In this article, we study d-reducibility and its injective counterpart, d_1 -reducibility. We prove a criterion for d-completeness in terms of fixed-point free functions and also show that a set is d-complete if and only if it is d_1 -complete. We also define a notion of (d,d_1) -cylinder, and, using this notion, prove that any noncomputable and nonsimple c.e. d-degree contains either only one or infinitely many d_1 -degrees.
A parameterized string (p-string) is a string that can contain two kinds of characters, static symbols and parameter characters. Parameterized pattern matching is a form of pattern matching that allows parameters to be renamed by applying a one-to-one function. The parameterized suffix array is a data structure that is useful in efficient parameterized pattern matching when accompanied by the parameterized longest common prefix (LCP) array. Reconstructing input from a given instance of a data structure is the task of determining whether the instance is valid or not, and if valid, producing a plausible set of data that it can represent. In this paper we consider parameterized suffix and LCP arrays and reconstruct a corresponding p-string that they can represent. In previous work, an algorithm can determine in O(n^2) time whether a p-string can be constructed to correspond to the input parameterized suffix and LCP arrays of size n [1]. In this work, we develop an algorithm that accomplishes this in Õ(n) time.
The Suffix Array is a classic text index enabling on-line pattern matching queries via simple binary search. The main drawback of the Suffix Array is that it takes linear space in the text’s length, even when the text itself is extremely compressible. Several works in the literature showed that the Suffix Array can be compressed, but they all rely on complex succinct data structures which in practice tend to exhibit poor cache locality and thus significantly slow down queries. In this paper, we propose a new simple and very efficient solution to this problem by presenting the Suffixient Array: a tiny subset of the Suffix Array sufficient to locate on-line one pattern occurrence (in general, all its Maximal Exact Matches) via binary search, provided that random access to the text is available. We prove that: (i) the Suffixient Array length χ is a strong repetitiveness measure, (ii) unlike most existing repetition-aware indexes such as the r -index, our new index is efficient in the I/O model, and (iii) Suffixient Arrays can be computed in linear time and compressed working space. We show experimentally that, when using well-established compressed random access data structures on repetitive collections, the Suffixient Array sA is simultaneously (i) faster and orders of magnitude smaller than the Suffix Array and (ii) smaller and one to two orders of magnitude faster than the r -index. With an average pattern matching query time as low as 3.5 ns per character, our new index gets very close to the ultimate lower bound: the RAM throughput of our workstation (1.18 ns per character).
A closed string u is either of length one or contains a non-empty border that occurs only as a prefix and as a suffix in u and nowhere else within u. In this paper, we present fast 𝒪(nlog n) time algorithms to compute all 𝒪(n^2) closed substrings by introducing a compact representation for all closed substrings of a string w[1..n], using only 𝒪(n log n) space. These simple and space-efficient algorithms also compute maximal closed strings. Furthermore, we compare the performance of these algorithms and identify classes of strings where each performs best. Finally, we show that the exact number of maximal closed substrings in a Fibonacci word f_n , for n ≥ 5 , is ≈( 1 + 1/ϕ ^2) F_n ≈ 1.382 F_n , where ϕ is the golden ratio and F_n = |f_n| .
Motivated by the strict latency constraints in high-throughput cloud storage, this article addresses the fundamental theoretical challenge of designing high-efficiency bin packing algorithms with near-linear-time complexity. While traditional approximation algorithms achieve high packing density, their super-linear runtime overheads often prohibit their use in real-time systems. To bridge this gap between time and space efficiency, we apply a novel grouping-based framework that discretizes the item space into K types. We present an online algorithm OnGP and an offline algorithm OffGP, both utilizing an equidistant grouping strategy. Theoretical analysis establishes that OnGP suppresses stochastic fluctuations via a resource pooling parameter, while OffGP reaches a matching fixed point with high probability for fixed K when m≥ K_c and admits complementary instance-dependent residual certificates. Extensive experiments further demonstrate that OffGP substantially reduces the stochastic component of waste on the tested instances. Results show that OnGP provides a robust online solution with efficiencies close to the Best Fit algorithm, while OffGP matches the near-optimal performance of Best Fit Decreasing within a 1.036% margin and is approximately 6.5× faster, confirming their value for time-critical applications.
