GRAIL: Scalable Reachability Index for Large Graphs Hilmi Yldrm 1 - PowerPoint PPT Presentation

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work GRAIL: Scalable Reachability Index for Large Graphs Hilmi Yıldırım 1 Vineet Chaoji 2 Mohammed J.Zaki 1 1 Rensselaer Polytechnic Institute Troy, NY 2 Yahoo! Labs Bangalore, India 14 September VLDB 2010

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Outline Problem Definition & Motivation Background Related Work Interval Labeling Our Approach : GRAIL Index Construction Querying Experiments Experimental Setup & Datasets Results and Comparison with Other Methods Sensitivity to Different Graph Types and Parameters Conclusion & Future Work

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Problem Definition Reachability Query : Given two vertices u and v in a directed acyclic graph G, is there a path between u and v ? • Simple in undirected graphs • Any directed graph can be transformed into a dag A • Query(B,I) C B D Reachable F G E H I • Query(D,B) Not Reachable J

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Motivation Traditional Applications • Class Hierarchies, GIS, dependency graphs Trending Applications • Semantic Web • Biological networks • Citation graphs Motivation • Existing methods do not scale for large and dense graphs

Motivation

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Related Work Construction Time Query Time Index Size O ( n 2 ) Opt. Tree Cover (Agrawal et al. 89) O ( nm ) O ( n ) GRIPP (Trissl et al. 07) O ( m + n ) O ( m − n ) O ( m + n ) O ( n + m + t 3 ) O ( n + t 2 ) Dual Labeling (Wang et al. 06) O (1) PathTree (Jin et al. 08) O ( mk ) O ( mk ) / O ( mn ) O ( nk ) O ( √ m ) O ( n √ m ) O ( n 4 ) 2HOP (Cohen et al. 03) O ( √ m ) O ( n √ m ) O ( n 3 ) HOPI (Schenkel et al. 05) GRAIL (this paper) O ( d ( n + m )) O ( d ) / O ( n + m ) O ( dn ) DFS/BFS Full Transitive Closure Construction Time O ( nm ) O (1) Query Time O (1) O ( n + m ) Index Size O ( n 2 ) O (1)

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 2 • e ( u ) is the post-order value of node u 3 4 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 8 9

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 2 • e ( u ) is the post-order value of node u 3 4 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 8 1] 9

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 2 • e ( u ) is the post-order value of node u 3 4 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 8 [1,1] 9

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 2 • e ( u ) is the post-order value of node u 3 4 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 8 [1,1] 9 2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 2 • e ( u ) is the post-order value of node u 3 4 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 2 • e ( u ) is the post-order value of node u 3 4 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 3] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 2 • e ( u ) is the post-order value of node u 3 4 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 [1,3] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 2 • e ( u ) is the post-order value of node u 3 [1,4] 4 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 [1,3] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 2 • e ( u ) is the post-order value of node u 3 [1,4] 4 [5,5] 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 [1,3] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 [1,6] 2 • e ( u ) is the post-order value of node u 3 [1,4] 4 [5,5] 5 • s ( u ) is the min of e ( v ) where u ⇒ v 6 7 [1,3] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a Tree 0 [1,10] Post-Order Labeling • Interval of u is [ s ( u ) , e ( u )] 1 [1,6] 2 [7,9] • e ( u ) is the post-order value of node u 3 [1,4] 4 [5,5] 5 [7,8] • s ( u ) is the min of e ( v ) where u ⇒ v 6 [7,7] 7 [1,3] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a DAG 0 1 2 3 4 5 6 7 8 9

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a DAG 0 1 2 3 4 5 6 7 8 9

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a DAG 0 [1,10] • False positives on DAGs such as 6 − > 9 1 [1,6] 2 [1,9] 3 [1,4] 4 [1,5] 5 [1,8] 6 [1,7] 7 [1,3] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a DAG 0 [1,10] • False positives on DAGs such as 6 − > 9 1 [1,6] 2 [1,9] • Variants of Interval Labeling – Tree Cover 3 [1,4] 4 [1,5] 5 [1,8] • Optimal Tree Cover • GRIPP • PathTree 6 [1,7] 7 [1,3] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a DAG 0 [1,10] • False positives on DAGs such as 6 − > 9 1 [1,6] 2 [7,9] • Variants of Interval Labeling – Tree Cover 3 [1,4] 4 [5,5] 5 [7,8] • Optimal Tree Cover • GRIPP • PathTree 6 [7,7] 7 [1,3] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a DAG 0 [1,10] • False positives on DAGs such as 6 − > 9 1 [1,6] 2 [7,9] • Variants of Interval Labeling – Tree Cover 3 [1,4] 4 [5,5] 5 [7,8] • Optimal Tree Cover • GRIPP • PathTree 6 [7,7] 7 [1,3] [1,1] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a DAG 0 [1,10] • False positives on DAGs such as 6 − > 9 1 [1,6] 2 [7,9] • Variants of Interval Labeling – Tree Cover 3 [1,4] 4 [5,5] 5 [7,8] • Optimal Tree Cover [1,1] • GRIPP • PathTree 6 [7,7] 7 [1,3] [1,1] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work Interval Labeling on a DAG 0 [1,10] • False positives on DAGs such as 6 − > 9 1 [1,6] 2 [7,9] • Variants of Interval Labeling [1,4] – Tree Cover 3 [1,4] 4 [5,5] 5 [7,8] • Optimal Tree Cover [1,1] • GRIPP • PathTree 6 [7,7] 7 [1,3] [1,1] 8 [1,1] 9 [2,2]

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work GRAIL : Graph Reachability Indexing via RAndomized Interval Labeling Key Observations • No false negatives. • Interval labeling is repeatable with different traversals.

Problem Definition & Motivation Background Our Approach : GRAIL Experiments Conclusion & Future Work GRAIL : Graph Reachability Indexing via RAndomized Interval Labeling Key Observations • No false negatives. • Interval labeling is repeatable with different traversals. GRAIL Index Construction • For each dimension of the index • Generate a randomized post-order labeling • Each label corresponds to a dimension of the hyperrectangle that node represents. • Each new dimension reduces the number of exceptions.