arXiv subjects

Darya Melnyk

Publications and source records attributed to Darya Melnyk.

22 records · Page 2Linked to original sources

arXiv2021

Local Mending

In this work we introduce the graph-theoretic notion of mendability: for each locally checkable graph problem we can define its mending radius, which captures the idea of how far one needs to modify a partial solution in order to "patch a hole." We explore how mendability is connected to the existence of efficient algorithms, especially in distributed, parallel, and fault-tolerant settings. It is easy to see that $O(1)$-mendable problems are also solvable in $O(\log^* n)$ rounds in the LOCAL model of distributed computing. One of the surprises is that in paths and cycles, a converse also holds in the following sense: if a problem $Π$ can be solved in $O(\log^* n)$, there is always a restriction $Π' \subseteq Π$ that is still efficiently solvable but that is also $O(1)$-mendable. We also explore the structure of the landscape of mendability. For example, we show that in trees, the mending radius of any locally checkable problem is $O(1)$, $Θ(\log n)$, or $Θ(n)$, while in general graphs the structure is much more diverse.

cs.DC↗

arXiv2021

Online k-Way Matching with Delays and the H-Metric

In this paper, we study $k$-Way Min-cost Perfect Matching with Delays - the $k$-MPMD problem. This problem considers a metric space with $n$ nodes. Requests arrive at these nodes in an online fashion. The task is to match these requests into sets of exactly $k$, such that the space and time cost of all matched requests are minimized. The notion of the space cost requires a definition of an underlying metric space that gives distances of subsets of $k$ elements. For $k>2$, the task of finding a suitable metric space is at the core of our problem: We show that for some known generalizations to $k=3$ points, such as the $2$-metric and the $D$-metric, there exists no competitive randomized algorithm for the $3$-MPMD problem. The $G$-metrics are defined for 3 points and allows for a competitive algorithm for the $3$-MPMD problem. For $k>3$ points, there exist two generalizations of the $G$-metrics known as $n$- and $K$-metrics. We show that neither the $n$-metrics nor the $K$-metrics can be used for the $k$-MPMD problem. On the positive side, we introduce the $H$-metrics, the first metrics to allow for a solution of the $k$-MPMD problem for all $k$. In order to devise an online algorithm for the $k$-MPMD problem on the $H$-metrics, we embed the $H$-metric into trees with an $O(\log n)$ distortion. Based on this embedding result, we extend the algorithm proposed by Azar et al. (2017) and achieve a competitive ratio of $O(\log n)$ for the $k$-MPMD problem.

cs.DS↗

arXiv2020

Space Complexity of Streaming Algorithms on Universal Quantum Computers

Universal quantum computers are the only general purpose quantum computers known that can be implemented as of today. These computers consist of a classical memory component which controls the quantum memory. In this paper, the space complexity of some data stream problems, such as PartialMOD and Equality, is investigated on universal quantum computers. The quantum algorithms for these problems are believed to outperform their classical counterparts. Universal quantum computers, however, need classical bits for controlling quantum gates in addition to qubits. Our analysis shows that the number of classical bits used in quantum algorithms is equal to or even larger than that of classical bits used in corresponding classical algorithms. These results suggest that there is no advantage of implementing certain data stream problems on universal quantum computers instead of classical computers when space complexity is considered.

quant-ph↗

arXiv2018

Byzantine Preferential Voting

In the Byzantine agreement problem, n nodes with possibly different input values aim to reach agreement on a common value in the presence of t < n/3 Byzantine nodes which represent arbitrary failures in the system. This paper introduces a generalization of Byzantine agreement, where the input values of the nodes are preference rankings of three or more candidates. We show that consensus on preferences, which is an important question in social choice theory, complements already known results from Byzantine agreement. In addition preferential voting raises new questions about how to approximate consensus vectors. We propose a deterministic algorithm to solve Byzantine agreement on rankings under a generalized validity condition, which we call Pareto-Validity. These results are then extended by considering a special voting rule which chooses the Kemeny median as the consensus vector. For this rule, we derive a lower bound on the approximation ratio of the Kemeny median that can be guaranteed by any deterministic algorithm. We then provide an algorithm matching this lower bound. To our knowledge, this is the first non-trivial multi-dimensional approach which can tolerate a constant fraction of Byzantine nodes.

cs.DC↗