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**Dynamic Matchings in Convex Bipartite Graphs**

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

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**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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**Matchings in General Graphs**

Deterministic O(E V) Micali, Vaziani ’80 Randomized O(V 2.476) Mucha, Sankowski ’04

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**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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**Convex Bipartite Graphs**

1 7 6 5 4 3 2 s(v) t(v) Y X v [4,7] [3,3] [1,2] [2,4]

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**Matchings in Convex Bipartite Graphs**

1 7 6 5 4 3 2 Y (time) X (jobs to schedule) [4,7] [3,3] [1,2] [2,4] Deterministic O(X+Y) Gabow, Tarjan ’85 Deterministic O(X) Steiner, Yeomans ’96

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**Matchings in Convex Bipartite Graphs: Glover’s Greedy Algorithm**

1 7 6 5 4 3 2 Y X [4,7] [3,3] [1,2] [2,4] 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**

Observation O(1) edge changes can change Ω(X) edges in the matching but only O(1) nodes in the matching need to change

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**Dynamic Convex Bipartite Graphs: Updates**

s(x) t(x) x y Insert(x,[s(x),t(x)]) Delete(x) Insert(y) Delete(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 Convex Bipartite Graphs: Queries**

Matched?(x) Matched?(y) Mate(x) Mate(y)

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**k = # updates since the last Mate query for the same node**

Result Updates O(log2 X) amortized Matched? O(1) worst-case Mate O( X ·log2 X) amortized O(min{k·log2 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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Related Work Perfect matchings in general graphs can be maintained in time O(V1.495) per edge update Sankowski ’04 Size of maximum matchings in general graphs can be maintained in time O(V1.495) per edge update Sankowski ‘07

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**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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**Case: s(x)=1 for all x (Jobscheduling view)**

{ n7 = 3 #scheduled jobs job length 1 2 3 4 5 6 7 8 9 10

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**Case: s(x)=1 for all x (Jobscheduling view)**

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 #scheduled jobs job length

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**Dynamizing Dekel-Sahni**

transfered(P) = transfered(R) U transfered’(R) Weight balanced search tree matched(P) = matched(L) U matched’(R) P matched(P) transfered’(R) and matched’(R) are computed from transfered(L) and matched(R) assuming all starting times equal min(R) transfered(P) L R y1 y2 y3 y4 ..... Updates: O(1) changes at each level, i.e. total O(log2 n) time

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**Conclusion Dynamic Matchings in Convex Bipartite Graphs**

Updates O(log2 X) amortized Matched? O(1) worst-case Mate O( X ·log2 X) amortized O(min{k·log2 X, X·log X}) worst-case Space O(Y+X·log X) Open problems...

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