Topic
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.
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01 Jan 2001
TL;DR: It is proved that the all-pairs shortest-paths problem on n-vertex networks can be solved in time O(n2 logn) with high probability with respect to various probability distributions on the set of inputs.
Abstract: We study both upper and lower bounds on the average-case complexity of shortestpaths algorithms. It is proved that the all-pairs shortest-paths problem on n-vertex networks can be solved in time O(n2 logn) with high probability with respect to various probability distributions on the set of inputs. Our results include the first theoretical analysis of the average behavior of shortest-paths algorithms with respect to the vertex-potential model, a family of probability distributions on complete networks with arbitrary real arc costs but without negative cycles. We also generalize earlier work with respect to the common uniform model, and we correct the analysis of an algorithm with respect to the endpoint-independent model. For the algorithm that solves the all-pairs shortest-paths problem on networks generated according to the vertex-potential model, a key ingredient is an algorithm that solves the single-source shortest-paths problem on such networks in time O(n) with high probability. All algorithms mentioned exploit that with high probability, the single-source shortest-paths problem can be solved correctly by considering only a rather sparse subset of the arc set. We prove a lower bound indicating the limitations of this approach. In a fairly general probabilistic model, any algorithm solving the single-source shortest-paths problem has to inspect Ω(n logn) arcs with high probability. Kurzzusammenfassung. In dieser Arbeit werden sowohl obere als auch untere Schranken für die average-case-Komplexität von Kürzeste-Wege-Algorithmen untersucht. Wir beweisen für verschiedene Wahrscheinlichkeitsverteilungen auf Netzwerken mit n Knoten, dass das all-pairs shortestpaths problem mit hoher Wahrscheinlichkeit in Zeit O(n2 logn) gelöst werden kann. Insbesondere können wir dieses Laufzeit für einen Algorithmus beweisen, dessen Eingaben gemäß des vertexpotential model erzeugt werden, einer Familie von Wahrscheinlichkeitsverteilungen auf vollständigen Netzwerke mit reellen Kantenkosten, die jedoch keine negative Kreise besitzen. Theoretische Ergebnisse für dieses Eingabemodell waren bislang nicht bekannt. Wir verallgemeinern außerdem frühere Arbeit bezüglich des uniform model und korrigieren die Laufzeit-Analyse eines Algorithmus bezüglich des endpoint-independent model. Der Algorithmus, der das all-pairs shortest-paths problem auf Netzwerken löst, die gemäß des vertex-potential model erzeugt werden, baut entscheidend darauf auf, dass wir auch einen Algorithmus entwickeln, der das single-source shortest-paths problem auf solchen Netzwerken mit hoher Wahrscheinlichkeit in Zeit O(n) löst. Alle bislang erwähnten Algorithmen nutzen aus, dass das single-source shortest-paths problem auch dann mit hoher Wahrscheinlichkeit korrekt gelöst werden kann, wenn wir nur einen Teil der Kantenmenge betrachten. Wir beweisen eine untere Schranke, die die Grenzen dieses Reduktionsansatzes belegt. Auf einer Klasse von Netzwerken mit ganzzahligen Kantenkosten muss jeder Algorithmus mit hoher Wahrscheinlichkeit Ω(n logn) Kanten inspizieren, um das single-source shortest-paths problem zu lösen.
7 citations
TL;DR: A simple mathematical algorithm to determine the complexity of software that includes control flow and data flow is described and it is shown that the determinant value fluctuates randomly whereas the new sum of product is a monotonic function that increases systematically with increasing complexity.
Abstract: This paper describes a simple mathematical algorithm to determine the complexity of software that includes control flow and data flow. Two techniques are analyzed using examples to determine the overall complexity. One of them computes the determinant of a square matrix represented as an N2 chart. The other technique that is new and proposed in this paper computes the sum of products of control flow and data flow. It is shown that the determinant value fluctuates randomly whereas the new sum of product is a monotonic function that increases systematically with increasing complexity. This complexity number can be used to determine the amount of effort (cost and time) required for development and verification of software and whether or not the software can be deployed to perform safety-critical functions with high assurance.
7 citations
TL;DR: This paper introduces wire length as a salient complexity measure for analyzing the circuit complexity of sensory processing in biological neural systems and presents new circuit design strategies for these benchmark problems that can be implemented within realistic complexity bounds.
7 citations
TL;DR: In this article, the authors studied the computational complexity of a matrix inversion formula with the intention of showing its improvement over the naive method of computing the inverse separately, and they did not claim that the matrix-inversion formula is their discovery.
Abstract: The purpose of our two-page communication was to study the computational complexity of a matrix inversion formula with the intention of showing its improvement over the naive method of computing the inverse separately. We did not intend to claim that the matrix inversion formula is our discovery. However, it is true that this point was not made clear in our short paper. >
7 citations