Dynamic Shortest Paths

1. Dynamic Shortest Paths Camil Demetrescu University of Rome “La Sapienza” Giuseppe F. Italiano University of Rome “Tor Vergata”

2. update weight of edge (u,v) to w Update(u,v,w): return distance from x to y Query(x,y): (or shortest path from x to y) Dynamic (All Pairs) Shortest Paths Given a weighteddirectedgraph G=(V,E,w),perform any intermixed sequence of the following operations:

3. Previous work on dynamic APSP # papers 65-69 70-74 75-79 80-84 85-89 90-94 95-99 00-

4. Graph Weight Update Query ~ reals O(n3) Update recomputingfrom scratch general O(1) ~ [0,C] O(n2.575 C0.681) King 99 general [0,C] O(n2.5 (C log n)0.5) O(1) Henzinger et al. 97 planar [0,C] O(n9/7 log(nC)) Fackcharoemphol, Rao 01 planar reals O(n9/5 log13/5 n) general reals ? < o(n3) O(1) D, Italiano 01 general S reals O(n2.5 (S log3 n)0.5) O(1) Previous work on fully dynamic APSP

5. New combinatorial properties of graphs: Uniform paths A new approach

6. Graph Weight Update Query King 99 general [0,C] O(n2.5 (C log n)0.5) O(1) Henzinger et al. 97 planar [0,C] O(n9/7 log(nC)) Fackcharoemphol, Rao 01 planar reals O(n9/5 log13/5 n) D, Italiano 01 general S reals O(n2.5 (S log3 n)0.5) O(1) D, Italiano 02 general reals O(n2 log n) O(1) A new fully dynamic algorithm

7. (n) (n) (n2) changes per update +1 -1 +1

8. A path is uniform if all of its subpaths are shortest paths x y πxy Shortest path Shortest path Not a shortest path Shortest path x y πxy Uniform paths UNIFORM NOT UNIFORM

9. Uniform paths Shortest paths Properties of Uniform paths Theorem I Shortest paths  Uniform paths

10. For the sake of simplicity, we assume that no two paths in the graph have the same weight. Theorem II Uniform paths πxy are internally vertex-disjoint π1 π2 x y π3 Properties of Uniform paths (Ties can be broken by adding a tiny fraction to the weight of each edge)

11. x y Properties of Uniform paths Theorem III There are at most n-1 uniform paths connecting x,y This is a consequence of vertex-disjointess…

12. We call dynamic graph a sequence of graphs <G0,…,Gk> such that, for any t, Gt-1and Gt differ in the weight of exactly one edge Dynamic graphs

13. A uniform path in Gt is appearing if it is not uniform in Gt-1 Gt-1 x y Gt x y Uniform paths in dynamic graphs

14. Gt x y A uniform path in Gt is disappearing if it is not uniform in Gt+1 Gt+1 x y Uniform paths in dynamic graphs

15. Theorem IV At most n uniform paths can appear after an increase At most 1 uniform path can disappear after an increase Corollary The amortized number of appearing uniform paths per update in an increase-only sequence is O(1) Uniform paths in dynamic graphs

16. What about fully dynamic sequences? 10 10 20 20 x x y y 100 30 30 The weights of uniform paths that disappear and then reappear do not change… 40 40 Uniform paths in dynamic graphs

17. We conjecture that the bound is O(1)… Uniform paths in dynamic graphs Theorem V For any pair (x,y), the amortized number of uniform paths πxyappearingwith a new weight per update in a fully dynamic sequence is O(log n)

18. The shortest path is the same, but has different weight: EASY EASY x y  The shortest path is different (update = decrease): x y - Shortest paths and edge weight updates How does a shortest path change after an update?

19. a b + HARD The shortest path is different (update = increase): If we look closer, we realize that the new shortest path from a to b was already uniform before the update! x y Shortest paths and edge weight updates

20. For each pair x,y, maintain in a data structure uniform pathsconnecting x to y Main idea: The combinatorial properties of uniform paths imply that only a small piece of information needs to be updated at each time… Are we done? A new approach to dynamic APSP NO

21. x x y y x y This path stays the same while flipping between uniform and non-uniform: We would like to have an update algorithm that pays only once for it over the whole sequence... How to pay only once?

22. may x y resurrect! x y This path remains a shortest path after the insertion This path is no longer a shortest path after the insertion… Looking at the substructure …but if we removed the edge it would get a shortest path again!

23. Zombies A path is azombieif it used to be a shortest path, and its edges have not been updated since then

24. shortest path x Uniform path πxy y shortest path shortest path or zombie x Potentiallyuniform path πxy y shortest path or zombie Potentially uniform paths Relaxed notion of uniformity: Subpaths do not need to be shortest at the same time

25. Theorem I Uniform paths  Potentially uniform paths Uniform paths Shortest paths Properties of potentially uniform paths Potentiallyuniform paths

26. Properties of potentially uniform paths Theorem II O(zn2) zombies at any time O(zn2) new potentially uniform paths per update

27. A lot! We can construct a dynamic graph with (n3) zombies at any time, amortized. 100 100 100 (n) (n) 90 90 100 90 80 80 (n) 70 = (n3) # zombies = (n2) +(n2) +(n2) (n) How many zombies can we have?

28. 100 90 100 90 80 100 100 100 100 70 100 100 (n) (n) 90 90 90 90 100 90 90 80 80 80 80 (n) 70 70 (n2) (n2) (n2) 0 (in general, O(n2)) Reducing # of zombies: Smoothing At each update we pick an edge with the maximum number of zombies passing through it, and we remove and reinsert it # zombies =

29. For each pair x,y, maintain in a data structure the potentially uniform paths connecting x to y Main idea: The combinatorial properties of potentially uniform paths imply that, if we do smoothing, we have only O(n2) new potentially uniform paths per update, amortized… A new approach to dynamic APSP (II) There exists and updatealgorithm that spendsO(log n) time per potentially uniform path

30. EASY + The shortest path is different (update = increase): If we maintain potentially uniform paths using priority queues, we can find this path in O(1) time! x y Handling the hard case

31. Uniform paths are the heart of dynamic shortest paths Combinatorially well-behaved in dynamic graphs New approach based on maintaining potentially uniform paths Applications to Railway Optimization Problems? Solves the problem in its full generality Conclusions