Results 1  10
of
27
An Algorithm for Exact Bounds on the Time Separation of Events in Concurrent Systems
 IEEE Transactions on Computers
, 1993
"... Determining the time separation of events is a fundamental problem in the analysis, synthesis, and optimization of concurrent systems. Applications range from logic optimization of asynchronous digital circuits to evaluation of execution times of programs for realtime systems. We present an efficie ..."
Abstract

Cited by 44 (7 self)
 Add to MetaCart
Determining the time separation of events is a fundamental problem in the analysis, synthesis, and optimization of concurrent systems. Applications range from logic optimization of asynchronous digital circuits to evaluation of execution times of programs for realtime systems. We present an efficient algorithm to find exact (tight) bounds on the separation time of events in an arbitrary process graph without conditional behavior. This result is more general than the methods presented in several previously published papers as it handles cyclic graphs and yields the tightest possible bounds on event separations. The algorithm is based on a functional decomposition technique that permits the implicit evaluation of an infinitely unfolded process graph. Examples are presented that demonstrate the utility and efficiency of the solution. The algorithm will form a basis for exploration of timingconstrained synthesis techniques. Index terms: Abstract algebra, asynchronous systems, concurrent ...
The genus of curves over finite fields with many rational points
 MANUSCRIPTA MATH
, 1996
"... We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2rational points reachs the HasseWeil upper bound. Under a hypothesis on nongaps at rational points we prove that maximal curves are F q 2isomorphic to y ..."
Abstract

Cited by 27 (10 self)
 Add to MetaCart
We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2rational points reachs the HasseWeil upper bound. Under a hypothesis on nongaps at rational points we prove that maximal curves are F q 2isomorphic to y q + y = x m for some m ∈ Z +.
The Frobenius Problem, Rational Polytopes, and FourierDedekind Sums
, 2003
"... We study the number of lattice points in integer dilates of the rational polytope P = (x1,..., xn) ∈ R n n∑ ≥0: xkak ≤ 1, where a1,..., an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1,..., an, fin ..."
Abstract

Cited by 27 (13 self)
 Add to MetaCart
We study the number of lattice points in integer dilates of the rational polytope P = (x1,..., xn) ∈ R n n∑ ≥0: xkak ≤ 1, where a1,..., an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1,..., an, find the largest value of t (the Frobenius number) such that m1a1 + · · · + mnan = t has no solution in positive integers m1,..., mn. This is equivalent to the problem of finding the largest dilate tP such that the facet n k=1 xkak = t} contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L(P, t): = #(tP ∩ Z n) and L(P ◦ , t):= #(tP ◦ ∩ Z n). Within the computations a Dedekindlike finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of k=1
Analysis of Shellsort and related algorithms
 ESA ’96: Fourth Annual European Symposium on Algorithms
, 1996
"... This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellso ..."
Abstract

Cited by 26 (0 self)
 Add to MetaCart
This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellsort and Shellsortbased networks; averagecase results; proposed probabilistic sorting networks based on the algorithm; and a list of open problems. 1 Shellsort The basic Shellsort algorithm is among the earliest sorting methods to be discovered (by D. L. Shell in 1959 [36]) and is among the easiest to implement, as exhibited by the following C code for sorting an array a[l],..., a[r]: shellsort(itemType a[], int l, int r) { int i, j, h; itemType v;
The polynomial part of a restricted partition function related to the Frobenius problem
, 2003
"... ..."
On Weierstrass points and optimal curves
, 1997
"... We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves. ..."
Abstract

Cited by 14 (9 self)
 Add to MetaCart
We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves.
Frobenius Problem and the Covering Radius of a Lattice, submitted
"... Abstract. Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this Ntuple is defined to be the largest positive integer that cannot be expressed as ∑N i=1 aixi where x1,...,xN are nonnegative integers. The condition that gcd(a1,...,aN) = 1 implies that such n ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
Abstract. Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this Ntuple is defined to be the largest positive integer that cannot be expressed as ∑N i=1 aixi where x1,...,xN are nonnegative integers. The condition that gcd(a1,...,aN) = 1 implies that such number exists. The general problem of determining the Frobenius number given N and a1,..., aN is NPhard, but there has been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating Frobenius number to the covering radius of the nulllattice of this Ntuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often. 1.
Hard Equality Constrained Integer Knapsacks
, 2005
"... We consider the following integer feasibility problem: “Given positive integer numbers a0, a1,..., an, with gcd(a1,..., an) = 1 and a = (a1,..., an), does there exist a vector x ∈ Z n ≥0 satisfying ax = a0? ” We prove that if the coefficients a1,..., an have a certain decomposable structure, then t ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
We consider the following integer feasibility problem: “Given positive integer numbers a0, a1,..., an, with gcd(a1,..., an) = 1 and a = (a1,..., an), does there exist a vector x ∈ Z n ≥0 satisfying ax = a0? ” We prove that if the coefficients a1,..., an have a certain decomposable structure, then the Frobenius number associated with a1,..., an, i.e., the largest value of a0 for which ax = a0 does not have a nonnegative integer solution, is close to a known upper bound. In the instances we consider, we take a0 to be the Frobenius number. Furthermore, we show that the decomposable structure of a1,..., an makes the solution of a lattice reformulation of our problem almost trivial, since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small. For branchandbound such instances are difficult to solve, since they are infeasible and have large values of a0/ai, 1 ≤ i ≤ n. We illustrate our results by some computational examples.
Multidimensional versions of a theorem of Fine and Wilf and a formula of
, 1661
"... Abstract. Let �v0,..., �vk be vectors in Z k which generate Z k. We show that a body V ⊂ Z k with the vectors �v0,..., �vk as edge vectors is an almost minimal set with the property that every function f: V → R with periods �v0,..., �vk is constant. For k = 1 the result reduces to the theorem of Fin ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
Abstract. Let �v0,..., �vk be vectors in Z k which generate Z k. We show that a body V ⊂ Z k with the vectors �v0,..., �vk as edge vectors is an almost minimal set with the property that every function f: V → R with periods �v0,..., �vk is constant. For k = 1 the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma. Suppose �0 is not a nontrivial linear combination of �v0,..., �vk with nonnegative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of �v0,..., �vk over Z≥0, but infinitely many points on each of its hyperfaces are not. For k =1theresult reduces to a formula of Sylvester corresponding to Frobenius ’ Coinchanging Problem in the case of coins of two denominations. 1.
The worst case in Shellsort and related algorithms
 Journal of Algorithms
, 1993
"... Abstract. We show that sorting a sufficiently long list of length N using Shellsort with m increments (not necessarily decreasing) requires at least N 1+c/ √ m comparisons in the worst case, for some constant c> 0. For m ≤ (log N / log log N) 2 we obtain an upper bound of the same form. We also prov ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Abstract. We show that sorting a sufficiently long list of length N using Shellsort with m increments (not necessarily decreasing) requires at least N 1+c/ √ m comparisons in the worst case, for some constant c> 0. For m ≤ (log N / log log N) 2 we obtain an upper bound of the same form. We also prove that Ω(N(log N / log log N) 2) comparisons are needed regardless of the number of increments. Our approach is general enough to apply to other sorting algorithms, including Shakersort, for which an even stronger result is proved. 1.