arXiv subjects

Joan Boyar

Publications and source records attributed to Joan Boyar.

33 records · Page 2Linked to original sources

arXiv2017

Online Bounded Analysis

Though competitive analysis is often a very good tool for the analysis of online algorithms, sometimes it does not give any insight and sometimes it gives counter-intuitive results. Much work has gone into exploring other performance measures, in particular targeted at what seems to be the core problem with competitive analysis: the comparison of the performance of an online algorithm is made to a too powerful adversary. We consider a new approach to restricting the power of the adversary, by requiring that when judging a given online algorithm, the optimal offline algorithm must perform as well as the online algorithm, not just on the entire final request sequence, but also on any prefix of that sequence. This is limiting the adversary's usual advantage of being able to exploit that it knows the sequence is continuing beyond the current request. Through a collection of online problems, including machine scheduling, bin packing, dual bin packing, and seat reservation, we investigate the significance of this particular offline advantage.

cs.DS↗

arXiv2017

Relaxing the Irrevocability Requirement for Online Graph Algorithms

Online graph problems are considered in models where the irrevocability requirement is relaxed. Motivated by practical examples where, for example, there is a cost associated with building a facility and no extra cost associated with doing it later, we consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we also consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. For Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, but for Vertex Cover the Late Accept model is sufficient and for Minimum Spanning Forest the Late Reject model is sufficient. The Matching problem has a competitive ratio of 2, but in the Late Accept/Reject model, its competitive ratio is 3/2.

cs.DS↗

arXiv2017

Adding Isolated Vertices Makes some Online Algorithms Optimal

An unexpected difference between online and offline algorithms is observed. The natural greedy algorithms are shown to be worst case online optimal for Online Independent Set and Online Vertex Cover on graphs with 'enough' isolated vertices, Freckle Graphs. For Online Dominating Set, the greedy algorithm is shown to be worst case online optimal on graphs with at least one isolated vertex. These algorithms are not online optimal in general. The online optimality results for these greedy algorithms imply optimality according to various worst case performance measures, such as the competitive ratio. It is also shown that, despite this worst case optimality, there are Freckle graphs where the greedy independent set algorithm is objectively less good than another algorithm. It is shown that it is NP-hard to determine any of the following for a given graph: the online independence number, the online vertex cover number, and the online domination number.

cs.DM↗

arXiv2016

Batch Coloring of Graphs

In graph coloring problems, the goal is to assign a positive integer color to each vertex of an input graph such that adjacent vertices do not receive the same color assignment. For classic graph coloring, the goal is to minimize the maximum color used, and for the sum coloring problem, the goal is to minimize the sum of colors assigned to all input vertices. In the offline variant, the entire graph is presented at once, and in online problems, one vertex is presented for coloring at each time, and the only information is the identity of its neighbors among previously known vertices. In batched graph coloring, vertices are presented in k batches, for a fixed integer k > 1, such that the vertices of a batch are presented as a set, and must be colored before the vertices of the next batch are presented. This last model is an intermediate model, which bridges between the two extreme scenarios of the online and offline models. We provide several results, including a general result for sum coloring and results for the classic graph coloring problem on restricted graph classes: We show tight bounds for any graph class containing trees as a subclass (e.g., forests, bipartite graphs, planar graphs, and perfect graphs), and a surprising result for interval graphs and k = 2, where the value of the (strict and asymptotic) competitive ratio depends on whether the graph is presented with its interval representation or not.

cs.DS↗

arXiv2016

On the List Update Problem with Advice

We study the online list update problem under the advice model of computation. Under this model, an online algorithm receives partial information about the unknown parts of the input in the form of some bits of advice generated by a benevolent offline oracle. We show that advice of linear size is required and sufficient for a deterministic algorithm to achieve an optimal solution or even a competitive ratio better than $15/14$. On the other hand, we show that surprisingly two bits of advice are sufficient to break the lower bound of $2$ on the competitive ratio of deterministic online algorithms and achieve a deterministic algorithm with a competitive ratio of $5/3$. In this upper-bound argument, the bits of advice determine the algorithm with smaller cost among three classical online algorithms, TIMESTAMP and two members of the MTF2 family of algorithms. We also show that MTF2 algorithms are $2.5$-competitive.

cs.DS↗

arXiv2016

The Advice Complexity of a Class of Hard Online Problems

The advice complexity of an online problem is a measure of how much knowledge of the future an online algorithm needs in order to achieve a certain competitive ratio. Using advice complexity, we define the first online complexity class, AOC. The class includes independent set, vertex cover, dominating set, and several others as complete problems. AOC-complete problems are hard, since a single wrong answer by the online algorithm can have devastating consequences. For each of these problems, we show that $\log\left(1+(c-1)^{c-1}/c^{c}\right)n=Θ(n/c)$ bits of advice are necessary and sufficient (up to an additive term of $O(\log n)$) to achieve a competitive ratio of $c$. The results are obtained by introducing a new string guessing problem related to those of Emek et al. (TCS 2011) and Böckenhauer et al. (TCS 2014). It turns out that this gives a powerful but easy-to-use method for providing both upper and lower bounds on the advice complexity of an entire class of online problems, the AOC-complete problems. Previous results of Halldórsson et al. (TCS 2002) on online independent set, in a related model, imply that the advice complexity of the problem is $Θ(n/c)$. Our results improve on this by providing an exact formula for the higher-order term. For online disjoint path allocation, Böckenhauer et al. (ISAAC 2009) gave a lower bound of $Ω(n/c)$ and an upper bound of $O((n\log c)/c)$ on the advice complexity. We improve on the upper bound by a factor of $\log c$. For the remaining problems, no bounds on their advice complexity were previously known.

cs.DS↗

arXiv2015

Constructive Relationships Between Algebraic Thickness and Normality

We study the relationship between two measures of Boolean functions; \emph{algebraic thickness} and \emph{normality}. For a function $f$, the algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero coefficients in the unique GF(2) polynomial representing $f$, and the normality is the largest dimension of an affine subspace on which $f$ is constant. We show that for $0 < ε<2$, any function with algebraic thickness $n^{3-ε}$ is constant on some affine subspace of dimension $Ω\left(n^{\fracε{2}}\right)$. Furthermore, we give an algorithm for finding such a subspace. We show that this is at most a factor of $Θ(\sqrt{n})$ from the best guaranteed, and when restricted to the technique used, is at most a factor of $Θ(\sqrt{\log n})$ from the best guaranteed. We also show that a concrete function, majority, has algebraic thickness $Ω\left(2^{n^{1/6}}\right)$.

cs.CC↗

arXiv2014

Cancellation-Free Circuits in Unbounded and Bounded Depth

We study the notion of "cancellation-free" circuits. This is a restriction of linear Boolean circuits (XOR circuits), but can be considered as being equivalent to previously studied models of computation. The notion was coined by Boyar and Peralta in a study of heuristics for a particular circuit minimization problem. They asked how large a gap there can be between the smallest cancellation-free circuit and the smallest linear circuit. We show that the difference can be a factor $Ω(n/\log^{2}n)$. This improves on a recent result by Sergeev and Gashkov who have studied a similar problem. Furthermore, our proof holds for circuits of constant depth. We also study the complexity of computing the Sierpinski matrix using cancellation-free circuits and give a tight $Ω(n\log n)$ lower bound.

cs.CC↗

arXiv2013

Online Bin Packing with Advice

We consider the online bin packing problem under the advice complexity model where the 'online constraint' is relaxed and an algorithm receives partial information about the future requests. We provide tight upper and lower bounds for the amount of advice an algorithm needs to achieve an optimal packing. We also introduce an algorithm that, when provided with log n + o(log n) bits of advice, achieves a competitive ratio of 3/2 for the general problem. This algorithm is simple and is expected to find real-world applications. We introduce another algorithm that receives 2n + o(n) bits of advice and achieves a competitive ratio of 4/3 + ε. Finally, we provide a lower bound argument that implies that advice of linear size is required for an algorithm to achieve a competitive ratio better than 9/8.

cs.DS↗

arXiv2013

The Frequent Items Problem in Online Streaming under Various Performance Measures

In this paper, we strengthen the competitive analysis results obtained for a fundamental online streaming problem, the Frequent Items Problem. Additionally, we contribute with a more detailed analysis of this problem, using alternative performance measures, supplementing the insight gained from competitive analysis. The results also contribute to the general study of performance measures for online algorithms. It has long been known that competitive analysis suffers from drawbacks in certain situations, and many alternative measures have been proposed. However, more systematic comparative studies of performance measures have been initiated recently, and we continue this work, using competitive analysis, relative interval analysis, and relative worst order analysis on the Frequent Items Problem.

cs.DS↗

arXiv2013

Relative Interval Analysis of Paging Algorithms on Access Graphs

Access graphs, which have been used previously in connection with competitive analysis and relative worst order analysis to model locality of reference in paging, are considered in connection with relative interval analysis. The algorithms LRU, FIFO, FWF, and FAR are compared using the path, star, and cycle access graphs. In this model, some of the expected results are obtained. However, although LRU is found to be strictly better than FIFO on paths, it has worse performance on stars, cycles, and complete graphs, in this model. We solve an open question from [Dorrigiv, Lopez-Ortiz, Munro, 2009], obtaining tight bounds on the relationship between LRU and FIFO with relative interval analysis.

cs.DS↗

arXiv2012

A Comparison of Performance Measures for Online Algorithms

This paper provides a systematic study of several proposed measures for online algorithms in the context of a specific problem, namely, the two server problem on three colinear points. Even though the problem is simple, it encapsulates a core challenge in online algorithms which is to balance greediness and adaptability. We examine Competitive Analysis, the Max/Max Ratio, the Random Order Ratio, Bijective Analysis and Relative Worst Order Analysis, and determine how these measures compare the Greedy Algorithm, Double Coverage, and Lazy Double Coverage, commonly studied algorithms in the context of server problems. We find that by the Max/Max Ratio and Bijective Analysis, Greedy is the best of the three algorithms. Under the other measures, Double Coverage and Lazy Double Coverage are better, though Relative Worst Order Analysis indicates that Greedy is sometimes better. Only Bijective Analysis and Relative Worst Order Analysis indicate that Lazy Double Coverage is better than Double Coverage. Our results also provide the first proof of optimality of an algorithm under Relative Worst Order Analysis.

cs.DS↗

arXiv2012

Access Graphs Results for LRU versus FIFO under Relative Worst Order Analysis

Access graphs, which have been used previously in connection with competitive analysis to model locality of reference in paging, are considered in connection with relative worst order analysis. In this model, FWF is shown to be strictly worse than both LRU and FIFO on any access graph. LRU is shown to be strictly better than FIFO on paths and cycles, but they are incomparable on some families of graphs which grow with the length of the sequences.

cs.DS↗

arXiv2012

Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators

We continue to study the notion of cancellation-free linear circuits. We show that every matrix can be computed by a cancellation- free circuit, and almost all of these are at most a constant factor larger than the optimum linear circuit that computes the matrix. It appears to be easier to prove statements about the structure of cancellation-free linear circuits than for linear circuits in general. We prove two nontrivial superlinear lower bounds. We show that a cancellation-free linear circuit computing the $n\times n$ Sierpinski gasket matrix must use at least 1/2 n logn gates, and that this is tight. This supports a conjecture by Aaronson. Furthermore we show that a proof strategy for proving lower bounds on monotone circuits can be almost directly converted to prove lower bounds on cancellation-free linear circuits. We use this together with a result from extremal graph theory due to Andreev to prove a lower bound of Ω(n^(2- ε)) for infinitely many $n \times n$ matrices for every $ε> 0$ for. These lower bounds for concrete matrices are almost optimal since all matrices can be computed with $O(n^2/\log n)$ gates.

cs.CC↗

arXiv2011

A Comparison of Performance Measures via Online Search

Though competitive analysis has been a very useful performance measure for the quality of online algorithms, it is recognized that it sometimes fails to distinguish between algorithms of different quality in practice. A number of alternative measures have been proposed, but, with a few exceptions, these have generally been applied only to the online problem they were developed in connection with. Recently, a systematic study of performance measures for online algorithms was initiated [Boyar, Irani, Larsen: Eleventh International Algorithms and Data Structures Symposium 2009], first focusing on a simple server problem. We continue this work by studying a fundamentally different online problem, online search, and the Reservation Price Policies in particular. The purpose of this line of work is to learn more about the applicability of various performance measures in different situations and the properties that the different measures emphasize. We investigate the following analysis techniques: Competitive, Relative Worst Order, Bijective, Average, Relative Interval, Random Order, and Max/Max. In addition to drawing conclusions on this work, we also investigate the measures' sensitivity to integral vs. real-valued domains, and as a part of this work, generalize some of the known performance measures. Finally, we have established the first optimality proof for Relative Interval Analysis.

cs.DS↗