Buffer Overflow Management with Class Segregation

We consider a new model for buffer management of network switches with Quality of Service (QoS) requirements. A stream of packets, each attributed with a value representing its Class of Service (CoS), arrives over time at a network switch and demands a further transmission. The switch is equipped with multiple queues of limited capacities, where each queue stores packets of one value only. The objective is to maximize the total value of the transmitted packets (i.e., the weighted throughput). We analyze a natural greedy algorithm, GREEDY, which sends in each time step a packet with the greatest value. For general packet values $(v_1 < \cdots < v_m)$, we show that GREEDY is $(1+r)$-competitive, where $r = \max_{1\le i \le m-1} \{v_i/v_{i+1}\}$. Furthermore, we show a lower bound of $2 - v_m / \sum_{i=1}^m v_i$ on the competitiveness of any deterministic online algorithm. In the special case of two packet values (1 and $\alpha > 1$), GREEDY is shown to be optimal with a competitive ratio of $(\alpha + 2)/(\alpha + 1)$

The Expected Competitive Ratio for Weighted Completion Time Scheduling

A set of n independent jobs is to be scheduled without preemption on m identical parallel machines. For each job j, a diffuse adversary chooses the distribution Fj of the random processing time Pj from a certain class of distributions Fj. The scheduler is given the expectation μj = E[Pj], but the actual duration is not known in advance. A positive weight wj is associated with each job j and all jobs are ready for execution at time zero. The scheduler determines a list of the jobs, which is then scheduled in a non-preemptive manner. The objective is to minimise the total weighted completion time ∑j wj Cj. The performance of an algorithm is measured with respect to the expected competitive ratio maxF ∈ F E[∑j wj Cj/OPT], where Cj denotes the completion time of job j and OPT the offline optimum value. We show a general bound on the expected competitive ratio for list scheduling algorithms, which holds for a class of so-called new-better-than-used processing time distributions. This class includes, among others, the exponential distribution. As a special case, we consider the popular rule weighted shortest expected processing time first (WSEPT) in which jobs are processed according to the non-decreasing μj/wj ratio. We show that it achieves E[WSEPT/OPT] ≤ 3 - 1/m for exponential distributed processing time

Optimal Algorithms for Train Shunting and Relaxed List Update Problems

This paper considers a Train Shunting problem which occurs in cargo train organizations: We have a locomotive travelling along a track segment and a collection of n cars, where each car has a source and a target. Whenever the train passes the source of a car, it needs to be added to the train, and on the target, the respective car needs to be removed. Any such operation at the end of the train incurs low shunting cost, but adding or removing truly in the interior requires a more complex shunting operation and thus yields high cost. The objective is to schedule the adding and removal of cars as to minimize the total cost. This problem can also be seen as a relaxed version of the well-known List Update problem, which may be of independent interest. We derive polynomial time algorithms for Train Shunting by reducing this problem to finding independent sets in bipartite graphs. This allows us to treat several variants of the problem in a generic way. Specifically, we obtain an algorithm with running time O(n^{5/2}) for the uniform case, where all low costs and all high costs are identical, respectively. Furthermore, for the non-uniform case we have running time of O(n^3). Both versions translate to a symmetric variant, where it is also allowed to add and remove cars at the front of the train at low cost. In addition, we formulate a dynamic program with running time O(n^4), which exploits the special structure of the graph. Although the running time is worse, it allows us to solve many extensions, e.g., prize-collection, economies of scale, and dependencies between consecutive stations

SRPT Is 1.86-competitive for Completion Time Scheduling

We consider the classical problem of scheduling preemptible jobs, that ar-rive over time, on identical parallel machines. The goal is to minimize the total completion time of the jobs. In standard scheduling notation of Graham et al. [5], this problem is denoted P | rj,pmtn | j cj. A pop-ular algorithm called SRPT, which always schedules the unfinished jobs with shortest remaining processing time, is known to be 2-competitive, see Phillips et al. [12, 13]. This is also the best known competitive ratio for any online algorithm. However, it is conjectured that the competitive ra-tio of SRPT is significantly less than 2. Even breaking the barrier of 2 is considered a significant step towards the final answer of this classical online problem. We improve on this open problem by showing that SRPT is 1.86-competitive. This result is obtained using the following method, which might be of general interest: We define two dependent random variables that sum up to the difference between the cost of an SRPT schedule and the cost of an optimal schedule. Then we bound the sum of the expected values of these random variables with respect to the cost of the optimal schedule, yielding the claimed competitiveness. Furthermore, we show a lower bound of 21/19 for SRPT, improving on the previously best known 12/11 due to Lu et al. [10]

Sobre uma seqüência de vértebras sacrocaudais de um dinossauro terópode da Formação Santana, Cretáceo Inferior, Nordeste do Brasil

Besides being rare, most theropod remains from fossil deposits of Brazil are incomplete. Up to date the Romualdo Member (Aptian/Albian) of the Santana Formation yielded six theropod specimens. To those we add the description of a sequence of three posterior sacral and six anterior caudal vertebrae with three chevrons (MN 4743-V). Differences between MN 4743-V and members of the major theropod clades such as Ceratosauria, Allosauroidea and Coelorusauria do not allow its assignment to one of those groups. Instead, MN 4743-V is referred to the Spinosauroidea (Spinosauridae plus Torvosauridae), based on the presence of three robust laminae below the transverse process of the anterior caudals, which delimit three fossae. The lack of paired processes on the chevrons suggest that within Spinosauroidea MN 4743-V is a member of the Spinosauridae. This is congruent with previous findings of spinosaurids in the Romualdo Member. MN 4743-V differs from the remaining specimens that present sacrocaudal elements indicating the co-existence of at least five theropod species in this deposit.Restos de terópodes de depósitos fossilíferos brasileiros são raros, sendo a maioria dos espécimens incompletos. O Membro Romualdo (Aptiano/Albiano) da Formação Santana forneceu até então seis exemplares. Somando-se a estes é descrito uma sequência de seis vértebras sacrais posteriores, seis anteriores com três chevrons (MN 4743-V). Diferenças marcantes entre MN 4743-V e alguns táxons dos principais grupos de Theropoda, como Ceratosauria, Allosauroidea e Coelurosauria não permitem que este material seja associado a algum destes clados. Por outro lado, MN 4743-V apresenta uma semelhança marcante com os Spinosauroidea (Spinosauridae+Torvosauridae), que é a presença de duas lâminas robustas sob o processo transverso, delimitando três fossas. A ausência de processos pareados na parte proximal do chevron sugere que, dentro de Spinosauroidea, MN 4743-V seja um membro de Spinosauridae, o que é congruente com achados prévios de espinossaurídeos no Membro Romualdo. MN 4743-V difere dos demais exemplares com elementos sacro-caudais encontrados no Membro Romualdo, indicando a presença de pelo menos cinco espécies de terópodes neste depósito

Balanced Interval Coloring

We consider the discrepancy problem of coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time $O(n \log n + kn \log k)$ for its construction. This is in particular interesting because many known results for discrepancy problems are non-constructive. This problem naturally models a load balancing scenario, where $n$ tasks with given start- and endtimes have to be distributed among $k$ servers. Our results imply that this can be done ideally balanced. When generalizing to $d$-dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for any $d \ge 2$ and any $k \ge 2$ it is NP-complete to decide if such a solution exists, which implies also NP-hardness of the respective minimization problem. In an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm.Comment: Accepted at STACS 201