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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

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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

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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

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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

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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]

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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)

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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)]

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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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