Combinatorial optimization: algorithms and complexity
Combinatorial optimization: algorithms and complexity
A data structure for dynamic trees
Journal of Computer and System Sciences
Faster scaling algorithms for network problems
SIAM Journal on Computing
SIAM Journal on Computing
Approximation algorithms for bipartite and non-bipartite matching in the plane
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications
SIAM Journal on Computing
A near-linear constant-factor approximation for euclidean bipartite matching?
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
A near linear time constant factor approximation for Euclidean bichromatic matching (cost)
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Efficient Sketches for Earth-Mover Distance, with Applications
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Maintaining a large matching and a small vertex cover
Proceedings of the forty-second ACM symposium on Theory of computing
Fully Dynamic Maximal Matching in O (log n) Update Time
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
A near-linear time ε-approximation algorithm for geometric bipartite matching
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
A sub-quadratic algorithm for bipartite matching of planar points with bounded integer coordinates
Proceedings of the twenty-ninth annual symposium on Computational geometry
Reprint of: Optimally solving a transportation problem using Voronoi diagrams
Computational Geometry: Theory and Applications
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For A, B ⊂ Rd, |A| + |B| = n, let a ε A have a demand da ε Z+ and b ε B have a supply sb ε Z+, [EQUATION] = U and let d(.,.) be a distance function. Suppose the diameter of A ∪ B is Δ under d(.,.), and ε 0 is a parameter. We present an algorithm that in O((n√Ulog2 n + U log U)Φ(n) log(ΔU/ε)) time computes a solution to the transportation problem on A, B which is within an additive error ε from the optimal solution. Here Φ(n) is the query and update time of a dynamic weighted nearest neighbor data structure under distance function d(.,.). Note that the (1/ε) appears only in the log term. As among various consequences we obtain, • For A, B ⊂ Rd and for the case where d(.,.) is a metric, an ε-approximation algorithm for the transportation problem in O((n√U log2 n + U log U)Φ(n) log (U/ε)) time. • For A, B ⊂ [Δ]d and the L1 and L∞ distance, exact algorithm for computing an optimal bipartite matching of A, B that runs in O(n3/2 log d+O(1) n log Δ) time. • For A, B ⊂ [Δ]2 and RMS distance, exact algorithm for computing an optimal bipartite matching of A, B that runs in O(n3/2+δ log Δ) time, for an arbitrarily small constant δ 0. For point sets, A, B ⊂ [Δ]d, for the Lp norm and for 0 α, β α)U of the demands are satisfied and whose cost is within (1 + β) of that of the optimal (complete) solution to the transportation problem with high probability. The insertion, deletion and update times are O(poly(log(nΔ)/αβ)), provided U = nO(1).