1,046 research outputs found

    On Range Searching with Semialgebraic Sets II

    Full text link
    Let PP be a set of nn points in Rd\R^d. We present a linear-size data structure for answering range queries on PP with constant-complexity semialgebraic sets as ranges, in time close to O(n11/d)O(n^{1-1/d}). It essentially matches the performance of similar structures for simplex range searching, and, for d5d\ge 5, significantly improves earlier solutions by the first two authors obtained in~1994. This almost settles a long-standing open problem in range searching. The data structure is based on the polynomial-partitioning technique of Guth and Katz [arXiv:1011.4105], which shows that for a parameter rr, 1<rn1 < r \le n, there exists a dd-variate polynomial ff of degree O(r1/d)O(r^{1/d}) such that each connected component of RdZ(f)\R^d\setminus Z(f) contains at most n/rn/r points of PP, where Z(f)Z(f) is the zero set of ff. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications

    Network Density of States

    Full text link
    Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues - the spectral density - is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.Comment: 10 pages, 7 figure

    Subquadratic Algorithms for Some 3Sum-Hard Geometric Problems in the Algebraic Decision Tree Model

    Get PDF
    We present subquadratic algorithms in the algebraic decision-tree model for several 3Sum-hard geometric problems, all of which can be reduced to the following question: Given two sets A, B, each consisting of n pairwise disjoint segments in the plane, and a set C of n triangles in the plane, we want to count, for each triangle ∆ ∈ C, the number of intersection points between the segments of A and those of B that lie in ∆. The problems considered in this paper have been studied by Chan (2020), who gave algorithms that solve them, in the standard real-RAM model, in O((n2/log2 n) logO(1) log n) time. We present solutions in the algebraic decision-tree model whose cost is O(n60/31+ε), for any ε > 0. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl (2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the order type of the lines, a “handicap” that turns out to be beneficial for speeding up our algorithm.SCOPUS: cp.pinfo:eu-repo/semantics/publishe

    Route Planning in Transportation Networks

    Full text link
    We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle

    Lower Bounds for Semialgebraic Range Searching and Stabbing Problems

    Get PDF
    In the semialgebraic range searching problem, we are to preprocess nn points in Rd\mathbb{R}^d s.t. for any query range from a family of constant complexity semialgebraic sets, all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, the problem can be solved using S(n)S(n) space and with Q(n)Q(n) query time with S(n)Qd(n)=O~(nd)S(n)Q^d(n) = \tilde{O}(n^d) and this trade-off is almost tight. Consequently, there exists low space structures that use O~(n)\tilde{O}(n) space with O(n11/d)O(n^{1-1/d}) query time and fast query structures that use O(nd)O(n^d) space with O(logdn)O(\log^{d} n) query time. However, for the general semialgebraic ranges, only low space solutions are known, but the best solutions match the same trade-off curve as the simplex queries. It has been conjectured that the same could be done for the fast query case but this open problem has stayed unresolved. Here, we disprove this conjecture. We give the first nontrivial lower bounds for semilagebraic range searching and related problems. We show that any data structure for reporting the points between two concentric circles with Q(n)Q(n) query time must use S(n)=Ω(n3o(1)/Q(n)5)S(n)=\Omega(n^{3-o(1)}/Q(n)^5) space, meaning, for Q(n)=O(logO(1)n)Q(n)=O(\log^{O(1)}n), Ω(n3o(1))\Omega(n^{3-o(1)}) space must be used. We also study the problem of reporting the points between two polynomials of form Y=i=0ΔaiXiY=\sum_{i=0}^\Delta a_i X^i where a0,,aΔa_0, \cdots, a_\Delta are given at the query time. We show S(n)=Ω(nΔ+1o(1)/Q(n)Δ2+Δ)S(n)=\Omega(n^{\Delta+1-o(1)}/Q(n)^{\Delta^2+\Delta}). So for Q(n)=O(logO(1)n)Q(n)=O(\log^{O(1)}n), we must use Ω(nΔ+1o(1))\Omega(n^{\Delta+1-o(1)}) space. For the dual semialgebraic stabbing problems, we show that in linear space, any data structure that solves 2D ring stabbing must use Ω(n2/3)\Omega(n^{2/3}) query time. This almost matches the linearization upper bound. For general semialgebraic slab stabbing problems, again, we show an almost tight lower bounds.Comment: Submitted to SoCG'21; this version: readjust the table and other minor change
    corecore