Version 5.0 The LEDA User Manual

• to compute maximum and minimum weighted matchings in bipartite graph,
• to check the optimality of matchings, and
• to scale edge weights, so as to avoid round-off errors in computations with the
number type double.
All functions for computing maximum or minimum weighted matchings provide a proof
of optimality in the form of a potential function pot; see the chapter on bipartite weighted
matchings of the LEDA book for a discussion of potential functions.
The functions in this section are template functions. The template parameter NT can be
instantiated with any number type. In order to use the template version of the function
the appropriate .h-file must be included.
#include
There are pre-instantiations for the number types int and double.The pre-instantiated
versions have the same function names except for the suffix _T. In order to use them
either
#include
or
#include
has to be included (the latter file includes the former). The connection between template
functions and pre-instantiated functions is discussed in detail in the section “Templates
for Network Algorithms” of the LEDA book. The function names of the pre-instantiated
versions and the template versions only differ by an additional suffix _T in the names of
the latter ones.
Special care should be taken when using the template functions with a number type NT
that can incur rounding error, e.g., the type double. The section “Algorithms on Weighted
Graphs and Arithmetic Demand” of the LEDA book contains a general discussion of this
issue. The template functions are only guaranteed to perform correctly if all arithmetic
performed is without rounding error. This is the case if all numerical values in the input
are integers (albeit stored as a number of type NT ) and if none of the intermediate
results exceeds the maximal integer representable by the number type (2 53 − 1 in the
case of doubles). All intermediate results are sums and differences of input values, in
particular, the algorithms do not use divisions and multiplications.
The algorithms have the following arithmetic demands. Let C be the maximal absolute
value of any edge cost. If all weights are integral then all intermediate values are bounded
by 3C in the case of maximum weight matchings and by 4nC in the case of the other
matching algorithms. Let f = 3 in the former case and let f = 4n in the latter case.