57 research outputs found
An algorithm to compute the transitive closure, a transitive approximation and a transitive opening of a fuzzy proximity
A method to compute the transitive closure, a transitive opening and a transitive approximation of a reflexive and symmetric fuzzy relation is given. Other previous methods in literature compute just the transitive closure, some transitive approximations or some transitive openings. The proposed algorithm computes the three different similarities that approximate a proximity for the computational cost of computing just one. The shape of the binary partition tree for the three output similarities are the same.Peer ReviewedPostprint (published version
A fast algorithm for All-Pairs-Shortest-Paths suitable for neural networks
Given a directed graph of nodes and edges connecting them, a common problem
is to find the shortest path between any two nodes. Here I show that the
shortest path distances can be found by a simple matrix inversion: If the edges
are given by the adjacency matrix then with a suitably small value of
the shortest path distances are I derive some bounds on useful
for a practical application. Even when the distance function is not globally
accurate across the entire graph, it still works locally to instruct pursuit of
the shortest path. In this mode, it also extends to weighted graphs with
positive edge weights. For a wide range of dense graphs this distance function
is computationally faster than the best available alternative. Finally I show
that this method leads naturally to a neural network solution of the
all-pairs-shortest-path problem.Comment: 11 pages, 4 figures, see also https://github.com/markusmeister/APS
Shortest Distances as Enumeration Problem
We investigate the single source shortest distance (SSSD) and all pairs
shortest distance (APSD) problems as enumeration problems (on unweighted and
integer weighted graphs), meaning that the elements -- where
and are vertices with shortest distance -- are produced and
listed one by one without repetition. The performance is measured in the RAM
model of computation with respect to preprocessing time and delay, i.e., the
maximum time that elapses between two consecutive outputs. This point of view
reveals that specific types of output (e.g., excluding the non-reachable pairs
, or excluding the self-distances ) and the order of
enumeration (e.g., sorted by distance, sorted row-wise with respect to the
distance matrix) have a huge impact on the complexity of APSD while they appear
to have no effect on SSSD.
In particular, we show for APSD that enumeration without output restrictions
is possible with delay in the order of the average degree. Excluding
non-reachable pairs, or requesting the output to be sorted by distance,
increases this delay to the order of the maximum degree. Further, for weighted
graphs, a delay in the order of the average degree is also not possible without
preprocessing or considering self-distances as output. In contrast, for SSSD we
find that a delay in the order of the maximum degree without preprocessing is
attainable and unavoidable for any of these requirements.Comment: Updated version adds the study of space complexit
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