Description Usage Arguments Details Value References See Also
A binary relation R is transitive, iff for all x, y, z we have xRy and yRz => xRz.
1 2 3 4 5 |
R |
an object coercible to a 0-1 (logical) square matrix, representing a binary relation on a finite set. |
rel_is_transitive
finds out if a given binary relation
is transitive. The algorithm has O(n^3) time complexity,
pessimistically, where
n is the number of rows in R
.
If R
contains missing values behind the diagonal,
the result will be NA
.
The transitive closure of a binary relation R,
determined by rel_closure_transitive
,
is the minimal superset of R such that it is transitive.
Here we use the well-known Warshall algorithm (1962),
which runs in O(n^3) time.
The transitive reduction,
see (Aho et al. 1972), of an acyclic binary relation R,
determined by rel_reduction_transitive
,
is a minimal unique subset R' of R,
such that the transitive closures of R and R' are equal.
The implemented algorithm runs in O(n^3) time.
Note that a transitive reduction of a reflexive relation
is also reflexive. Moreover, some kind of transitive reduction
(not necessarily minimal) is also determined in
rel_reduction_hasse
– it is useful for
drawing Hasse diagrams.
The rel_closure_transitive
and
rel_reduction_transitive
functions
return a logical square matrix. dimnames
of R
are preserved.
On the other hand, rel_is_transitive
returns
a single logical value.
Aho A.V., Garey M.R., Ullman J.D., The Transitive Reduction of a Directed Graph, SIAM Journal on Computing 1(2), 1972, pp. 131-137.
Warshall S., A theorem on Boolean matrices, Journal of the ACM 9(1), 1962, pp. 11-12.
Other binary_relations: check_comonotonicity
,
pord_nd
, pord_spread
,
pord_weakdom
, rel_graph
,
rel_is_antisymmetric
,
rel_is_asymmetric
,
rel_is_cyclic
,
rel_is_irreflexive
,
rel_is_reflexive
,
rel_is_symmetric
,
rel_is_total
,
rel_reduction_hasse
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