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Dynamic Matchings in Convex Bipartite Graphs Irit Katriel Brown University Loukas Georgiadis Hewlett-Packard Laboratories Kristoffer Arnsfelt Hansen University.

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Presentation on theme: "Dynamic Matchings in Convex Bipartite Graphs Irit Katriel Brown University Loukas Georgiadis Hewlett-Packard Laboratories Kristoffer Arnsfelt Hansen University."— Presentation transcript:

1 Dynamic Matchings in Convex Bipartite Graphs Irit Katriel Brown University Loukas Georgiadis Hewlett-Packard Laboratories Kristoffer Arnsfelt Hansen University of Chicago Gerth Stølting Brodal University of Aarhus MFCS 2007, Hotel Růže, Český Krumlov, Czech Republic, August 26-31, 2007

2 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 2 Outline of Talk Definitions: Graphs and matchings Definitions: Convex bipartite graphs Glovers algorithm for convex bipartite graphs Definitions: Matchings in dynamic convex bipartite graphs Result Ingredients of the solution Conclusion

3 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 3 Matchings in General Graphs Deterministic O(E V) Micali, Vaziani ’80 Randomized O(V 2.476 ) Mucha, Sankowski ’04

4 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 4 Matchings in Bipartite Graphs Y X Deterministic O(E V) Hopcroft, Karp ’73 Randomized O(V 2.476 ) Mucha, Sankowski ’04

5 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 5 Convex Bipartite Graphs Y X 1 76 5432 v t(v)s(v)s(v) [4,7][3,3][1,2][2,4][1,2]

6 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 6 Matchings in Convex Bipartite Graphs Y X Deterministic O(X+Y) Gabow, Tarjan ’85 Deterministic O(X) Steiner, Yeomans ’96 1 76 5432 [4,7][3,3][1,2][2,4][1,2] (time) (jobs to schedule)

7 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 7 Matchings in Convex Bipartite Graphs: Glover’s Greedy Algorithm Y X 1 76 5432 [4,7][3,3][1,2][2,4][1,2] For the current time y maintain a priority queue of the t(v) where y [s(v),t(v)]

8 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 8 Dynamic Matchings in Convex Bipartite Graphs Observation O(1) edge changes can change Ω(X) edges in the matching but only O(1) nodes in the matching need to change Y X

9 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 9 Dynamic Convex Bipartite Graphs: Updates Insert (x,[s(x),t(x)]) Delete (x) s(x)s(x)t(x)t(x) x Insert (y) Delete (y) y Note: Cannot delete/insert a y that equals s(x) or t(x) for some x Observation: Updates preserve convexity

10 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 10 Dynamic Convex Bipartite Graphs: Queries Matched? (x) Matched? (y) Mate (x) Mate (y) x y

11 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 11 Result Updates O(log 2 X) amortized Matched? O(1) worst-case Mate O( X ·log 2 X) amortized O(min{k·log 2 X, X·log X}) worst-case Space O(Y+X·log X) k = # updates since the last Mate query for the same node

12 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 12 Related Work Perfect matchings in general graphs can be maintained in time O(V 1.495 ) per edge update Sankowski ’04 Size of maximum matchings in general graphs can be maintained in time O(V 1.495 ) per edge update Sankowski ‘07

13 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 13 Ingredients of Our Solution Special data structure for the case s(x)=1 for all x Dynamic version of Dekel-Sahni’s parallel algorithm (divide-and-conquer) Mate queries are handled by lazy construction of the matching

14 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 14 { n 7 = 3 Case: s(x)=1 for all x (Jobscheduling view) 17981023456 job length #scheduled jobs

15 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 15 Store vertical distance from each to diagonal Insert job length i : –Find first j ≥ i with on diagonal –Decrement distance by one for i..j-1 O(log n) time using augmented binary search tree Case: s(x)=1 for all x (Jobscheduling view) job length #scheduled jobs

16 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 16 Dynamizing Dekel-Sahni y1y1 y2y2 y3y3 y4y4..... Weight balanced search tree matched(P) P LR transfered(P) matched(P) = matched(L) U matched’(R) transfered(P) = transfered(R) U transfered’(R) transfered’(R) and matched’(R) are computed from transfered(L) and matched(R) assuming all starting times equal min(R) Updates: O(1) changes at each level, i.e. total O(log 2 n) time

17 Dynamic Matchings in Convex Bipartite Graphs Brodal, Georgiadis, Hansen, Katriel 17 Conclusion Dynamic Matchings in Convex Bipartite Graphs Updates O(log 2 X) amortized Matched? O(1) worst-case Mate O( X ·log 2 X) amortized O(min{k·log 2 X, X·log X}) worst-case Space O(Y+X·log X) Open problems...


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