Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

Chan, Timothy M.; Tsakalidis, Konstantinos

doi:10.1007/s00454-016-9784-4

Cited by 39 publications

(70 citation statements)

References 32 publications

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“…As mentioned above, the geometric core of Chan's data structure consists of an efficient construction of small-size shallow cuttings of a particularly favorable kind, which we refer to as vertical shallow cuttings [12,15]. To define these constructs, we first recall the notion of a level in an arrangement of n function graphs in three dimensions.…”

Section: Our Results and Techniquesmentioning

confidence: 99%

“…Techniques for computing vertical shallow cuttings for planes, and the conflict lists associated with their prisms [15,27] heavily rely on the fact that if a plane intersects a semi-unbounded prism τ it must intersect a vertical edge of τ . This does not necessarily hold for general functions, and we therefore need to use a somewhat different approach, that results in cuttings of slightly suboptimal size, but only by (small) polylogarithmic factors.…”

Section: Approximate K-levelsmentioning

confidence: 99%

“…Since |H (i) | ≤ β i−1 n, the overall size of D is O(n log n). 13 We can construct each cutting Λ j , in each of the substructures D (i) , using the algorithm of Chan and Tsakalidis [15], in O(|H (i) | log n) time. Summing over j, we can construct D (i) in O(|H (i) | log 2 n) time, and summing over i, using the fact that |H (i) | decreases geometrically with i, we get a total of O(n log 2 n) running time.…”

Section: A Static Structurementioning

confidence: 99%

“…In SODA 2006, Chan [13] presented an ingenious construction, in which both the (amortized) update time and the (worst-case) query time are polylogarithmic, for the case of planes. More precisely, Chan's data structure (combined with the recent deterministic construction of shallow cuttings by Chan and Tsakalidis [15]) supports insertions in O(log 3 n) amortized time, deletions in O(log 6 n) amortized time, and queries in O(log 2 n) worst-case time. However, so far it has still been unknown whether a similar result (with polylogarithmic update time) is possible for arbitrary bivariate functions of constant description complexity with linear envelope complexity.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications

Kaplan¹,

Mulzer²,

Roditty³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions that includes Lp-norms and additively weighted Euclidean distances, and for general (convex, pairwise disjoint) sites that have constant description complexity (line segments, disks, etc.). Our data structure has a polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat and Sharir that required O(n ε ) time for an update and O(log n) time for a query [1]. Our data structure has numerous applications, and in all of them it gives faster algorithms, typically reducing an O(n ε ) factor in the bounds to polylogarithmic. To further demonstrate its effectiveness, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph.To obtain this data structure, we combine and extend various techniques and obtain several side results that are of independent interest. Our data structure depends on the existence and an efficient construction of "vertical" shallow cuttings in arrangements of bivariate algebraic functions. We prove that an appropriate level in an arrangement of a random sample of a suitable size provides such a cutting. To compute it efficiently, we develop a randomized incremental construction algorithm for finding the lowest k levels in an arrangement of bivariate algebraic functions (we mostly consider here collections of functions whose lower envelope * A full version is available on the arXiv [20] has linear complexity, as is the case in the dynamic nearestneighbor context). To analyze this algorithm, we improve a longstanding bound on the combinatorial complexity of the vertical decomposition of these levels. Finally, to obtain our structure, we plug our vertical shallow cutting construction into Chan's algorithm for efficiently maintaining the lower envelope of a dynamic set of planes in R 3 . While doing this, we also revisit Chan's technique and present a variant that uses a single binary counter, with a simpler analysis and an improved amortized deletion time.

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Section: Our Results and Techniquesmentioning

confidence: 99%

Section: Approximate K-levelsmentioning

confidence: 99%

Section: A Static Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications

Kaplan¹,

Mulzer²,

Roditty³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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show abstract

“…Also, if a problem can handle both updates and queries, then we can formalize both as instance-updates in Gn as shown in Example 6.5.13 The best known algorithm for planar nearest neighbor is by Chan[Cha10] which has polylogarithmic amortized update time. The algorithm is randomized, but it is later derandomized using the result by Chan and Tsakalidis[CT16].…”

mentioning

confidence: 99%

Coarse-Grained Complexity for Dynamic Algorithms

Bhattacharya¹,

Nanongkai²,

Saranurak³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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To date, the only way to argue polynomial lower bounds for dynamic algorithms is via fine-grained complexity arguments. These arguments rely on strong assumptions about specific problems such as the Strong Exponential Time Hypothesis (SETH) and the Online Matrix-Vector Multiplication Conjecture (OMv). While they have led to many exciting discoveries, dynamic algorithms still miss out some benefits and lessons from the traditional "coarse-grained" approach that relates together classes of problems such as P and NP. In this paper we initiate the study of coarse-grained complexity theory for dynamic algorithms. Below are among questions that this theory can answer.What if dynamic Orthogonal Vector (OV) is easy in the cell-probe model? A research program for proving polynomial unconditional lower bounds for dynamic OV in the cell-probe model is motivated by the fact that many conditional lower bounds can be shown via reductions from the dynamic OV problem (e.g. [Abboud, V.-Williams, FOCS 2014]). Since the cell-probe model is more powerful than word RAM and has historically allowed smaller upper bounds (e.g. [Larsen, Williams, SODA 2017; Chakraborty, Kamma, Larsen, STOC 2018]), it might turn out that dynamic OV is easy in the cell-probe model, making this research direction infeasible. Our theory implies that if this is the case, there will be very interesting algorithmic consequences: If dynamic OV can be maintained in polylogarithmic worst-case update time in the cell-probe model, then so are several important dynamic problems such as k-edge connectivity, (1 + ǫ)-approximate mincut, (1 + ǫ)-approximate matching, planar nearest neighbors, Chan's subset union and 3-vs-4 diameter. The same conclusion can be made when we replace dynamic OV by, e.g., subgraph connectivity, single source reachability, Chan's subset union, and 3-vs-4 diameter.Lower bounds for k-edge connectivity via dynamic OV? The ubiquity of reductions from dynamic OV raises a question whether we can prove conditional lower bounds for, e.g., k-edge connectivity, approximate mincut, and approximate matching, via the same approach. Our theory provides a method to refute such possibility (the so-called non-reducibility). In particular, we show that there are no "efficient" reductions (in both cell-probe and word RAM models) from dynamic OV to k-edge connectivity under an assumption about the classes of dynamic algorithms whose analogue in the static setting is widely believed. We are not aware of any existing assumptions that can play the same role. (The NSETH of Carmosino et al. [ITCS 2016] is the closest one, but is not enough.) To show similar results for other problems, one only need to develop efficient randomized verification protocols for such problems.

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Simplex Range Searching and Its Variants: A Review

Agarwal

2017

A Journey Through Discrete Mathematics

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Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

Cited by 39 publications

References 32 publications

Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications

Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications

Coarse-Grained Complexity for Dynamic Algorithms

Simplex Range Searching and Its Variants: A Review

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