We consider problems of the following type: given a graph G, how many edges are needed in worst case for sparse subgraph H that approximately preserves distances between set node pairs P? Examples include pairwise spanners, distance preservers, reachability etc. There has been trend area simple constructions based on hitting technique, followed by somewhat more complicated improve over bounds o...

The notion of a probabilistic metric  space  corresponds to thesituations when we do not know exactly the  distance.  Probabilistic Metric space was  introduced by Karl Menger. Alsina, Schweizer and Sklar gave a general definition of  probabilistic normed space based on the definition of Menger [1]. In this note we study the PN spaces which are  topological vector spaces and the open mapping an...