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We define a new model for algorithms to reach Byzantine Agreement. It allows one to measure the complexity more accurately, to differentiate between processor faults, and to include communication link failures. A deterministic algorithm is presented that exhibits early stopping by phase 2f + 3 in the worst case, where f is the actual number of faults, under less stringent conditions than the ones of previous algorithms. Its average performance can also easily be analysed making realistic assumptions on random distribution of faults. We show that it stops with high probability after a small number of phases.

Following a suggestion of Pratt, we consider propositional dynamic logic in which programs are nondeterministic finite automata over atomic programs and tests (i.e., flowcharts), rather than regular expressions. While the resulting version of PDL, call it APDL, is clearly equivalent in expressive power to PDL, it is also (in the worst case) exponentially more succinct. In particular, deciding its validity problem by reducing it to that of PDL leads to a double exponential time procedure, although PDL itself is decidable in exponential time. We present an elementary combined proof of the completeness of a simple axiom system for APDL and decidability of the validity problem in exponential time. The results are thus stronger than those for PDL, since PDL can be encoded in APDL with no additional cost, and the proofs simpler, since induction on the structure of programs is virtually eliminated. Our axiom system for APDL relates to the PDL system just as Floyd's proof method for partial correctness relates to Hoare's.

Let P₁,…, P_k be pairwise non-intersecting simple polygons with a total of n vertices and s start vertices. A start vertex, in general, is a vertex both of which neighbors have larger x coordinate. We present an algorithm for triangulating P₁,…, P_k in time O(n + s log s). s may be viewed as a measure of non-convexity. In particular, s is always bounded by the number of concave angles + 1, and is usually much smaller. We also describe two new applications of triangulation. Given a triangulation of the plane with respect to a set of k pairwise non-intersecting simple polygons, then the intersection of this set with a convex polygon Q can be computed in time linear with respect to the combined number of vertices of the k + 1 polygons. Such a result had only be known for two convex polygons. The other application improves the bound on the number of convex parts into which a polygon can be decomposed.

The class NC consists of problems solvable very fast (in time polynomial in log n) in parallel with a feasible (polynomial) number of processors. Many natural problems in NC are known; in this paper an attempt is made to identify important subclasses of NC and give interesting examples in each subclass. The notion of NC¹-reducibility is introduced and used throughout (problem R is NC¹-reducible to problem S if R can be solved with uniform log-depth circuits using oracles for S). Problems complete with respect to this reducibility are given for many of the subclasses of NC. A general technique, the “parallel greedy algorithm,” is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in NC² (solvable by uniform Boolean circuits of depth O(log² n) and polynomial size). The class LOGCFL is given a new characterization in terms of circuit families. The class DET of problems reducible to integer determinants is defined and many examples given. A new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph. This paper is a revised version of S. A. Cook, (1983, in “Proceedings 1983 Intl. Found. Comut. Sci. Conf.,” Lecture Notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York).

The complexity of priority queue operations is analyzed with respect to the cell probe computational model of A. Yao (J. Assoc. Comput. Mach. 28, No. 3 (1981), 615–628). A method utilizing families of hash functions is developed which permits priority queue operations to be implemented in constant worst-case time provided that a size constraint is satisfied. The minimum necessary size of a family of hash functions for computing the rank function is estimated and contrasted with the minimum size required for perfect hashing.

We present an algorithm which given an arbitrary A-free context-free grammar produces an equivalent context-free grammar in 2 Greibach normal form. The upper bound on the size of the resulting grammar in terms of the size of the initially given grammar is given. Our algorithm consists of an elementary construction, while the upper bound on the size of the resulting grammar is not bigger than the bounds known for other algorithms for converting context-free grammars into equivalent context-free grammars in Greibach normal form.

Decomposing a polygon into simple shapes is a basic problem in computational geometry, with applications in pattern recognition and integrated circuit manufacture. Here we examine the special case of covering a rectilinear polygon (or polyomino) with the minimum number of rectangles, with overlapping allowed. The problem is NP-hard. However, we give here an O(v²) algorithm for constructing a minimum rectangle cover, when the polygon is vertically convex. (Here v is the number of vertices.) The problem is first reduced to a 1-dimensional interval “basis” problem. In showing our algorithm produces an optimal cover we give a new proof of a minimum basis-maximum independent set duality theorem first proved by E. Györi (J. Combin Theory Ser. B 37, No. 1, 1–9).

A set of polygons is called c-oriented if the edges of all polygons are oriented in a constant number of previously defined directions. The intersection searching problem is studied for such objects, namely: Given a set of c-oriented polygons P and a c-oriented query polygon q, find all polygons in P that intersect q. It is shown that this problem can be solved in O(log² n + t) time with O(n log n) space and O(n log² n) preprocessing, where n is the cardinality of P and t the number of answers to a query. Furthermore, the solution is extended to the cases in which P is a semidynamic or dynamic set of polygons. Whereas planar intersection searching can be carried out more efficiently for orthogonal objects (e.g., rectangles) it is expensive for arbitrary polygons. This suggests that the c-oriented solution be used in appropriate areas of application, for instance, in VLSI-design.