Encoding Two-Dimensional Range Top-k Queries Revisited

We consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering Top-k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For 2 x n arrays, we first give upper and lower bounds on space for answering sorted and unsorted 3-sided Top-k queries. For m x n arrays, with m <=n and k <=mn, we obtain (m lg{(k+1)n choose n}+4nm(m-1)+o(n))-bit encoding for answering sorted 4-sided Top-k queries. This improves the min{(O(mn lg{n}),m^2 lg{(k+1)n choose n} + m lg{m}+o(n))}-bit encoding of Jo et al. [CPM, 2016] when m = o(lg{n}). This is a consequence of a new encoding that encodes a 2 x n array to support sorted 4-sided Top-k queries on it using an additional 4n bits, in addition to the encodings to support the Top-k queries on individual rows. This new encoding is a non-trivial generalization of the encoding of Jo et al. [CPM, 2016] that supports sorted 4-sided Top-2 queries on it using an additional 3n bits. We also give almost optimal space encodings for 3-sided Top-k queries, and show lower bounds on encodings for 3-sided and 4-sided Top-k queries

Encoding Two-Dimensional Range Top-k Queries

We consider various encodings that support range top-k queries on a two-dimensional array containing elements from a total order. For an m x n array, we first propose an almost optimal encoding for answering one-sided top-k queries, whose query range is restricted to [1 ... m][1 .. a], for 1 <= a <= n. Next, we propose an encoding for the general top-k queries that takes m^2 * lg(binom((k+1)n)(n)) + m * lg(m) + o(n) bits. This generalizes the one-dimensional top-k encoding of Gawrychowski and Nicholson [ICALP, 2015]. Finally, for a 2 x n array, we obtain a 2 lg(binom(3n)(n)) + 3n + o(n)-bit encoding for answering top-2 queries

Approximate Query Processing over Static Sets and Sliding Windows

Indexing of static and dynamic sets is fundamental to a large set of applications such as information retrieval and caching. Denoting the characteristic vector of the set by B, we consider the problem of encoding sets and multisets to support approximate versions of the operations rank(i) (i.e., computing sum_{j = i}) queries. We study multiple types of approximations (allowing an error in the query or the result) and present lower bounds and succinct data structures for several variants of the problem. We also extend our model to sliding windows, in which we process a stream of elements and compute suffix sums. This is a generalization of the window summation problem that allows the user to specify the window size at query time. Here, we provide an algorithm that supports updates and queries in constant time while requiring just (1+o(1)) factor more space than the fixed-window summation algorithms

Lagrangian Energetics and Vertical Dispersion in Stably Stratified Turbulence

The vertical dispersion of fluid particles in stably stratified homogeneous turbulence with mean shear is investigated. An analysis framework which describes the associated flow energetics in the Lagrangian frame is developed. This provides a more clear and consistent interpretation of the behavior of the mean square vertical displacement which can be related to the total potential energy (TPE) of a given set of fluid particles. The analysis considers TPE in terms of the available potential energy (APE), associated with the nonequilibrium displacement, and the reference potential energy (RPE), associated with the change in particle equilibrium height, i.e., the equilibrium displacement. The corresponding evolution equations describe the key sequence of processes. As fluid particles move away from their equilibrium height, vertical kinetic energy is converted (reversibly) to APE. This establishes nonequilibrium displacement and increases TPE. Without molecular diffusion, gravity will reduce the vertical velocity and the particle will eventually return to its original equilibrium height; APE is converted back to KE in this reversible process. With molecular diffusion, fluid particles will change their density, and therefore their equilibrium height, such to reduce density fluctuation; i.e., some of the APE will be dissipated and converted to RPE. Thus, molecular diffusion acts to preserve displacements and reduce the reconversion of PE to KE. In this manner, fluid particles can move further away from their original equilibrium level and the mean square vertical displacement can grow without limit.The evolution equations are integrated in time and give a relation for mean square vertical displacement. At long time, the RPE will dominate the TPE; mean square vertical displacement is then a measure of the total APE dissipated by the flow. The significance of this with respect to the total energy dissipated is given by the cumulative mixing efficiency, Ωc, which depends on the strength of stratification. In the case of decaying turbulence, mean square vertical displacement evolves to a constant value, proportional to Ωc. In the case of stationary turbulence, the (constant) rate of growth of mean square vertical displacement is proportional to Ωc. The analysis is demonstrated using direct numerical simulations of homogeneous shear flows with decaying, stationary, and growing turbulence. Results for the latter case show mean square vertical displacement to continually increase, and to have a reduced dependence on and reduced values of Ωc as the strength of stratification decreases. In general, simulation results are in agreement with the analysis and confirm that,in homogeneous stratified flows with mean shear, an effective time scale for verticaldispersion at long time is that of the turbulence decay time

Encoding Two-Dimensional Range Top-k Queries

We consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering Top-k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an m×n array, with m≤n, we first propose an encoding for answering 1-sided Top-k queries, whose query range is restricted to [1…m][1…a], for 1≤a≤n. Next, we propose an encoding for answering for the general (4-sided) Top-k queries that takes (mlg((k+1)nn)+2nm(m−1)+o(n)) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial O(nmlgn)-bit encoding, our encoding takes less space when m=o(lgn). In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided Top-k queries, which show that our upper bound results are almost optimal

Succinct Encodings for Families of Interval Graphs

We consider the problem of designing succinct data structures for interval graphs with n vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time. Towards showing succinctness, we first show that at least nlog2n−2nlog2log2n−O(n) bits are necessary to represent any unlabeled interval graph G with n vertices, answering an open problem of Yang and Pippenger (Proc Am Math Soc Ser B 4(1):1–3, 2017). This is augmented by a data structure of size nlog2n+O(n) bits while supporting not only the above queries optimally but also capable of executing various combinatorial algorithms (like proper coloring, maximum independent set etc.) on interval graphs efficiently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit interval graphs, k-improper interval graphs, and circular-arc graphs, and design succinct data structures for these graph classes as well along with supporting queries on them efficiently

Approximate query processing over static sets and sliding windows

Indexing of static and dynamic sets is fundamental to a large set of applications such as information retrieval and caching. Denoting the characteristic vector of the set by B, we consider the problem of encoding sets and multisets to support approximate versions of the operations rank(i) (i.e., computing ∑ j≤i B[ j]) and select(i) (i.e., finding min{p | rank(p) ≥ i}) queries. We study multiple types of approximations (allowing an error in the query or the result) and present lower bounds and succinct data structures for several variants of the problem. We also extend our model to sliding windows, in which we process a stream of elements and compute suffix sums. This is a generalization of the window summation problem that allows the user to specify the window size at query time. Here, we provide an algorithm that supports updates and queries in constant time while requiring just (1 + o(1)) factor more space than the fixed-window summation algorithm