special structure)  Suitable for massively parallel and distributed computation  History of exploration (i.e., recursive call stack, visited vertices)  Requires a global view (random access) of the graph must be kept  dependences / communication for visiting a remote vertex when calculating on a distributed system  How to address huge graphs? 11 12 C. Kessler, IDA, Linköpings universitet. C. Kessler, IDA, Linköpings universitet. 2

  3. Pregel Pregel [Malevicz et al . 2010]  A framework to process/query large distributed graphs  Proprietary (by Google) Kneiphof  Pregel is the ”MapReduce” for graphs island Pregel  Iterative computations  Sequence of BSP supersteps Sequence of BSP supersteps  Each BSP superstep is basically a composition of the By Bogdan Giuşcă - Public MapReduce phases (Map, Combine, Sort, Reduce) domain (PD), based on the N image, CC BY-SA 3.0, https://commons.wikimedia.org  Attempts to utilize all servers available /w/index.php?curid=112920 K E by partitioning and distributing the graph A multigraph S Leonhard Euler  Good for computations touching all vertices / edges 1736: There is no Eulertour (1707-1783), (path traversing each bridge  Bad for computations touching only few vertices / edges Swiss exactly once) for Königsberg mathematician nor any other topology with >2 nodes having odd degree. 13 14 C. Kessler, IDA, Linköpings universitet. Source: Wikipedia C. Kessler, IDA, Linköpings universitet. Pregel Example: Maximum vertex value Programming Model States of a vertex Vote to halt  Graph Vertices For now, assume 1 BSP a b c d  Each with unique identifier (String) Active Inactive processing node per vertex  Each with a user-defined value  Each with a state in { Active, Inactive } 4 5 3 2 Message received Initial states / values  Initially (before superstep 1), every vertex is active Superstep 0:  Graph Edges send my value to all neighbors;  Each edge can have a user-defined value (e.g., weight) receive from all neighbors  One (conceptual) BSP process assigned to each vertex Superstep 1:  Not to the edges, by the way… maximize over all received values;  Iteratively executing supersteps while active if larger than my value, 5 5 5 3 update it and  Two-sided communication with send () and receive () calls send new value to all neighbors;  Graph can be dynamic receive from neighbors;  Vertices and/or edges can be added or removed in each superstep else vote for halt;  Algorithm termination 5 5 5 5 Superstep 2:  When all vertices are simultaneously inactive … and there are no messages in transit Superstep 3:  Otherwise, go for another superstep … 5 5 5 5 15 16 C. Kessler, IDA, Linköpings universitet. C. Kessler, IDA, Linköpings universitet. Exercise: Connected Components Example: Maximum vertex value based on Maximum (with local accumulation) Now: multiple BSP processes Server 0 Server 1 per server node a b c d  message aggregation, local combining 4 5 3 2 0 Initial states / values 0 4 Superstep 0: 4 send my value to all neighbors; receive from all neighbors 1 1 Superstep 1: 5 maximize over all received values; 5 if larger than my value, 5 5 5 3 2 update it and send new value to all neighbors; 2 receive from neighbors; 6 3 else vote for halt; 6 3 5 5 5 5 Superstep 2: … Superstep 3: Initialization … 5 5 5 5 17 18 C. Kessler, IDA, Linköpings universitet. C. Kessler, IDA, Linköpings universitet. 3

Recommend

More recommend