scholarly journals Optimal Online Algorithms for File-Bundle Caching and Generalization to Distributed Caching

2021 ◽  
Vol 6 (1) ◽  
pp. 1-23
Author(s):  
Tiancheng Qin ◽  
S. Rasoul Etesami
Keyword(s):  
Competitive Ratio ◽  
General Framework ◽  
Online Algorithm ◽  
Deterministic Algorithm ◽  
Cache Size ◽  
Distributed Caching ◽  
Minimum Number ◽  
Paging Problem ◽  
Cache Miss ◽  
Special Case

We consider a generalization of the standard cache problem called file-bundle caching, where different queries (tasks), each containing l ≥ 1 files, sequentially arrive. An online algorithm that does not know the sequence of queries ahead of time must adaptively decide on what files to keep in the cache to incur the minimum number of cache misses. Here a cache miss refers to the case where at least one file in a query is missing among the cache files. In the special case where l = 1, this problem reduces to the standard cache problem. We first analyze the performance of the classic least recently used (LRU) algorithm in this setting and show that LRU is a near-optimal online deterministic algorithm for file-bundle caching with regard to competitive ratio. We then extend our results to a generalized ( h,k )-paging problem in this file-bundle setting, where the performance of the online algorithm with a cache size k is compared to an optimal offline benchmark of a smaller cache size h < k . In this latter case, we provide a randomized O ( l ln k / k-h )-competitive algorithm for our generalized ( h, k )-paging problem, which can be viewed as an extension of the classic marking algorithm . We complete this result by providing a matching lower bound for the competitive ratio, indicating that the performance of this modified marking algorithm is within a factor of 2 of any randomized online algorithm. Finally, we look at the distributed version of the file-bundle caching problem where there are m ≥ 1 identical caches in the system. In this case, we show that for m = l + 1 caches, there is a deterministic distributed caching algorithm that is ( l 2 + l )-competitive and a randomized distributed caching algorithm that is O ( l ln ( 2l + 1)-competitive when l ≥ 2. We also provide a general framework to devise other efficient algorithms for the distributed file-bundle caching problem and evaluate the performance of our results through simulations.

scholarly journals Online Clique Clustering

Algorithmica ◽  
2019 ◽  
Vol 82 (4) ◽  
pp. 938-965
Author(s):  
Marek Chrobak ◽  
Christoph Dürr ◽  
Aleksander Fabijan ◽  
Bengt J. Nilsson
Keyword(s):  
Objective Function ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Optimal Solution ◽  
Deterministic Algorithm ◽  
Input Graph ◽  
Online Clustering ◽  
Objective Value ◽  
Asymptotic Competitive Ratio ◽  
Better Than

Abstract Clique clustering is the problem of partitioning the vertices of a graph into disjoint clusters, where each cluster forms a clique in the graph, while optimizing some objective function. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The goal is to maintain a clustering with an objective value close to the optimal solution. For the variant where we want to maximize the number of edges in the clusters, we propose an online algorithm based on the doubling technique. It has an asymptotic competitive ratio at most 15.646 and a strict competitive ratio at most 22.641. We also show that no deterministic algorithm can have an asymptotic competitive ratio better than 6. For the variant where we want to minimize the number of edges between clusters, we show that the deterministic competitive ratio of the problem is $$n-\omega (1)$$n-ω(1), where n is the number of vertices in the graph.

Online Machine Scheduling with Family Setups

2016 ◽  
Vol 33 (04) ◽  
pp. 1650027
Author(s):  
Lele Zhang ◽  
Andrew Wirth
Keyword(s):  
Lower Bound ◽  
Single Machine ◽  
Competitive Analysis ◽  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Machine Scheduling ◽  
Processing Times ◽  
Family Setups ◽  
Special Case

We consider the problem of online scheduling a single machine with family setups under job availability. A setup must be scheduled when the next job comes from a different family from the last completed one, if any. The aim is to minimize the total completion time of all jobs. For the special case of identical processing times, we provide a lower bound for the competitive ratio and an online algorithm with its competitive analysis.

Online Job Dispatching and Scheduling to Minimize Job Completion Time and to Meet Deadlines

2018 ◽  
Vol 18 (04) ◽  
pp. 1850012
Author(s):  
YUPENG LI
Keyword(s):  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Performance Metrics ◽  
Case Analysis ◽  
Deterministic Algorithm ◽  
Worst Case Analysis ◽  
Worst Case ◽  
Job Dispatching ◽  
Shed Light

In this paper, we study the problem of job dispatching and scheduling, where each job consists of a set of tasks. Each task is processed by a set of machines simultaneously. We consider two important performance metrics, the average job completion time (JCT), and the number of deadline-aware jobs that meet their deadlines. The goal is to minimize the former and maximize the latter. We first propose OneJ to minimize the job completion time (JCT) when there is exactly one single job in the system. Then, we propose an online algorithm called MultiJ, taking OneJ as a subroutine, to minimize the average JCT, and prove it has a good competitive ratio. We then derive another online algorithm QuickJ to maximize the number of jobs that can meet their deadlines. We show that QuickJ is competitive via a worst case analysis. We also conjecture that the competitive ratio of QuickJ is likely to be the best one that any deterministic algorithm can achieve. We also shed light on several important merits of MultiJ and QuickJ, such as no severe coordination overhead, scalability, work conservation, and no job starvation.

scholarly journals Algorithms for Energy Conservation in Heterogeneous Data Centers

2021 ◽  
pp. 75-89
Author(s):  
Susanne Albers ◽  
Jens Quedenfeld
Keyword(s):  
Data Center ◽  
Competitive Ratio ◽  
Online Algorithm ◽  
Data Centers ◽  
Heterogeneous Data ◽  
Deterministic Algorithm ◽  
Low Load ◽  
Save Energy ◽  
Feasible Solutions ◽  
Discrete Setting

AbstractPower consumption is the major cost factor in data centers. It can be reduced by dynamically right-sizing the data center according to the currently arriving jobs. If there is a long period with low load, servers can be powered down to save energy. For identical machines, the problem has already been solved optimally by [25] and [1].In this paper, we study how a data-center with heterogeneous servers can dynamically be right-sized to minimize the energy consumption. There are d different server types with various operating and switching costs. We present a deterministic online algorithm that achieves a competitive ratio of 2d as well as a randomized version that is 1.58d-competitive. Furthermore, we show that there is no deterministic online algorithm that attains a competitive ratio smaller than 2d. Hence our deterministic algorithm is optimal. In contrast to related problems like convex body chasing and convex function chasing [17, 30], we investigate the discrete setting where the number of active servers must be an integral, so we gain truly feasible solutions.

scholarly journals Online Makespan Scheduling with Job Migration on Uniform Machines

Algorithmica ◽  
2021 ◽  
Author(s):  
Matthias Englert ◽  
David Mezlaf ◽  
Matthias Westermann
Keyword(s):  
Competitive Ratio ◽  
Online Algorithm ◽  
Parallel Machines ◽  
Scheduling Algorithm ◽  
Input Sequence ◽  
Job Migration ◽  
Average Completion Time ◽  
Uniform Machine ◽  
Uniform Machines ◽  
Completion Times

AbstractIn the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and $$\approx 1.4659$$ ≈ 1.4659 . They show that $$k = O(m)$$ k = O ( m ) is sufficient to achieve this bound and no $$k = o(n)$$ k = o ( n ) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a $$\delta = \varTheta (1)$$ δ = Θ ( 1 ) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than $$1.4659 + \delta $$ 1.4659 + δ with $$k = o(n)$$ k = o ( n ) . We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and $$\approx 1.7992$$ ≈ 1.7992 with $$k = O(m)$$ k = O ( m ) . We also show that $$k = \varOmega (m)$$ k = Ω ( m ) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

scholarly journals Deferred On-Line Bipartite Matching

10.37236/5756 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Jakub Kozik ◽  
Grzegorz Matecki
Keyword(s):  
Competitive Ratio ◽  
Bipartite Graphs ◽  
Deterministic Algorithm ◽  
The Other ◽  
Bipartite Matching ◽  
Classical Version ◽  
New Model ◽  
On Line

We present a new model for the problem of on-line matching on bipartite graphs. Suppose that one part of a graph is given, but the vertices of the other part are presented in an on-line fashion. In the classical version, each incoming vertex is either irrevocably matched to a vertex from the other part or stays unmatched forever. In our version, an algorithm is allowed to match the new vertex to a group of elements (possibly empty). Later on, the algorithm can decide to remove some vertices from the group and assign them to another (just presented) vertex, with the restriction that each element belongs to at most one group. We present an optimal (deterministic) algorithm for this problem and prove that its competitive ratio equals $1-\pi/\cosh(\frac{\sqrt{3}}{2}\pi)\approx 0.588$.

A BEST POSSIBLE ONLINE ALGORITHM FOR SCHEDULING TO MINIMIZE MAXIMUM FLOW-TIME ON BOUNDED BATCH MACHINES

2014 ◽  
Vol 31 (04) ◽  
pp. 1450030 ◽  
Author(s):  
CHENGWEN JIAO ◽  
WENHUA LI ◽  
JINJIANG YUAN
Keyword(s):  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Unit Length ◽  
Maximum Flow ◽  
Flow Time ◽  
Release Time ◽  
Batch Machine ◽  
The Difference ◽  
Over Time

We consider online scheduling of unit length jobs on m identical parallel-batch machines. Jobs arrive over time. The objective is to minimize maximum flow-time, with the flow-time of a job being the difference of its completion time and its release time. A parallel-batch machine can handle up to b jobs simultaneously as a batch. Here, the batch capacity is bounded, that is b < ∞. In this paper, we provide a best possible online algorithm for the problem with a competitive ratio of [Formula: see text].

SEMI-ONLINE MACHINE COVERING

2007 ◽  
Vol 24 (03) ◽  
pp. 373-382 ◽  
Author(s):  
SHENG-YI CAI
Keyword(s):  
Competitive Ratio ◽  
Completion Time ◽  
Online Algorithm ◽  
Parallel Machines ◽  
Identical Parallel Machines ◽  
Processing Times ◽  
First Case ◽  
Machine Covering

This paper investigates two different semi-online versions of the machine covering, which is the problem of assigning a set of jobs to a system of m(m ≥ 3) identical parallel machines so as to maximize the earliest machine completion time. In the first case, we assume that the largest processing times is known in advance. In the second case, we assume that the total processing times of all jobs is known in advance. For each version we propose a semi-online algorithm and investigate its competitive ratio. The competitive ratio of each algorithm is [Formula: see text], which is shown to be the best possible competitive ratio for each semi-online problem.

Online peak-aware energy scheduling with untrusted advice

2021 ◽  
Vol 1 (1) ◽  
pp. 59-77
Author(s):  
Russell Lee ◽  
Jessica Maghakian ◽  
Mohammad Hajiesmaili ◽  
Jian Li ◽  
Ramesh Sitaraman ◽  
...  
Keyword(s):  
Competitive Ratio ◽  
Energy Demand ◽  
Large Scale ◽  
Performance Metrics ◽  
Randomized Algorithm ◽  
Deterministic Algorithm ◽  
Pareto Optimal ◽  
Energy Prices ◽  
Worst Case ◽  
Cost Of Energy

This paper studies the online energy scheduling problem in a hybrid model where the cost of energy is proportional to both the volume and peak usage, and where energy can be either locally generated or drawn from the grid. Inspired by recent advances in online algorithms with Machine Learned (ML) advice, we develop parameterized deterministic and randomized algorithms for this problem such that the level of reliance on the advice can be adjusted by a trust parameter. We then analyze the performance of the proposed algorithms using two performance metrics: robustness that measures the competitive ratio as a function of the trust parameter when the advice is inaccurate, and consistency for competitive ratio when the advice is accurate. Since the competitive ratio is analyzed in two different regimes, we further investigate the Pareto optimality of the proposed algorithms. Our results show that the proposed deterministic algorithm is Pareto-optimal, in the sense that no other online deterministic algorithms can dominate the robustness and consistency of our algorithm. Furthermore, we show that the proposed randomized algorithm dominates the Pareto-optimal deterministic algorithm. Our large-scale empirical evaluations using real traces of energy demand, energy prices, and renewable energy generations highlight that the proposed algorithms outperform worst-case optimized algorithms and fully data-driven algorithms.

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