Online Conflict-Free Colouring for Hypergraphs

2009 ◽  
Vol 19 (4) ◽  
pp. 493-516 ◽  
Author(s):  
A. BAR-NOY ◽  
P. CHEILARIS ◽  
S. OLONETSKY ◽  
S. SMORODINSKY
Keyword(s):  
High Probability ◽  
Online Algorithm ◽  
Asymptotically Optimal ◽  
Unit Disks

We provide a framework for online conflict-free colouring of any hypergraph. We introduce the notion of a degenerate hypergraph, which characterizes hypergraphs that arise in geometry. We use our framework to obtain an efficient randomized online algorithm for conflict-free colouring of any k-degenerate hypergraph with n vertices. Our algorithm uses O(k log n) colours with high probability and this bound is asymptotically optimal. Moreover, our algorithm uses O(k log k log n) random bits with high probability. We introduce algorithms that are allowed to perform a few recolourings of already coloured points. We provide deterministic online conflict-free colouring algorithms for points on the line with respect to intervals and for points on the plane with respect to half-planes (or unit disks) that use O(log n) colours and perform a total of at most O(n) recolourings.

scholarly journals Balanced Parallel Exploration of Orthogonal Regions

Algorithms ◽  
2019 ◽  
Vol 12 (5) ◽  
pp. 104
Author(s):  
Wyatt Clements ◽  
Costas Busch ◽  
Limeng Pu ◽  
Daniel Smith ◽  
Hsiao-Chun Wu
Keyword(s):  
Mobile Agents ◽  
Online Algorithm ◽  
Total Work ◽  
Entire Area ◽  
Asymptotically Optimal ◽  
Face To Face ◽  
Exploration Time ◽  
Unknown Area ◽  
Agent Coordination ◽  
Multiple Mobile Agents

We consider the use of multiple mobile agents to explore an unknown area. The area is orthogonal, such that all perimeter lines run both vertically and horizontally. The area may consist of unknown rectangular holes which are non-traversable internally. For the sake of analysis, we assume that the area is discretized into N points allowing the agents to move from one point to an adjacent one. Mobile agents communicate through face-to-face communication when in adjacent points. The objective of exploration is to develop an online algorithm that will explore the entire area while reducing the total work of all k agents, where the work is measured as the number of points traversed. We propose splitting the exploration into two alternating tasks, perimeter and room exploration. The agents all begin with the perimeter scan and when a room is found they transition to room scan after which they continue with perimeter scan until the next room is found and so on. Given the total traversable points N, our algorithm completes in total O ( N ) work with each agent performing O ( N / k ) work, namely the work is balanced. If the rooms are hole-free the exploration time is also asymptotically optimal, O ( N / k ) . To our knowledge, this is the first agent coordination algorithm that considers simultaneously work balancing and small exploration time.

scholarly journals MAXIMIZING THE AREA OF OVERLAP OF TWO UNIONS OF DISKS UNDER RIGID MOTION

2009 ◽  
Vol 19 (06) ◽  
pp. 533-556 ◽  
Author(s):  
SERGIO CABELLO ◽  
MARK DE BERG ◽  
PANOS GIANNOPOULOS ◽  
CHRISTIAN KNAUER ◽  
RENÉ VAN OOSTRUM ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
High Probability ◽  
Rigid Motion ◽  
Maximum Area ◽  
Empty Intersection ◽  
Rigid Motions ◽  
Unit Disks

Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m ≥ n. We consider the problem of finding a translation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1 - ∊)-approximation algorithms for translations and for rigid motions, which run in O((nm/∊2) log (m/∊)) and O((n2m2/∊3) log m)) time, respectively. For rigid motions, we can also compute a (1 - ∊)-approximation in O((m2n4/3Δ1/3/∊3) log n log m) time, where Δ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1 - ∊)-approximation algorithm for rigid motions that runs in O((m2/∊4) log 2(m/∊) log m) time and succeeds with high probability. Our results generalize to the case where A and B consist of possibly intersecting disks of different radii, provided that (i) the ratio of the radii of any two disks in A ∪ B is bounded, and (ii) within each set, the maximum number of disks with a non-empty intersection is bounded.

RELIABLE INTERNET-BASED MASTER-WORKER COMPUTING IN THE PRESENCE OF MALICIOUS WORKERS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250002 ◽  
Author(s):  
ANTONIO FERNÁNDEZ ANTA ◽  
CHRYSSIS GEORGIOU ◽  
LUIS LÓPEZ ◽  
AGUSTÍN SANTOS
Keyword(s):  
Lower Bounds ◽  
Distributed System ◽  
High Probability ◽  
Upper Bounds ◽  
The Internet ◽  
Decision Strategy ◽  
Single Task ◽  
Asymptotically Optimal ◽  
A Value ◽  
Probability Of Success

We consider a Master-Worker distributed system where a master processor assigns, over the Internet, tasks to a collection of n workers, which are untrusted and might act maliciously. In addition, a worker may not reply to the master, or its reply may not reach the master, due to unavailabilities or failures of the worker or the network. Each task returns a value, and the goal is for the master to accept only correct values with high probability. Furthermore, we assume that the service provided by the workers is not free; for each task that a worker is assigned, the master is charged with a work-unit. Therefore, considering a single task assigned to several workers, our objective is to have the master processor to accept the correct value of the task with high probability, with the smallest possible amount of work (number of workers the master assigns the task). We probabilistically bound the number of faulty processors by assuming a known probability p < 1/2 of any processor to be faulty. Our work demonstrates that it is possible to obtain, with provable analytical guarantees, high probability of correct acceptance with low work. In particular, we first show lower bounds on the minimum amount of (expected) work required, so that any algorithm accepts the correct value with probability of success 1 - ε, where ε ≪ 1 (e.g., 1/n). Then we develop and analyze two algorithms, each using a different decision strategy, and show that both algorithms obtain the same probability of success 1 - ε, and in doing so, they require similar upper bounds on the (expected) work. Furthermore, under certain conditions, these upper bounds are asymptotically optimal with respect to our lower bounds.

scholarly journals Scheduling in the Random-Order Model

Algorithmica ◽  
2021 ◽  
Author(s):  
Susanne Albers ◽  
Maximilian Janke
Keyword(s):  
Competitive Ratio ◽  
High Probability ◽  
Online Algorithm ◽  
Fundamental Problem ◽  
Random Order ◽  
Random Permutation ◽  
Worst Case Analysis ◽  
Makespan Minimization ◽  
Order Model ◽  
Worst Case

AbstractMakespan minimization on identical machines is a fundamental problem in online scheduling. The goal is to assign a sequence of jobs to m identical parallel machines so as to minimize the maximum completion time of any job. Already in the 1960s, Graham showed that Greedy is $$(2-1/m)$$ ( 2 - 1 / m ) -competitive. The best deterministic online algorithm currently known achieves a competitive ratio of 1.9201. No deterministic online strategy can obtain a competitiveness smaller than 1.88. In this paper, we study online makespan minimization in the popular random-order model, where the jobs of a given input arrive as a random permutation. It is known that Greedy does not attain a competitive factor asymptotically smaller than 2 in this setting. We present the first improved performance guarantees. Specifically, we develop a deterministic online algorithm that achieves a competitive ratio of 1.8478. The result relies on a new analysis approach. We identify a set of properties that a random permutation of the input jobs satisfies with high probability. Then we conduct a worst-case analysis of our algorithm, for the respective class of permutations. The analysis implies that the stated competitiveness holds not only in expectation but with high probability. Moreover, it provides mathematical evidence that job sequences leading to higher performance ratios are extremely rare, pathological inputs. We complement the results by lower bounds, for the random-order model. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 4/3. Moreover, no deterministic online algorithm can attain a competitiveness smaller than 3/2 with high probability.

scholarly journals Red/LeD: An Asymptotically Optimal and Scalable Online Algorithm for Service Caching at the Edge

2018 ◽  
Vol 36 (8) ◽  
pp. 1857-1870 ◽  
Author(s):  
Tao Zhao ◽  
I.-Hong Hou ◽  
Shiqiang Wang ◽  
Kevin Chan
Keyword(s):  
Online Algorithm ◽  
Asymptotically Optimal ◽  
Red Led

Online Dynamic B-Matching

2021 ◽  
Vol 48 (3) ◽  
pp. 99-108
Author(s):  
Marcin Bienkowski ◽  
David Fuchssteiner ◽  
Jan Marcinkowski ◽  
Stefan Schmid
Keyword(s):  
Network Performance ◽  
Online Algorithm ◽  
Temporal Structure ◽  
Algorithmic Problem ◽  
Maximum Weight ◽  
Matching Problem ◽  
Asymptotically Optimal ◽  
Datacenter Networks ◽  
Maximum Weight Matching ◽  
Theoretical Results

This paper initiates the study of online algorithms for the maximum weight b-matching problem, a generalization of maximum weight matching where each node has at most b≥1 adjacent matching edges. The problem is motivated by emerging optical technologies which allow to enhance datacenter networks with reconfigurable matchings, providing direct connectivity between frequently communicating racks. These additional links may improve network performance, by leveraging spatial and temporal structure in the workload. We show that the underlying algorithmic problem features an intriguing connection to online paging (a.k.a. caching), but introduces a novel challenge. Our main contribution is an online algorithm which is O(b)- competitive; we also prove that this is asymptotically optimal. We complement our theoretical results with extensive trace-driven simulations, based on real-world datacenter workloads as well as synthetic traffic traces.

scholarly journals Perturbed-History Exploration in Stochastic Multi-Armed Bandits

2019 ◽  
Author(s):  
Branislav Kveton ◽  
Csaba Szepesvári ◽  
Mohammad Ghavamzadeh ◽  
Craig Boutilier
Keyword(s):  
High Probability ◽  
Online Algorithm ◽  
State Of The Art ◽  
Average Reward ◽  
Regret Minimization

We propose an online algorithm for cumulative regret minimization in a stochastic multi-armed bandit. The algorithm adds O(t) i.i.d. pseudo-rewards to its history in round t and then pulls the arm with the highest average reward in its perturbed history. Therefore, we call it perturbed-history exploration (PHE). The pseudo-rewards are carefully designed to offset potentially underestimated mean rewards of arms with a high probability. We derive near-optimal gap-dependent and gap-free bounds on the n-round regret of PHE. The key step in our analysis is a novel argument that shows that randomized Bernoulli rewards lead to optimism. Finally, we empirically evaluate PHE and show that it is competitive with state-of-the-art baselines.

Quick Consult: Quick Consult: Acromioplasty

2002 ◽  
Vol 7 (3) ◽  
pp. 4-5
Keyword(s):  
Glasgow Coma Scale ◽  
Outcome Assessment ◽  
High Probability ◽  
Vegetative State ◽  
Behavioral Impairment ◽  
Whole Person ◽  
Ordinal Scales ◽  
Probability Of Death ◽  
The Individual ◽  
Assessment Scales

Abstract Different jurisdictions use the AMA Guides to the Evaluation of Permanent Impairment (AMA Guides) for different purposes, and this article reviews a specific jurisdictional definition in the Province of Ontario of catastrophic impairment that incorporates the AMA Guides. In Ontario, a whole person impairment (WPI) exceeding 54% or a mental or behavioral impairment of Class 4 or 5 qualifies the individual for catastrophic benefits, and individuals who do not meet the test receive a lesser benefit. By inference, this establishes a parity threshold among dissimilar injuries and dissimilar outcome assessment scales for benefits. In Ontario, the Glasgow Coma Scale (GCS) identifies patients who have a high probability of death or of severely disabled survival. The GCS recognizes gradations of vegetative state and disability, but translating the gradations for rating individual impairment on ordinal scales into a method of assessing percentage impairments cannot be done reliably, as explained in the AMA Guides, Fifth Edition. The AMA Guides also notes that mental and behavioral impairment in Class 4 (marked impairment) or 5 (extreme impairment) indicates “catastrophic impairment” by significantly impeding useful functioning (Class 4) or significantly impeding useful functioning and implying complete dependency on another person for care (Class 5). Translating the AMA Guides guidelines into ordinal scales cannot be done reliably.

The Effect of Outcome Probability on Generalization in Predictive Learning

2019 ◽  
Vol 66 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Hadar Ram ◽  
Dieter Struyf ◽  
Bram Vervliet ◽  
Gal Menahem ◽  
Nira Liberman
Keyword(s):  
High Probability ◽  
Lower Probability ◽  
Perceptual Similarity ◽  
Learning Paradigm ◽  
Predictive Learning ◽  
Lightning Bolt ◽  
Outcome Probability

Abstract. People apply what they learn from experience not only to the experienced stimuli, but also to novel stimuli. But what determines how widely people generalize what they have learned? Using a predictive learning paradigm, we examined the hypothesis that a low (vs. high) probability of an outcome following a predicting stimulus would widen generalization. In three experiments, participants learned which stimulus predicted an outcome (S+) and which stimulus did not (S−) and then indicated how much they expected the outcome after each of eight novel stimuli ranging in perceptual similarity to S+ and S−. The stimuli were rings of different sizes and the outcome was a picture of a lightning bolt. As hypothesized, a lower probability of the outcome widened generalization. That is, novel stimuli that were similar to S+ (but not to S−) produced expectations for the outcome that were as high as those associated with S+.

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