Distributed Graph Algorithms
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This document presents a distributed architecture for computing all connected sub-graphs of a given graph, addressing the challenge of managing exponential numbers of sub-graphs efficiently. The proposed solution employs a master-slave architecture with task queues and bloom filters for uniqueness checks, ensuring scalability and fault tolerance. Simulation results demonstrate the effectiveness of the architecture in processing tasks across multiple machines and highlight opportunities for future improvements.
Distributed Graph Algorithms
- 1. distributed graph algorithms Generalized Architecture For Some Graph Problems Abhilash Kumar and Saurav Kumar November 10, 2015 Indian Institute of Technology Kanpur
- 3. problem statement ∙ Compute all connected sub-graphs of a given graph, in a distributed environment 2
- 4. problem statement ∙ Compute all connected sub-graphs of a given graph, in a distributed environment ∙ Develop a generalized architecture to solve similar graph problems 2
- 5. motivation
- 6. motivation ∙ Exponential number of connected sub-graphs of a given graph 4
- 7. motivation ∙ Exponential number of connected sub-graphs of a given graph ∙ Necessity to build distributed systems which utilize the worldwide plethora of distributed resources 4
- 8. approach
- 9. approach Insights ∙ Connected sub-graphs exhibit sub-structure 6
- 10. approach Insights ∙ Connected sub-graphs exhibit sub-structure ∙ Extend smaller sub-graphs by adding an outgoing edge to generate larger sub-graphs 6
- 11. approach Insights ∙ Connected sub-graphs exhibit sub-structure ∙ Extend smaller sub-graphs by adding an outgoing edge to generate larger sub-graphs ∙ Base cases are sub-graphs represented by all the edges of the graph 6
- 12. approach Algorithm to compute all connected sub-graphs ∙ Initialize: 7
- 13. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q 7
- 14. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G 7
- 15. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G ∙ Create a sub-graph G’ representing the edge 7
- 16. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G ∙ Create a sub-graph G’ representing the edge ∙ Push G’ to Q 7
- 17. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G ∙ Create a sub-graph G’ representing the edge ∙ Push G’ to Q ∙ Process: 7
- 18. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G ∙ Create a sub-graph G’ representing the edge ∙ Push G’ to Q ∙ Process: ∙ while Q is not empty 7
- 19. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G ∙ Create a sub-graph G’ representing the edge ∙ Push G’ to Q ∙ Process: ∙ while Q is not empty ∙ G = Q.pop() 7
- 20. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G ∙ Create a sub-graph G’ representing the edge ∙ Push G’ to Q ∙ Process: ∙ while Q is not empty ∙ G = Q.pop() ∙ Save G 7
- 21. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G ∙ Create a sub-graph G’ representing the edge ∙ Push G’ to Q ∙ Process: ∙ while Q is not empty ∙ G = Q.pop() ∙ Save G ∙ For each outgoing edge E of G G’ = G U E if G’ has not been seen yet Push G’ to Q 7
- 22. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G ∙ Create a sub-graph G’ representing the edge ∙ Push G’ to Q ∙ Process: ∙ while Q is not empty ∙ G = Q.pop() ∙ Save G ∙ For each outgoing edge E of G G’ = G U E if G’ has not been seen yet Push G’ to Q 7
- 23. approach Algorithm to compute all connected sub-graphs ∙ Initialize: ∙ Queue Q ∙ For each edge in G ∙ Create a sub-graph G’ representing the edge ∙ Push G’ to Q ∙ Process: ∙ while Q is not empty ∙ G = Q.pop() ∙ Save G ∙ For each outgoing edge E of G G’ = G U E if G’ has not been seen yet Push G’ to Q 7
- 24. approach Figure: Generating initial sub-graphs from a given graph 8
- 25. approach Figure: Extending a sub-graph to generate new sub-graphs 9
- 26. approach Figure: Consider only unique sub-graphs generated for further processing 10
- 27. architecture
- 28. architecture Master-Slave Architecture ∙ Commonly used approach for parallel and distributed applications 12
- 29. architecture Master-Slave Architecture ∙ Commonly used approach for parallel and distributed applications ∙ Message passing to communicate over TCP 12
- 30. architecture Master-Slave Architecture ∙ Commonly used approach for parallel and distributed applications ∙ Message passing to communicate over TCP ∙ Master assigns tasks to slaves and finally collects the results 12
- 31. architecture Master-Slave Architecture ∙ Commonly used approach for parallel and distributed applications ∙ Message passing to communicate over TCP ∙ Master assigns tasks to slaves and finally collects the results ∙ A Task object represents a sub-graph which contains all necessary information to process that sub-graph 12
- 32. architecture Master-Slave Architecture ∙ Commonly used approach for parallel and distributed applications ∙ Message passing to communicate over TCP ∙ Master assigns tasks to slaves and finally collects the results ∙ A Task object represents a sub-graph which contains all necessary information to process that sub-graph ∙ A slave may request a task from other slaves when its task queue is empty and processing ends when all task queues are empty 12
- 33. architecture Task, Queue and Bloom filter ∙ A task has these information: 13
- 34. architecture Task, Queue and Bloom filter ∙ A task has these information: ∙ A list of vertices that are already in the sub-graph 13
- 35. architecture Task, Queue and Bloom filter ∙ A task has these information: ∙ A list of vertices that are already in the sub-graph ∙ A list of edges that can be extended in the next step 13
- 36. architecture Task, Queue and Bloom filter ∙ A task has these information: ∙ A list of vertices that are already in the sub-graph ∙ A list of edges that can be extended in the next step ∙ Task Queue 13
- 37. architecture Task, Queue and Bloom filter ∙ A task has these information: ∙ A list of vertices that are already in the sub-graph ∙ A list of edges that can be extended in the next step ∙ Task Queue ∙ Each slave has a task queue 13
- 38. architecture Task, Queue and Bloom filter ∙ A task has these information: ∙ A list of vertices that are already in the sub-graph ∙ A list of edges that can be extended in the next step ∙ Task Queue ∙ Each slave has a task queue ∙ Slave picks up a task from its task queue and processes it 13
- 39. architecture Task, Queue and Bloom filter ∙ A task has these information: ∙ A list of vertices that are already in the sub-graph ∙ A list of edges that can be extended in the next step ∙ Task Queue ∙ Each slave has a task queue ∙ Slave picks up a task from its task queue and processes it ∙ Newly generated unique tasks are pushed into the task queue 13
- 40. architecture Task, Queue and Bloom filter ∙ A task has these information: ∙ A list of vertices that are already in the sub-graph ∙ A list of edges that can be extended in the next step ∙ Task Queue ∙ Each slave has a task queue ∙ Slave picks up a task from its task queue and processes it ∙ Newly generated unique tasks are pushed into the task queue ∙ Bloom filter 13
- 41. architecture Task, Queue and Bloom filter ∙ A task has these information: ∙ A list of vertices that are already in the sub-graph ∙ A list of edges that can be extended in the next step ∙ Task Queue ∙ Each slave has a task queue ∙ Slave picks up a task from its task queue and processes it ∙ Newly generated unique tasks are pushed into the task queue ∙ Bloom filter ∙ We use Bloom filter to check uniqueness of the newly generated tasks (i.e. sub-graphs) 13
- 42. architecture Task, Queue and Bloom filter ∙ A task has these information: ∙ A list of vertices that are already in the sub-graph ∙ A list of edges that can be extended in the next step ∙ Task Queue ∙ Each slave has a task queue ∙ Slave picks up a task from its task queue and processes it ∙ Newly generated unique tasks are pushed into the task queue ∙ Bloom filter ∙ We use Bloom filter to check uniqueness of the newly generated tasks (i.e. sub-graphs) ∙ Bloom filter is also distributed so that none of the servers get loaded 13
- 43. architecture Bloom Filter Vs Hashing ∙ Used bloom filter because its very space efficient 14
- 44. architecture Bloom Filter Vs Hashing ∙ Used bloom filter because its very space efficient ∙ Space required to get error probability of p is −n × ln p (ln 2)2 bits 14
- 45. architecture Bloom Filter Vs Hashing ∙ Used bloom filter because its very space efficient ∙ Space required to get error probability of p is −n × ln p (ln 2)2 bits ∙ Error probability can be reduced with very little extra space 14
- 46. architecture Bloom Filter Vs Hashing ∙ Used bloom filter because its very space efficient ∙ Space required to get error probability of p is −n × ln p (ln 2)2 bits ∙ Error probability can be reduced with very little extra space ∙ Hashing can be used to make the algorithm deterministic 14
- 47. architecture Bloom Filter Vs Hashing ∙ Used bloom filter because its very space efficient ∙ Space required to get error probability of p is −n × ln p (ln 2)2 bits ∙ Error probability can be reduced with very little extra space ∙ Hashing can be used to make the algorithm deterministic ∙ Bloom filter can also be parallelized whereas Hashing cannot be. 14
- 48. architecture How to use this architecture? ∙ Two functions required: initialize and process 15
- 49. architecture How to use this architecture? ∙ Two functions required: initialize and process ∙ Initialize generates initial tasks. Master randomly assigns these tasks to the slaves. 15
- 50. architecture How to use this architecture? ∙ Two functions required: initialize and process ∙ Initialize generates initial tasks. Master randomly assigns these tasks to the slaves. ∙ Process defines a procedure that will generate new tasks from a given task (extend sub-graph in our case) 15
- 51. architecture How to use this architecture? ∙ Two functions required: initialize and process ∙ Initialize generates initial tasks. Master randomly assigns these tasks to the slaves. ∙ Process defines a procedure that will generate new tasks from a given task (extend sub-graph in our case) 15
- 52. architecture How to use this architecture? ∙ Two functions required: initialize and process ∙ Initialize generates initial tasks. Master randomly assigns these tasks to the slaves. ∙ Process defines a procedure that will generate new tasks from a given task (extend sub-graph in our case) 15
- 53. architecture Fitting the connected sub-graph problem ∙ Initialize creates all the tasks (sub-graphs) with one edge. 16
- 54. architecture Fitting the connected sub-graph problem ∙ Initialize creates all the tasks (sub-graphs) with one edge. ∙ Process takes a connected sub-graph and extends it by adding all extend-able edges, one at a time 16
- 55. simulation
- 56. simulation Simulation for testing ∙ Used 2 machines, say H and L. 18
- 57. simulation Simulation for testing ∙ Used 2 machines, say H and L. ∙ H: 24 core, 200 GB, Xeon E5645 @ 2.40GHz 18
- 58. simulation Simulation for testing ∙ Used 2 machines, say H and L. ∙ H: 24 core, 200 GB, Xeon E5645 @ 2.40GHz ∙ L: 4 core, 8 GB, i5-3230M CPU @ 2.60GHz 18
- 59. simulation Simulation for testing ∙ Used 2 machines, say H and L. ∙ H: 24 core, 200 GB, Xeon E5645 @ 2.40GHz ∙ L: 4 core, 8 GB, i5-3230M CPU @ 2.60GHz ∙ Opened multiple ports (6 on H, 2 on L) to mimic 8 slave servers. 18
- 60. simulation Simulation for testing ∙ Used various combinations of number of slaves on H and L 19
- 61. simulation Simulation for testing ∙ Used various combinations of number of slaves on H and L ∙ Used 2 tree graphs G(14, 13) and G(16, 15): easy to match results 19
- 62. simulation Simulation for testing ∙ Used various combinations of number of slaves on H and L ∙ Used 2 tree graphs G(14, 13) and G(16, 15): easy to match results ∙ Collected data for number of tasks processed by each slave and number of hash-check queries made by each slave. 19
- 63. simulation Simulation for testing ∙ Used various combinations of number of slaves on H and L ∙ Used 2 tree graphs G(14, 13) and G(16, 15): easy to match results ∙ Collected data for number of tasks processed by each slave and number of hash-check queries made by each slave. ∙ Collected total running time data for both graphs, including the cases of network fault. 19
- 64. results
- 65. results Figure: Number of hash check queries vs number of slaves for G(14, 13) 21
- 66. results Figure: Distribution of number of tasks processed by slaves for G(14, 13) 22
- 67. results Figure: Distribution of number of tasks processed by slaves for G(14, 13) 23
- 68. results Figure: Distribution of number of tasks processed by slaves for G(14, 13) 24
- 69. results Figure: Number of hash check queries vs number of slaves for G(16, 15) 25
- 70. results Figure: Distribution of number of tasks processed by slaves for G(16, 15) 26
- 71. results Figure: Distribution of number of tasks processed by slaves for G(16, 15) 27
- 72. results Figure: Distribution of number of tasks processed by slaves for G(16, 15) 28
- 73. results Actual Running Time ∙ Network faults happened, specially due to fewer physical machines 29
- 74. results Actual Running Time ∙ Network faults happened, specially due to fewer physical machines ∙ The architecture recovers from these faults, but a lot of time is consumed 29
- 75. results Actual Running Time ∙ Network faults happened, specially due to fewer physical machines ∙ The architecture recovers from these faults, but a lot of time is consumed ∙ For G(14, 13), running time ranged from 15s to 91s 29
- 76. results Actual Running Time ∙ Network faults happened, specially due to fewer physical machines ∙ The architecture recovers from these faults, but a lot of time is consumed ∙ For G(14, 13), running time ranged from 15s to 91s ∙ For G(15, 14), running time ranged from 255s to 447s 29
- 77. results Actual Running Time ∙ Network faults happened, specially due to fewer physical machines ∙ The architecture recovers from these faults, but a lot of time is consumed ∙ For G(14, 13), running time ranged from 15s to 91s ∙ For G(15, 14), running time ranged from 255s to 447s ∙ These are the cases when process function doesn’t do additional computation per subgraph. 29
- 78. results Figure: Running time when process does addition computation(10ms) 30
- 79. advantages
- 81. advantages Advantages ∙ Highly scalable ∙ More slaves can be added easily 32
- 82. advantages Advantages ∙ Highly scalable ∙ More slaves can be added easily ∙ Performance increases with number of slaves 32
- 83. advantages Advantages ∙ Highly scalable ∙ More slaves can be added easily ∙ Performance increases with number of slaves ∙ Even distribution of tasks: efficient machines process more tasks 32
- 84. advantages Advantages ∙ Highly scalable ∙ More slaves can be added easily ∙ Performance increases with number of slaves ∙ Even distribution of tasks: efficient machines process more tasks ∙ Architecture is very reusable 32
- 85. advantages Advantages ∙ Highly scalable ∙ More slaves can be added easily ∙ Performance increases with number of slaves ∙ Even distribution of tasks: efficient machines process more tasks ∙ Architecture is very reusable ∙ Many other problems can be solved using this architecture 32
- 86. advantages Advantages ∙ Highly scalable ∙ More slaves can be added easily ∙ Performance increases with number of slaves ∙ Even distribution of tasks: efficient machines process more tasks ∙ Architecture is very reusable ∙ Many other problems can be solved using this architecture ∙ Only need to provide 2 functions: initialize and process 32
- 87. advantages Advantages ∙ Highly scalable ∙ More slaves can be added easily ∙ Performance increases with number of slaves ∙ Even distribution of tasks: efficient machines process more tasks ∙ Architecture is very reusable ∙ Many other problems can be solved using this architecture ∙ Only need to provide 2 functions: initialize and process ∙ Network fault tolerant 32
- 88. advantages Other problems that can be solved using this paradigm ∙ Generating all cliques, paths, cycles, sub-trees, spanning sub-trees 33
- 89. advantages Other problems that can be solved using this paradigm ∙ Generating all cliques, paths, cycles, sub-trees, spanning sub-trees ∙ Can also solve few classical NP problems like finding all maximal cliques and TSP 33
- 90. future works
- 91. future works Further improvements ∙ Implement parallelized bloom filter 35
- 92. future works Further improvements ∙ Implement parallelized bloom filter ∙ Parallely solving tasks in a slave (on powerful servers) 35
- 93. future works Further improvements ∙ Implement parallelized bloom filter ∙ Parallely solving tasks in a slave (on powerful servers) ∙ Handle slave/master failures 35
- 94. future works Further improvements ∙ Implement parallelized bloom filter ∙ Parallely solving tasks in a slave (on powerful servers) ∙ Handle slave/master failures ∙ Using file I/O to store task queue for large problems 35
- 95. future works Further improvements ∙ Implement parallelized bloom filter ∙ Parallely solving tasks in a slave (on powerful servers) ∙ Handle slave/master failures ∙ Using file I/O to store task queue for large problems ∙ Exploring this paradigm to solve other problems 35
- 96. conclusion
- 97. conclusion Conclusion ∙ The algorithm is very efficient, total computation is not greater than m * T, where T is the minimum computation required to find all sub-graphs and m is number of edges. 37
- 98. conclusion Conclusion ∙ The algorithm is very efficient, total computation is not greater than m * T, where T is the minimum computation required to find all sub-graphs and m is number of edges. ∙ In practice time complexity is c*T where c is much smaller. Bound on c can be improved to min(m, log T). 37
- 99. conclusion Conclusion ∙ The algorithm is very efficient, total computation is not greater than m * T, where T is the minimum computation required to find all sub-graphs and m is number of edges. ∙ In practice time complexity is c*T where c is much smaller. Bound on c can be improved to min(m, log T). ∙ As we are interested in finding all connected sub-graph, T better not be very large. 37
- 100. conclusion Conclusion ∙ The algorithm is very efficient, total computation is not greater than m * T, where T is the minimum computation required to find all sub-graphs and m is number of edges. ∙ In practice time complexity is c*T where c is much smaller. Bound on c can be improved to min(m, log T). ∙ As we are interested in finding all connected sub-graph, T better not be very large. ∙ The architecture help us solve this problem in much scalable manner and significantly reduces the time of computation provided good infrastructure and better implementation. 37
- 101. Questions? Implementation of the algorithm and the architecture available at github.com/abhilak/DGA Slides created using Beamer(mtheme) and plot.ly on ShareLaTeX 38
- 102. Thank You 39