Cache-Oblivious Implicit Predecessor Dictionaries with the Working Set Property

In this paper we present an implicit dynamic dictionary with the working-set property, supporting insert(e) and delete(e) in O(log n) time, predecessor(e) in O(log l_{p(e)}) time, successor(e) in O(log l_{s(e)}) time and search(e) in O(log min(l_{p(e)},l_{e}, l_{s(e)})) time, where n is the number of elements stored in the dictionary, l_{e} is the number of distinct elements searched for since element e was last searched for and p(e) and s(e) are the predecessor and successor of e, respectively. The time-bounds are all worst-case. The dictionary stores the elements in an array of size n using no additional space. In the cache-oblivious model the log is base B and the cache-obliviousness is due to our black box use of an existing cache-oblivious implicit dictionary. This is the first implicit dictionary supporting predecessor and successor searches in the working-set bound. Previous implicit structures required O(log n) time.Comment: An extended abstract is accepted at STACS 2012, this is the full version of that paper with the same name "Cache-Oblivious Implicit Predecessor Dictionaries with the Working-Set Property", Symposium on Theoretical Aspects of Computer Science 201

PArtially Persistent Data Structures of Bounded Degree with Constant Update Time

The problem of making bounded in-degree and out-degree data structures partially persistent is considered. The node copying method of Driscoll et al. is extended so that updates can be performed in worst-case constant time on the pointer machine model. Previously it was only known to be possible in amortised constant time [Driscoll89]. The result is presented in terms of a new strategy for Dietz and Raman's dynamic two player pebble game on graphs. It is shown how to implement the strategy, and the upper bound on the required number of pebbles is improved from 2b + 2d + O(sqrt(b)) to d + 2b, where b is the bound of the in-degree and d the bound of the out-degree. We also give a lower bound that shows that the number of pebbles depends on the out-degree d

Soft Sequence Heaps

Chazelle [JACM00] introduced the soft heap as a building block for efficient minimum spanning tree algorithms, and recently Kaplan et al. [SOSA2019] showed how soft heaps can be applied to achieve simpler algorithms for various selection problems. A soft heap trades-off accuracy for efficiency, by allowing $\epsilon N$ of the items in a heap to be corrupted after a total of $N$ insertions, where a corrupted item is an item with artificially increased key and $0 < \epsilon \leq 1/2$ is a fixed error parameter. Chazelle's soft heaps are based on binomial trees and support insertions in amortized $O(\lg(1/\epsilon))$ time and extract-min operations in amortized $O(1)$ time. In this paper we explore the design space of soft heaps. The main contribution of this paper is an alternative soft heap implementation based on merging sorted sequences, with time bounds matching those of Chazelle's soft heaps. We also discuss a variation of the soft heap by Kaplan et al. [SICOMP2013], where we avoid performing insertions lazily. It is based on ternary trees instead of binary trees and matches the time bounds of Kaplan et al., i.e. amortized $O(1)$ insertions and amortized $O(\lg(1/\epsilon))$ extract-min. Both our data structures only introduce corruptions after extract-min operations which return the set of items corrupted by the operation.Comment: 16 pages, 3 figure

School-Aged Children With Severe Disabilities And Limited Verbal Language: Exploring The Importance Of Social Interactions And Friendship, And Its Application In Practice

Social interactions and friendships with peers have been found to be essential to children’s and adolescents’ development, learning, and overall quality of life. However, research shows children and adolescents with severe disabilities and limited verbal language have fewer friendships and quality social interaction with peers than those without disabilities. This still occurs today, despite the implementation of the Least Restrictive Environment (LRE) legislation in 1978. One of the goals of the LRE was to maximize the opportunities children and adolescents with disabilities have when it comes to social interactions with peers in the general education classroom. However, as the general education classroom is not always deemed the most appropriate learning environment for students with severe needs, they receive very little of their education outside of their specialized classrooms. Knowing how important social interaction and peer relationships are when it comes to development, learning, and overall quality of life, it becomes essential to understand why these research findings are not being implemented into practice on a larger scale. Thus, this three-article dissertation aims to explore how past, current, and future research on social interactions and friendship impact how we educate students with disabilities and limited verbal language. Further, this study aims to provide educators, policymakers, and researchers a holistic understanding of why social inclusion and peer relationships are essential to how well students with disabilities and limited verbal language succeed in school, their sense of belonging, and their overall quality of life

Fast Algorithm for Partial Covers in Words

A factor $u$ of a word $w$ is a cover of $w$ if every position in $w$ lies within some occurrence of $u$ in $w$. A word $w$ covered by $u$ thus generalizes the idea of a repetition, that is, a word composed of exact concatenations of $u$. In this article we introduce a new notion of $\alpha$-partial cover, which can be viewed as a relaxed variant of cover, that is, a factor covering at least $\alpha$ positions in $w$. We develop a data structure of $O(n)$ size (where $n=|w|$) that can be constructed in $O(n\log n)$ time which we apply to compute all shortest $\alpha$-partial covers for a given $\alpha$. We also employ it for an $O(n\log n)$-time algorithm computing a shortest $\alpha$-partial cover for each $\alpha=1,2,\ldots,n$

Deterministic Cache-Oblivious Funnelselect

In the multiple-selection problem one is given an unsorted array $S$ of $N$ elements and an array of $q$ query ranks $r_1<\cdots<r_q$, and the task is to return, in sorted order, the $q$ elements in $S$ of rank $r_1, \ldots, r_q$, respectively. The asymptotic deterministic comparison complexity of the problem was settled by Dobkin and Munro [JACM 1981]. In the I/O model an optimal I/O complexity was achieved by Hu et al. [SPAA 2014]. Recently [ESA 2023], we presented a cache-oblivious algorithm with matching I/O complexity, named funnelselect, since it heavily borrows ideas from the cache-oblivious sorting algorithm funnelsort from the seminal paper by Frigo, Leiserson, Prokop and Ramachandran [FOCS 1999]. Funnelselect is inherently randomized as it relies on sampling for cheaply finding many good pivots. In this paper we present deterministic funnelselect, achieving the same optional I/O complexity cache-obliviously without randomization. Our new algorithm essentially replaces a single (in expectation) reversed-funnel computation using random pivots by a recursive algorithm using multiple reversed-funnel computations. To meet the I/O bound, this requires a carefully chosen subproblem size based on the entropy of the sequence of query ranks; deterministic funnelselect thus raises distinct technical challenges not met by randomized funnelselect. The resulting worst-case I/O bound is $O\bigl(\sum_{i=1}^{q+1} \frac{\Delta_i}{B} \cdot \log_{M/B} \frac{N}{\Delta_i} + \frac{N}{B}\bigr)$, where $B$ is the external memory block size, $M\geq B^{1+\epsilon}$ is the internal memory size, for some constant $\epsilon>0$, and $\Delta_i = r_{i} - r_{i-1}$ (assuming $r_0=0$ and $r_{q+1}=N + 1$)