Average-case complexity

About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.

Algorithm simplification through object orientation

Abstract: The object oriented (O-O) approach is claimed to have a number of advantages. Some support to these claims appeared during an O-O redesign of a legacy CAD system. A surprisingly simple and efficient solution algorithm was discovered for a change propagation problem. The analysis of the case employs the new concepts of implementation and extension complexity, which indicate the amount of code (software costs) required for the implementation and for later extensions. These two complexities are functions of the problem complexity expressed by the number N of object types employed to model the problem domain. Moving from the old system to the new O-O system reduced the implementation complexity from O N ( ) 2 to O N ( ) , the

Average-case analysis of unification algorithms

Abstract: Unification in first-order languages is a central operation in symbolic computation and logic programming. Many unification algorithms have been proposed in the past, however there is no consensus on which algorithm is the best to use in practice. While Paterson and Wegman's linear unification algorithm has the lowest time complexity in the worst case, it requires an important overhead to be implemented. This is true also, although less importantly, for Martelli and Montanari's algorithm [MM82], and Robinson's algorithm [Rob71] is finally retained in many applications despite its exponential worst-case time complexity. There are many explanations for that situation: one important argument is that in practice unification subproblems are not independent, and linear unification algorithms do not perform well on sequences of unify-deunify operations [MU86]. In this paper we present average case complexity theoretic arguments. We first show that the family of unifiable pairs of binary trees is exponentially negligible with respect to the family of arbitrary pairs of binary trees formed over l binary function symbols, c constants and v variables. We analyze the different reasons for failure and get asymptotical and numerical evaluations. We then extend the previous results of [DL89] to these families of trees, we show that a slight modification of Herbrand-Robinson's algorithm has a constant average cost on random pairs of trees. On the other hand, we show that various variants of Martelli and Montanari's algorithm all have a linear average cost on random pairs of trees. The point is that failures by clash are not sufficient to lead to a constant average cost, an efficient occur check (i.e. without a complete traversal of subterms) is necessary. In the last section we extend the results on the probability of the occur check in presence of an unbounded number of variables.

A fast algorithm for determining the linear complexity of a binary sequence with period 2 n p m

Abstract: An efficient algorithm for determining the linear complexity and the minimal polynomial of a binary sequence with period 2 n p m is proposed and proved, where 2 is a primitive root modulo p 2 . The new algorithm generalizes the algorithm for computing the linear complexity of a binary sequence with period 2 n and the algorithm for computing the linear complexity of a binary sequence with period p n , where 2 is a primitive root modulo p 2 .

On the linear complexity of cascaded sequences

Abstract: In the papers [1][2] Kjeldsen derived very interesting properties of cascade couple sequence generators. For applications in ciphering we are interested to know the linear complexity of such sequences. In the following we first consider the examples 1 and 2.

Instruction sequence based non-uniform complexity classes

Abstract: We present an approach to non-uniform complexity in which single-pass instruction sequences play a key part, and answer various questions that arise from this approach. We introduce several kinds of non-uniform complexity classes. One kind includes a counterpart of the well-known non-uniform complexity class P/poly and another kind includes a counterpart of the well-known non-uniform complexity class NP/poly. Moreover, we introduce a general notion of completeness for the non-uniform complexity classes of the latter kind. We also formulate a counterpart of the well-known complexity theoretic conjecture that NP is not included in P/poly. We think that the presented approach opens up an additional way of investigating issues concerning non-uniform complexity.