Threesomes, Degenerates, and Love Triangles

Grønlund, Allan; Pettie, Seth

doi:10.1145/3185378

Cited by 50 publications

(61 citation statements)

References 65 publications

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“…Indeed, for a long time it was conjectured that no subquadratic decision tree exists for 3SUM [57] and an (n 2 ) lower bound is known in a restricted linear decision tree model [4,40]. However, in a recentand in our opinion quite astonishing-result, Grønlund and Pettie showed that if we allow only slightly more powerful algebraic decision trees than in the previous lower bounds, one can decide 3SUM non-uniformly in O(n 2−ε ) steps, for some fixed ε > 0 [52]. They also show that this leads to a general subquadratic algorithm for 3SUM, a situation very similar to the Fréchet distance as described in the present paper.…”

Section: Recent Developmentsmentioning

confidence: 93%

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Buchin

Meulemans

et al. 2017

Discrete Comput Geom

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Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fréchet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original O(n 2 log n) algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here, n denotes the number of edges on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time O(n 2 √ log n(log log n) 3/2 ) on a pointer machine and in time O(n 2 (log log n) 2 ) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth O(n 2−ε ), for some ε > 0. We believe that this reveals an intriguing new aspect of this well-studied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Fréchet distance on a word RAM.

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Section: Recent Developmentsmentioning

confidence: 93%

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Buchin

Meulemans

et al. 2017

Discrete Comput Geom

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“…There are now known to be O(n 2 / poly(log n)) algorithms for both integer inputs [5] and real inputs [47]. The Integer 3SUM Conjecture asserts that the problem requires Ω(n 2−o(1) ) time, even if A ⊂ {−n 3 , .…”

Section: Introductionmentioning

confidence: 99%

“…Of course, the plausibility of the 3SUM and OMv conjectures continue to be actively scrutinized. Stronger forms of the 3SUM and OMv conjectures have already been refuted; see [47,59].…”

Section: Introductionmentioning

confidence: 99%

Connectivity Oracles for Graphs Subject to Vertex Failures

Duan¹,

Pettie²

2017

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

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We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. Our deterministic structure processes a batch of d ≤ d failed vertices inÕ(d 3 ) time and thereafter answers connectivity queries in O(d) time. It occupies space O(d m log n). We develop a randomized Monte Carlo version of our data structure with update timeÕ(d 2 ), query time O(d), and spaceÕ(m) for any d . This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures.Our data structures are based on a new decomposition theorem for an undirected graph G = (V, E), which is of independent interest. It states that for any terminal set U ⊆ V we can remove a set B of |U |/(s − 2) vertices such that the remaining graph contains a Steiner forest for U − B with maximum degree s. IntroductionThe dynamic subgraph model [19,21,36,39,37,42,65] is a constrained dynamic graph model. Rather than allow the graph to evolve in completely arbitrary ways (via an unbounded sequence of edge insertions and deletions), there is assumed to be a fixed ideal graph G = (V, E) that can be preprocessed in advance. The ideal graph is susceptible only to the failure of edges/vertices and their subsequent recovery, possibly with a bound d on the number of failures at one time. Queries naturally answer questions about the current failure-free subgraph. This model is useful because it more accurately represents the behavior of many real-world networks: changes to the underlying topology are relatively rare but transient failures very common. More importantly, this model offers the algorithm designer the freedom to explore exotic graph rep-

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“…The conjecture that it cannot be solved in O(n 2−ε ) time (in the real or integer setting) has been used as a basis for proving conditional lower bounds for numerous problems from a variety of areas (computational geometry, data structures, string algorithms, and so on). See previous papers (such as [22]) for more background.…”

mentioning

confidence: 99%

“…In a surprising breakthrough, Grønlund and Pettie [22] discovered the first subquadratic algorithms for 3SUM. They showed the decision-tree complexity of the problem is O(n 3/2 √ log n), and gave a randomized O((n 2 / log n)(log log n) 2 )-time algorithm and a deterministic O((n 2 / log 2/3 n)(log log n) 2/3 )-time algorithm in the standard real-RAM model.…”

mentioning

confidence: 99%

More Logarithmic-Factor Speedups for 3SUM, (median,+)-Convolution, and Some Geometric 3SUM-Hard Problems

Chan

2018

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

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We present an algorithm that solves the 3SUM problem for n real numbers in O((n 2 / log 2 n)(log log n) O(1) ) time, improving previous solutions by about a logarithmic factor. Our framework for shaving off two logarithmic factors can be applied to other problems, such as (median,+)-convolution/matrix multiplication and algebraic generalizations of 3SUM. We also obtain the first subquadratic results on some 3SUM-hard problems in computational geometry, for example, deciding whether (the interiors of) a constant number of simple polygons have a common intersection.

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Threesomes, Degenerates, and Love Triangles

Cited by 50 publications

References 65 publications

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Connectivity Oracles for Graphs Subject to Vertex Failures

More Logarithmic-Factor Speedups for 3SUM, (median,+)-Convolution, and Some Geometric 3SUM-Hard Problems

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