45,697 research outputs found

    Four-Dimensional Dominance Range Reporting in Linear Space

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    In this paper we study the four-dimensional dominance range reporting problem and present data structures with linear or almost-linear space usage. Our results can be also used to answer four-dimensional queries that are bounded on five sides. The first data structure presented in this paper uses linear space and answers queries in O(log^{1+?} n + k log^? n) time, where k is the number of reported points, n is the number of points in the data structure, and ? is an arbitrarily small positive constant. Our second data structure uses O(n log^? n) space and answers queries in O(log n+k) time. These are the first data structures for this problem that use linear (resp. O(n log^? n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(n log^? n)-space data structure supports queries in O(n^? + k) time (Bentley and Mauer, 1980). Our results can be generalized to d ? 4 dimensions. For example, we can answer d-dimensional dominance range reporting queries in O(log log n (log n/log log n)^{d-3} + k) time using O(n log^{d-4+?} n) space. Compared to the fastest previously known result (Chan, 2013), our data structure reduces the space usage by O(log n) without increasing the query time

    Shared-Constraint Range Reporting

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    Orthogonal range reporting is one of the classic and most fundamental data structure problems. (2,1,1) query is a 3 dimensional query with two-sided constraint on the first dimension and one sided constraint on each of the 2nd and 3rd dimension. Given a set of N points in three dimension, a particular formulation of such a (2,1,1) query (known as four-sided range reporting in three-dimension) asks to report all those K points within a query region [a, b]X(-infinity, c]X[d, infinity). These queries have overall 4 constraints. In Word-RAM model, the best known structure capable of answering such queries with optimal query time takes O(N log^{epsilon} N) space, where epsilon>0 is any positive constant. It has been shown that any external memory structure in optimal I/Os must use Omega(N log N/ log log_B N) space (in words), where B is the block size [Arge et al., PODS 1999]. In this paper, we study a special type of (2,1,1) queries, where the query parameters a and c are the same i.e., a=c. Even though the query is still four-sided, the number of independent constraints is only three. In other words, one constraint is shared. We call this as a Shared-Constraint Range Reporting (SCRR) problem. We study this problem in both internal as well as external memory models. In RAM model where coordinates can only be compared, we achieve linear-space and O(log N+K) query time solution, matching the best-known three dimensional dominance query bound. Whereas in external memory, we present a linear space structure with O(log_B N + log log N + K/B) query I/Os. We also present an I/O-optimal (i.e., O(log_B N+K/B) I/Os) data structure which occupies O(N log log N)-word space. We achieve these results by employing a novel divide and conquer approach. SCRR finds application in database queries containing sharing among the constraints. We also show that SCRR queries naturally arise in many well known problems such as top-k color reporting, range skyline reporting and ranked document retrieval

    Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles

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    In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a \emph{fat} rectangle. Our two-dimensional data structure uses O(n)O(n) words and supports queries in O(log⁥log⁥U+k)O(\log\log U +k) time, where nn is the number of points in the data structure, UU is the size of the universe and kk is the number of points in the query range. Our three-dimensional data structure needs O(nlog⁥ΔU)O(n\log^{\varepsilon}U) words of space and answers queries in O(log⁥log⁥U+k)O(\log \log U + k) time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses O(n)O(n) space and answers stabbing queries in O(log⁥Ulog⁥log⁥U+k)O(\log U\log\log U +k) time.Comment: extended version of a WADS'19 pape

    Join-Reachability Problems in Directed Graphs

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    For a given collection G of directed graphs we define the join-reachability graph of G, denoted by J(G), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of G. Our goal is to compute an efficient representation of J(G). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for G. In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of G. This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs

    Decoding Hidden Markov Models Faster Than Viterbi Via Online Matrix-Vector (max, +)-Multiplication

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    In this paper, we present a novel algorithm for the maximum a posteriori decoding (MAPD) of time-homogeneous Hidden Markov Models (HMM), improving the worst-case running time of the classical Viterbi algorithm by a logarithmic factor. In our approach, we interpret the Viterbi algorithm as a repeated computation of matrix-vector (max⁥,+)(\max, +)-multiplications. On time-homogeneous HMMs, this computation is online: a matrix, known in advance, has to be multiplied with several vectors revealed one at a time. Our main contribution is an algorithm solving this version of matrix-vector (max⁥,+)(\max,+)-multiplication in subquadratic time, by performing a polynomial preprocessing of the matrix. Employing this fast multiplication algorithm, we solve the MAPD problem in O(mn2/log⁥n)O(mn^2/ \log n) time for any time-homogeneous HMM of size nn and observation sequence of length mm, with an extra polynomial preprocessing cost negligible for m>nm > n. To the best of our knowledge, this is the first algorithm for the MAPD problem requiring subquadratic time per observation, under the only assumption -- usually verified in practice -- that the transition probability matrix does not change with time.Comment: AAAI 2016, to appea

    I/O-Efficient Planar Range Skyline and Attrition Priority Queues

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    In the planar range skyline reporting problem, we store a set P of n 2D points in a structure such that, given a query rectangle Q = [a_1, a_2] x [b_1, b_2], the maxima (a.k.a. skyline) of P \cap Q can be reported efficiently. The query is 3-sided if an edge of Q is grounded, giving rise to two variants: top-open (b_2 = \infty) and left-open (a_1 = -\infty) queries. All our results are in external memory under the O(n/B) space budget, for both the static and dynamic settings: * For static P, we give structures that answer top-open queries in O(log_B n + k/B), O(loglog_B U + k/B), and O(1 + k/B) I/Os when the universe is R^2, a U x U grid, and a rank space grid [O(n)]^2, respectively (where k is the number of reported points). The query complexity is optimal in all cases. * We show that the left-open case is harder, such that any linear-size structure must incur \Omega((n/B)^e + k/B) I/Os for a query. We show that this case is as difficult as the general 4-sided queries, for which we give a static structure with the optimal query cost O((n/B)^e + k/B). * We give a dynamic structure that supports top-open queries in O(log_2B^e (n/B) + k/B^1-e) I/Os, and updates in O(log_2B^e (n/B)) I/Os, for any e satisfying 0 \le e \le 1. This leads to a dynamic structure for 4-sided queries with optimal query cost O((n/B)^e + k/B), and amortized update cost O(log (n/B)). As a contribution of independent interest, we propose an I/O-efficient version of the fundamental structure priority queue with attrition (PQA). Our PQA supports FindMin, DeleteMin, and InsertAndAttrite all in O(1) worst case I/Os, and O(1/B) amortized I/Os per operation. We also add the new CatenateAndAttrite operation that catenates two PQAs in O(1) worst case and O(1/B) amortized I/Os. This operation is a non-trivial extension to the classic PQA of Sundar, even in internal memory.Comment: Appeared at PODS 2013, New York, 19 pages, 10 figures. arXiv admin note: text overlap with arXiv:1208.4511, arXiv:1207.234

    Affective norms for italian words in older adults: Age differences in ratings of valence, arousal and dominance

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    In line with the dimensional theory of emotional space, we developed affective norms for words rated in terms of valence, arousal and dominance in a group of older adults to complete the adaptation of the Affective Norms for English Words (ANEW) for Italian and to aid research on aging. Here, as in the original Italian ANEW database, participants evaluated valence, arousal, and dominance by means of the Self-Assessment Manikin (SAM) in a paper-and-pencil procedure. We observed high split-half reliabilities within the older sample and high correlations with the affective ratings of previous research, especially for valence, suggesting that there is large agreement among older adults within and across-languages. More importantly, we found high correlations between younger and older adults, showing that our data are generalizable across different ages. However, despite this across-ages accord, we obtained age-related differences on three affective dimensions for a great number of words. In particular, older adults rated as more arousing and more unpleasant a number of words that younger adults rated as moderately unpleasant and arousing in our previous affective norms. Moreover, older participants rated negative stimuli as more arousing and positive stimuli as less arousing than younger participants, thus leading to a less-curved distribution of ratings in the valence by arousal space. We also found more extreme ratings for older adults for the relationship between dominance and arousal: older adults gave lower dominance and higher arousal ratings for words rated by younger adults with middle dominance and arousal values. Together, these results suggest that our affective norms are reliable and can be confidently used to select words matched for the affective dimensions of valence, arousal and dominance across younger and older participants for future research in aging. Figure

    Four not six: revealing culturally common facial expressions of emotion

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    As a highly social species, humans generate complex facial expressions to communicate a diverse range of emotions. Since Darwin’s work, identifying amongst these complex patterns which are common across cultures and which are culture-specific has remained a central question in psychology, anthropology, philosophy, and more recently machine vision and social robotics. Classic approaches to addressing this question typically tested the cross-cultural recognition of theoretically motivated facial expressions representing six emotions, and reported universality. Yet, variable recognition accuracy across cultures suggests a narrower cross-cultural communication, supported by sets of simpler expressive patterns embedded in more complex facial expressions. We explore this hypothesis by modelling the facial expressions of over 60 emotions across two cultures, and segregating out the latent expressive patterns. Using a multi-disciplinary approach, we first map the conceptual organization of a broad spectrum of emotion words by building semantic networks in two cultures. For each emotion word in each culture, we then model and validate its corresponding dynamic facial expression, producing over 60 culturally valid facial expression models. We then apply to the pooled models a multivariate data reduction technique, revealing four latent and culturally common facial expression patterns that each communicates specific combinations of valence, arousal and dominance. We then reveal the face movements that accentuate each latent expressive pattern to create complex facial expressions. Our data questions the widely held view that six facial expression patterns are universal, instead suggesting four latent expressive patterns with direct implications for emotion communication, social psychology, cognitive neuroscience, and social robotics
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