Date 1214 Oct. 1987
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[Front cover]
Publication Year: 1987 , Page(s): C1 
Table of contents
Publication Year: 1987 , Page(s): xi  xiv 

Machtey Award
Publication Year: 1987 , Page(s): v 
Polytope range searching and integral geometry
Publication Year: 1987 , Page(s): 1  10First Page of the ArticleView full abstract» 
An output sensitive algorithm for computing visibility graphs
Publication Year: 1987 , Page(s): 11  19
Cited by: Papers (14)  Patents (1)The visibility graph of a set of nonintersecting polygonal obstacles in the plane is an undirected graph whose vertices are the vertices of the obstacles and whose edges are pairs of vertices (u, v) such that the open line segment between u and v does not intersect any of the obstacles. The visibility graph is an important combinatorial structure in computational geometry and is used in applications such as solving visibility problems and computing shortest paths. An algorithm is presented that computes the visibility graph of s set of obstacles in time O(E + n log n), where E is the number of edges in the visibility graph and n is the total number of vertices in all the obstacles. View full abstract»

Delaunay graphs are almost as good as complete graphs
Publication Year: 1987 , Page(s): 20  26
Cited by: Papers (1)Let S be any set of N points in the plane and let DT(S) be the graph of the Delaunay triangulation of S. For all points a and b of S, let d(a, b) be the Euclidean distance from a to b and let DT(a, b) be the length of the shortest path in DT(S) from a to b. We show that there is a constant c(≤ 1+√5/2 π ≈ 5.08) independent of S and N such that DT(a, b)/d(a, b) ≪ c. View full abstract»

On the lower envelope of bivariate functions and its applications
Publication Year: 1987 , Page(s): 27  37
Cited by: Papers (1)We consider the problem of obtaining sharp (nearly quadratic) bounds for the combinatorial complexity of the lower envelope (i.e. pointwise minimum) of a collection of n bivariate (or generally multivariate) continuous and "simple" functions, and of designing efficient algorithms for the calculation of this envelope. This problem generalizes the wellstudied univariate case (whose analysis is based on the theory of DavenportSchinzel sequences), but appears to be much more difficult and still largely unsolved. It is a central problem that arises in many areas in computational and combinatorial geometry, and has numerous applications including generalized planar Voronoi diagrams, hidden surface elimination for intersecting surfaces, purely translational motion planning, finding common transversals of polyhedra, and more. In this abstract we provide several partial solutions and generalizations of this problem, and apply them to the problems mentioned above. The most significant of our results is that the lower envelope of n triangles in three dimensions has combinatorial complexity at most O(n2α(n)) (where α(n) is the extremely slowly growing inverse of Ackermann's function), that this bound is tight in the worst case, and that this envelope can be calculated in time O(n2α(n)). View full abstract»

A new algebraic method for robot motion planning and real geometry
Publication Year: 1987 , Page(s): 39  48
Cited by: Papers (11)We present an algorithm which solves the findpath or generalized movers' problem in single exponential sequential time. This is the first algorithm for the problem whose sequential time bound is less than double exponential. In fact, the combinatorial exponent of the algorithm is equal to the number of degrees of freedom, making it worstcase optimal, and equaling or improving the time bounds of many special purpose algorithms. The algorithm accepts a formula for a semialgebraic set S describing the set of free configurations and produces a onedimensional skeleton or "roadmap" of the set, which is connected within each connected component of S. Additional points may be linked to the roadmap in linear time. Our method draws from results of singularity theory, and in particular makes use of the notion of stratified sets as an efficient alternative to cell decomposition. We introduce an algebraic tool called the multivariate resultant which gives a necessary and sufficient condition for a system of homogeneous polynomials to have a solution, and show that it can be computed in polynomial parallel time. Among the consequences of this result are new methods for quantifier elimination and an improved gap theorem for the absolute value of roots of a system of polynomials. View full abstract»

New lower bound techniques for robot motion planning problems
Publication Year: 1987 , Page(s): 49  60
Cited by: Papers (66)We present new techniques for establishing lower bounds in robot motion planning problems. Our scheme is based on path encoding and uses homotopy equivalence classes of paths to encode state. We first apply the method to the shortest path problem in 3 dimensions. The problem is to find the shortest path under an Lp metric (e.g. a euclidean metric) between two points amid polyhedral obstacles. Although this problem has been extensively studied, there were no previously known lower bounds. We show that there may be exponentially many shortest path classes in singlesource multipledestination problems, and that the singlesource singledestination problem is NPhard. We use a similar proof technique to show that two dimensional dynamic motion planning with bounded velocity is NPhard. Finally we extend the technique to compliant motion planning with uncertainty in control. Specifically, we consider a point in 3 dimensions which is commanded to move in a straight line, but whose actual motion may differ from the commanded motion, possibly involving sliding against obstacles. Given that the point initially lies in some start region, the problem of finding a sequence of commanded velocities which is guaranteed to move the point to the goal is shown to be nondeterministic exponential time hard, making it the first provably intractable problem in robotics. View full abstract»

Learning onecounter languages in polynomial time
Publication Year: 1987 , Page(s): 61  67
Cited by: Papers (1)We demonstrate that the class of languages accepted by deterministic onecounter machines, or DOCAs (a natural subset of the contextfree languages), is learnable in polynomial time. Our learning protocol is based upon Angluin's concept of a "minimally adequate teacher" who can answer membership queries about a concept and provide counterexamples to incorrect hypothesized concepts. We also demonstrate that the problem of testing DOCAs for equivalence may be solved in polynomial time, answering a question posed by Valiant and Paterson. View full abstract»

Learning quickly when irrelevant attributes abound: A new linearthreshold algorithm
Publication Year: 1987 , Page(s): 68  77
Cited by: Papers (7)Valiant and others have studied the problem of learning various classes of Boolean functions from examples. Here we discuss online learning of these functions. In online learning, the learner responds to each example according to a current hypothesis. Then the learner updates the hypothesis, if necessary, based on the correct classification of the example. One natural measure of the quality of learning in the online setting is the number of mistakes the learner makes. For suitable classes of functions, online learning algorithms are available that make a bounded number of mistakes, with the bound independent of the number of examples seen by the learner. We present one such algorithm, which learns disjunctive Boolean functions, and variants of the algorithm for learning other classes of Boolean functions. The algorithm can be expressed as a linearthreshold algorithm. A primary advantage of this algorithm is that the number of mistakes that it makes is relatively little affected by the presence of large numbers of irrelevant attributes in the examples; we show that the number of mistakes grows only logarithmically with the number of irrelevant attributes. At the same time, the algorithm is computationaUy time and space efficient. View full abstract»

Diversitybased inference of finite automata
Publication Year: 1987 , Page(s): 78  87
Cited by: Papers (2)We present a new procedure for inferring the structure of a finitestate automaton (FSA) from its input/output behavior, using access to the automaton to perform experiments. Our procedure uses a new representation for FSA's, based on the notion of equivalence between testa. We call the number of such equivalence classes the diversity of the automaton; the diversity may be as small as the logarithm of the number of states of the automaton. The size of our representation of the FSA, and the running time of our procedure (in some case provably, in others conjecturally) is polynomial in the diversity and ln(1/ε), where ε is a given upper bound on the probability that our procedure returns an incorrect result. (Since our procedure uses randomization to perform experiments, there is a certain controllable chance that it will return an erroneous result.) We also present some evidence for the practical efficiency of our approach. For example, our procedure is able to infer the structure of an automaton based on Rubik's Cube (which has approximately 1019 states) in about 2 minutes on a DEC Micro Vax. This automaton is many orders of magnitude larger than possible with previous techniques, which would require time proportional at least to the number of global states. (Note that in this example, only a small fraction (1014) of the global states were even visited.) View full abstract»

Incomparability in parallel computation
Publication Year: 1987 , Page(s): 89  98
Cited by: Papers (4)We consider the relative power of concurrentwrite PRAMs when the number of processors (and input variables) is fixed at n, and infinite shared memory is allowed. Several different models (COMMON, ARBITRARY, PRIORITY) have been used for algorithm design in the literature; these models differ in their method of writeconflict resolution. Recent work in separating these models ([FRW1,2,3], [LY]) has relied on further restrictions (limiting the size of memory or the power of processors); the only unrestricted results known concern the element distinctness problem ([FMW], [RSSW]). In this paper we contribute further unrestricted results. We consider the COLLISION model, a natural generalization of the Ethernet ([G]). Our main result is a lower bound of Ω(logloglogn) steps on COLLISION for a problem that can be done in O(1) steps on ARBITRARY. We use this result, together with a reduction performed by means of Ramsey's Theorem, to show that the powers of COMMON and COLLISION are incomparable. We also introduce a new and natural model, TOLERANT, and show that it is strictly less powerful than COLLISION and incomparable with COMMON. The proofs involved use combinatorial arguments, including Turán's Theorem for graphs and the ErdösRado intersecting set theorem. View full abstract»

Threshold circuits of bounded depth
Publication Year: 1987 , Page(s): 99  110
Cited by: Papers (28)  Patents (1)We examine a powerful model of parallel computation: polynomial size threshold circuits of bounded depth (the gates compute threshold functions with polynomial weights). Lower bounds are given to separate polynomial size threshold circuits of depth 2 from polynomial size threshold circuits of depth 3, and from probabilistic polynomial size threshold circuits of depth 2. We also consider circuits of unreliable threshold gates, circuits of imprecise threshold gates and threshold quantifiers. View full abstract»

Complete and incomplete randomized NP problems
Publication Year: 1987 , Page(s): 111  117
Cited by: Papers (3)First Page of the ArticleView full abstract» 
Generic oracles and oracle classes
Publication Year: 1987 , Page(s): 118  126
Cited by: Papers (23)In this paper, we examine various complexity issues relative to an oracle for a generic set in order to determine which are the more "natural" conjectures for these issues. Generic oracle results should be viewed as parallels to random oracle results, as in [BG]; the two are in many ways related, but, as we shall exhibit, not equivalent. Looking at computation relative to a generic oracle is in some ways a better reflection of computation without an oracle; for example, whereas adding a random oracle allows a deterministic polynomialtime machine to solve any problem in BPP, adding a generic oracle will not help solve any recursive problem faster than it could be solved without an oracle. Generic sets were first introduced by Cohen as a tool for proving independence results in set theory [Co]. Their recursion theoretic properties have also been explored in depth; for example, see [J] and [Ku2]. Some related work using forcing and/or generic sets as tools in oracle constructions can be found in [Ku3], [Do], [P], and [ASFH]. However, this is to our knowledge the first knowledge the first thorough examination of complexity relative to a generic Oracle. View full abstract»

Functional decomposition of polynomials
Publication Year: 1987 , Page(s): 127  131
Cited by: Papers (1)First Page of the ArticleView full abstract» 
Factoring polynomials over finite fields
Publication Year: 1987 , Page(s): 132  137We propose a new deterministic method of factoring polynomials over finite fields. Assuming the Generalized Riemann Hypothesis (GRH), we obtain, in polynomial time, the factorization of any polynomial with a bounded number of irreducible factors. Other consequences include a polynomial time algorithm to find a nontrivial factor of any completely splitting even degree polynomial when a quadratic nonresidue in the field is given. View full abstract»

Multiplicative complexity of polynomial multiplication over finite fields
Publication Year: 1987 , Page(s): 138  140Let Mq(n) denote the number of multiplications required to compute the coefficients of the product of two polynomials of degree n over a qelement field by means of bilinear algorithms. It is shown that Mq(n) ≥ 3n  o(n). In particular, if q/2 ≪ n ≤ q + 1, we establish the tight bound Mq(n) = 3n + 1  ⌊q/2⌋. The technique we use can be applied to analysis of algorithms for multiplication of polynomials modulo a polynomial as well. View full abstract»

The multiplicative complexity of quadratic Boolean functions
Publication Year: 1987 , Page(s): 141  150Let the multiplicative complexity L(f) of a boolean function f be the minimal number of ∧gates that are sufficient to evaluate f by circuits over the basis ∧,⊕,1. We give a polynomial time algorithm which for quadratic boolean forms f=⊕i≠jaijxixj determines L(f) from the coefficients aij. Two quadratic forms f,g have the same complexity L(f) = L(g) iff they are isomorphic by a linear isomorphism. We also determine the multiplicative complexity of pairs of quadratic boolean forms. We give a geometric interpretation to the complexity L(f1,f2) of pairs of quadratic forms. View full abstract»

Cascading divideandconquer: A technique for designing parallel algorithms
Publication Year: 1987 , Page(s): 151  160
Cited by: Papers (3)We present techniques for parallel divideandconquer, resulting in improved parallel algorithms for a number of problems. The problems for which we give improved algorithms include intersection detection, trapezoidal decomposition (hence, polygon triangulation), and planar point location (hence, Voronoi diagram construction). We also give efficient parallel algorithms for fractional cascading, 3dimensional maxima, 2set dominance counting, and visibility from a point. All of our algorithms run in O(log n) time with either a linear or sublinear number of processors in the CREW PRAM model. View full abstract»

A new parallel algorithm for the maximal independent set problem
Publication Year: 1987 , Page(s): 161  165
Cited by: Papers (2)A new parallel algorithm for the maximal independent set problem (MIS) is constructed. It runs in O(log4 n) time when implemented on a linear number of EREWprocessors. This is the first deterministic algorithm for MIS whose running time is polylogarithmic and whose processortime product is optimal up to a polylogarithmic factor. View full abstract»

The matching problem for bipartite graphs with polynomially bounded permanents is in NC
Publication Year: 1987 , Page(s): 166  172
Cited by: Papers (4)It is shown that the problem of deciding and constructing a perfect matching in bipartite graphs G with the polynomial permanents of their n × n adjacency matrices A (perm(A) = nO(1)) are in the deterministic classes NC2 and NC3, respectively. We further design an NC3 algorithm for the problem of constructing all perfect matchings (enumeration problem) in a graph G with a permanent bounded by O(nk). The basic step was the development of a new symmetric functions method for the decision algorithm and the new parallel technique for the matching enumerator problem. The enumerator algorithm works in O(log3 n) parallel time and O(n3k+5.5 Â¿ log n) processors. In the case of arbitrary bipartite graphs it yields an 'optimal' (up to the log n factor) parallel time algorithm for enumerating all the perfect matchings in a graph. It entails also among other things an efficient NC3algorithm for computing small (polynomially bounded) arithmetic permanents, and a sublinear parallel time algorithm for enumerating all the perfect matchings in graphs with permanents up to 2nε. View full abstract»

Some polynomial and Toeplitz matrix computations
Publication Year: 1987 , Page(s): 173  184
Cited by: Papers (2)First Page of the ArticleView full abstract»