A Batch Algorithm for Maintaining a Topological Order David J. Pearce and Paul H.J. Kelly Victoria University of Wellington, New Zealand & Imperial College, London

Topological Orders

Updating Topological Orders

Affected Region

Updating Topological Orders Affected Region

Updating Topological Orders Unit Change Problem : Pearce, et al. 1990: Alpern, et al. 1996: Marchetti-Spaccamela, et al. 2006: Ajwani, et al. 2009: Bender, et al. 2008: Haeupler, et al. 2005: Katriel, et al. 2007: Liu, et al.

Batch Problem Definition: Given a batch of edge insertions, update the topological order Q) Can we do better than by processing them one at a time?

Batch Problem Definition: Given a batch of edge insertions, update the topological order Q) Can we do better than by processing them one at a time? A) Yes!

Previous MNR Algorithm Discovery – DFS to identify nodes reachable from tail in affected region

Previous MNR Algorithm Discovery – DFS to identify nodes reachable from tail in affected region

Previous MNR Algorithm Discovery – DFS to identify nodes reachable from tail in affected region

Previous MNR Algorithm Discovery – DFS to identify nodes reachable from tail in affected region

Previous MNR Algorithm Discovery – DFS to identify nodes reachable from tail in affected region

Previous MNR Algorithm Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

Previous MNR Algorithm Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

Previous MNR Algorithm Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

Previous MNR Algorithm Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

Previous MNR Algorithm Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

Previous MNR Algorithm Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

MNR Algorithm Shift – marked nodes up the order, unmarked nodes down the order (in affected region)

Our Novel Batch Algorithm Observation – edge insertions which overlap can be processed independently of others

Our Novel Batch Algorithm Observation – edge insertions which overlap can be processed independently of others Affected Region

Our Novel Batch Algorithm Discovery – identify “target node” for each node reachable from an insertion in affected region

Our Novel Batch Algorithm Discovery – identify “root node” for each node reachable from an insertion in affected region

Our Novel Batch Algorithm Discovery – identify “root node” for each node reachable from an insertion in affected region

Our Novel Batch Algorithm Discovery – identify “root node” for each node reachable from an insertion in affected region

Our Novel Batch Algorithm Discovery – identify “root node” for each node reachable from an insertion in affected region

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Our Novel Batch Algorithm Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target

Results on Sparse Random Graphs

Results on Denser Random Graphs

Applications Extensions: –Incremental Cycle Detection Applications: –Pointer Analysis –Software Model Checking –Constraint-based Local Search Algorithms –Dead-lock Detection –Machine Learning –Multiple Sequence Alignment

Conclusion That’s all folks!