doc: document simple hull

diff --git a/doc/implementation.tex b/doc/implementation.tex
index 0257d06..663f46f 100644 (file)
--- a/doc/implementation.tex
+++ b/doc/implementation.tex
@@ -102,6 +102,55 @@ $$
 $$
 \end{definition}
 
+\section{Simple Hull}\label{s:simple hull}
+
+It is sometimes useful to have a single
+basic set or basic relation that contains a given set or relation.
+For rational sets, the obvious choice would be to compute the
+(rational) convex hull.  For integer sets, the obvious choice
+would be the integer hull.
+However, {\tt isl} currently does not support an integer hull operation
+and even if it did, it would be fairly expensive to compute.
+The convex hull operation is supported, but it is also fairly
+expensive to compute given only an implicit representation.
+
+Usually, it is not required to compute the exact integer hull,
+and an overapproximation of this hull is sufficient.
+The ``simple hull'' of a set is such an overapproximation
+and it is defined as the (inclusion-wise) smallest basic set
+that is described by constraints that are translates of
+the constraints in the input set.
+This means that the simple hull is relatively cheap to compute
+and that the number of constraints in the simple hull is no
+larger than the number of constraints in the input.
+\begin{definition}[Simple Hull of a Set]
+The {\em simple hull} of a set
+$S = \bigcup_{1 \le i \le v} S_i$, with
+$$
+S : \Z^n \to 2^{\Z^d} : \vec s \mapsto
+S(\vec s) =
+\left\{\, \vec x \in \Z^d \mid \exists \vec z \in \Z^e :
+\bigvee_{1 \le i \le v}
+A_i \vec x + B_i \vec s + D_i \vec z + \vec c_i \geq \vec 0 \,\right\}
+$$
+is the set
+$$
+H : \Z^n \to 2^{\Z^d} : \vec s \mapsto
+S(\vec s) =
+\left\{\, \vec x \in \Z^d \mid \exists \vec z \in \Z^e :
+\bigwedge_{1 \le i \le v}
+A_i \vec x + B_i \vec s + D_i \vec z + \vec c_i + \vec K_i \geq \vec 0
+\,\right\}
+,
+$$
+with $\vec K_i$ the (component-wise) smallest non-negative integer vectors
+such that $S \subseteq H$.
+\end{definition}
+The $\vec K_i$ can be obtained by solving a number of
+LP problems, one for each element of each $\vec K_i$.
+If any LP problem is unbounded, then the corresponding constraint
+is dropped.
+
 \section{Coalescing}\label{s:coalescing}
 
 See \shortciteN{Verdoolaege2009isl}, for now.

diff --git a/doc/user.pod b/doc/user.pod
index 0fab92b..e2f496d 100644 (file)
--- a/doc/user.pod
+++ b/doc/user.pod
@@ -893,6 +893,24 @@ basic set or relation.
 If the input set or relation has any existentially quantified
 variables, then the result of these operations is currently undefined.
 
+=item * Simple hull
+
+       __isl_give isl_basic_set *isl_set_simple_hull(
+               __isl_take isl_set *set);
+       __isl_give isl_basic_map *isl_map_simple_hull(
+               __isl_take isl_map *map);
+
+These functions compute a single basic set or relation
+that contains the whole input set or relation.
+In particular, the output is described by translates
+of the constraints describing the basic sets or relations in the input.
+
+=begin latex
+
+(See \autoref{s:simple hull}.)
+
+=end latex
+
 =item * Affine hull
 
        __isl_give isl_basic_set *isl_basic_set_affine_hull(

diff --git a/include/isl_map.h b/include/isl_map.h
index ac146ae..69c2098 100644 (file)
--- a/include/isl_map.h
+++ b/include/isl_map.h
@@ -149,6 +149,7 @@ __isl_give isl_basic_map *isl_basic_map_universe(__isl_take isl_dim *dim);
 __isl_give isl_basic_map *isl_basic_map_universe_like(
                __isl_keep isl_basic_map *bmap);
 struct isl_basic_map *isl_basic_map_convex_hull(struct isl_basic_map *bmap);
+__isl_give isl_basic_map *isl_map_simple_hull(__isl_take isl_map *map);
 
 __isl_give isl_basic_map *isl_basic_map_intersect_domain(
                __isl_take isl_basic_map *bmap,

diff --git a/include/isl_set.h b/include/isl_set.h
index f6ebe4b..d76bb52 100644 (file)
--- a/include/isl_set.h
+++ b/include/isl_set.h
@@ -211,7 +211,7 @@ __isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set);
 __isl_give isl_set *isl_set_detect_equalities(__isl_take isl_set *set);
 __isl_give isl_basic_set *isl_set_affine_hull(__isl_take isl_set *set);
 __isl_give isl_basic_set *isl_set_convex_hull(__isl_take isl_set *set);
-struct isl_basic_set *isl_set_simple_hull(struct isl_set *set);
+__isl_give isl_basic_set *isl_set_simple_hull(__isl_take isl_set *set);
 struct isl_basic_set *isl_set_bounded_simple_hull(struct isl_set *set);
 __isl_give isl_set *isl_set_recession_cone(__isl_take isl_set *set);