Analysis Of Expected Time Research Articles

Optimization-Time Analysis for Cybersecurity

A mathematical framework to reason about time resilience in cybersecurity is here introduced. We first consider an attacker who is able to mount several multi-stage attacks on the organization: the defender’s objective is to select an optimal portfolio of security controls, within a given budget, to withstand the highest number of attacks. The mathematical model is a Markov chain with an initial state called the safe state, intermediate states for all possible attacks (each attack state denoting a probabilistic attack graph), and a sink state denoting a successful attack. The overall defence problem is formulated as a bi-level multi-objective optimization, i.e., the defender selects an optimal portfolio of security controls to mitigate an optimal attacker. In order to determine the probability of success of an attack two cases will be considered: (a) the expected probability of success and (b) the highest probability of success. We refer to these two cases as expected-time analysis and worst-case time analysis, respectively. To solve precisely these bi-level optimizations strong duality and Mixed Integer Linear Programming are used. We then extend the framework to investigate resilience in terms of the total duration of the attacks; variations of the previous optimizations are presented to this purpose. Finally numerical evaluations are provided to compare the results obtained from the expected-time analysis and the worst-case time analysis.

Computing inversion pair cardinality through partition-based sorting

In this paper, we introduce a new randomized, partition-based algorithm for the problem of computing the number of inversion pairs in an unsorted array of n numbers. The algorithm runs in expected time O(n · log n) and uses O(n) extra space. The expected time analysis of the new algorithm is different from the analyses existing in the literature, in that it explicitly uses inversion pairs. The problem of determining the inversion pair cardinality of an array finds applications in a number of design domains, including but not limited to real-time scheduling and operations research. At the heart of our algorithm is a new inversion pair conserving partition procedure that is different from existing partition procedures such as Hoare-partition and Lomuto-partition. Although the algorithm is not fully adaptive, we believe that it is the first step towards the design of an adaptive, partition-based sorting algorithm whose running time is proportional to the number of inversion pairs in the input.

Expected time analysis for Delaunay point location

We consider point location in Delaunay triangulations with the aid of simple data structures. In particular, we analyze methods in which a simple data structure is used to first locate a point close to the query point. For points uniformly distributed on the unit square, we show that the expected point location complexities are Θ( n ) for the Green–Sibson rectilinear search, Θ( n 1/3) for Jump and Walk, Θ( n 1/4) for BinSearch and Walk (which uses a 1-dimensional search tree), Θ( n 0.056…) for search based on a random 2-d tree, and Θ(log n) for search aided by a 2-d median tree.

Analysis of range search for random k-d trees

We analyze the expected time complexity of range searching with k-d trees in all dimensions when the data points are uniformly distributed in the unit hypercube. The partial match results of Flajolet and Puech are reproved using elementary probabilistic methods. In addition, we give asymptotic expected time analysis for orthogonal and convex range search, as well as nearest neighbor search. We disprove a conjecture by Bentley that nearest neighbor search for a given random point in the k-d tree can be done in O(1) expected time.

Records, the maximal layer, and uniform distributions in monotone sets

Consider a nondecreasing nonnegative integrable function f on [0, 1]. Draw an independent sample ( X 1, Y 1), …, ( X n , Y n ) of size n from the uniform distribution under f, and let N n be the number of records in the sample, where ( X i , Y i ) is a record when, for all j ≠ i, either X j > X i or Y j < Y i . We study the dependence upon f of the constant C in the asymptotic formula E N n ∼ C√ n, and show that whenever ∫ 0 1 f′ > 0, N n /E N n → 1 in probability. The results are related to the expected time analysis of algorithms for finding the collection of all records (i.e., the maximal layer).

Expected time analysis of interpolation merge — a simple new merging algorithm

Binary merge is the best general-purpose merging algorithm known to date. Binary merge consists of two components: a first component in which an array index is incremented by a number nearly equal to the ratio of the sizes of the two arrays being merged followed by a second component, binary search. In this paper we formulate a simple algorithm called interpolation merge, where the binary search component is replaced with linear search, and analyze its expected behavior over data drawn from a uniform distribution. Our results, both theoretical and experimental, indicate a constant factor (≈0.75) speed-up over straight two-way merge. Further, our analysis of interpolation merge, which uses a mechanism of incremental indexing similar to that in binary merge, will hopefully lead to a better understanding of the latter algorithm. Currently, no significant results are known about the expected behavior of binary merge over data drawn from any standard probability distribution.