Economical Caching

We study the management of buffers and storages in environments with unpredictably varying prices in a competitive analysis. In the economical caching problem, there is a storage with a certain capacity. For each time step, an online algorithm is given a price from the interval $[1,alpha]$, a consumption, and possibly a buying limit. The online algorithm has to decide the amount to purchase from some commodity, knowing the parameter $alpha$ but without knowing how the price evolves in the future. The algorithm can purchase at most the buying limit. If it purchases more than the current consumption, then the excess is stored in the storage; otherwise, the gap between consumption and purchase must be taken from the storage. The goal is to minimize the total cost. Interesting applications are, for example, stream caching on mobile devices with different classes of service, battery management in micro hybrid cars, and the efficient purchase of resources. First we consider the simple but natural class of algorithms that can informally be described as memoryless. We show that these algorithms cannot achieve a competitive ratio below $sqrt{alpha}$. Then we present a more sophisticated deterministic algorithm achieving a competitive ratio of [textstyle frac{1}{Wleft(frac{1-alpha}{ealpha}right)+1} in left[frac{sqrt{alpha}}{sqrt{2}}, frac{sqrt{alpha}+1}{sqrt{2}} right] enspace, ] where $W$ denotes the Lambert~W function. We prove that this algorithm is optimal and that not even randomized online algorithms can achieve a better competitive ratio. On the other hand, we show how to achieve a constant competitive ratio if the storage capacity of the online algorithm exceeds the storage capacity of an optimal offline algorithm by a factor of $log alpha$

Online makespan scheduling with job migration on uniform machines

In 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 reassign up to k jobs to different machines in the final assignment. For m identical machines, Albers and Hellwig (Algorithmica, 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 ~~ 1.4659. They show that k = O(m) is sufficient to achieve this bound and no 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 = Theta(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 with 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 ~~ 1.7992 with k = O(m). We also show that k = Omega(m) is necessary to achieve a competitive ratio below 2. Our algorithm is based on a subtle 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

Online packet scheduling for CIOQ and buffered crossbar switches

We consider the problem of online packet scheduling in Combined Input and Output Queued (CIOQ) and buffered crossbar switches. In the widely used CIOQ switches, packet buffers (queues) are placed at both input and output ports. An N×N CIOQ switch has N input ports and N output ports, where each input port is equipped with N queues, each of which corresponds to an output port, and each output port is equipped with only one queue. In each time slot, arbitrarily many packets may arrive at each input port, and only one packet can be transmitted from each output port. Packets are transferred from the queues of input ports to the queues of output ports through the internal fabric. Buffered crossbar switches follow a similar design, but are equipped with additional buffers in their internal fabric. In either model, our goal is to maximize the number or, in case the packets have weights, the total weight of transmitted packets. Our main objective is to devise online algorithms that are both competitive and efficient. We improve the previously known results for both switch models, both for unweighted and weighted packets. For unweighted packets, Kesselman and Rosén (J. Algorithms 60(1):60–83, 2006) give an online algorithm that is 3-competitive for CIOQ switches. We give a faster, more practical algorithm achieving the same competitive ratio. In the buffered crossbar model, we also show 3-competitiveness, improving the previously known ratio of 4. For weighted packets, we give 5.83- and 14.83-competitive algorithms with an elegant analysis for CIOQ and buffered crossbar switches, respectively. This improves upon the previously known ratios of 6 and 16.24

Comparison-based FIFO buffer management in QoS switches

The following online problem arises in network devices, e.g., switches, with quality of service (QoS) guarantees. In each time step, an arbitrary number of packets arrive at a single FIFO buffer and only one packet can be transmitted. Packets may be kept in the buffer of limited size and, due to the FIFO property, the sequence of transmitted packets has to be a subsequence of the arriving packets. The differentiated service concept is implemented by attributing each packet with a non-negative value corresponding to its service level. A buffer management algorithm can reject arriving packets and preempt buffered packets. The goal is to maximize the total value of transmitted packets. We study comparison-based buffer management algorithms, i.e., algorithms that make their decisions based solely on the relative order between packet values with no regard to the actual values. This kind of algorithms proves to be robust in the realm of QoS switches. Kesselman et al. (SIAM J. Comput., 2004) present a comparison-based algorithm that is 2-competitive. For a long time, it has been an open problem whether a comparison-based algorithm exists with a competitive ratio below 2. We present a lower bound of 1 + 1/√2 ≈ 1.707 on the competitive ratio of any deterministic comparison-based algorithm and give an algorithm that matches this lower bound in the case of monotonic sequences, i.e., packets arrive in a non-decreasing order according to their values

Polylogarithmic guarantees for generalized reordering buffer management

In the Generalized Reordering Buffer Management Problem (GRBM) a sequence of items located in a metric space arrives online, and has to be processed by a set of k servers moving within the space. In a single step the first b still unprocessed items from the sequence are accessible, and a scheduling strategy has to select an item and a server. Then the chosen item is processed by moving the chosen server to its location. The goal is to process all items while minimizing the total distance travelled by the servers. This problem was introduced in [Chan, Megow, Sitters, van Stee TCS 12] and has been subsequently studied in an online setting by [Azar, Englert, Gamzu, Kidron STACS 14]. The problem is a natural generalization of two very well-studied problems: the k-server problem for b=1 and the Reordering Buffer Management Problem (RBM) for k=1. In this paper we consider the GRBM problem on a uniform metric in the online version. We show how to obtain a competitive ratio of O(log k(log k+loglog b)) for this problem. Our result is a drastic improvement in the dependency on b compared to the previous best bound of O(√b log k), and is asymptotically optimal for constant k, because Ω(log k + loglog b) is a lower bound for GRBM on uniform metrics

Breaking the Barrier Of 2 for the Competitiveness of Longest Queue Drop

We consider the problem of managing the buffer of a shared-memory switch that transmits packets of unit value. A shared-memory switch consists of an input port, a number of output ports, and a buffer with a specific capacity. In each time step, an arbitrary number of packets arrive at the input port, each packet designated for one output port. Each packet is added to the queue of the respective output port. If the total number of packets exceeds the capacity of the buffer, some packets have to be irrevocably rejected. At the end of each time step, each output port transmits a packet in its queue and the goal is to maximize the number of transmitted packets. The Longest Queue Drop (LQD) online algorithm accepts any arriving packet to the buffer. However, if this results in the buffer exceeding its memory capacity, then LQD drops a packet from the back of whichever queue is currently the longest, breaking ties arbitrarily. The LQD algorithm was first introduced in 1991, and is known to be 2-competitive since 2001. Although LQD remains the best known online algorithm for the problem and is of practical interest, determining its true competitiveness is a long-standing open problem. We show that LQD is 1.707-competitive, establishing the first (2-?) upper bound for the competitive ratio of LQD, for a constant ? > 0

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on real world Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every $p\in\mathbb{N}$, a family of $L_p$ instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which $n$ points are placed uniformly at random in the unit square $[0,1]^2$. We consider a more advanced model in which the points can be placed independently according to general distributions on $[0,1]^d$, for an arbitrary $d\ge 2$. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number $n$ of points and the maximal density $\phi$ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of $\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})$. When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to $\tilde{O}(n^{4+1/3-1/d}\cdot\phi^{8/3})$. If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by $\tilde{O}(n^{4-1/d}\cdot\phi)$. In addition, we prove an upper bound of $O(\sqrt[d]{\phi})$ on the expected approximation factor with respect to all $L_p$ metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with $\phi=1$ and a smoothed analysis with Gaussian perturbations of standard deviation $\sigma$ with $\phi\sim1/\sigma^d$.Comment: An extended abstract of this work has appeared in the Proc. of the 18th ACM-SIAM Symposium on Discrete Algorithms. The results of this extended abstract have been split into two articles (Algorithmica 2014) and (ACM Transactions on Algorithms 2016). This report is an updated version of the first journal article, in which two minor errors in the proofs of Lemma 8 and Lemma 9 have been correcte

Reachability in two-dimensional unary vector addition systems with states is NL-complete

Blondin et al. showed at LICS 2015 that two-dimensional vector addition systems with states have reachability witnesses of length exponential in the number of states and polynomial in the norm of vectors. The resulting guess-and-verify algorithm is optimal (PSPACE), but only if the input vectors are given in binary. We answer positively the main question left open by their work, namely establish that reachability witnesses of pseudo-polynomial length always exist. Hence, when the input vectors are given in unary, the improved guess-and-verify algorithm requires only logarithmic space

An O(log k)-competitive algorithm for generalized caching

In the generalized caching problem, we have a set of pages and a cache of size k. Each page p has a size wpe1 and fetching cost cp for loading the page into the cache. At any point in time, the sum of the sizes of the pages stored in the cache cannot exceed k. The input consists of a sequence of page requests. If a page is not present in the cache at the time it is requested, it has to be loaded into the cache incurring a cost of cp. We give a randomized O(log k)-competitive online algorithm for the generalized caching problem, improving the previous bound of O(log2 k) by Bansal, Buchbinder, and Naor (STOC'08). This improved bound is tight and of the same order as the known bounds for the classic problem with uniform weights and sizes. We use the same LP based techniques as Bansal et al. but provide improved and slightly simplified methods for rounding fractional solutions online