M. Reingold. On the optimality of some set algorithms. Journal of the ACM, 19:649-- 659, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with the speed of f(n) The required time is expressed by the number of constant time operations needed to complete the algorithm. Before carrying out such an analysis, we have to define what kind of operations can be used in the algorithm. In the generally accepted algebraic decision tree model [Rei72], the following operation type is allowed: an algebraic function is evaluated and according to its result a decision is made to select the next operation. In this framework, the computation can be visualised by a binary tree that is divided into two branches by every single decision. In the ....

M. Reingold. On the optimality of some set algorithms. Journal of the ACM, 19:649-- 659, 1972.

....point lies to the right or the left of a line given by two other points. For such problems, comparison model lower bounds are not interesting (because a lower bound of 1 trivially holds in that model ) Therefore the majority of the lower bounds are proved in the algebraic computation tree model [Rei72, Rab72, BO83, PS85] In [PS85] three fundamental prototype problems are identified for this model; Sorting, Element Distinctness and Extreme Points (the problem of computing if a set of points in the plane all are vertices of their convex hull) These problems all have complexity Omega Gamma N ....

E. M. Reingold, On the optimality of some set algorithms, J. ACM 19 (1972), 649--659.

....is absolutely necessary. Algorithms that depend on additional operations are not really comparable to other algorithms that use only algebraic operations. Furthermore, if we restrict our algorithms to the algebraic model, we often can prove nontrivial lower bounds using the algebraic decision tree [13, 77, 34, 69, 67]. Formally, an algebraic decision tree (see [13] is a binary tree T with a function that assigns 6 ffl to any leaf vertex v an output Y ES or NO; ffl to any vertex v with exactly one child an operational instruction of the form f v : f v 1 ffi f v 2 or f v : q f v 1 , where v i is an ....

E. M. Reingold. On the optimality of some set algorithms. Journal of the ACM, 19:649--659, 1972.

....point lies to the right or the left of a line given by two other points. For such problems, comparison model lower bounds are not interesting (because a lower bound of 1 trivially holds in that model ) Therefore the majority of the lower bounds are proved in the algebraic computation tree model [Rei72, Rab72, BO83, PS85] In [PS85] three fundamental prototype problems are identified for this model; Sorting, Element Distinctness and Extreme Points (the problem of computing if a set of points in the plane all are vertices of their convex hull) These problems all have complexity Omega Gamma N ....

E. M. Reingold, On the optimality of some set algorithms, J. ACM 19 (1972), 649--659.

.... possible to use standard methods from sorting algorithms to set up 135 in time O(nlog n) a data structure which will permit this test to be carried out in time O(log n) giving an algorithm with total cost O(nlog n) We note that set containment is known to require at least nlog n comparisons [107], so this bound is in some sense optimal. By Lemma 5.4.1 a database entails a conjunctive query 8 just in case it entails each path of 8. Combining this with Lemma 5.4.2, we obtain the following characterization of entailment of conjunctive queries: D entails 8 just in case every path of 8 is a ....

E.M. Reingold. On the optimality of some set algorithms. Journal of the ACM, 19:649--659, 1972.

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