1. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Department of Information Technology Indian Institute of Information Technology and Management Gwalior Optimized Graph Search Using Multi-Level Graph Clustering Rahul Kala, Department of Information Technology Kala, Rahul, Shukla, Anupam & Tiwari, Ritu (2009) Optimized Graph Search using Multi Level Graph Clustering, Proceedings of the Springer International Conference on Contemporary Computing, IC3'09, pp 103-114, Noida, India

2. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Breadth First Search Depth First Search Iterative Deepening Search A* Algorithm Heuristic Search Algorithm Graph Searching Algorithms Multi Neuron Heuristic Search

3. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 The Basic Idea Pre-Cluster the graph at various levels

4. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Multi Level Graph Clustering - I

5. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Multi Level Graph Clustering - II

6. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Multi Level Graph Clustering - III

7. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Multi Level Graph Clustering - IV

8. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Multi Level Graph Clustering - V

9. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Clustering Algorithm g ← original graph for i = 1 to A g ← makeCluster(g) Is there change in g No Yes break Add g to graphs

10. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Making Clusters 12 34 5 7 8 6 9 9 5 5 10

11. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Making Clusters Algorithm Make Clusters() Step1: While more clusters are possible Step2: c ← getNextCluster() Step3:for each vertex v in c Step4:Delete v from graph and delete all edges from/to it Step5:Add a new unique vertex v 2 Step6:Add edges from/to v 2 that were deleted from the graph Step7:Add information of cluster v 2 to set of clusters in the particular level

12. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Star Vertex

13. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Selecting Nodes of Cluster getNextCluster() Step1: Find the vertex v in graph with maximum edges Step2: If the maximum edges are less than α then return null Step3: c ← all vertices that are at a maximum distance of 2 units away from v Step4: Sort c in order of decreasing number of edges of vertices Step5: Select any 3 vertices v 1, v 2, v 3 in c such that all 3 vertices are connected to β common vertices Step6: c 2 ← all vertices in c that are connected to v 1 and v 2 and v 3 Step7: Add all vertices in c to c 2 that are connected to at least 4 vertices already present in c 2 Step8: Return c 2

14. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Department of Information Technology Indian Institute of Information Technology and Management Gwalior Graph Search

15. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Point Search Source Goal Source Goal Source Goal Source and Goal

16. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Point Algorithm Level ← 0 While there are graphs in current level Is source member of any cluster of this level Source ← Cluster number where it was found Is goal member of any cluster of this level Goal ← Cluster number where it was found Add source, destination to point set Level ← Level + 1 Yes No

17. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Search Algorithm Source Goal Source Goal Source Goal Source and Goal

18. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Search Algorithm Search() Step1: Solution ← null Step2: For each (source, destination) in point set Step3:Solution 2 ← start + all vertices in solution + destination Step4:If any vertex in solution 2 is a cluster of the higher level Step5:Replace that vertex with the star vertex of that cluster Step6:Solution ← null Step7:For all adjacent vertices (v 1,v 2 ) in Solution 2, taken in order Step8:Solution ← Solution + bfs(current level graph,v 1,v 2 ) – v 2 Step9:Solution ← Solution + destination

19. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Adding and Modifying Nodes Needs to be done onceTime ExpensiveTemporary Addition

20. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Applications

21. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Analogy in Social Networking

22. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Comparisons Proposed Algorithm Search Time LessAlmost Best Results Adding/ Modifying Expensive Work for special Graphs Other Graph Algorithms Search Time MoreBest ResultsNo extra penaltyWork for all graphs

23. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Results

24. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Department of Information Technology Indian Institute of Information Technology and Management Gwalior Conclusions

25. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Future Scope Validation against actual data Weighted Graphs Clustering Criterion Tradeoff between loss of result quality with time Type of graphs

26. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 More interesting algorithms at:

27. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 References 1. Anders Karl-Heinrich, ‘A Hierarchical Graph-Clustering Approach to find Groups of Objects’, In ICA Commission on Map Generalization, 5th Workshop on Progress in Automated Map Generalization (2003) 2. Arcaute Esteban, Chen Ning, Kumar Ravi, Liben-Nowell David, Mahdian Mohammad, Nazerzadeh Hamid, Xu Ying, ‘Deterministic Decentralized Search in Random Graphs’, Proceedings of the 5th Workshop on Algorithms and Models for the Web-Graph, (2007) 3. Brandes Ulrik, Gaertler Marco, Wagner Dorothea, ‘Experiments on Graph Clustering Algorithms’, In Proceedings of the 11th Annual European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 2832. 568--579 (2003) 4. Craswell Nick, Szummer Martin, ‘Random Walks on the Click Graph’, SIGIR Conf Research and Development in Information Retrieval, 239—246 (2007) 5. Goldberg Andrew V., Harrelson Chris, ‘Computing the Shortest Path: A* Search Meets Graph Theory’, In Proceedings of SODA, 156—165 (2005) 6. Gunter Simon, Bunke Horst, ‘Validation indices for graph clustering’, Pattern Recognition Letters 24, 1107--1113 (2003) 7. He Hao, Wang Haixun, Yang Jun, Yu Philip S,’ BLINKS: Ranked Keyword Searches on Graphs’, in the ACM International Conference on Management of Data (SIGMOD), Beijing, China.(2007)

28. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 8. Hlaoui Adel, Wang Shengrui, ‘A Graph Clustering Algorithm with Applications to Content-Based Image Retrieval’, Proceedings of the Second International Conference on Machine Learning and Cybernetics, Xi’an, (2003) 9.Kacholia Varun, Pandit Shashank, Chakrabarti Soumen, Sudarshan S., Desai Rushi, Karambelkar Hrishikesh, ‘Bidirectional Expansion For Keyword Search on Graph Databases’ ACM Proceedings of the 31st international conference on Very large data bases Trondheim, Norway (2005) 10. Najork Marc, Wiener Janet L., ‘Breadth-First Search Crawling Yields High-Quality Pages’, ACM Proceedings of the 10th international conference on World Wide Web Hong Kong, (2001) 11. Rattigan Matthew J, Maier Marc, Jensen David, ‘Graph Clustering with Network Structure Indices’, ACM Proceedings of the 24th international conference on Machine learning Corvalis, Oregon, (2007) 12. Tadikonda Satish K., Sonka Milan, Collins Steve M, ‘Efficient Coronary Border Detection Using Heuristic Graph Searching’ ieeexplore 13. Wang Yonggu, Li Xiaojuan, ‘Social Network Analysis of Interaction in Online Learning Communities’ ICALT 2007. Seventh IEEE International Conference on Advanced Learning Technologies, (2007) 14. Yushi Jing, Shumeet Baluja, ‘PageRank for Product Image Search’, WWW 2008 / Refereed Track: Rich Media, (2008) 15. Zhou Rong, Hansen Eric A, ‘Sparse-Memory Graph Search’, 18th International Joint Conference on Artificial Intelligence, Acapulco, Mexico (2003)

29. Department of Information Technology Indian Institute of Information Technology and Management Gwalior IC3’09 - 18 th August, 2009 Department of Information Technology Indian Institute of Information Technology and Management Gwalior Thank You