DOCSLIB.ORG

Integrating Column-Oriented Storage and Query Processing Techniques Into Graph Database Management Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Integrating Column-Oriented Storage and Query Processing Techniques Into Graph Database Management Systems Integrating Column-Oriented Storage and Query Processing Techniques Into Graph Database Management Systems Pranjal Gupta, Amine Mhedhbi, Semih Salihoglu University of Waterloo {pranjal.gupta, amine.mhedhbi, semih.salihoglu}@uwaterloo.ca ABSTRACT such, these columnar techniques are relevant for improving We revisit column-oriented storage and query processing the performance and scalability of GDBMSs. techniques in the context of contemporary graph database In this paper, we revisit columnar storage and query pro- management systems (GDBMSs). Similar to column-orien- cessing techniques in the context of GDBMSs. Specifically, ted RDBMSs, GDBMSs support read-heavy analytical work- we focus on an in-memory GDBMS setting and discuss the loads that however have fundamentally different data access applicability of columnar storage and compression techniques patterns than traditional analytical workloads. We first de- for storing different components of graphs [17, 19, 58, 66], rive a set of desiderata for optimizing storage and query and block-based query processing [20, 27]. Despite their sim- processors of GDBMS based on their access patterns. We ilarities, workloads in GDBMS and column-oriented RDBMSs then present the design of columnar storage, compression, also have fundamentally different access patterns. For ex- and query processing techniques based on these desiderata. ample, workloads in GDBMSs contain large many-to-many In addition to showing direct integration of existing tech- joins, which are not frequent in column-oriented RDBMSs. niques from columnar RDBMSs, we also propose novel ones This calls for redesigning columnar techniques in the context that are optimized for GDBMSs. These include a novel list- of GDBMSs. The contributions of this paper are as follows. based query processor, which avoids expensive data copies Guidelines and Desiderata: We begin in Section3 by ana- of traditional block-based processors under many-to-many lyzing the properties of data access patterns in GDBMSs. joins and avoids materializing adjacency lists in intermedi- For example, we observe that different components of data ate tuples, a new data structure we call single-indexed edge stored in GDBMSs can have some structure and the order property pages and an accompanying edge ID scheme, and a in which operators access vertex and edge properties often new application of Jacobson’s bit vector index for compress- follow the order of edges in adjacency lists. This analysis ing NULL values and empty lists. We integrated our tech- instructs a set of guidelines and desiderata for designing the niques into the GraphflowDB in-memory GDBMS. Through physical data layout and query processor of a GDBMS. extensive experiments, we demonstrate the scalability and Columnar Storage: Section4 explores the application of query performance benefits of our techniques. columnar data structures for storing different components of GDBMSs. While existing columnar structures can directly be used for storing vertex properties and many-to-many (n- 1. INTRODUCTION n) edges, we observe that using a straightforward columnar Contemporary GDBMSs are data management software structure, which we call edge columns, to store properties such as Neo4j [9], Neptune [1], TigerGraph [13], and Graph- of n-n edges is suboptimal as it does not guarantee sequen- flowDB [44, 50] that adopt the property graph data model [10]. tial access when reading edge properties in either forward or In this model, application data is represented as a set of ver- backward directions. An alternative, which we call double- tices, which represent the entities in the application, directed indexed property CSRs, can achieve sequential access in both arXiv:2103.02284v1 [cs.DB] 3 Mar 2021 edges, which represent the relationships between entities, directions but requires duplicating edge properties, which and key-value properties on the vertices and edges. can be undesirable as graph-structured data often contain GDBMSs have lately gained popularity to support a wide orders of magnitude more edges than vertices. We then de- range of analytical applications, from fraud detection and scribe an alternative design point, single-directional property risk assessment in financial services to recommendations in pages, that avoids duplication and achieves good locality e-commerce and social networks [57]. These applications when reading properties of edges in one direction and still have workloads that search for patterns in a graph-structured guarantees random access in the other when using a new database, which often requires reading large amounts of edge ID scheme that we describe. Our new ID schemes allow data. In the context of RDBMSs, column-oriented sys- for extensive compression when storing them in adjacency tems [11, 41, 58, 63] employ a set of read-optimized storage, lists without decompression overheads. Lastly, as a new ap- indexing, and query processing techniques to support tradi- plication of vertex columns, we show that single cardinality tional analytical applications, such as business intelligence edges and edge properties, i.e. those with one-to-one (1-1), and reporting, that also process large amounts of data. As one-to-many (1-n) or many-to-one (n-1) cardinalities, are stored more efficiently with vertex columns instead of the structures we described above for n-n edges. 1 name: “peter” name: “alice” age: 17 Columnar Compression: In Section5, we review existing since: 2015 age: 45 Vertex labels - gender: M gender: F columnar compression techniques, such as dictionary encod- PERSON ing, that satisfy our desiderata and can be directly applied doj: 2014 doj: 2019 ORG to GDBMSs. We next show that existing techniques for Name: “UofT” compressing NULL values in columns from references [17, name: “UW” doj: 2006 estd: 1885 estd: 1934 19] by Abadi et al. lead to very slow accesses to arbitrary since: 2011 Edge labels - doj: 2006 non-NULL values. We then review Jacobson’s bit vector in- since: 1999 FOLLOWS since: 1992 since: 2006 since: doj: 1980 dex [42, 43] to support constant time rank queries, which has since: since:2009 2012 STUDYAT found several prior applications e.g., recently in a range filter name: “jenny” name: “bob” WORKAT age: 23 age: 54 gender: M structure in databases [61], in information retrieval [37, 52] gender: F since: 2003 ★ doj: date of joining and computational geometry [30, 53]. We show how to en- hance one of Abadi’s schemes with an adaptation of Jacob- Figure 1: Running example graph. son’s index to make it suitable for compressing NULL values and empty adjacency lists in GDBMSs. Our final structure (iii) edge properties. In every native GDBMS we are aware allows constant-time access to NULL or non-NULL values of, the topology is stored in data structures that organize with a small increase in storage overhead per entry compared data in adjacency lists [28], such as in compressed sparse to prior techniques. row (CSR) format. Typically, given the ID of a vertex v, List-based Processing: In Section6, we observe that tradi- the system can in constant-time access v’s adjacency list, tional Volcano-style [38] tuple-at-a-time processors, which which contains a list of (edge ID, neighbour ID) pairs. Typ- are used in some GDBMSs, do not benefit from processing ically, an adjacency list of v is further clustered by the blocks of data in tight loops but also has the advantage of edge label which enables traversing the neighbourhood of avoiding expensive data copies when performing many-to- v based on a particular label efficiently. Vertex and edge many joins, e.g. in long path queries. On the other hand, properties can be stored in a number of ways. For exam- columnar RDBMSs have block-based [20, 64] query proces- ple, some systems use a separate key-value store, such as sors that process fixed-length blocks of data in tight loops, DGraph [2] and JanusGraph [5], and some use a variant of which achieves better CPU and cache utility. However, interpreted attribute layout [25], where records consist of se- block-based processing results in expensive data copies un- rialized variable-sized key-value properties. These records der many-to-many joins. To address this, we propose a new can be located consecutively in disk or memory or have block-based processor we call list-based processor (LBP), pointers to each other, as in Neo4j. which modifies traditional block-based processors in two ways Queries in GDBMSs consist of a subgraph pattern Q that to tailor them for GDBMSs: (i) Instead of representing the describes the joins in the query (similar to SQL’s FROM) intermediate tuples processed by operators as a single group and optionally predicates on these patterns with final group- of equal-sized blocks, we represent them as multiple groups by-and-aggregation operations. We assume a GDBMS with of blocks. We call these list groups. LBP avoids expensive a query processor that uses variants of the following rela- data copies by flattening blocks of some groups into sin- tional operators, as in many GDBMSs, e.g., Neo4j [9], Mem- gle values when performing many-to-many joins. Flattening graph [6], or GraphflowDB: does not require any data copies and happens by setting an SCAN: Scans a set of vertices from the graph. index field of a list group. (ii) Instead of fixed-length blocks, JOIN (e.g. EXPAND in Neo4j and Memgraph, EXTEND in Graph- LBP uses variable length blocks that take the lengths of ad- flowDB): Performs an index nested loop join using the ad- jacency lists that are represented in the intermediate tuples. jacency list index to match an edge of Q. Takes as input a Because adjacency lists are already stored in memory con- partial match t that has matched k of the query edges in Q. secutively, this allows us to avoid materializing adjacency For each t, extends t by matching an unmatched query edge lists during join processing, improving query performance.
Load more

Recommended publications
  • Practical Parallel Hypergraph Algorithms
    Practical Parallel Hypergraph Algorithms Julian Shun [email protected] MIT CSAIL Abstract v While there has been signicant work on parallel graph pro- 0 cessing, there has been very surprisingly little work on high- e0 performance hypergraph processing. This paper presents v0 v1 v1 a collection of ecient parallel algorithms for hypergraph processing, including algorithms for betweenness central- e1 ity, maximal independent set, k-core decomposition, hyper- v2 trees, hyperpaths, connected components, PageRank, and v2 v3 e single-source shortest paths. For these problems, we either 2 provide new parallel algorithms or more ecient implemen- v3 tations than prior work. Furthermore, our algorithms are theoretically-ecient in terms of work and depth. To imple- (a) Hypergraph (b) Bipartite representation ment our algorithms, we extend the Ligra graph processing Figure 1. An example hypergraph representing the groups framework to support hypergraphs, and our implementa- , , , , , , and , , and its bipartite repre- { 0 1 2} { 1 2 3} { 0 3} tions benet from graph optimizations including switching sentation. between sparse and dense traversals based on the frontier size, edge-aware parallelization, using buckets to prioritize processing of vertices, and compression. Our experiments represented as hyperedges, can contain an arbitrary number on a 72-core machine and show that our algorithms obtain of vertices. Hyperedges correspond to group relationships excellent parallel speedups, and are signicantly faster than among vertices (e.g., a community in a social network). An algorithms in existing hypergraph processing frameworks. example of a hypergraph is shown in Figure 1a. CCS Concepts • Computing methodologies → Paral- Hypergraphs have been shown to enable richer analy- lel algorithms; Shared memory algorithms.
    [Show full text]
  • Practical Parallel Hypergraph Algorithms
    Practical Parallel Hypergraph Algorithms Julian Shun [email protected] MIT CSAIL Abstract v0 While there has been significant work on parallel graph pro- e0 cessing, there has been very surprisingly little work on high- v0 v1 v1 performance hypergraph processing. This paper presents a e collection of efficient parallel algorithms for hypergraph pro- 1 v2 cessing, including algorithms for computing hypertrees, hy- v v 2 3 e perpaths, betweenness centrality, maximal independent sets, 2 v k-core decomposition, connected components, PageRank, 3 and single-source shortest paths. For these problems, we ei- (a) Hypergraph (b) Bipartite representation ther provide new parallel algorithms or more efficient imple- mentations than prior work. Furthermore, our algorithms are Figure 1. An example hypergraph representing the groups theoretically-efficient in terms of work and depth. To imple- fv0;v1;v2g, fv1;v2;v3g, and fv0;v3g, and its bipartite repre- ment our algorithms, we extend the Ligra graph processing sentation. framework to support hypergraphs, and our implementations benefit from graph optimizations including switching between improved compared to using a graph representation. Unfor- sparse and dense traversals based on the frontier size, edge- tunately, there is been little research on parallel hypergraph aware parallelization, using buckets to prioritize processing processing. of vertices, and compression. Our experiments on a 72-core The main contribution of this paper is a suite of efficient machine and show that our algorithms obtain excellent paral- parallel hypergraph algorithms, including algorithms for hy- lel speedups, and are significantly faster than algorithms in pertrees, hyperpaths, betweenness centrality, maximal inde- existing hypergraph processing frameworks.
    [Show full text]
  • Adjacency Matrix
    CSE 373 Graphs 3: Implementation reading: Weiss Ch. 9 slides created by Marty Stepp http://www.cs.washington.edu/373/ © University of Washington, all rights reserved. 1 Implementing a graph • If we wanted to program an actual data structure to represent a graph, what information would we need to store? for each vertex? for each edge? 1 2 3 • What kinds of questions would we want to be able to answer quickly: 4 5 6 about a vertex? about edges / neighbors? 7 about paths? about what edges exist in the graph? • We'll explore three common graph implementation strategies: edge list , adjacency list , adjacency matrix 2 Edge list • edge list : An unordered list of all edges in the graph. an array, array list, or linked list • advantages : 1 2 3 easy to loop/iterate over all edges 4 5 6 • disadvantages : hard to quickly tell if an edge 7 exists from vertex A to B hard to quickly find the degree of a vertex (how many edges touch it) 0 1 2 3 4 56 7 8 (1, 2) (1, 4) (1, 7) (2, 3) 2, 5) (3, 6)(4, 7) (5, 6) (6, 7) 3 Graph operations • Using an edge list, how would you find: all neighbors of a given vertex? the degree of a given vertex? whether there is an edge from A to B? 1 2 3 whether there are any loops (self-edges)? • What is the Big-Oh of each operation? 4 5 6 7 0 1 2 3 4 56 7 8 (1, 2) (1, 4) (1, 7) (2, 3) 2, 5) (3, 6)(4, 7) (5, 6) (6, 7) 4 Adjacency matrix • adjacency matrix : An N × N matrix where: the non-diagonal entry a[i,j] is the number of edges joining vertex i and vertex j (or the weight of the edge joining vertex i and vertex j).
    [Show full text]
  • Introduction to Graphs
    Multidimensional Arrays & Graphs CMSC 420: Lecture 3 Mini-Review • Abstract Data Types: • Implementations: • List • Linked Lists • Stack • Circularly linked lists • Queue • Doubly linked lists • Deque • XOR Doubly linked lists • Dictionary • Ring buffers • Set • Double stacks • Bit vectors Techniques: Sentinels, Zig-zag scan, link inversion, bit twiddling, self- organizing lists, constant-time initialization Constant-Time Initialization • Design problem: - Suppose you have a long array, most values are 0. - Want constant time access and update - Have as much space as you need. • Create a big array: - a = new int[LARGE_N]; - Too slow: for(i=0; i < LARGE_N; i++) a[i] = 0 • Want to somehow implicitly initialize all values to 0 in constant time... Constant-Time Initialization means unchanged 1 2 6 12 13 Data[] = • Access(i): if (0≤ When[i] < count and Where[When[i]] == i) return Where[] = 6 13 12 Count = 3 Count holds # of elements changed Where holds indices of the changed elements. When[] = 1 3 2 When maps from index i to the time step when item i was first changed. Access(i): if 0 ≤ When[i] < Count and Where[When[i]] == i: return Data[i] else: return DEFAULT Multidimensional Arrays • Often it’s more natural to index data items by keys that have several dimensions. E.g.: • (longitude, latitude) • (row, column) of a matrix • (x,y,z) point in 3d space • Aside: why is a plane “2-dimensional”? Row-major vs. Column-major order • 2-dimensional arrays can be mapped to linear memory in two ways: 1 2 3 4 5 1 2 3 4 5 1 1 2 3 4 5 1 1 5 9 13 17 2 6 7 8 9 10 2 2 6 10 14 18 3 11 12 13 14 15 3 3 7 11 15 19 4 16 17 18 19 20 4 4 8 12 16 20 Row-major order Column-major order Addr(i,j) = Base + 5(i-1) + (j-1) Addr(i,j) = Base + (i-1) + 4(j-1) Row-major vs.
    [Show full text]
  • 9 the Graph Data Model
    CHAPTER 9 ✦ ✦ ✦ ✦ The Graph Data Model A graph is, in a sense, nothing more than a binary relation. However, it has a powerful visualization as a set of points (called nodes) connected by lines (called edges) or by arrows (called arcs). In this regard, the graph is a generalization of the tree data model that we studied in Chapter 5. Like trees, graphs come in several forms: directed/undirected, and labeled/unlabeled. Also like trees, graphs are useful in a wide spectrum of problems such as com- puting distances, finding circularities in relationships, and determining connectiv- ities. We have already seen graphs used to represent the structure of programs in Chapter 2. Graphs were used in Chapter 7 to represent binary relations and to illustrate certain properties of relations, like commutativity. We shall see graphs used to represent automata in Chapter 10 and to represent electronic circuits in Chapter 13. Several other important applications of graphs are discussed in this chapter. ✦ ✦ ✦ ✦ 9.1 What This Chapter Is About The main topics of this chapter are ✦ The definitions concerning directed and undirected graphs (Sections 9.2 and 9.10). ✦ The two principal data structures for representing graphs: adjacency lists and adjacency matrices (Section 9.3). ✦ An algorithm and data structure for finding the connected components of an undirected graph (Section 9.4). ✦ A technique for finding minimal spanning trees (Section 9.5). ✦ A useful technique for exploring graphs, called “depth-first search” (Section 9.6). 451 452 THE GRAPH DATA MODEL ✦ Applications of depth-first search to test whether a directed graph has a cycle, to find a topological order for acyclic graphs, and to determine whether there is a path from one node to another (Section 9.7).
    [Show full text]
  • List Representation: Adjacency List – N Rows of the Adjacency Matrix Are
    List Representation: Adjacency List { n rows of the adjacency matrix are repre- sented as n linked lists { Declare an Array of size n e.g. A[1:::n] of Lists { A[i] is a pointer to the edges in E(G) starting at vertex i. Undirected graph: data in each list cell is a number indicating the adjacent vertex Weighted graph: data in each list cell is a pair (vertex, weight) 1 a b c e a [b,2] [c,6] [e,4] b a b [a,2] c a e d c [a,6] [e,3] [d,1] d c d [c,1] e a c e [a,4] [c,3] Properties: { the degree of any node is the number of elements in the list { O(e) to check for adjacency for e edges. { O(n + e) space for graph with n vertices and e edges. 2 Search and Traversal Techniques Trees and Graphs are models on which algo- rithmic solutions for many problems are con- structed. Traversal: All nodes of a tree/graph are exam- ined/evaluated, e.g. | Evaluate an expression | Locate all the neighbours of a vertex V in a graph Search: only a subset of vertices(nodes) are examined, e.g. | Find the first token of a certain value in an Abstract Syntax Tree. 3 Types of Traversals Binary Tree traversals: | Inorder, Postorder, Preorder General Tree traversals: | With many children at each node Graph Traversals: | With Trees as a special case of a graph 4 Binary Tree Traversal Recall a Binary tree | is a tree structure in which there can be at most two children for each parent | A single node is the root | The nodes without children are leaves | A parent can have a left child (subtree) and a right child (subtree) A node may be a record of information | A node is visited when it is considered in the traversal | Visiting a node may involve computation with one or more of the data fields at the node.
    [Show full text]
  • Package 'Igraph'
    Package ‘igraph’ February 28, 2013 Version 0.6.5-1 Date 2013-02-27 Title Network analysis and visualization Author See AUTHORS file. Maintainer Gabor Csardi <[email protected]> Description Routines for simple graphs and network analysis. igraph can handle large graphs very well and provides functions for generating random and regular graphs, graph visualization,centrality indices and much more. Depends stats Imports Matrix Suggests igraphdata, stats4, rgl, tcltk, graph, Matrix, ape, XML,jpeg, png License GPL (>= 2) URL http://igraph.sourceforge.net SystemRequirements gmp, libxml2 NeedsCompilation yes Repository CRAN Date/Publication 2013-02-28 07:57:40 R topics documented: igraph-package . .5 aging.prefatt.game . .8 alpha.centrality . 10 arpack . 11 articulation.points . 15 as.directed . 16 1 2 R topics documented: as.igraph . 18 assortativity . 19 attributes . 21 autocurve.edges . 23 barabasi.game . 24 betweenness . 26 biconnected.components . 28 bipartite.mapping . 29 bipartite.projection . 31 bonpow . 32 canonical.permutation . 34 centralization . 36 cliques . 39 closeness . 40 clusters . 42 cocitation . 43 cohesive.blocks . 44 Combining attributes . 48 communities . 51 community.to.membership . 55 compare.communities . 56 components . 57 constraint . 58 contract.vertices . 59 conversion . 60 conversion between igraph and graphNEL graphs . 62 convex.hull . 63 decompose.graph . 64 degree . 65 degree.sequence.game . 66 dendPlot . 67 dendPlot.communities . 68 dendPlot.igraphHRG . 70 diameter . 72 dominator.tree . 73 Drawing graphs . 74 dyad.census . 80 eccentricity . 81 edge.betweenness.community . 82 edge.connectivity . 84 erdos.renyi.game . 86 evcent . 87 fastgreedy.community . 89 forest.fire.game . 90 get.adjlist . 92 get.edge.ids . 93 get.incidence . 94 get.stochastic .
    [Show full text]
  • An XML-Based Description of Structured Networks
    Massimo Ancona, Walter Cazzola, Sara Drago, and Francesco Guido. An XML-Based Description of Structured Networks. In Proceedings of Interna- tional Conference Communications 2004, pages 401–406, Bucharest, Romania, June 2004. IEEE Press. An XML-Based Description of Structured Networks Massimo Ancona£ Walter Cazzola† Sara Drago£ Francesco Guido‡ £ DISI University of Genova, e-mail: fancona,[email protected] † DICo University of Milano, e-mail: [email protected] ‡ Marconi Selenia Communication, e-mail: [email protected] Abstract In this paper we present an XML-based formalism to describe hierarchically organized communication networks. After a short overview of existing graph description languages, we discuss the advantages and disadvantages of their application in network optimization. We conclude by extending one of these formalisms with features supporting the description of the relationship between the optimized logical layout of a network and its physical counterpart. Elements for describing traffic parameters are also given. 1 Introduction The problem of network design and optimization has a strategical importance especially for military networks. In this case, design for fault tolerance and security plays a role more and more crucial than the equivalent importance required for commercial and research networks. The reengineering of a large communication network is a complex problem, which consists of different aspects that can be strongly affected by the way of describing data. The plainest way to describe a communication network is to model the relationship among sites and links by means of a weighted undirected graph, but unless some more assumptions are taken, this approach can raise several problems. A first issue is encountered when an analysis and a visualization of the network has to be realized: practical networks include hundreds or often thousands of nodes and links, so that even a simple description and documentation of the network structure is hard to maintain and update.
    [Show full text]
  • Note: Nonlinear Data Structures Do Not Have Their Elements in a Sequence
    CS231 - Fall 2017 Graphs • Graph is a non-linear data structure (like a tree). Note: nonlinear data structures do not have their elements in a sequence. • But trees are somewhat orderly. Root is connected to its children. The children are connected to their children…all the way to the leaves. • Graph can be linked in any pattern. • A graph consists of vertices (nodes) and edges (arcs). o Name the vertices: {A,B,C,D,E} o Name the edges: {(A,B), (A,E), (A,D),…} • Applications: o roads between cities o communication networks o oil pipelines o personal relationships between people o interval graphs o rooms in caves with passages between them • Directed vs. Undirected o Graphs can be directed or undirected. o In undirected graphs, edges have no specific direction. Edges are always “two-way” o In directed graphs (also called digraphs), edges have a direction 1 o A self-edge a.k.a. a loop is an edge of the form (a,a) • Graph Connectivity o A graph is connected when there is a path between every pair of vertices. o In a connected graph, there are no unreachable vertices. o A graph that is not connected is disconnected. o A graph G is said to be disconnected if there exist two nodes in G such that no path in G has those nodes as endpoints. • Weighted graphs o In a weighed graph, each edge has a weight a.k.a. cost o Some graphs allow negative weights; some not • Paths and Cycles 2 o Path from one vertex to another is a sequence of vertices, each a neighbor of the previous one.
    [Show full text]
  • Chapter 12 Graphs and Their Representation
    Chapter 12 Graphs and their Representation 12.1 Graphs and Relations Graphs (sometimes referred to as networks) offer a way of expressing relationships between pairs of items, and are one of the most important abstractions in computer science. Question 12.1. What makes graphs so special? What makes graphs special is that they represent relationships. As you will (or might have) discover (discovered already) relationships between things from the most abstract to the most concrete, e.g., mathematical objects, things, events, people are what makes everything inter- esting. Considered in isolation, hardly anything is interesting. For example, considered in isolation, there would be nothing interesting about a person. It is only when you start consid- ering his or her relationships to the world around, the person becomes interesting. Challenge yourself to try to find something interesting about a person in isolation. You will have difficulty. Even at a biological level, what is interesting are the relationships between cells, molecules, and the biological mechanisms. Question 12.2. Trees captures relationships too, so why are graphs more interesting? Graphs are more interesting than other abstractions such as tree, which can also represent certain relationships, because graphs are more expressive. For example, in a tree, there cannot be cycles, and multiple paths between two nodes. Question 12.3. What do we mean by a “relationship”? Can you think of a mathematical way to represent relationships. 231 232 CHAPTER 12. GRAPHS AND THEIR REPRESENTATION Alice Bob Josefa Michel Figure 12.1: Friendship relation f(Alice, Bob), (Bob, Alice), (Bob, Michel), (Michel, Bob), (Josefa, Michel), (Michel, Josefa)g as a graph.
    [Show full text]
  • Data Structure for Representing a Graph: Combination of Linked List and Hash Table
    Data structure for representing a graph: combination of linked list and hash table Maxim A. Kolosovskiy Altai State Technical University, Russia [email protected] Abstract In this article we discuss a data structure, which combines advantages of two different ways for representing graphs: adjacency matrix and collection of adjacency lists. This data structure can fast add and search edges (advantages of adjacency matrix), use linear amount of memory, let to obtain adjacency list for certain vertex (advantages of collection of adjacency lists). Basic knowledge of linked lists and hash tables is required to understand this article. The article contains examples of implementation on Java. 1. Introduction There are two most common ways for representing a graph: Adjacency matrix Collection of adjacency lists Let’s look at comparison table of these ways: Adjacency Collection of matrix adjacency lists Memory complexity (optimal – O(|E|)) O(|V|2) O(|E|) Add new edge (optimal – O(1)) O(1) O(1) Remove edge (optimal – O(1)) O(1) O(|K|) Search edge (optimal – O(1)) O(1) O(|K|) Enumeration of the vertices adjacent to u O(|V|) O(|K|) (optimal – O(|K|)) Table 1. Comparison table of two ways for representing graphs. V – set of vertices, E – set of edges, K – set of vertices adjacent to vertex u. We assume that |V|<|E|. Bold font – optimal complexity. Operations with a graph represented by an adjacency matrix are faster. But if a graph is large we can’t use such big matrix to represent a graph, so we should use collection of adjacency lists, which is more compact.
    [Show full text]
  • Analyzing the Facebook Friendship Graph
    Analyzing the Facebook friendship graph S. Catanese1, P. De Meo2, E. Ferrara3, G. Fiumara1 1Dept. of Physics, Informatics Section, University of Messina 2Dept. of Computer Sciences, Vrije Universiteit Amsterdam 3Dept. of Mathematics, University of Messina 1st International Workshop on Mining the Future Internet 20 September 2010, Berlin Catanese, De Meo, Ferrara, Fiumara () Analyzing the Facebook friendship graph MIFI 2010, 20/09/2010, Berlin 1 / 42 Outline 1 Motivation Main objective The Basic Problem Classic Work 2 Our Results/Contribution Data Extraction and Cleaning Data Analysis Main Results 3 Future Issues Catanese, De Meo, Ferrara, Fiumara () Analyzing the Facebook friendship graph MIFI 2010, 20/09/2010, Berlin 2 / 42 Outline 1 Motivation Main objective The Basic Problem Classic Work 2 Our Results/Contribution Data Extraction and Cleaning Data Analysis Main Results 3 Future Issues Catanese, De Meo, Ferrara, Fiumara () Analyzing the Facebook friendship graph MIFI 2010, 20/09/2010, Berlin 3 / 42 Main objective Analyze the Facebook friendship graph using: I a wrapper (for extraction, cleaning and normalization of data) I a tool for graph visualization and analysis I developed by some of us Catanese, De Meo, Ferrara, Fiumara () Analyzing the Facebook friendship graph MIFI 2010, 20/09/2010, Berlin 4 / 42 Main objective Analyze the Facebook friendship graph using: I a wrapper (for extraction, cleaning and normalization of data) I a tool for graph visualization and analysis I developed by some of us Catanese, De Meo, Ferrara, Fiumara
    [Show full text]