Deterministic Online Algorithm Research Articles

Managing Multiple Mobile Resources

We extend the Mobile Server problem introduced in Feldkord and Meyer auf der Heide (TOPC 6(3), 14:1–14:17 2019) to a model where k identical mobile resources, here named servers, answer requests appearing at points in the Euclidean space. To reduce communication costs, the positions of the servers can be adapted by a limited distance ms per round for each server. The costs are measured similarly to the classical Page Migration problem: i.e., answering a request induces costs proportional to the distance to the nearest server, and moving a server induces costs proportional to the distance multiplied with a weight D. We show that, in our model, no online algorithm can have a constant competitive ratio: i.e., one which is independent of the input length n, even if an augmented moving distance of (1 + δ)ms is allowed for the online algorithm. Therefore we investigate a restriction of the power of the adversary dictating the sequence of requests: We demand locality of requests: i.e., that consecutive requests come from points in the Euclidean space with distance bounded by some constant mc. We show constant lower bounds on the competitiveness in this setting (independent of n, but dependent on k, ms and mc). On the positive side, we present a deterministic online algorithm with bounded competitiveness when an augmented moving distance and locality of requests is assumed. Our algorithm simulates any given algorithm for the classical k-Page Migration problem as guidance for its servers and extends it by a greedy move of one server in every round. The resulting competitive ratio is polynomial in the number of servers k, the ratio between mc and ms, the inverse of the augmentation factor 1/δ and the competitive ratio of the simulated k-Page Migration algorithm. We also show how to directly adapt the Double Coverage algorithm (Chrobak et al. SIAM J. Discrete Math. 4(2), 172–181 11) for the k-Server problem to receive an algorithm with improved competitiveness on the line.

Approximating the Canadian Traveller Problem with Online Randomization

In this paper, we study online algorithms for the Canadian Traveller Problem defined by Papadimitriou and Yannakakis in 1991. This problem involves a traveller who knows the entire road network in advance, and wishes to travel as quickly as possible from a source vertex s to a destination vertex t, but discovers online that some roads are blocked (e.g., by snow) once reaching them. Achieving a bounded competitive ratio for the problem is PSPACE-complete. Furthermore, if at most k roads can be blocked, the optimal competitive ratio for a deterministic online algorithm is \(2k+1\), while the only randomized result known so far is a lower bound of \(k+1\). We show, for the first time, that a polynomial time randomized algorithm can outperform the best deterministic algorithms when there are at least two blockages, and surpass the lower bound of \(2k+1\) by an o(1) factor. Moreover, we prove that the randomized algorithm can achieve a competitive ratio of \(\big (1+ \frac{\sqrt{2}}{2} \big )k + \sqrt{2}\) in pseudo-polynomial time. The proposed techniques can also be exploited to implicitly represent multiple near-shortest s-t paths.

Cost-Driven Data Caching in Edge-based Content Delivery Networks

In this paper, we studied a data caching problem in edge-based CDNs to facilitate the content delivery to serve a sequence of requests, off-line and online, with minimum costs as a goal based on a semi-homo cost model. To this end, we first designed an O(mn \log(mn)) time and space optimal proactive off-line algorithm,called pro-caching, by reducing the problem to a simple shortest path problem in a directed weighted network graph, and then extended the idea of anticipatory caching to develop an 2-competitive reactive online algorithm, called re-caching, for this problem and showed its tightness by proving that no deterministic online algorithm can do better than 2-o(1) in its worst case. Finally, to combine the advantages of both algorithms, we also presented a hybrid algorithm, called hy-caching, to fully utilize the power and benefits of edge-based CDNs while reducing their service costs. Our results improve the previous results not only in the cost model being used but also in the time complexity, competitive ratio, and the quality of the solutions. We provably achieve these results with our deep insights into the problem and the careful analysis, together with an empirical evaluation.

Online Joint Placement and Allocation of Virtual Network Functions With Heterogeneous Servers

Network function virtualization (NFV) is a promising virtualization technology that has the potential to significantly reduce the expenses and improve service agility. The NFV makes it possible for Internet service providers (ISPs) to employ various virtual network functions (VNFs) without installing new equipments. One of the most attractive approaches in the NFV technology is the so-called joint placement and allocation of virtual network functions (JPA-VNFs), which considers the balance between VNF investment with Quality of Services (QoS). We introduce a novel capability function to measure the potential of locating VNF instances for each server in the proposed OJPA-HS model. This model allows the servers in the network to be heterogeneous, at the same time combines and generalizes many classical JPA-VNF models. Despite its NP-hardness, we present a provable best-possible deterministic online algorithm based on dynamic programming (DP). To conquer the high complexity of DP, we propose two additional randomized heuristics, Las Vegas (LV) and Monte Carlo (MC) randomized algorithms, which perform even as good as DP with much smaller complexity. Besides, MC is a promising heuristic in practice as it has the advantage to deal with the big data environment. Extensive numerical experiments are constructed for the proposed algorithms in this article.

Deterministic Sparse Suffix Sorting in the Restore Model

Given a text T of length n , we propose a deterministic online algorithm computing the sparse suffix array and the sparse longest common prefix array of T in O( c √ lg n + m lg m lg n lg * n ) time with O( m ) words of space under the premise that the space of T is rewritable, where m ≤ n is the number of suffixes to be sorted (provided online and arbitrarily), and c is the number of characters with m ≤ c ≤ n that must be compared for distinguishing the designated suffixes.

Dynamic speed scaling minimizing expected energy consumption for real-time tasks

This paper proposes a discrete time Markov decision process approach to compute the optimal on-line speed scaling policy to minimize the energy consumption of a single processor executing a finite or infinite set of jobs with real-time constraints. We provide several qualitative properties of the optimal policy: monotonicity with respect to the jobs parameters, comparison with on-line deterministic algorithms. Numerical experiments in several scenarios show that our proposition performs well when compared with off-line optimal solutions and out-performs on-line solutions oblivious to statistical information on the jobs.

Online unit covering in Euclidean space

We revisit the online Unit Covering problem in higher dimensions: Given a set of n points in Rd, that arrive one by one, cover the points by balls of unit radius, so as to minimize the number of balls used. In this paper, we work in Rd using the Euclidean distance.(I) We give an online deterministic algorithm with competitive ratio O(1.321d), thereby improving on the previous record, O(2ddlog⁡d), due to Charikar et al. (2004), by an exponential factor. In particular, the competitive ratios are 5 in the plane and 12 in 3-space (the previous ratios were 7 and 21, respectively). For d=3, the ratio of our online algorithm matches the ratio of the current best offline algorithm for the same problem due to Biniaz et al. (2017), which is remarkable (and rather unusual).(II) We show that the competitive ratio of every deterministic online algorithm for Unit Covering in Rd under the L2 norm is at least d+1 for every d≥1. This greatly improves upon the previous best lower bound, Ω(log⁡d/log⁡log⁡log⁡d), due to Charikar et al. (2004).(III) We generalize the above result to Unit Covering in Rd under the LC norm, where C is a centrally symmetric convex body, via the illumination number.(IV) We obtain lower bounds of 4 and 5 for the competitive ratio of any deterministic algorithm for online Unit Covering in R2 and R3, respectively; the previous best lower bounds were 3 for both cases.(V) When the input points are from the square or hexagonal lattice in R2, we give deterministic online algorithms for Unit Covering with an optimal competitive ratio of 3. For the cubic lattice in R3, we give a deterministic online algorithm with a competitive ratio of 5.

An optimal online algorithm for scheduling with general machine cost functions

We present two deterministic online algorithms for the problem of scheduling with a general machine cost function. In this problem, every machine has a cost that is given by a nonnegative cost function, and the objective function is the sum of makespan and the cost of the purchased machines. In previous work by Imreh, it was shown that no deterministic algorithm has competitive ratio below 2, and an algorithm whose competitive ratio does not exceed $$(3+\sqrt{5})/2 \approx 2.618$$ was presented. Our first algorithm imitates an optimal offline solution with respect to the number of machines used, and we show that its competitive ratio is exactly 2.5. Then, we modify our algorithm by imitating a preemptive optimal offline solution instead of a non-preemptive one. This leads to the design of a 2-competitive algorithm, which is the best possible.

Online scheduling to minimize maximum weighted flow-time on a bounded parallel-batch machine

An online scheduling problem on a bounded parallel-batch machine to minimize the maximum weighted flow-time is considered in this paper. Jobs arrive over time with the identical processing time. The maximum ratio between the weights of any two jobs is w. The parallel-batch machine can process at most b jobs simultaneously as a batch, and the jobs in a batch have the same starting time and the same completion time. For this problem, a deterministic online algorithm is presented. The algorithm is showed to be the best possible with a competitive ratio of $$\frac{\sqrt{4w+1}+1}{2}$$ when $$w\in [1,2]$$ , and to have a competitive ratio not greater than w when $$w\in (2,+\infty )$$ .

Cloud Scheduling with Discrete Charging Units

We consider a scheduling problem for running jobs on machines rented from the cloud. Cloud service providers such as Amazon EC2 and Google Cloud offer machines to rent on demand, and charge the rental usage by a specific interval of time, say at an hourly rate. This pricing model creates an interesting optimization problem called Interval Scheduling with Discrete Charging Units (ISDCU) which assigns jobs to run on the machines with the objective of minimizing the rental cost. In this paper, we study the problem of ISDCU where each machine can process a maximum of $g$g jobs simultaneously. We focus on interval jobs where each job must be assigned to a machine upon its arrival and run for a required processing length. We show that ISDCU is NP-hard even for the case of $g = 1$g=1. We also show that no deterministic online algorithm can achieve a competitive ratio better than $\max \lbrace 2, g\rbrace$max{2,g} in the non-clairvoyant setting, and better than $\max \lbrace 3/2, g\rbrace$max{3/2,g} in the clairvoyant setting. Lastly, we develop and analyze several online algorithms, most of which achieve a competitive ratio of $O(g)$O(g).

On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a one-pass deterministic online algorithm is $1/2$, which is achieved by any greedy algorithm. Durr et al. recently presented a $2$-pass algorithm called Category-Advice that achieves approximation ratio $3/5$. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the $k$-pass Category-Advice algorithm for all $k \ge 1$, and show that the approximation ratio converges to the inverse of the golden ratio $2/(1+\sqrt{5}) \approx 0.618$ as $k$ goes to infinity. The convergence is extremely fast --- the $5$-pass Category-Advice algorithm is already within $0.01\%$ of the inverse of the golden ratio. We then consider a natural greedy algorithm in the online stochastic IID model---MinDegree. This algorithm is an online version of a well-known and extensively studied offline algorithm MinGreedy. We show that MinDegree cannot achieve an approximation ratio better than $1-1/e$, which is guaranteed by any consistent greedy algorithm in the known IID model. Finally, following the work in Besser and Poloczek, we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model.

The ( h,k )-Server Problem on Bounded Depth Trees

We study the k -server problem in the resource augmentation setting, i.e., when the performance of the online algorithm with k servers is compared to the offline optimal solution with h ≤ k servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic k -server algorithms are roughly (1+1/ϵ)-competitive when k =(1+ϵ) h , for any ϵ > 0. Surprisingly, however, no o ( h )-competitive algorithm is known even for HSTs of depth 2 and even when k / h is arbitrarily large. We obtain several new results for the problem. First, we show that the known k -server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio Ω ( h ) irrespective of the value of k , even for depth-2 HSTs. Similarly, the Work Function Algorithm, which is believed to be optimal for all metric spaces when k = h , has competitive ratio Ω ( h ) on depth-3 HSTs even if k =2 h . Our main result is a new algorithm that is O (1)-competitive for constant depth trees, whenever k =(1+ϵ) h for any ϵ > 0. Finally, we give a general lower bound that any deterministic online algorithm has competitive ratio at least 2.4 even for depth-2 HSTs and when k / h is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the ( h , k )-server problem.

Online Algorithm for Leasing Wireless Channels in a Three-Tier Spectrum Sharing Framework

The three-tier spectrum sharing framework (3-TSF) is a spectrum sharing model adopted by the Federal Communications Commission. According to this model, under-utilized federal spectrum like the Citizens Broadband Radio Service band is released for shared use where the highest preference is given to Tier-1 followed by Tier-2 (T2) and then Tier-3 (T3). In this paper, we study how a wireless operator, who is interested in maximizing its profit, can strategically operate as a T2 and/or a T3 user. T2 is characterized by paid but ”almost” guaranteed and interference-free channel access while T3 access is free but has the lesser guarantee and also faces channel interference. So the operator has to optimally decide between paid but better channel quality and free but uncertain channel quality. Also, the operator has to make these decisions without knowing future market variables like customer demand or channel availability. The main contribution of this paper is a deterministic online algorithm for leasing channels that has finite competitive ratio, low time complexity, and that does not rely on the knowledge of market statistics. Such algorithms are desirable in the early stages of the deployment of 3-TSF because the knowledge of market statistics may be rather inaccurate. We use tools from the ski-rental literature to design the online algorithm. The online optimization problem for leasing channels is a novel generalization of the ski-rental problem. We, therefore, make fundamental contributions to the ski-rental literature, the applications of which extend beyond this paper. We also conduct simulations using synthetic traces to compare our online algorithm with the benchmark and state-of-the-art algorithms.

Quantum Online Algorithms with Respect to Space and Advice Complexity

Online computation is a well-known area of computer science. We introduce quantum online algorithms and investigate them with respect to a competitive ratio in two points of view: space complexity and advice complexity. We start with exploring a model with restricted memory and show that quantum online algorithms can be better than classical ones (deterministic or randomized) for sublogarithmic space (memory), and they can be better than deterministic online algorithms without restriction for memory. Additionally, we consider the polylogarithmic space case and show that in this case, quantum online algorithms can be better than deterministic ones as well. Another point of view to the online algorithms model is advice complexity. So, we introduce quantum online algorithms with a quantum channel with an adviser. Firstly, we show that quantum algorithms have at least the same computational power as classical ones have. We give some examples of quantum online algorithms with advice. Secondly, we show that if we allow to use shared entangled qubits (EPR-pairs), then a quantum online algorithm can use half as many advise qubits compared to a classical one. We apply this approach to the well-known Paging Problem.

Competitive clustering of stochastic communication patterns on a ring

This paper studies a fundamental dynamic clustering problem. The input is an online sequence of pairwise communication requests between n nodes (e.g., tasks or virtual machines). Our goal is to minimize the communication cost by partitioning the communicating nodes into ell clusters (e.g., physical servers) of size k (e.g., number of virtual machine slots). We assume that if the communicating nodes are located in the same cluster, the communication request costs 0; if the nodes are located in different clusters, the request is served remotely using inter-cluster communication, at cost 1. Additionally, we can migrate: a node from one cluster to another at cost alpha ge 1. We initiate the study of a stochastic problem variant where the communication pattern follows a fixed distribution, set by an adversary. Thus, the online algorithm needs to find a good tradeoff between benefitting from quickly moving to a seemingly good configuration (of low inter-cluster communication costs), and the risk of prematurely ending up in a configuration which later turns out to be bad, entailing high migration costs. Our main technical contribution is a deterministic online algorithm which is O(log {n})-competitive with high probability (w.h.p.), for a specific but fundamental class of problems: namely on ring graphs. We also provide first insights in slightly more general models, where the adversary is not restricted to a fixed distribution or the ring.

On-line scheduling with monotone subsequence constraints

In this paper, we study an on-line scheduling problem that is motivated by applications such as carpooling. The ride requests arrive on-line over-list and specify the service locations (among m locations). The goal is to determine a schedule that maximizes the number of satisfied requests using k servers. We consider two variants of the problem with respect to constraints on the service location: in the monotone service direction variant, the service locations of the requests assigned to a server must be monotonic; in the strict monotone service direction variant, the service locations must be strictly monotonic. We present lower bounds on the competitive ratio for both variants. For the monotone service direction variant, we give an optimal straightforward algorithm if k≥m and we prove that no deterministic on-line algorithm can achieve a constant competitive ratio if k<m. For the strict monotone service direction variant, we give a lower bound max⁡{2(m−1)m,1} on the competitive ratio of barely deterministic algorithms if k>2, and a lower bound of max⁡{m2,1} if k≤2. We propose a Balanced Interval Algorithm for the strict monotone service direction variant if k>2 and report some numerical experiments to evaluate the efficiency of this algorithm.

Minimum cost perfect matching with delays for two sources

We study a version of the online min-cost perfect matching with delays (MPMD) problem recently introduced by Emek et al. (STOC 2016). In this problem, requests arrive in a continuous time online fashion and should be matched to each other. Each request emerges from one out of nsources, with metric inter-source distances. The algorithm is allowed to delay the matching of requests, but with a cost: when matching two requests, it pays the distance between their respective sources and the time each request has waited from its arrival until it was matched. In this paper, we consider the special case of n=2 sources that captures the essence of the match-or-wait challenge (cf. rent-or-buy). It turns out that even for this degenerate metric space, the problem is far from trivial. Our results include a deterministic 3-competitive online algorithm for this problem, a proof that no deterministic online algorithm can have competitive ratio smaller than 3, and a proof that the same lower bound applies also for the restricted family of memoryless randomized algorithms.

Online production planning to maximize the number of on-time orders

We consider a production planning problem with two planning periods. Detailed planning occurs in the first period, when complete information is known about a set of orders that are initially available. An additional set of orders becomes available at the start of the second planning period. The objective is to maximize the number of on-time orders. We derive an upper bound on the competitive ratio of any deterministic online algorithm, relative to the performance of an algorithm with perfect information about the second set of orders. This ratio depends on the relative lengths of the two planning periods. We also describe an efficient algorithm that delivers a solution which asymptotically achieves this upper bound ratio as the number of jobs becomes large.

Online delivery route recommendation in spatial crowdsourcing

With the emergence of many crowdsourcing platforms, crowdsourcing has gained much attention. Spatial crowdsourcing is a rapidly developing extension of the traditional crowdsourcing, and its goal is to organize workers to perform spatial tasks. Route recommendation is an important concern in spatial crowdsourcing. In this paper, we define a novel problem called the Online Delivery Route Recommendation (OnlineDRR) problem, in which the income of a single worker is maximized under online scenarios. It is proved that no deterministic online algorithm for this problem has a constant competitive ratio. We propose an algorithm to balance three influence factors on a worker’s choice in terms of which task to undertake next. In order to overcome its drawbacks resulting from the dynamic nature of tasks, we devise an extended version which attaches gradually increased importance to the destination of the worker over time. Extensive experiments are conducted on both synthetic and real-world datasets and the results prove the algorithms proposed in this paper are effective and efficient.

Online buffer management for transmitting packets with processing cycles

We study an online buffer management problem under the model introduced by Azar and Gilon (2015) [5] recently. Unit-sized packets arrive and are kept in a First-In-First-Out buffer of size B in an online fashion at a network server. Each packet is associated with an arrival time, a value and a processing cycle time in the buffer. The density of a packet is defined to be the ratio of its value to processing time. It is assumed that every packet can be transmitted only after its processing cycle is completed and only the packet at the head of the buffer can be processed. A packet is allowed to be preempted and then discarded from the buffer. But, the value of a packet is attained only if it is successfully transmitted. Under the model, the objective of online buffer management is to maximize the total value of transmitted packets. This model finds applications to packet scheduling in communication networks.In this study, we consider the model with constant density from a theoretical perspective. We first propose some lower bounds for the problem. Azar and Gilon obtained a 4-competitive algorithm for the online buffer management problem for packets with constant density. Here, we present a (2+1B−1)-competitive algorithm for the case B>1 as well as its generalization to the multi-buffer model. Moreover, we prove that the competitive ratio of a deterministic online algorithm is at least four when the buffer size is one. We also conduct experiments to demonstrate the superior performance of the proposed online algorithm against the previous approach.