Theory of Computing Systems最新文献

The amortized step complexity of an implementation measures its performance as a whole, rather than the performance of individual operations. Specifically, the amortized step complexity of an implementation is the average number of steps performed by invoked operations, in the worst case, taken over all possible executions. The point contention of an execution, denoted by (dot{c}), measures the maximal number of precesses simultaneously active in the execution. Ruppert (2016) showed that the amortized step complexity of known lock-free implementations for many shared data structures includes an additive factor linear in the point contention (dot{c}). This paper shows that there is no lock-free implementation with (o(min {dot{c}, sqrt{log log n}})) amortized RMR complexity of queues, stacks or heaps from reads, writes, comparison primitives (such as compare &swap) and LL/SC, where n is the total number of the processes in the system. In addition, the paper shows a (Omega (min {dot{c}, log log n})) lower bound on the amortized step complexity for shared linked lists, skip lists, search trees and other pointer-based data structures. These lower bounds mean that the additive factor linear in (dot{c}) is inherent for these implementations, provided that the point contention is small compared to the number of processes in the system (i.e. (dot{c}in O(sqrt{log log n})) or (dot{c}in O(log log n))).

Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time (k^{O(k + d)} s^{O(1)}) for inputs of size s, whereas Temporal Cluster Editing is (textsf {W[1]})-hard with respect to k even if (d = 3).

We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in (O(m+n)) time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs.

We investigate several geometric problems of finding tours and cycle covers with minimum turn cost, which have been studied in the past, with complexity, approximation results, and open problems dating back to work by Arkin et al. in 2001. Many new practical applications have spawned variants: For full coverage, all points have to be covered, for subset coverage, specific points have to be covered, and for penalty coverage, points may be left uncovered by incurring a penalty. We show that finding a minimum-turn (full) cycle cover is NP-hard even in 2-dimensional grid graphs, solving the long-standing open Problem 53 in The Open Problems Project edited by Demaine, Mitchell and O’Rourke. We also prove NP-hardness of finding a subset cycle cover of minimum turn cost in thin grid graphs, for which Arkin et al. gave a polynomial-time algorithm for full coverage; this shows that their boundary techniques cannot be applied to compute exact solutions for subset and penalty variants. We also provide a number of positive results. In particular, we establish the first constant-factor approximation algorithms for all considered subset and penalty problem variants for grid-based instances, based on LP/IP techniques. These geometric versions allow many possible edge directions (and thus, turn angles, such as in hexagonal grids or higher-dimensional variants); our approximation factors improve the combinatorial ones of Arkin et al.

We combine Solomonoff’s approach to universal prediction with algorithmic statistics and suggest to use the computable measure that provides the best “explanation” for the observed data (in the sense of algorithmic statistics) for prediction. In this way we keep the expected sum of squares of prediction errors bounded (as it was for the Solomonoff’s predictor) and, moreover, guarantee that the sum of squares of prediction errors is bounded along any Martin-Löf random sequence. An extended abstract of this paper was presented at the 16th International Computer Science Symposium in Russia (CSR 2021) (Milovanov 2021).

The power word problem for a group (varvec{G}) asks whether an expression (varvec{u_1^{x_1} cdots u_n^{x_n}}), where the (varvec{u_i}) are words over a finite set of generators of (varvec{G}) and the (varvec{x_i}) binary encoded integers, is equal to the identity of (varvec{G}). It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over (varvec{G})). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group (varvec{G}) is (varvec{textsf{uNC}^{1}})-many-one reducible to the power word problem for a finite-index subgroup of (varvec{G}). For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is (varvec{textsf{AC} ^0})-Turing-reducible to the word problem for the free group (varvec{F_2}) and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups (varvec{mathcal {C}}) without order two elements, the uniform power word problem in a graph product can be solved in (varvec{textsf{AC} ^0[textsf{C}_=textsf{L} ^{{{,textrm{UPowWP},}}(mathcal {C})}]}), where (varvec{{{,textrm{UPowWP},}}(mathcal {C})}) denotes the uniform power word problem for groups from the class (varvec{mathcal {C}}). As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is (varvec{textsf{NP}})-complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated in Stober and Weiß (2022a) is true, our proof relies on this additional assumption.