Nonpreemptive scheduling of periodic tasks in uni- and multiprocessor systems

In this paper we study the problem of scheduling a set of periodic tasks nonpreemptively in hard-real-time systems, where it is critical for all requests of the tasks to be processed in time. A taskT is characterized by itsarrival time A, itsperiod P, and itsexecution time C. Starting fromA, a new request ofT arrives in everyP units of time, requestingC units of processing time, and itsdeadline coincides with the arrival of the next request ofT. All requests must be processed nonpreemptively to meet their corresponding deadlines. We show that the problem of testing the feasibility of a given task set {T ¹,T ²,,T ⁿ} satisfyingP ₁₊₁=k_i p_i, wherek _i; is an integer 1 for 1i n–1, on a single processor is NP-hard in the strong sense, even if all tasks have the same arrival time. For task sets satisfyingP ⁱ⁺¹=K Pⁱ, whereK is an integer 2 for 1 i n–1 and all tasks have the same arrival time, we present linear-time (in the number of requests) optimal scheduling algorithms as well as linear-time (in the number of tasks, i.e.,n) algorithms for testing feasibility in both uniprocessor and multiprocessor systems. We also extend our results to more general task sets.     

A faster divide-and-conquer algorithm for constructing delaunay triangulations

An easily implemented modification to the divide-and-conquer algorithm for computing the Delaunay triangulation ofn sites in the plane is presented. The change reduces its (n logn) expected running time toO(n log logn) for a large class of distributions that includes the uniform distribution in the unit square. Experimental evidence presented demonstrates that the modified algorithm performs very well forn2¹⁶, the range of the experiments. It is conjectured that the average number of edges it creates—a good measure of its efficiency—is no more than twice optimal forn less than seven trillion. The improvement is shown to extend to the computation of the Delaunay triangulation in theL _p metric for 1<p.This research was supported by National Science Foundation Grants DCR-8352081 and DCR-8416190.     

A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the orderk Voronoi diagrams, can be done inO(n logn+k ³n) expected time and O(k²n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k ^(d+1)/2+1 n ^(d+1)/2) and space complexityO(k ^(d+1)/2 n ^(d+1)/2). The algorithm is simple and experimental results are given.This work has been supported in part by the ESPRIT Basic Research Action No. 3075 (ALCOM).     

Applications of the crossing number

LetG be a graph ofn vertices that can be drawn in the plane by straight-line segments so that nok+1 of them are pairwise crossing. We show thatG has at mostc _k nlog^2k–2 n edges. This gives a partial answer to a dual version of a well-known problem of Avital-Hanani, Erdós, Kupitz, Perles, and others. We also construct two point sets {p ₁,,p _n }, {q ₁,,q _n } in the plane such that any piecewise linear one-to-one mappingfR ²R ² withf(pi)=qi (1in) is composed of at least (n ²) linear pieces. It follows from a recent result of Souvaine and Wenger that this bound is asymptotically tight. Both proofs are based on a relation between the crossing number and the bisection width of a graph.The first author was supported by NSF Grant CCR-91-22103, PSC-CUNY Research Award 663472, and OTKA-4269. An extended abstract of this paper was presented at the 10th Annual ACM Symposium on Computational Geometry, Stony Brook, NY, 1994.     

On the Noninteger Polyhedron Vertices of the Three-index Axial Transportation Problem

For the three-index axial transportation polyhedron defined by the integer vector, existence of noninteger vertices was proved. In particular, the three-index n × m × k axial transportation polyhedron having vertices with r fractional components was shown to exist for and only for any number r {4,6,7,...,(n, m, k)}, where (n, m, k) = min{n, m + k - 2} + m + k - 2, n m k 3.     

On selecting thek largest with median tests

LetW _itk(n) be the minimax complexity of selecting thek largest elements ofn numbersx ₁,x ₂,...,x _n by pairwise comparisonsx _i :x _j . It is well known thatW ₂(n) =n–2+ [lgn], andW _k (n) = n + (k–1)lg n +O(1) for all fixed k 3. In this paper we studyW _k (n), the minimax complexity of selecting thek largest, when tests of the form Isx _i the median of {x _i ,x _j ,x _t }? are also allowed. It is proved thatW₂(n) =n–2+ [lgn], andW _k (n) =n + (k–1)lg₂ n +O(1) for all fixedk3.This research was supported in part by the National Science Foundation under Grant No. DCR-8308109.     

A 2n−2 step algorithm for routing in ann ×n array with constant-size queues

In this paper we describe a deterministic algorithm for solving any 1–1 packet-routing problem on ann ×n mesh in 2n–2 steps using constant-size queues. The time bound is optimal in the worst case. The best previous deterministic algorithm for this problem required time 2n+(n/q) using queues of size (q) for any 1qn, and the best previous randomized algorithm required time 2n+(logn) using constant-size queues.This research was supported by the Clear Center at UTD, DARPA Contracts N00014-91-J-1698 and N00014-92-J-1799, Air Force Contract F49620-92-J-0125, Army Contract DAAL-03-86-K-0171, an NSF Presidential Young Investigator Award with matching funds from AT&T and IBM, and by the Texas Advanced Research Program under Grant No. 3972. A preliminary version of this paper appeared in [5].     

On lower bounds of the closeness between complexity classes

We show that if an NP-m-hard set is the union of a set in P_btt(Sparse) and the setA, then NP P_dtt(A). Since co-NP, R, and FewP are closed under _dtt ^P -reductions, so no NP-m-hard set is the union of a set in P_btt(Sparse) and a set in co-NP (resp. R, FewP) unless NP = co-NP (resp. NP = R, NP = FewP). We also study the distance between many important complexity classes. LetA,B be two sets. The distance function dist _A,B is defined as in [S1] such that dist _A,B (n) is the number of strings of length n inA B = (A – B) (B – A). We show that there exists a setA in p_{(k + 1)–tt}(Sparse) such that, for everyB in P _k–tt(Sparse), dist _A,B has an exponential lower bound.This research was supported in part by HTP 863.     

Primality testing with fewer random bits

In the usual formulations of the Miller-Rabin and Solovay-Strassen primality testing algorithms for a numbern, the algorithm chooses candidatesx ₁,x ₂, ...,x _k uniformly and independently at random from _n , and tests if any is a witness to the compositeness ofn. For either algorithm, the probabilty that it errs is at most 2^–k .In this paper, we study the error probabilities of these algorithms when the candidates are instead chosen asx, x+1, ..., x+k–1, wherex is chosen uniformly at random from _n . We prove that fork=[1/2log₂ n], the error probability of the Miller-Rabin test is no more thann ^–1/2+o(1), which improves on the boundn ^–1/4+o(1) previously obtained by Bach. We prove similar bounds for the Solovay-Strassen test, but they are not quite as strong; in particular, we only obtain a bound ofn ^–1/2+o(1) if the number of distinct prime factors ofn iso(logn/loglogn).     

A relationship between difference hierarchies and relativized polynomial hierarchies

Chang and Kadin have shown that if the difference hierarchy over NP collapses to levelk, then the polynomial hierarchy (PH) is equal to thekth level of the difference hierarchy over ₂ ^p . We simplify their poof and obtain a slightly stronger conclusion: if the difference hierarchy over NP collapses to levelk, then PH collapses to (P _(k–1) ^NP )^NP, the class of sets recognized in polynomial time withk – 1 nonadaptive queries to a set in NP^NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any classC that has _m ^p -complete sets and is closed under _conj ^p -and _m ^NP -reductions (alternatively, closed under _disj ^p -and _m ^co-NP -reductions), if the difference hierarchy overC collapses to levelk, then PH ^C = (P _(k–1)–tt ^NP ) ^C . Then we show that the exact counting class C_P is closed under _disj ^p - and _m ^co-NP -reductions. Consequently, if the difference hierarchy over C_P collapses to levelk, then PH^PP(= PH^C_P) is equal to (P _(k–1)–tt ^NP )^PP. In contrast, the difference hierarchy over the closely related class PP is known to collapse.Finally we consider two ways of relativizing the bounded query class P _k–tt ^NP : the restricted relativization P _k–tt ^{NP C} and the full relativization (P _k–tt ^NP ) ^C . IfC is NP-hard, then we show that the two relativizations are different unless PH ^C collapses.Richard Beigel was supported in part by NSF Grants CCR-8808949 and CCR-8958528. Richard Chang was supported in part by NSF Research Grant CCR 88-23053. This work was done while Mitsunori Ogiwara was at the Department of Information Science, Tokyo Institute of Technology, Tokyo, Japan.     

A forbidden subposet characterization of an order — dimension inequality

Dushnik and Miller defined the dimension of a partially ordered setX, DimX, as the minimum number of linear extensions ofX whose intersection is the partial ordering onX. The concept of dimension for a partially ordered set has applications to preference structures and the theory of measurement. Hiraguchi proved that DimX [|X|/2] when |X| 4. Bogart, Trotter, and Kimble gave a forbidden subposet characterization of Hiraguchi's inequality by constructing for eachn 2 the minimum collection _n of posets such that if [|X|/2] =n 2, then DimX < n unlessX contains one of the posets in _n . Recently Trotter gave a simple proof of Hiraguchi's inequality based on the following theorem. IfA is an antichain ofX and |X – A| =n 2, then DimX n. In this paper we give a forbidden subposet characterization of this last inequality.     

On the Quadratic Convergence of the Levenberg-Marquardt Method without Nonsingularity Assumption

Recently, Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by using _k=||F(x_k)||, where [1,2], instead of _k=||F(x_k)||² as the Levenberg-Marquardt parameter. If ||F(x)|| provides a local error bound for the system of nonlinear equations F(x)=0, it is shown that the sequence {x_k} generated by the new method converges to a solution quadratically, which is stronger than dist(x_k,X^*)0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.     

Undirecteds-t connectivity in polynomial time and sublinear space

Thes-t connectivity problem for undirected graphs is to decide whether two designated vertices,s andt, are in the same connected component. This paper presents the first known deterministic algorithms solving undirecteds-t connectivity using sublinear space and polynomial time. Our algorithms provide a nearly smooth time-space tradeoff between depth-first search and Savitch's algorithm. Forn vertex,m edge graphs, the simplest of our algorithms uses spaceO(s),n ^1/2log₂ nsnlog₂ n, and timeO(((m+n)n ² log² n)/s). We give a variant of this method that is faster at the higher end of the space spectrum. For example, with space (nlogn), its time bound isO((m+n)logn), close to the optimal time for the problem. Another generalization uses less space, but more time: spaceO(n ^1/logn), for 2log₂ n, and timen ^O(). For constant the time remains polynomial.     

Variable Precision Algorithm for the Numerical Computation of the Fermi–Dirac Function ℱj(x) of Order j=−3/2

The purpose of this technical note is to present a piecewise Chebyshev expansion for the numerical computation of the Fermi–Dirac function _–3/2(x), –<x<. The variable precision algorithm we given automatically adjusts the degrees of the Chebyshev expansions so that _–3/2(x) can be efficiently computed to d significant decimal digits of accuracy, for a user specified value of d in the range 1d15.     

Constructing H4, a Fast Depth-Size Optimal Parallel Prefix Circuit

Given n values x ₁, x ₂,...,x _n and an associative binary operation , the prefix problem is to compute x ₁x ₂x _i, 1in. Prefix circuits are combinational circuits for solving the prefix problem. For any n-input prefix circuit D with depth d and size s, if d+s=2n–2, then D is depth-size optimal. In general, a prefix circuit with a small depth is faster than one with a large depth. For prefix circuits with the same depth, a prefix circuit with a smaller fan-out occupies less area and is faster in VLSI implementation. This paper is on constructing parallel prefix circuits that are depth-size optimal with small depth and small fan-out. We construct a depth-size optimal prefix circuit H4 with fan-out 4. It has the smallest depth among all known depth-size optimal prefix circuits with a constant fan-out; furthermore, when n136, its depth is less than, or equal to, those of all known depth-size optimal prefix circuits with unlimited fan-out. A size lower bound of prefix circuits is also derived. Some properties related to depth-size optimality and size optimality are introduced; they are used to prove that H4 is depth-size optimal.     

Unextendible Product Bases and Locally Unconvertible Bound Entangled States

Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs S and T the associated bound entangled states _S and _T cannot be converted to each other by LOCC, unless S and T coincide up to local unitaries. More specifically, there exists a finite precision (S,T) > 0 such that for any LOCC protocol mapping _S into a probabilistic ensemble (p, ), the fidelity between _T and any possible final state satisfies F(_T, ) = 1 - (S,T).PACS: 03.65.Bz; 03.67.-a; 89.70+c.     

Systems of Strings with High Mutual Complexity

Consider a binary string x ₀ of Kolmogorov complexity K(x ₀) n. The question is whether there exist two strings x ₁ and x ₂ such that the approximate equalities K(x _i x _j ) n and K(x _i x _j , x _k ) n hold for all 0 i, j, k 2, i j k, i k. We prove that the answer is positive if we require the equalities to hold up to an additive term O(log K(x ₀)). It becomes negative in the case of better accuracy, namely, O(log n).     

To the Metrical Rigidity of Binary Codes

A code C in the n-dimensional metric space E ⁿ over GF(2) is called metrically rigid if each isometry I : C E ⁿ can be extended to an isometry of the whole space E ⁿ . For n large enough, metrical rigidity of any length-n binary code that contains a 2-(n, k, )-design is proved. The class of such codes includes, for instance, all families of uniformly packed codes of large enough lengths that satisfy the condition d – 2, where d is the code distance and is the covering radius.     

Approximate minimum weight matching on points ink-dimensional space

We study the problem of finding a minimum weight complete matching in the complete graph on a set V ofn points ink-dimensional space. The points are the vertices of the graph and the weight of an edge between any two points is the distance between the points under someL _q,-metric. We give anO((2c _q )^{1.5k –1.5k} ((n, n))^0.5 n ^1.5(logn)^2.5) algorithm for finding an almost minimum weight complete matching in such a graph, wherec _q =6k ^1/q for theL _q -metric, is the inverse Ackermann function, and 1. The weight of the complete matching obtained by our algorithm is guaranteed to be at most (1 + ) times the weight of a minimum weight complete matching.This research was supported by a fellowship from the Shell Foundation.     

Gradually intractable problems and nondeterministic log-space lower bounds

The paper places five different problems (thek-pebble game problem, two problems aboutk finite automata, the reachability problem for Petri nets withk tokens, and the teachability problem for graphs whose k-dimensional edge sets are described by Cartesian products ofk factors) into the hierarchyNL _k of problems solvable by nondeterministic Turing machines ink-log₂ n space (and binary tape alphabet, to avoid tape speed-up). The results, when combined with the conjecture thatNL _i contains problems that requireO(n ^k ) deterministic time, show that these problems, while inP for every fixed value ofk, have polynomial deterministic time complexities with the degree of the polynomial growing linearly with the parameterk, and hence are, in this sense, gradually intractable.