Selection and sorting with limited storage

AbstractWhen selecting from, or sorting, a file stored on a read-only tape and the internal storage is rather limited, several passes of the input tape may be required. We study the relation between the amount of internal storage available and the number of passes required to select the Kth highest of N inputs. We show, for example, that to find the median in two passes requires at least ω(N12) and at most O(N12log N) internal storage. For probabilistic methods, θ(N12) internal storage is necessary and sufficient for a single pass method which finds the median with arbitrarily high probability

Selection from read-only memory with limited workspace

Given an unordered array of $N$ elements drawn from a totally ordered set and an integer $k$ in the range from $1$ to $N$, in the classic selection problem the task is to find the $k$-th smallest element in the array. We study the complexity of this problem in the space-restricted random-access model: The input array is stored on read-only memory, and the algorithm has access to a limited amount of workspace. We prove that the linear-time prune-and-search algorithm---presented in most textbooks on algorithms---can be modified to use $\Theta(N)$ bits instead of $\Theta(N)$ words of extra space. Prior to our work, the best known algorithm by Frederickson could perform the task with $\Theta(N)$ bits of extra space in $O(N \lg^{*} N)$ time. Our result separates the space-restricted random-access model and the multi-pass streaming model, since we can surpass the $\Omega(N \lg^{*} N)$ lower bound known for the latter model. We also generalize our algorithm for the case when the size of the workspace is $\Theta(S)$ bits, where $\lg^3{N} \leq S \leq N$. The running time of our generalized algorithm is $O(N \lg^{*}(N/S) + N (\lg N) / \lg{} S)$, slightly improving over the $O(N \lg^{*}(N (\lg N)/S) + N (\lg N) / \lg{} S)$ bound of Frederickson's algorithm. To obtain the improvements mentioned above, we developed a new data structure, called the wavelet stack, that we use for repeated pruning. We expect the wavelet stack to be a useful tool in other applications as well.Comment: 16 pages, 1 figure, Preliminary version appeared in COCOON-201

Succinct Dictionary Matching With No Slowdown

The problem of dictionary matching is a classical problem in string matching: given a set S of d strings of total length n characters over an (not necessarily constant) alphabet of size sigma, build a data structure so that we can match in a any text T all occurrences of strings belonging to S. The classical solution for this problem is the Aho-Corasick automaton which finds all occ occurrences in a text T in time O(|T| + occ) using a data structure that occupies O(m log m) bits of space where m <= n + 1 is the number of states in the automaton. In this paper we show that the Aho-Corasick automaton can be represented in just m(log sigma + O(1)) + O(d log(n/d)) bits of space while still maintaining the ability to answer to queries in O(|T| + occ) time. To the best of our knowledge, the currently fastest succinct data structure for the dictionary matching problem uses space O(n log sigma) while answering queries in O(|T|log log n + occ) time. In this paper we also show how the space occupancy can be reduced to m(H0 + O(1)) + O(d log(n/d)) where H0 is the empirical entropy of the characters appearing in the trie representation of the set S, provided that sigma < m^epsilon for any constant 0 < epsilon < 1. The query time remains unchanged.Comment: Corrected typos and other minor error

Improved ESP-index: a practical self-index for highly repetitive texts

While several self-indexes for highly repetitive texts exist, developing a practical self-index applicable to real world repetitive texts remains a challenge. ESP-index is a grammar-based self-index on the notion of edit-sensitive parsing (ESP), an efficient parsing algorithm that guarantees upper bounds of parsing discrepancies between different appearances of the same subtexts in a text. Although ESP-index performs efficient top-down searches of query texts, it has a serious issue on binary searches for finding appearances of variables for a query text, which resulted in slowing down the query searches. We present an improved ESP-index (ESP-index-I) by leveraging the idea behind succinct data structures for large alphabets. While ESP-index-I keeps the same types of efficiencies as ESP-index about the top-down searches, it avoid the binary searches using fast rank/select operations. We experimentally test ESP-index-I on the ability to search query texts and extract subtexts from real world repetitive texts on a large-scale, and we show that ESP-index-I performs better that other possible approaches.Comment: This is the full version of a proceeding accepted to the 11th International Symposium on Experimental Algorithms (SEA2014

On-line construction of position heaps

We propose a simple linear-time on-line algorithm for constructing a position heap for a string [Ehrenfeucht et al, 2011]. Our definition of position heap differs slightly from the one proposed in [Ehrenfeucht et al, 2011] in that it considers the suffixes ordered from left to right. Our construction is based on classic suffix pointers and resembles the Ukkonen's algorithm for suffix trees [Ukkonen, 1995]. Using suffix pointers, the position heap can be extended into the augmented position heap that allows for a linear-time string matching algorithm [Ehrenfeucht et al, 2011].Comment: to appear in Journal of Discrete Algorithm

Succinct representation of triangulations with a boundary

We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangulations with a boundary we propose a succinct representation of the combinatorial information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports the navigation between adjacent triangles in constant time (as well as other standard operations). For triangulations with $m$ faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g-1)lg m bits (which is negligible as long as g << m/lg m)

Quantum computation with mesoscopic superposition states

We present a strategy to engineer a simple cavity-QED two-bit universal quantum gate using mesoscopic distinct quantum superposition states. The dissipative effect on decoherence and amplitude damping of the quantum bits are analyzed and the critical parameters are presented.Comment: 9 pages, 5 Postscript and 1 Encapsulated Postscript figures. To be published in Phys. Rev.

Fixed Linear Crossing Minimization by Reduction to the Maximum Cut Problem

Many real-life scheduling, routing and locating problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the so-called fixed linear crossing number problem (FLCNP). We show that this NP-hard problem can be reduced to the well-known maximum cut problem. The latter problem was intensively studied in the literature; practically efficient exact algorithms based on the branch-and-cut technique have been developed. By an experimental evaluation on a variety of graphs, we prove that using this reduction for solving FLCNP compares favorably to earlier branch-and-bound algorithms

Multipartite entangled coherent states

We propose a scheme for generating multipartite entangled coherent states via entanglement swapping, with an example of a physical realization in ion traps. Bipartite entanglement of these multipartite states is quantified by the concurrence. We also use the $N$--tangle to compute multipartite entanglement for certain systems. Finally we establish that these results for entanglement can be applied to more general multipartite entangled nonorthogonal states.Comment: 7 pages, two figures. We added more detail discussions on the generation of multipartite entangled coherent states and multipartite entangelemen