Doug Ierardi

Mitra: A Scalable Continuous Media Server

Mitra is a scalable storage manager that supports the display of continuous media data types, e.g., audio and video clips. It is a software based system that employs off-the-shelf hardware components. Its present hardware platform is a cluster of multi-disk workstations, connected using an ATM switch. Mitra supports the display of a mix of media types. To reduce the cost of storage, it supports a hierarchical organization of storage devices and stages the frequently accessed objects on the magnetic disks. For the number of displays to scale as a function of additional disks, Mitra employs staggered striping. It implements three strategies to maximize the number of simultaneous displays supported by each disk. First, the EVEREST file system allows different files (corresponding to objects of different media types) to be retrieved at different block size granularities. Second, the FIXB algorithm recognizes the different zones of a disk and guarantees a continuous display while harnessing the average disk transfer rate. Third, Mitra implements the Grouped Sweeping Scheme (GSS) to minimize the impact of disk seeks on the available disk bandwidth.In addition to reporting on implementation details of Mitra, we present performance results that demonstrate the scalability characteristics of the system. We compare the obtained results with theoretical expectations based on the bandwidth of participating disks. Mitra attains between 65% to 100% of the theoretical expectations.

Counting Points on Curves over Finite Fields

We consider the problem of counting the number of points on a plane curve, defined by a homogeneous polynomialF(x,y,z) ?Fqx,y,z, which are rational over a ground field Fq. More precisely, we show that if we are given a projective plane curve C of degreen, and if C has only ordinary multiple points, then one can compute the number of Fq-rational points on C in randomized time (logq)?where ?=nO(1). Since our algorithm actually computes the characteristic polynomial of the Frobenius endomorphism on the Jacobian of C, it ,follows that we may also compute (1) the number of Fq-rational points on the smooth projective model of C, (2) the number of Fq-rational points on the Jacobian of C, and (3) the number of Fqm-rational points on C in any given finite extension Fqmof the ground field, each in a similar time bound.

Efficient Algorithms for the Riemann-Roch Problem and for Addition in the Jacobian of a Curve

We study the effective Riemann-Roch problem of computing a basis for the linear space L(D) associated with a divisor D on a plane curve C and its applications. Our presentation begins with an extension of Noethers algorithm for this problem to plane curves with singularities. Like the original, this algorithm has a worst-case complexity of ?(n!|D|), where n is the degree of the curve C.We next consider representations of divisors based on Chow forms. Using these, we produce a factorization-free polynomial-time algorithm for the effective Riemann-Roch problem, which improves the complexity of Noethers algorithm by an order of magnitude. We also present further improvements which, for curves of bounded degree, yield an algorithm with complexity which is linear in the size of the given divisor. Finally, we consider applications of this problem: to the arithmetic of the Jacobian of a plane curve, to the construction of rational functions on a curve with prescribed zeroes and poles, and to the construction of curves with prescribed intersections.

The complexity of oblivious plans for orienting and distinguishing polygonal parts

In the problem ofparts feeding we are given a class of feasible operations for reorienting a part, and we are asked to find a fixed sequence of these operations which is guaranteed to bring the part into a known goal orientation from any possible initial orientation. Goldberg addressed this problem in [2], and showed that, for planar polygonal parts, there is always a sequence of simple operations which can be performed by a simple parallel-jaw gripper, which is guaranteed to orient the part (up to symmetry) without the use of any sensor information; he also conjectured thatO(n) steps are sufficient.In this paper we prove Goldbergs conjecture by explicitly constructing plans of at most2n − 1 steps for orienting polygonal parts in this model. We also give a lower bound on the number of steps required for such plans to show that this upper bound is tight.Finally, we extend these results to the problem ofdistinguishing among a finite set of parts using minimal sensing. Specifically, we assume that we are given a set of known polygonal parts, and a parallel-jaw gripper able to sense the distance between its jaws upon closure. We construct a simple oblivious plan of linear complexity which, when presented with a polygonal part, determines the index of this part.

Counting rational points on curves over finite fields

We consider the problem of counting the number of points on a plane curve, given by a homogeneous polynomial F/spl isin/F/sub p/[x, y, z], which is rational over the ground field F/sub p/. More precisely, we show that if we are given a projective plane curve C of degree n, and if C has only ordinary multiple points, then one can compute the number of F/sub p/-rational points on C in randomized time (log p)/sup /spl Delta// where /spl Delta/=(degF)/sup O(1/). The complexity of this construction improves previously known bounds for this problem by at least an order of magnitude.<<ETX>>

An on-line algorithm to optimize file layout in a dynamic environment

We describe an algorithm to manage the storage and layout of files cached on mechanical devices, such as magnetic disk drives. The algorithms respond in an on-line manner to maintain a dynamically changing working set of disk-resident files with fewer than ⌜lg n⌝ breaks for each disk-resident file of n blocks.

2d-bubblesorting in average time O (√ N lg N )*

We study the complexity of a natural generalization of the parallel odd-even transposition sort to 2-dimensional meshconnected parallel processors, and prove that 0(/~~ steps suffice to sort a random N-element permutation on an ~ x m-processor mesh with probability at least 1N-’, for any fixed c >0. This answer a problem posed by Leightonl in [Lei92]. Odd-even transposition sort on a linear array of processors is a simple oblivious sorting algorithm. In extending the bubblesorting paradigm to the mesh, we impose a snake-like row-major ordering on the rows — so that adjacent rows are sorted in opposite directions — and have the processors repeatedly cycle through a small set of local comparisons. It is easy to see that such an algorithm actually does sort in O(N) time in the worst case, since a Id-array is embedded in the snake-like ordering of the mesh’s procesears. However, as Scherson et al. note in [SSS86], the performance of such an algorithm is also Q(N) in the worst case — substantially inferior to the 3@+ o(filg N) time of known optimal sorting algorithms for the mesh, such as [SS86]. In fact, in the worst case, a bubblesort is generally outperformed even by simple sub-optimal algorithms, such aa ShearSort [SSS86], which has a running time of @(filg N). One exception is the algont hm of [Sch88], which sketches a short–periodic sorting algorithm that does achieve time 0(~ Ig N) on a mesh equipped with wrap-around connections. The average case behavior of odd-even transposition sort on an array has been well studied. [Lei92] Recently, several studies have analyzed the average behavior of local iterative sorting algorithms on a 2-dimensional mesh. In [Sav93], fer example, Savari showed that two such algorithms require E)(N) time on average. Aydin and Ierardi [A193] show that a number of other natural 2d-bubblesorting algorithms —

Efficient algorithms for the Riemann-Roch problem and for addition in the Jacobian of a curve

Several computational problems concerning the construction of rational functions and intersecting curves over a given curve are studied. The first problem is to construct a rational function with prescribed zeros and poles over a given curve. More precisely, let C be a smooth projective curve and assume as given an affine plane model F(x,y)=0 for C, a finite set of points P/sub i/=(X/sub i/, Y/sub i/) with F (X/sub i/, Y/sub i/)=0 and natural numbers n/sub i/, and a finite set of points Q/sub i/=(X/sub j/, Y/sub j/) with F(X/sub j/, Y/sub j/)=0 and natural numbers m/sub j/. The problem is to decide whether there is a rational function which has zeros at each point P/sub i/ of order n/sub i/, poles at each Q/sub j/ of order m/sub j/, and no zeros or poles anywhere else on C. One would also like to construct such a rational function if one exists. An efficient algorithm for solving this problem when the given plane curve has only ordinary multiple points is given.<<ETX>>

Time-optimal trajectories of a rod in the plane subject to velocity constraints

We consider the problem of moving a line segment (a “rod” or “ladder”) in the plane between two given placements when subject to the constraint that no point on the line segment may exceed a given velocity bound. Specifically, we consider those trajectories which minimize the total time between given initial and goal placements, and provide a complete characterization of all solution, together with explicit constructions for each of the various cases encountered.

Stably Placing Piecewise Smooth Objects

Given a piecewise smooth object, its stable poses consist of all the orientations into which other initial orientations of the object will eventually converge under dissipative forces. The capture region for each stable pose is the set of initial orientations converging to the stable pose in question.