Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (τ -covers). We deal with two types of combinatorial questions which arise from this study. 1. Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. 2. We study the additivit...

We begin with trying to find the shortest path in a directed acyclic graph (DAG). Recall that a DAG has directed edges and contains no cycles. Recall the definition of a topological sort: Definition 1. Let G = (V,E) be a graph. Let v1, . . . ,vn be an ordering of the vertices in V . v1, . . . ,vn are in topologically sorted order if for all edges (vi,v j) ∈ E, i < j. If G is a DAG, then it is a...

Many applications like pointer analysis and incremental compilation require maintaining a topological ordering of the nodes of a directed acyclic graph (DAG) under dynamic updates. All known algorithms for this problem are either only analyzed for worst-case insertion sequences or only evaluated experimentally on random DAGs. We present the first average-case analysis of online topological orde...

In this paper, we consider the exploration of topological environments by a robot with weak sensory capabilities. We assume only that the robot can recognize when it has reached a vertex, and can assign a cyclic ordering to the edges leaving a vertex with reference to the edge it arrived from. Given this limited sensing capability, and without the use of any markers or additional information, w...

Active systems, from bacterial suspensions to cellular monolayers, are continuously driven out of equilibrium by local injection of energy from their constituent elements and exhibit turbulent-like and chaotic patterns. Here we demonstrate both theoretically and through numerical simulations, that the crossover between wet active systems, whose behaviour is dominated by hydrodynamics, and dry a...

The traveling salesman problem with precedence constraints (TSPPC) is one of the most difficult combinatorial optimization problems. In this paper, an efficient genetic algorithm (GA) to solve the TSPPC is presented. The key concept of the proposed GA is a topological sort (TS), which is defined as an ordering of vertices in a directed graph. Also, a new crossover operation is developed for the...

We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in Õ(n) time, independent of the number of edges m inserted. For dense DAGs, this is an improvement over the previously best result of O(min{m 32 log n,m 32 +n log n}) by Katriel et al. We also provide an empirical comparison of our algorithm with ...

Many ranking methods have been proposed so far. However, there is yet no method that can always give a satisfactory solution to every situation; some are counterintuitive, not discriminating; some use only the local information of fuzzy values; some produce different ranking for the same situation. For overcoming the above problems, we propose a new method for ranking fuzzy quantities based on ...

We study the following fundamental realization problem of directed acyclic graphs (dags). Given a sequence S := ( a1 b1 ) , . . . , ( an bn ) with ai, bi ∈ Z+0 , does there exist a dag (no parallel arcs allowed) with labeled vertex set V := {v1, . . . , vn} such that for all vi ∈ V indegree and outdegree of vi match exactly the given numbers ai and bi, respectively? Recently this decision probl...