Abstract. Using an orderly algorithm, the Steiner triple systems of order 19 are classified; there are 11,084,874,829 pairwise nonisomorphic such designs. For each design, the order of its automorphism group and the number of Pasch configurations it contains are recorded; 2,591 of the designs are anti-Pasch. There are three main parts of the classification: constructing an initial set of blocks, the seeds; completing the seeds to triple systems with an algorithm for exact cover; and carrying out isomorph rejection of the final triple systems. Isomorph rejection is based on the graph canonical labeling software nauty supplemented with a vertex invariant based on Pasch configurations. The possibility of using the (strongly regular) block graphs of these designs in the isomorphism tests is utilized. The aforementioned value is in fact a lower bound on the number of pairwise nonisomorphic strongly regular graphs with parameters (57, 24,11,9).
Let F n q be a vector space of dimension n over the finite field F q . A q-analog of a Steiner system (also known as a q-Steiner system), denoted S q (t,k,n), is a set S of k-dimensional subspaces of F n q such that each t-dimensional subspace of F n q is contained in exactly one element of S. Presently, q-Steiner systems are known only for t = 1, and in the trivial cases t = k and k = n. In this paper, the first nontrivial q-Steiner systems with t 2 are constructed. Specifically, several nonisomorphic q-Steiner systems S 2 (2, 3, 13) are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of GL(13, 2). This approach leads to an instance of the exact cover problem, which turns out to have many solutions.
LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of O(1), Θ(log * n), or Θ(n), and the design of optimal algorithms can be fully automated.This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: O(1), Θ(log * n), and Θ(n). However, given an LCL problem it is undecidable whether its complexity is Θ(log * n) or Θ(n) in 2-dimensional grids.Nevertheless, if we correctly guess that the complexity of a problem is Θ(log * n), we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form A • S k , where A is a finite function, S k is an algorithm for finding a maximal independent set in kth power of the grid, and k is a constant.Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations. arXiv:1702.05456v2 [cs.DC] 24 May 2017 1.1 Problem setting: LCL problems on grids 92 33 77 57 49 26 71 79 8 62 48 24 31 21 15 30 60 67 0 5 17 95 23 47 87 80 25 38 20 64 45 61 91 51 69 1 74 55 3 98 88 99 58 53 63 40 16 2 39Grids. In this work, we study distributed algorithms in a setting where the underlying input graph is a grid. Specifically, we consider the complexity of locally checkable labelling problems, or LCL problems, in the standard LOCAL model of distributed complexity, and consider graphs that are toroidal two-dimensional n × n grids with a consistent orientation; we focus on the two-dimensional case for concreteness, but most of our results generalise to d-dimensional grids of arbitrary dimensions. This setting occupies a middle ground between the wellunderstood directed n-cycles [10,32], where all solvable LCL problems are known to have deterministic time complexity either O(1), Θ(log * n) or Θ(n), and the more complicated setting of general n-vertex graphs, where intermediate problems with time complexities such as Θ(log n) are known to exist, even for bounded-degree graphs. Grid-like systems with local dynamics also occur frequently in the study of real-world phenomena. However, grids have so far not been systematically studied from a distributed computing perspective.LOCAL model and LCL problems. In the LOCAL model of distributed computing, nodes are labelled with unique numerical identifiers with O(log n) bits. A time-t algorithm in this model is simply a mapping from radius-t neighbourhoods to local outputs; equivalently, it can be interpreted as a message-passing algorithm in which the nodes exchange messages for t synchronous rounds and then announce their local outputs.LCL problems are graph problems for which the feasibility of a solution can be verified by checking the solution for each O(1)-radius neighbourhood; if all local neighbourhoods look valid, the s...
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