Baruch Schieber

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.

Approximating minimum feedback sets and multicuts in directed graphs

Abstract. This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set (FES) problem. In the {FVS} (resp. FES) problem, one is given a directed graph with weights (each of which is at least one) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NP-hard problems and have many applications. We also consider a generalization of these problems: subset-fvs and subset-fes, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NP-hard even when |X|=2 . We present approximation algorithms for the subset-fvs and subset-fes problems. The first algorithm we present achieves an approximation factor of O(log2|X|) . The second algorithm achieves an approximation factor of O(min{log τ* log log τ*, log n log log n)} , where τ* is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the subset-fes and subset-fvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1+ɛ) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.

Competitive paging with locality of reference

The Sleator-Tarjan competitive analysis of paging (Comm. ACM28 (1985), 202-208) gives us the ability to make strong theoretical statements about the performance of paging algorithms without making probabilistic assumptions on the input. Nevertheless practitioners voice reservations about the model, citing its inability to discern between LRU and FIFO (algorithms whose performances differ markedly in practice), and the fact that the theoretical comptitiveness of LRU is much larger than observed in practice, In addition, we would like to address the following important question: given some knowledge of a program?s reference pattern, can we use it to improve paging performance on that program? We address these concerns by introducing an important practical element that underlies the philosophy behind paging: locality of reference. We devise a graph-theoretical model, the access graph, for studying locality of reference. We use it to prove results that address the practical concerns mentioned above, In addition, we use our model to address the following questions: How well is LRU likely to perform on a given program? Is there a universal paging algorithm that achieves (nearly) the best possible paging performance on every program? We do so without compromising the benefits of the Sleator-Tarjan model, while bringing it closer to practice.

Divide-and-conquer approximation algorithms via spreading metrics

We present a novel divide-and-conquer paradigm for approximating NP-hard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divide-and-conquer approach is applicable. Second, a fractional spreading metric is computable in polynomial time. The spreading metric assigns lengths to either edges or vertices of the input graph, such that all subgraphs for which the optimization problem is nontrivial have large diameters. In addition, the spreading metric provides a lower bound, <inline-equation><f> <g>t</g></f> </inline-equation>, on the cost of solving the optimization problem. We present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is <italic>O</italic>(min{log <inline-equation><f> <g>t</g>,</f> </inline-equation>log log <inline-equation><f> <g>t</g></f> </inline-equation>, log <italic>k</italic> log log <italic>k</italic>}) where <italic>k</italic> denotes the number of “interesting” vertices in the problem instance, and is at most the number of vertices. We present seven problems that can be formulated to fit the paradigm. For all these problems our algorithm improves previous results. The problems are: (1) linear arrangement; (2) embedding a graph in a <italic>d</italic>-dimensional mesh; (3) interval graph completion; (4) minimizing storage-time product; (5) subset feedback sets in directed graphs and multicuts in circular networks; (6) symmetric multicuts in directed networks; (7) balanced partitions and <italic>p</italic>-separators (for small values of <italic>p</italic>) in directed graphs.

Buffer Overflow Management in QoS Switches

We consider two types of buffering policies that are used in network switches supporting Quality of Service (QoS). In the FIFO type, packets must be transmitted in the order in which they arrive; the constraint in this case is the limited buffer space. In the bounded-delay type, each packet has a maximum delay time by which it must be transmitted, or otherwise it is lost. We study the case of overloads resulting in packet loss. In our model, each packet has an intrinsic value, and the goal is to maximize the total value of transmitted packets. Our main contribution is a thorough investigation of some natural greedy algorithms in various models. For the FIFO model we prove tight bounds on the competitive ratio of the greedy algorithm that discards packets with the lowest value when an overflow occurs. We also prove that the greedy algorithm that drops the earliest packets among all low-value packets is the best greedy algorithm. This algorithm can be as much as 1.5 times better than the tail-drop greedy policy, which drops the latest lowest-value packets. In the bounded-delay model we show that the competitive ratio of any on-line algorithm for a uniform bounded-delay buffer is bounded away from 1, independent of the delay size. We analyze the greedy algorithm in the general case and in three special cases: delay bound 2, link bandwidth 1, and only two possible packet values. Finally, we consider the off-line scenario. We give efficient optimal algorithms and study the relation between the bounded-delay and FIFO models in this case.

Parallel construction of a suffix tree with applications

Many string manipulations can be performed efficiently on suffix trees. In this paper a CRCW parallel RAM algorithm is presented that constructs the suffix tree associated with a string ofn symbols inO(logn) time withn processors. The algorithm requires Θ(n2) space. However, the space needed can be reduced toO(n1+ɛ) for any 0< ɛ ≤1, with a corresponding slow-down proportional to 1/ɛ. Efficient parallel procedures are also given for some string problems that can be solved with suffix trees.

Minimizing Service and Operation Costs of Periodic Scheduling

We study the problem of scheduling activities of several types under the constraint that, at most, a fixed number of activities can be scheduled in any single time slot. Any given activity type is associated with a service cost and an operating cost that increases linearly with the number of time slots since the last service of this type. The problem is to find an optimal schedule that minimizes the long-run average cost per time slot. Applications of such a model are the scheduling of maintenance service to machines, multi-item replenishment of stock, and minimizing the mean response time in Broadcast Disks. Broadcast Disks recently gained a lot of attention because they were used to model backbone communications in wireless systems, Teletext systems, and Web caching in satellite systems. The first contribution of this paper is the definition of a general model that combines into one several important previous models. We prove that an optimal cyclic schedule for the general problem exists, and we establish the NP-hardness of the problem. Next, we formulate a nonlinear program that relaxes the optimal schedule and serves as a lower bound on the cost of an optimal schedule. We present an efficient algorithm for finding a near-optimal solution to the nonlinear program. We use this solution to obtain several approximation algorithms. 1 A 9/8 approximation for a variant of the problem that models the Broadcast Disks application. The algorithm uses some properties of “Fibonacci sequences.” Using this sequence, we present a 1.57-approximation algorithm for the general problem. 2 A simple randomized algorithm and a simple deterministic greedy algorithm for the problem. We prove that both achieve approximation factor of 2. To the best of our knowledge this is the first worst-case analysis of a widely used greedy heuristic for this problem.

Navigating in unfamiliar geometric terrain

Consider a robot that has to travel from a start location s to a target t in an environment with opaque obstacles that lie in its way. The robot always knows its current absolute position and that of the target. It does not, however, know the positions and extents of the obstacles in advance; rather, it finds out about obstacles as it encounters them. We compare the distance walked by the robot in going from .s to t to the length of the shortest path between s and t in the scene. We describe and analyze robot strategies that minimize this ratio for different kinds of scenes. In particular, we consider the cases of rectangular obstacles aligned with the axes, rectangular obstacles in more general orientations, and wider classes of convex bodies both in two and three dimensions. We study scenes with non-convex obstacles, which are related to the study of maze-traversal. We also show scenes where randomized algorithms are provably better than deterministic algorithms. 1. Motivation and Results Practical work on robot motion planning falls into two categories: motion planning through a known scene, in which the robot has a complete map of the environment, and motion planning through an unknown scene in which an autonomous robot must find its way through a new environment (see, for example, [6, 8, 9, 13, 15] and references therein). Virtually all previous theoretical work ([23] and ref“Laboratory for Computer Science, MIT, Cambridge, MA 02139. Supported in part by an NSF graduate fellowship. Part of the work was done while this author was visiting IBM T. J. Watson Research Center. avrim’dtheory. lcs. mit. edu tIBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY 10598. {pragh, sbar]@ibm. com Permission to copy without fee all or part of ths material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and ifs date appear, and notice is given that copying is by perrmsslon of the Asoclation for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. @ 1991 ACM 089791-397-3/91/0004/0494

Highly parallelizable problems

1.50 erences therein) has focused on the former problem. Papadimitriou and Yannakakis [17] studied the latter problem, which is also the subject of this paper: the design and evaluation of strategies for navigation in an unknown environment. The unfamiliar environment may be either a warehouse or a factory floor whose contents are frequently moved, or a remote terrain such as Mars [21]. The design and evaluation of algorithms for such navigation is a natural algorithmic problem that deserves more theoretical study.

Optimal doubly logarithmic parallel algorithms based on finding all nearest smaller values

of Results. We establish that several problems are highly parallelizable. For each of these problems, we design an optimal 0 (loglogn ) time parallel algorithm on the Common CRCW PRAM model which is the weakest among the CRCW PRAM models. These problems include: 0 all nearest smaller values, l preprocessing for answering range maxima queries, l several problems in Computational Geometry, l string matching. Until recently, such algorithms were known only for finding the maximum and merging.