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Improved dynamic reachability algorithms for directed graphs Liam Roditty and Uri Zwick Tel Aviv University

Dynamic reachability Transitive closure matrix The dynamic graph Operations Delete (1,5) (4,1) Insert (5,1) (5,2) (5,4) Reach? (1,4) Delete (2,3) (6,7) (8,5) 1 1 1 1 8 1 1 1 1 7 1 1 1 1 6 1 1 1 1 5 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 4 2 1 5 6 8 7 3 4 2 1 5 6 8 7 3 4 2 1 5 6 8 7 3 4 2 1 5 6 8 7

Decremental reachability - Results Baswana Hariharan Sen ’02 mn 4/3 1 Monte Carlo General RZ ’02 mn 1 Las Vegas General Demetrescu Italiano ’00 n 3 1 Deterministic General La Poutré van Leeuwen ’87 FMNZ ’01 m 2 1 Deterministic General Henzinger King ’95 mn log 2 n n log n Monte Carlo General Italiano ’88 mn 1 Deterministic DAGs Authors Total update time Query time Algorithm Graphs

Fully dynamic reachability - Results Roditty ’03 n 2 1 Deterministic General Demetrescu Italiano ’00 n 2 1 Deterministic General King ’99 n 2 log n 1 Deterministic General Authors Amortized update time Query time Algorithm Graphs

Fully dynamic reachability - Results Demetrescu, Italiano ’00 n 1.58 n 0.58 Monte Carlo DAGs m 0.58 n m 0.43 Monte Carlo General RZ ’02 mn 1/2 n 1/2 Deterministic General m 0.58 n n log n Monte Carlo General Henzinger King ’95 mn 1/2 log 2 n n log n Monte Carlo General RZ ’02 m n log n Deterministic DAGs Authors Amortized update time Query time Algorithm Graphs

Decremental maintenance of a reachability tree in a DAG – Italiano ’s algorithm Every edge is only examined once! Total complexity is O(m) per tree.

Decremental maintenance of a reachability tree in a general graph Frigioni, Miller, Nanni and Zaroliagis ’01 The graph induced on the Strongly Connected Components (SCCs) of a graph is a DAG. Maintain a reachability tree of SCCs ! If a deleted edge connects two different SCCs, use Italiano’s algorithm. If a deleted edge is in a SCC, and the SCC remains strongly connected, do nothing.

How do we maintain the SCCs? ,[object Object],[object Object],[object Object]

Decremental maintenance of a BFS tree in a general graph Even, Shiloach ’81 / Henzinger, King ’95 Every edge is only examined once per level! Total complexity is O(mn).

Detecting the decomposition of a SCC ,[object Object],[object Object],[object Object],w

When a SCC decomposes w w w 4 w 2 w 1 w 3 Total cost: mn + m 1 n 1 +m 2 n 2 +m 3 n 3 +m 4 n 4 + … = O(mn) ???

Choice of representatives w Choose a RANDOM representative !!! Expected running time is then O(mn) !!! w w

Decremental SCCs - Analysis Let be the expected total running time.

Fully dynamic reachability (after Henzinger-King ’95) G Decremental data structure … v 1 v 2 v t Initialize a decremental data structure O(mn) time Insert(E v ) – build/rebuild In(v) and Out(v). O(m) time. Reach?(u,v) – Query the decremental data structure and each pair of trees. O(t) time Delete(E’) – Update the decremental data structure and rebuild all trees. O(mt) time. When t=n 1/2 , restart. Amortized cost per update – O(mn 1/2 ) Worst-case query time – O(n 1/2 )

Decremental reachability – Open problems ,[object Object],[object Object],[object Object],[object Object]

Fully dynamic reachability – Open problems ,[object Object],[object Object]

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Dynamic uri

1. Improved dynamic reachability algorithms for directed graphs Liam Roditty and Uri Zwick Tel Aviv University

2. Dynamic reachability Transitive closure matrix The dynamic graph Operations Delete (1,5) (4,1) Insert (5,1) (5,2) (5,4) Reach? (1,4) Delete (2,3) (6,7) (8,5) 1 1 1 1 8 1 1 1 1 7 1 1 1 1 6 1 1 1 1 5 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 4 2 1 5 6 8 7 3 4 2 1 5 6 8 7 3 4 2 1 5 6 8 7 3 4 2 1 5 6 8 7

3. Decremental reachability - Results Baswana Hariharan Sen ’02 mn 4/3 1 Monte Carlo General RZ ’02 mn 1 Las Vegas General Demetrescu Italiano ’00 n 3 1 Deterministic General La Poutré van Leeuwen ’87 FMNZ ’01 m 2 1 Deterministic General Henzinger King ’95 mn log 2 n n log n Monte Carlo General Italiano ’88 mn 1 Deterministic DAGs Authors Total update time Query time Algorithm Graphs

4. Fully dynamic reachability - Results Roditty ’03 n 2 1 Deterministic General Demetrescu Italiano ’00 n 2 1 Deterministic General King ’99 n 2 log n 1 Deterministic General Authors Amortized update time Query time Algorithm Graphs

5. Fully dynamic reachability - Results Demetrescu, Italiano ’00 n 1.58 n 0.58 Monte Carlo DAGs m 0.58 n m 0.43 Monte Carlo General RZ ’02 mn 1/2 n 1/2 Deterministic General m 0.58 n n log n Monte Carlo General Henzinger King ’95 mn 1/2 log 2 n n log n Monte Carlo General RZ ’02 m n log n Deterministic DAGs Authors Amortized update time Query time Algorithm Graphs

6. Decremental maintenance of a reachability tree in a DAG – Italiano ’s algorithm Every edge is only examined once! Total complexity is O(m) per tree.

7. Decremental maintenance of a reachability tree in a general graph Frigioni, Miller, Nanni and Zaroliagis ’01 The graph induced on the Strongly Connected Components (SCCs) of a graph is a DAG. Maintain a reachability tree of SCCs ! If a deleted edge connects two different SCCs, use Italiano’s algorithm. If a deleted edge is in a SCC, and the SCC remains strongly connected, do nothing.

8. When a SCC decomposes

10. Decremental maintenance of a BFS tree in a general graph Even, Shiloach ’81 / Henzinger, King ’95 Every edge is only examined once per level! Total complexity is O(mn).

11.

12. When a SCC decomposes w w w 4 w 2 w 1 w 3 Total cost: mn + m 1 n 1 +m 2 n 2 +m 3 n 3 +m 4 n 4 + … = O(mn) ???

13. Choice of representatives w Choose a RANDOM representative !!! Expected running time is then O(mn) !!! w w

14. Decremental SCCs - Analysis Let be the expected total running time.

15. Decremental SCCs - Analysis

16. Fully dynamic reachability (after Henzinger-King ’95) G Decremental data structure … v 1 v 2 v t Initialize a decremental data structure O(mn) time Insert(E v ) – build/rebuild In(v) and Out(v). O(m) time. Reach?(u,v) – Query the decremental data structure and each pair of trees. O(t) time Delete(E’) – Update the decremental data structure and rebuild all trees. O(mt) time. When t=n 1/2 , restart. Amortized cost per update – O(mn 1/2 ) Worst-case query time – O(n 1/2 )

17.

18.

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