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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
1
23
4
76
8
5
9
Topological Orders
2 3 4 5 6 7 81 9
1
23
4
76
8
5
9
Updating Topological Orders
1
23
4
76
8
5
9
2 3 4 5 6 7 81 9
Updating Topological Orders
2 3 4 5 6 7 81 9
Affected Region
1
23
4
76
8
5
9
Updating Topological Orders
2 3 4 5 6 7 81 9
Affected Region
Updating Topological Orders
Unit Change Problem
2 3 4 5 6 7 81 92004: 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.
1
23
4
76
8
5
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1 2 3 4 5 6 7 8 9
1
23
4
76
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5
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1 2 3 4 5 6 7 8 9
1
23
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76
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1 2 3 45 6 7 8 9
1 2 3 4 5 6 7 8 9
1
23
4
76
8
5
9
1 2 3 45 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 45 6 7 89
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
2 3 4 5 6 7 81 9
Discovery – DFS to identify nodes reachable from tail in affected region
Previous MNR Algorithm
2 3 4 5 6 7 81 9
Discovery – DFS to identify nodes reachable from tail in affected region
Previous MNR Algorithm
2 3 4 5 6 7 81 9
Discovery – DFS to identify nodes reachable from tail in affected region
Previous MNR Algorithm
2 3 4 5 6 7 81 9
Discovery – DFS to identify nodes reachable from tail in affected region
Previous MNR Algorithm
2 3 4 5 6 7 81 9
Discovery – DFS to identify nodes reachable from tail in affected region
Previous MNR Algorithm
2 3 4 5 6 7 81
9
Shift – marked nodes up the order, unmarked nodes down the order (in affected region)
Previous MNR Algorithm
2 3 4 5 6 7 81
9
Shift – marked nodes up the order, unmarked nodes down the order (in affected region)
Previous MNR Algorithm
2 3 4 5 6 7 81
9
Shift – marked nodes up the order, unmarked nodes down the order (in affected region)
Previous MNR Algorithm
2 3 4 5 6 7 81
9
Shift – marked nodes up the order, unmarked nodes down the order (in affected region)
Previous MNR Algorithm
2 3 45 6 7 81
9
Shift – marked nodes up the order, unmarked nodes down the order (in affected region)
Previous MNR Algorithm
2 3 45 6 7 81
9
Shift – marked nodes up the order, unmarked nodes down the order (in affected region)
MNR Algorithm
2 3 45 6 7 81 9
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 Affected Region
Our Novel Batch Algorithm
Discovery – identify “target node” for each node reachable from an insertion in affected region
45 67 2 13
Our Novel Batch Algorithm
Discovery – identify “root node” for each node reachable from an insertion in affected region
45 67 2 13
Our Novel Batch Algorithm
Discovery – identify “root node” for each node reachable from an insertion in affected region
45 67 2 13
Our Novel Batch Algorithm
Discovery – identify “root node” for each node reachable from an insertion in affected region
45 67 2 134 44
Our Novel Batch Algorithm
Discovery – identify “root node” for each node reachable from an insertion in affected region
45 67 2 134 44 1 1
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
45 67 2 134 44 1 1
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
45 6
7
2 134 4
4
1 1
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
45 6
7
2 134 4
4
1 1
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
45 6
7 2
134 4
4 1
1
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
45 6
7 2
134 4
4 1
1
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
45 6
7 2
1
3
4 4
4 1 1
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
45 6
7 2
1
3
4 4
4 1 1
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
45 6
7
21
3
4 4
4 1
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
45 6
7
21 34 4
4
Our Novel Batch Algorithm
Shift – marked nodes up order and unmarked nodes down order; marked nodes moved just past their target
4 5 6 721 3
Results on Sparse Random Graphs
Results on Sparse Random Graphs
Results on Denser 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!
