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Several methods for generating random Steiner triple systems (STSs) have been proposed in the literature, such as Stinson's hill-climbing algorithm and Cameron's algorithm, but these are not yet completely understood. Those algorithms, as…
Storage architectures ranging from minimum bandwidth regenerating encoded distributed storage systems to declustered-parity RAIDs can be designed using dense partial Steiner systems in order to support fast reads, writes, and recovery of…
The pasch configuration and Steiner triple systems
We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).
Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs,…
We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a…
The paper addresses design/building frameworks for some kinds of tree-like and hierarchical structures of systems. The following approaches are examined: (1) expert-based procedures, (2) hierarchical clustering; (3) spanning problems (e.g.,…
String diagrams are an increasingly popular algebraic language for the analysis of graphical models of computations across different research fields. Whereas string diagrams have been thoroughly studied as semantic structures, much less…
Alignments, i.e., position-wise comparisons of two or more strings or ordered lists are of utmost practical importance in computational biology and a host of other fields, including historical linguistics and emerging areas of research in…
Many code families such as low-density parity-check codes, fractional repetition codes, batch codes and private information retrieval codes with low storage overhead rely on the use of combinatorial block designs or derivatives thereof. In…
In this article, we construct a Steiner system with the parameters $S(3,6,42)$, settling one of the smallest open parameter sets of Steiner $3$-designs. Furthermore, we establish the existence of rotational Steiner quadruple systems on $46$…
We study a class of combinatorial designs called Kirkman systems, and we show that infinitely many Kirkman systems are well-distributed in a precise sense. Steiner triple systems of order $n$ can achieve a minimum block sum of $n$. Kirkman…
Tiering is an essential technique for building large-scale information retrieval systems. While the selection of documents for high priority tiers critically impacts the efficiency of tiering, past work focuses on optimizing it with respect…
Euclidean Steiner trees are relevant to model minimal networks in real-world applications ubiquitously. In this paper, we study the feasibility of a hierarchical approach embedded with bundling operations to compute multiple and mutually…
The $\mathscr{P}$-position sets of some combinatorial games have special combinatorial structures. For example, the $\mathscr{P}$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system…
A partial Steiner triple system of order $u$ is a pair $(U,\mathcal{A})$ where $U$ is a set of $u$ elements and $\mathcal{A}$ is a set of triples of elements of $U$ such that any two elements of $U$ occur together in at most one triple. If…
Recommender systems are one of the most applied methods in machine learning and find applications in many areas, ranging from economics to the Internet of things. This article provides a general overview of modern approaches to recommender…
Well-structured systems, aka WSTSs, are computational models where the set of possible configurations is equipped with a well-quasi-ordering which is compatible with the transition relation between configurations. This structure supports…
We consider a distributed computing system in which a master node coordinates $N$ workers to evaluate a function over $n$ input files, where this function accepts general decomposition. In particular, we focus on the general case where the…
Compositionality is a key strategy for addressing combinatorial complexity and the curse of dimensionality. Recent work has shown that compositional solutions can be learned and offer substantial gains across a variety of domains, including…