Ari Freund

A unified approach to approximating resource allocation and scheduling

We present a general framework for solving resource allocation and scheduling problems. Given a resource of fixed size, we present algorithms that approximate the maximum throughput or the minimum loss by a constant factor. Our approximation factors apply to many problems, among which are: (i) real-time scheduling of jobs on parallel machines, (ii) bandwidth allocation for sessions between two endpoints, (iii) general caching, (iv) dynamic storage allocation, and (v) bandwidth allocation on optical line and ring topologies. For some of these problems we provide the first constant factor approximation algorithm. Our algorithms are simple and efficient and are based on the local-ratio technique. We note that they can equivalently be interpreted within the primal-dual schema.

On-Line Load Balancing in a Hierarchical Server Topology

In a hierarchical server environment jobs are to be assigned in an on-line fashion to a collection of servers which form a hierarchy of capability: each job requests a specific server meeting its needs, but the system is free to assign it either to that server or to any other server higher in the hierarchy. Each job carries a certain load, which it imparts to the server it is assigned to. The goal is to find a competitive assignment in which the maximum total load on a server is minimized. We consider the linear hierarchy in which the servers are totally ordered in terms of their capabilities. We investigate several variants of the problem. In the unweighted (as opposed to weighted) problem all jobs have unit weight. In the fractional (as opposed to integral) model a job may be assigned to several servers, each receiving some fraction of its weight. Finally, temporary (as opposed to permanent) jobs may depart after being active for some finite duration of time. We show an optimal e-competitive algorithm for the unweighted integral permanent model. The same algorithm is (e+1)-competitive in the weighted case. Its fractional version is e-competitive even if temporary jobs are allowed. For the integral model with temporary jobs we show an algorithm which is 4-competitive in the unweighted case and 5-competitive in the weighted case. We show a lower bound of e for the unweighted case (both integral and fractional). This bound is valid even with respect to randomized algorithms. We also show a lower bound of 3 for the unweighted integral model when temporary jobs are allowed. We generalize the problem and consider hierarchies in which the servers form a tree. In the tree hierarchy, any job assignable to a node is also assignable to the nodes ancestors. We show a deterministic algorithm which is 4-competitive in the unweighted case and 5-competitive in the weighted case, where only permanent jobs are allowed. Randomizing this algorithm improves its competitiveness to e and e+1, respectively. We also show an

Algorithmic aspects of bandwidth trading

\Omega(\sqrt{n})

Harnessing machine learning to improve the success rate of stimuli generation

lower bound when temporary jobs are allowed.

Competitive on-line switching policies

We study algorithmic problems that are motivated by bandwidth trading in next-generation networks. Typically, bandwidth trading involves sellers (e.g., network operators) interested in selling bandwidth pipes that offer to buyers a guaranteed level of service for a specified time interval. The buyers (e.g., bandwidth brokers) are looking to procure bandwidth pipes to satisfy the reservation requests of end-users (e.g., Internet subscribers). Depending on what is available in the bandwidth exchange, the goal of a buyer is to either spend the least amount of money so as to satisfy all the reservations made by its customers, or to maximize its revenue from whatever reservations can be satisfied. We model this as a real-time nonpreemptive scheduling problem in which machine types correspond to bandwidth pipes and jobs correspond to end-user reservation requests. Each job specifies a time interval during which it must be processed, and a set of machine types on which it can be executed. If necessary, multiple machines of a given type may be allocated, but each must be paid for. Finally, each job has associated with it a revenue, which is realized if the job is scheduled on some machine. There are two versions of the problem that we consider. In the cost minimization version, the goal is to minimize the total cost incurred for scheduling all jobs, and in the revenue maximization version the goal is to maximize the revenue of the jobs that are scheduled for processing on a given set of machines. We consider several variants of the problems that arise in practical scenarios, and provide constant factor approximations.

New algorithms for related machines with temporary jobs

The initial state of a design under verification has a major impact on the ability of stimuli generators to successfully generate the requested stimuli. For complexity reasons, most stimuli generators use sequential solutions without planning ahead. Therefore, in many cases, they fail to produce a consistent stimuli due to an inadequate selection of the initial state. We propose a new method, based on machine learning techniques, to improve generation success by learning the relationship between the initial state vector and generation success. We applied the proposed method in two different settings, with the objective of improving generation success and coverage in processor and system level generation. In both settings, the proposed method significantly reduced generation failures and enabled faster coverage

A lower bound of 8/ 7+ 1 k-1 on the integrality ratio of the Calinescu-Karloff-Rabani relaxation for multiway cut

Consider the following problem. A switch connecting n input channels to a single output channel must deliver all incoming messages through this channel. Messages are composed of packets , and in each time slot the switch can deliver a single packet from one of the input queues to the output channel. In order to prevent packet loss, a buffer is maintained for each input channel. The goal of a switching policy is to minimize the maximum buffer size. The setting is on-line; decisions must be made based on the current state without knowledge of future events. This general scenario models multiplexing tasks in various systems such as communication networks, cable modem systems, and traffic control. Traditionally, researchers analyzed the performance of a given policy assuming some distribution on the arrival rates of messages at the input queues, or assuming that the service rate is at least the aggregate of all the input rates. We use competitive analysis, avoiding any prior assumptions on the input. We show O(log n )-competitive switching policies for the problem and demonstrate matching lower bounds.

Practical aggregation of semantical program properties for machine learning based optimization

We consider the on-line problem of assigning temporary jobs to related machines. In this model machines have speeds and the jobs are weighted and may be temporary (i.e. may expire after some unknown nite amount of time). The cost of an assignment is the maximum load on a machine at any time. We present two new algorithms: a 6 + 2 p 5 10:47-competitive deterministic algorithm and a 9:572-competitive randomized algorithm. The previously known best upper bound is achieved by a 20-competitive deterministic algorithm whose randomized version is 5e = 13:59-competitive.

Combinatorial Interpretations of Dual Fitting and Primal Fitting

Abstract Given an edge-weighted graph and a subset of k vertices called terminals, a multiway cut is a partition of the vertices into k components, each containing exactly one terminal. The multiway cut problem is to find a multiway cut minimizing the sum of the weights of edges with endpoints in different components. Recently, Călinescu et al. described an approximation algorithm based on a geometric embedding of the graphs vertices into R k . We present a lower bound of 8/(7+ 1 k−1 ) on the integrality ratio of this relaxation.

Improved Subquadratic 3SUM

Iterative search combined with machine learning is a promising approach to design optimizing compilers harnessing the complexity of modern computing systems. While traversing a program optimization space, we collect characteristic feature vectors of the program, and use them to discover correlations across programs, target architectures, data sets, and performance. Predictive models can be derived from such correlations, effectively hiding the time-consuming feedback-directed optimization process from the application programmer. One key task of this approach, naturally assigned to compiler experts, is to design relevant features and implement scalable feature extractors, including statistical models that filter the most relevant information from millions of lines of code. This new task turns out to be a very challenging and tedious one from a compiler construction perspective. So far, only a limited set of ad-hoc, largely syntactical features have been devised. Yet machine learning is only able to discover correlations from information it is fed with: it is critical to select topical program features for a given optimization problem in order for this approach to succeed. We propose a general method for systematically generating numerical features from a program. This method puts no restrictions on how to logically and algebraically aggregate semantical properties into numerical features. We illustrate our method on the difficult problem of selecting the best possible combination of 88 available optimizations in GCC. We achieve 74% of the potential speedup obtained through iterative compilation on a wide range of benchmarks and four different general-purpose and embedded architectures. Our work is particularly relevant to embedded system designers willing to quickly adapt the optimization heuristics of a mainstream compiler to their custom ISA, microarchitecture, benchmark suite and workload. Our method has been integrated with the publicly released MILEPOST GCC [14].