Large Scale Graph Algorithms
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1 Large Scale Graph Algorithms A Guide to Web Research: Lecture 2 Yury Lifshits Steklov Institute of Mathematics at St.Petersburg Stuttgart, Spring / 34
2 Talk Objective To pose an abstract computational problem on graphs that has a huge list of applications in web technologies 2 / 34
3 Outline 1 Family of Problems: Finding Strongest Connection Problem Statement and Applications Variations of Strongest Connection Problem 3 / 34
4 Outline 1 Family of Problems: Finding Strongest Connection Problem Statement and Applications Variations of Strongest Connection Problem 2 Max-Intersection Problem Statement and Naive Solutions Hierarchical Schema Solution 3 / 34
5 Outline 1 Family of Problems: Finding Strongest Connection Problem Statement and Applications Variations of Strongest Connection Problem 2 Max-Intersection Problem Statement and Naive Solutions Hierarchical Schema Solution 3 Concluding Remarks Overview of Related Research Open Problems 3 / 34
6 Part I Family of Problems: Finding Strongest Connection Problem statement Applications Variations of the problem 4 / 34
7 Strongest Connection Problem (SCP) BASIC SETTINGS: a class of graphs G, a class of paths P INPUT: a graph G G Allowed time for preprocessing: o( G 2 ) QUERY: a (new) vertex v TASK: to find a vertex u G that has maximal number of P-paths from v to u Allowed time for query processing: o( G ) 5 / 34
8 Homogeneous Graph / 2-Step Paths Graph of coauthoring Coauthor suggest in DBLP The most common coauthor of my coauthors
9 Homogeneous Graph / 2-Step Paths Graph of coauthoring v Coauthor suggest in DBLP The most common coauthor of my coauthors
10 Homogeneous Graph / 2-Step Paths u Graph of coauthoring v Coauthor suggest in DBLP The most common coauthor of my coauthors 6 / 34
11 Directed Graph / 2-Step Paths Graph of hyperlinks Advanced option for Google search: link-based similar website The website that is most often co-cited with the given one
12 Directed Graph / 2-Step Paths Graph of hyperlinks v Advanced option for Google search: link-based similar website The website that is most often co-cited with the given one
13 Directed Graph / 2-Step Paths u Graph of hyperlinks v Advanced option for Google search: link-based similar website The website that is most often co-cited with the given one 7 / 34
14 Bipartite Graph / 2-Step Paths People Bands Last.fm similar music bands The band that is most often co-listened with the given one In general: any content-based similarity, keyword-similarity, any co-occurrence similarity
15 Bipartite Graph / 2-Step Paths People Bands v Last.fm similar music bands The band that is most often co-listened with the given one In general: any content-based similarity, keyword-similarity, any co-occurrence similarity
16 Bipartite Graph / 2-Step Paths People Bands v u Last.fm similar music bands The band that is most often co-listened with the given one In general: any content-based similarity, keyword-similarity, any co-occurrence similarity 8 / 34
17 Homogeneous-Bipartite Graph / 2-Step Paths Friendship graph Recommended items Social recommendations in networks like Facebook System recommends things that are popular among my friends
18 Homogeneous-Bipartite Graph / 2-Step Paths Friendship graph v Recommended items Social recommendations in networks like Facebook System recommends things that are popular among my friends
19 Homogeneous-Bipartite Graph / 2-Step Paths Friendship graph v Recommended items u Social recommendations in networks like Facebook System recommends things that are popular among my friends 9 / 34
20 Bipartite Graph / 3-Step Paths New girlfriend suggest: Boys Girls Amazon.com recommendations Subscription recommendations for FeedBurner, Google Reader Items that have the largest number of co-occurrences with my items
21 Bipartite Graph / 3-Step Paths New girlfriend suggest: Boys v Girls Amazon.com recommendations Subscription recommendations for FeedBurner, Google Reader Items that have the largest number of co-occurrences with my items
22 Bipartite Graph / 3-Step Paths New girlfriend suggest: Boys v Girls u Amazon.com recommendations Subscription recommendations for FeedBurner, Google Reader Items that have the largest number of co-occurrences with my items 10 / 34
23 Tripartite 3-Graph / 2-Step Paths Folksonomy is a set of triples < user, tag, object > Users Tags v Objects u Similar websites in Del.icio.us, similar pictures in Flicr Largest number of common tags Largest number of common users Largest number of common pairs < user, tag > 11 / 34
24 Tripartite 3-Graph / 2-Step Paths Folksonomy is a set of triples < user, tag, object > Users Tags v Objects u Similar websites in Del.icio.us, similar pictures in Flicr Largest number of common tags Largest number of common users Largest number of common pairs < user, tag > 11 / 34
25 Tripartite 3-Graph / 2-Step Paths Folksonomy is a set of triples < user, tag, object > Users Tags v Objects u Similar websites in Del.icio.us, similar pictures in Flicr Largest number of common tags Largest number of common users Largest number of common pairs < user, tag > 11 / 34
26 Multicolor-Multiparty Graph / k-step Paths Semantic search: Most popular drink that is available on bars that are visited by my friends Friendship graph Bar visiting Drinks in menu
27 Multicolor-Multiparty Graph / k-step Paths Semantic search: Most popular drink that is available on bars that are visited by my friends Friendship graph v Bar visiting Drinks in menu
28 Multicolor-Multiparty Graph / k-step Paths Semantic search: Most popular drink that is available on bars that are visited by my friends Friendship graph v Bar visiting Drinks in menu u 12 / 34
29 Variations of Strongest Connection Problem Directed/undirected graphs Weights on edges/vertices Task: offline, on-line, all-to-all Task: one best connection, k best connections Graph and weights are evolving with time 13 / 34
30 Claim Computing strongest connection is probably the most important algorithmic problem related to web technologies 14 / 34
31 Claim Computing strongest connection is probably the most important algorithmic problem related to web technologies Personal opinion of Yury Lifshits 14 / 34
32 Solution Variations Usual alternatives to exact algorithm: Approximate algorithms Randomized algorithms Input graph (or query) belongs to a certain distribution. Average complexity analysis Introducing additional assumptions Introducing additional input-complexity parameter Modifying the computation task Heuristics Look to particular cases (subproblems) 15 / 34
33 Part II Max-Intersection Problem Statement and naive solutions Hierarchical schema solution This section represents a work-in-progress joint research with Benjamin Hoffmann and Dirk Nowotka 16 / 34
34 Statement of Max-Intersection Problem In set notation: Input: Family F of n sets, f F f k Time for preprocessing: n polylog(n) poly(k) Query: a set f new, f new k Task: Find f i F that maximizes f new f i Time for query processing: polylog(n) poly(k) 17 / 34
35 Statement of Max-Intersection Problem In set notation: Input: Family F of n sets, f F f k Time for preprocessing: n polylog(n) poly(k) Query: a set f new, f new k Task: Find f i F that maximizes f new f i Time for query processing: polylog(n) poly(k) or at most o(n) 17 / 34
36 Statement of Max-Intersection Problem In set notation: Input: Family F of n sets, f F f k Time for preprocessing: n polylog(n) poly(k) Query: a set f new, f new k Task: Find f i F that maximizes f new f i Time for query processing: polylog(n) poly(k) or at most o(n) In bipartite graph notation: Input: Bipartite documents-terms graph, D = n, d D d k Query: a document d new, d new k Task: Find d i D that has maximal number of common terms with d new 17 / 34
37 Applications of Max-Intersection (1/2) Homogeneous graphs: References in scientific papers: (1) maximal number of co-occurrences in reference list (2) maximal intersection of reference lists Social networks (e.g. LinkedIn): a person that has maximal connections with my direct neighborhood Collaboration networks (e.g. DBLP): given a scientist, to find another one with maximal overlapping of coauthors-list 18 / 34
38 Applications of Max-Intersection (1/2) Homogeneous graphs: References in scientific papers: (1) maximal number of co-occurrences in reference list (2) maximal intersection of reference lists Social networks (e.g. LinkedIn): a person that has maximal connections with my direct neighborhood Collaboration networks (e.g. DBLP): given a scientist, to find another one with maximal overlapping of coauthors-list Bipartite graphs: Websites Words graph: find a website with maximal intersection of used terms with the given one Music Bands Listeners graph: find a band that has maximal intersection of listeners with the given one 18 / 34
39 Applications of Max-Intersection (2/2) Tripartite graphs: Long Search Queries Web Dictionary Websites: given a query to find a website with maximal number of query terms Advertisement Description Keywords Websites (e.g. AdSense Matching): find a website with maximal number of terms form advertisement description PC Members Keywords Submissions: find a paper that has maximal number of terms that belong to expertise of the given PC member 19 / 34
40 Inverted Index (1/2) Let us use documents-terms notation Inverted index approach: Preprocessing. For every term produce a list of all documents that contain it Complexity: O(n k) Query d new = {t 1,..., t k }. Retrieve document lists for all terms of query. Check all documents in all these k lists and return the one with maximal intersection with d new Worst case complexity: Ω(n) 20 / 34
41 Inverted Index (1/2) Let us use documents-terms notation Inverted index approach: Preprocessing. For every term produce a list of all documents that contain it Complexity: O(n k) Query d new = {t 1,..., t k }. Retrieve document lists for all terms of query. Check all documents in all these k lists and return the one with maximal intersection with d new Worst case complexity: Ω(n) Let T max be the maximal degree of terms. Then the query complexity is O(k T max ) 20 / 34
42 Inverted Index (2/2) Rare-Term Requirement Cheating: modify the Max-Intersection problem New Task: Given the document d new, find a document d i such that 1 It has a joint rare term (term that occurs in at most r documents) with d new 2 The intersection with d new is maximal among all documents satisfying (1) 21 / 34
43 Inverted Index (2/2) Rare-Term Requirement Cheating: modify the Max-Intersection problem New Task: Given the document d new, find a document d i such that 1 It has a joint rare term (term that occurs in at most r documents) with d new 2 The intersection with d new is maximal among all documents satisfying (1) Observation Inverted index can handle queries in O(r k) time now 21 / 34
44 Inverted Set-Index Assume that k is extremely small, say k = O(log log n) Inverted set-index approach: Preprocessing. Write down all term subsets of all documents. Sort all these subsets in lexicographical order Complexity: O(n 2 k ) Query d new = {t 1,..., t k }. For every subset of query terms search it in the inverted set-index. Return the document that corresponds to the maximal subset founded in index Complexity: O(2 k (k + log n)) 22 / 34
45 Hierarchical Schema Table of terms: k levels Level i is divided to 2 i 1 cells Every cell contains k terms Random nature of D and d new : Choose random cell on the bottom level Mark all cells that are above it Choose one random term in every marked cell
46 Hierarchical Schema Table of terms: k levels Level i is divided to 2 i 1 cells Every cell contains k terms Random nature of D and d new : Choose random cell on the bottom level Mark all cells that are above it Choose one random term in every marked cell 23 / 34
47 Magic Levels (1/2) Assume that there are 2 k such random documents in D Notation: magic levels q = k, log k+1 q = k log k Theorem With very high probability there exists d D that has the same terms from top q ε levels 24 / 34
48 Magic Levels (2/2) Theorem With very high probability there are no d D that has at least q + ε common elements with d new 25 / 34
49 Algorithm for Hierarchical Schema Preprocessing: Encode every document as a 2k 1 sequence, every odd element lies in range [1..k], every even is 0 or 1 Construct a lexicographic tree for all encodings Query processing: Find the largest prefix-match between d new and documents from D 26 / 34
50 Algorithm for Hierarchical Schema Preprocessing: Encode every document as a 2k 1 sequence, every odd element lies in range [1..k], every even is 0 or 1 Construct a lexicographic tree for all encodings Query processing: Find the largest prefix-match between d new and documents from D By two theorems above with very high probability maximal prefix-match is very close to maximal intersection 26 / 34
51 Part III Concluding Remarks Overview of related research Open problems 27 / 34
52 Overview of Related Research Famous computational problems that need scalable algorithms: Nearest neighbors in vector spaces Nearest neighbors in abstract metric spaces Connection subgraph problem Collaborative filtering Mining association rules Indexing with errors 28 / 34
53 Overview of Related Research Famous computational problems that need scalable algorithms: Nearest neighbors in vector spaces Nearest neighbors in abstract metric spaces Connection subgraph problem Collaborative filtering Mining association rules Indexing with errors Common approach: heuristical algorithm + experimental validation 28 / 34
54 Overview of Related Research Famous computational problems that need scalable algorithms: Nearest neighbors in vector spaces Nearest neighbors in abstract metric spaces Connection subgraph problem Collaborative filtering Mining association rules Indexing with errors Common approach: heuristical algorithm + experimental validation Alternative: randomized model of input + probabilistic analysis 28 / 34
55 Overview of Related Research Famous computational problems that need scalable algorithms: Nearest neighbors in vector spaces Nearest neighbors in abstract metric spaces Connection subgraph problem Collaborative filtering Mining association rules Indexing with errors Common approach: heuristical algorithm + experimental validation Alternative: randomized model of input + probabilistic analysis Alternative: realistic assumption about input + exact algorithm 28 / 34
56 Algorithms for Max-Intersection Algorithmic open problems: 1 Max-Intersection for bounded tree-width graphs 2 Max-Intersection in configuration model 3 Max-Intersection in preferential attachment model 29 / 34
57 Algorithms for Max-Intersection Algorithmic open problems: 1 Max-Intersection for bounded tree-width graphs 2 Max-Intersection in configuration model 3 Max-Intersection in preferential attachment model Conceptual open problem: 1 Find simple-but-realistic assumptions allowing required exact solution of Max-Intersection 29 / 34
58 Algorithms for Max-Intersection Algorithmic open problems: 1 Max-Intersection for bounded tree-width graphs 2 Max-Intersection in configuration model 3 Max-Intersection in preferential attachment model Conceptual open problem: 1 Find simple-but-realistic assumptions allowing required exact solution of Max-Intersection Long-term goal: to develop theoretical framework for scalability analysis of algorithms 29 / 34
59 Data Structure Complexity On-line inclusion problem Input: Family F of 2 k subsets of [1..k 2 ] Data storage after preprocessing: 2 k poly(k) Query: a set f new [1..k 2 ] Task: decide whether f F : Time for query processing: poly(k) f new f 30 / 34
60 Data Structure Complexity On-line inclusion problem Input: Family F of 2 k subsets of [1..k 2 ] Data storage after preprocessing: 2 k poly(k) Query: a set f new [1..k 2 ] Task: decide whether f F : Time for query processing: poly(k) f new f Conjecture: the on-line inclusion problem can not be solved within such time/space constraints 30 / 34
61 Call for participation Know a relevant reference? Have an idea? Find a mistake? Solved one of these problems? Knock to my office Write to me yura@logic.pdmi.ras.ru Join our informal discussions Participate in writing a follow-up paper 31 / 34
62 Highlights Strongest Connection family, including Max-Intersection v u 32 / 34
63 Highlights Strongest Connection family, including Max-Intersection v u Open problems: Max-Intersection in complex-networks models Data structure complexity of on-line inclusion problem 32 / 34
64 Highlights Strongest Connection family, including Max-Intersection v u Open problems: Max-Intersection in complex-networks models Data structure complexity of on-line inclusion problem Vielen Dank für Ihre Aufmerksamkeit! Fragen? 32 / 34
65 References (1/2) Course homepage Y. Lifshits Web research: open problems J. Zobel and A. Moffat Inverted files for text search engines C. Faloutsos, K.S. McCurley, A. Tomkins Fast discovery of connection subgraphs M.E.J. Newman The structure and function of complex networks 33 / 34
66 References (2/2) P.N. Yianilos Data structures and algorithms for nearest neighbor search in general metric spaces J. Kleinberg Two algorithms for nearest-neighbor search in high dimensions R. Agrawal and R. Srikant Fast algorithms for mining association rules in large databases M. O Connors J. Herlocker Clustering items for collaborative filtering m.pdf 34 / 34
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