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This document describes a 3-round MapReduce algorithm for finding small dense subgraphs within a large graph. In Round 1, it computes the graph density by counting edges and nodes. Round 2 partitions the graph and prunes edges with degrees lower than the density. Round 3 iteratively removes the lowest degree node to find the smallest subgraph. The algorithm was tested on real graphs and found optimal solutions with runtimes of minutes or less depending on the graph size and pruning threshold.
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- 1. Finding Small Dense Subgraphs Big Data Computing 2014 – 2015 Pavel Popa – Rui Madeira
- 2. Algorithm 3 Rounds of map-reduce Based on graph pruning to eliminate the edges that least contribute to the best solution Graph pruning is done in parallel based on graph partitioning Data structures prepared to have optimal access time to nodes 1
- 3. Round 1 – Computing graph density Mapper: Goes through all the graph outputting all the edges to the same Reducer Reducer: Goes through the list received calculating the graph density by storing the number of edges and nodes Outputs the complete list of edges and the graph density 2
- 4. Round 2 – Partitioning and Pruning the graph Mapper: Goes through all the edges and for each one it ouputs a <SubgraphID, edge> key value pair SubgraphID randomly chose in [0, graph density) Reducer: Each reducer takes care of a subgraph Goes through the edges list, if both endpoints of the edge have degree > graph density, it outputs it 3
- 5. Subgraph 0 Subgraph 1 Round 2 – Partitioning and Pruning the graph 4
- 6. Round 3 – Finding the smallest subgraph Mapper: Outputs all the edges to the same reducer Reducer: Iterates until there are no nodes in the graph It removes the node with minimum degree at each step It calculates the new graph density and number of nodes If it’s better, stores the number of nodes and the graph 5
- 7. Implementation details Importance of the number of graph partitions Balance between more parallel processing and loss of information Importance of the threshold for pruning Balance between processing time and loss of information In the last round min-heap was used to have optimal acess to the graph nodes, easily extract the minimum and reorder by decreasing the keys (the degree of the node) of its neighbors 6
- 8. Running times on Amazon AWS Round 1 Round 2 Round 3 as-skitter 1 minute 1 minute 48 seconds web-BerkStan 8 minutes 1 minute 44 seconds loc-gowalla 57 seconds 58 seconds 42 seconds For the last round the time depends on the rho, however the time variations between values are minimal 7
- 9. Results rho = 2 rho = 3 rho = 4 as-skitter 5 - 10 7 - 21 9 - 36 web-BerkStan 5 - 10 7 - 21 9 - 36 loc-gowalla 5 - 10 7 - 21 9 - 36 Results obtained with the first correct run of the algorithm. No improvement was made since an optimal solution was found 8