Near-optimal Linear Decision Trees for k-SUM and Related Problems

Kane, Daniel M.; Lovett, Shachar; Moran, Shay

doi:10.1145/3285953

Cited by 26 publications

(27 citation statements)

References 25 publications

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“…Using folklore meet-inthe-middle algorithms, k-SUM can be solved in time O(n k/2 ) if k is odd, and in time O(n k/2 log n) if k is even. Recently, Kane, Lovett, and Moran [26] showed that it can be solved in time O(n log 2 n) in the linear decision tree model, improving on previous polynomial bounds [13,19].…”

Section: Introductionmentioning

confidence: 87%

“…Finally, we consider the nonuniform decision tree complexity, also known as query complexity, of the two problems. By applying a recent result of Kane, Lovett, and Moran [26], we can bound the number of algebraic tests that are required to detect copies of P in an input set S. In fact, if the pattern P is a fixed parameter, that is, when P is not part of the input, but known at the algorithm design time, then the decision tree in the statement above only involves linear tests.…”

Section: Corollary 3 There Exists Anmentioning

confidence: 99%

“…Theorem 16 (Kane, Lovett, Moran [26]). The k-SUM problem on n elements can be solved by a linear decision tree of height O(n log 2 n) in which all the queries are 2k-sparse and have only {−1, 0, 1} coefficients.…”

Section: Algebraic Decision Tree Complexitymentioning

confidence: 99%

See 2 more Smart Citations

Geometric Pattern Matching Reduces to k -SUM

Aronov

Cardinal

2021

Discrete Comput Geom

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We prove that some exact geometric pattern matching problems reduce in linear time to k-SUM when the pattern has a fixed size k. This holds in the real RAM model for searching for a similar copy of a set of k ≥ 3 points within a set of n points in the plane, and for searching for an affine image of a set of k ≥ d + 2 points within a set of n points in d-space.As corollaries, we obtain improved real RAM algorithms and decision trees for the two problems. In particular, they can be solved by algebraic decision trees of near-linear height. ACM Subject ClassificationTheory of computation → Design and analysis of algorithms; Theory of computation → Computational geometry Keywords and phrases Geometric pattern matching, k-SUM problem, Linear decision trees

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Section: Introductionmentioning

confidence: 87%

Section: Corollary 3 There Exists Anmentioning

confidence: 99%

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Geometric Pattern Matching Reduces to k -SUM

Aronov

Cardinal

2021

Discrete Comput Geom

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“…For example, the x-coordinates of the O(n 2 ) intersections of a set of n lines can be sorted using O(n 2 ) comparisons [58], since these O(n 2 ) values come from O(n) real numbers. (Another example from Fredman's original paper is the well-known X + Y sorting problem, although for the decision tree complexity of that particular problem, simpler [58] and better [45] methods were later found.) We show that this type of result is not limited to the sorting problem alone, and that logarithmic-factor shavings in the decision tree setting are actually not difficult to obtain for many other problems, including point location of multiple query points in multiple subdivisions, assuming that the query points and subdivision vertices originate from O(n) real numbers.…”

Section: Second Approach Via Decision Treesmentioning

confidence: 99%

Hopcroft's Problem, Log-Star Shaving, 2D Fractional Cascading, and Decision Trees

Chan,

Zheng

2021

Preprint

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We revisit Hopcroft's problem and related fundamental problems about geometric range searching. Given n points and n lines in the plane, we show how to count the number of point-line incidence pairs or the number of point-above-line pairs in O(n 4/3 ) time, which matches the conjectured lower bound and improves the best previous time bound of n 4/3 2 O(log * n) obtained almost 30 years ago by Matoušek.We describe two interesting and different ways to achieve the result: the first is randomized and uses a new 2D version of fractional cascading for arrangements of lines; the second is deterministic and uses decision trees in a manner inspired by the sorting technique of . The second approach extends to any constant dimension.Many consequences follow from these new ideas: for example, we obtain an O(n 4/3 )-time algorithm for line segment intersection counting in the plane, O(n 4/3 )-time randomized algorithms for bichromatic closest pair and Euclidean minimum spanning tree in three or four dimensions, and a randomized data structure for halfplane range counting in the plane with O(n 4/3 ) preprocessing time and space and O(n 1/3 ) query time.

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“…Our model is also related to linear decision tree complexity, see, e.g., [7,14], though such lower bounds typically involve just seeing a threshold applied to Mv i , and typically M is a vector. In our case, we observe the entire output vector Mv i .…”

Section: Introductionmentioning

confidence: 99%

Querying a Matrix through Matrix-Vector Products

Sun

Woodruff

Yang

et al. 2021

ACM Trans. Algorithms

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We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products , namely, the algorithm chooses vectors v 1 , ⃛ , v q , and observes Mv 1 , ⃛ , Mv q . Here the v i can be randomized as well as chosen adaptively as a function of Mv 1 , ⃛ , Mv i-1 . Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.

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Near-optimal Linear Decision Trees for k-SUM and Related Problems

Cited by 26 publications

References 25 publications

Geometric Pattern Matching Reduces to k -SUM

Geometric Pattern Matching Reduces to k -SUM

Hopcroft's Problem, Log-Star Shaving, 2D Fractional Cascading, and Decision Trees

Querying a Matrix through Matrix-Vector Products

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