#include <iostream>
#include <ostream>
#include <vector>
+#include <cassert>
#include "simplicialcomplex.h"
#include "booleanmatrix.h"
namespace libstick {
-template<class T>
-std::ostream& operator<<(std::ostream &, const std::vector<T> &);
-
-
/** Persistence computers various persistent homology information on a
* simplex_order of a simplicial_complex. */
template<int MAXDIM, class IT, class VT>
class persistence {
public:
- typedef typename simplicial_complex<MAXDIM, IT, VT>::simplex_order simplex_order;
- typedef typename simplex_order::boundary_matrix boundary_matrix;
+ typedef simplicial_complex<MAXDIM, IT, VT> scomplex;
+ typedef typename scomplex::simplex_order simplex_order;
+ typedef boolean_colmatrix<IT> boundary_matrix;
typedef boolean_colmatrix<IT> transformation_matrix;
+ typedef std::vector<IT> lowestones_matrix;
- /** A point in a diagram comprises birth- and death-index. These are the
- * indices of the simplices of 'order' that give birth and death to the
- * class. */
+ /** A point in a persistence diagram comprises birth- and death-index.
+ * These are the indices of the simplices of 'order' that give birth
+ * and death to the class. */
struct diagram_point {
+ /** 'birth'-th simplex in 'order' gave birth to this cycle. */
IT birth;
+ /** 'death'-th simplex in 'order' killed this cycle. If 'death' is
+ * zero, the cycle never dies. Otherwise, 'death' is always larger
+ * than 'birth'. */
IT death;
};
+ /** Save the births and deaths of simplices of a fixed dimension, say p. */
+ struct diagram {
+ /** Saves the sequence of birth/death-points of p-simplices, sorted
+ * by birth-index. If death-index is zero, the corresponding
+ * homology class never dies. */
+ std::vector<diagram_point> births;
+
+
+ /** Gives the p-th Betti number, i.e., the number of immortal
+ * p-dimensional homology classes. */
+ size_t betti() const {
+ size_t b = 0;
+
+ for (unsigned i=0; i < births.size(); ++i)
+ if (births[i].death == 0)
+ b++;
+
+ return b;
+ }
+
+ /** Gives the number of p-dimensional homology classes that are
+ * born by simplex 'from' or earlier and die after simplex 'to' is
+ * inserted. */
+ size_t persistent_betti(IT from, IT to) const {
+ assert(from <= to);
+ size_t b = 0;
+
+ for (unsigned i=0; i < births.size(); ++i) {
+ // All following simplices are born too late.
+ if (births[i].birth > from)
+ break;
+ if (births[i].death > to || births[i].death == 0)
+ b++;
+ }
+
+ return b;
+ }
+ };
public:
/** Create a new peristence object on the given simplex_order */
order(order),
bm(0),
rbm(0),
+ lowestones(0),
tm(0)
- {
+ {
reset();
}
/** Reset all results gained from 'order'. */
void reset() {
done_matrices = false;
+ done_diagrams = false;
}
/** Get boundary matrix 'bm' of 'order', compute reduces boundary
bm = order.get_boundary_matrix();
rbm = bm;
- tm = create_unit_matrix<transformation_matrix>(bm.size());
- reduce_boundary_matrix(rbm, tm);
+ tm = create_unit_matrix<transformation_matrix>(bm.width());
+ lowestones = lowestones_matrix(bm.width());
+
+ // Make every column reduced, i.e., it is a zero-column or contains only one 1.
+ for (unsigned c=0; c < rbm.width(); ++c) {
+ //if (c % 100 == 0)
+ //std::cout << "c = " << c << " (" << (100.0*float(c)/rbm.width()) << " %)" << std::endl;
+ // Reduce as long as we need to reduce
+ while (rbm.get_column(c).size() > 0) {
+ // (r, c) is the lowest one of col
+ const IT r = rbm.get_column(c).back();
+ // This column contains its lowest one on the same row
+ const IT c2 = lowestones[r];
+ assert(c2 < c);
+
+ // There is no valid c2, hence we have completely reduced
+ if (c2 == 0)
+ break;
+ // Reduce col by column c2
+ else {
+ assert(rbm.get(r, c));
+ rbm.add_column(c, rbm.get_column(c2));
+ tm.add_column(c, tm.get_column(c2));
+ assert(!rbm.get(r, c));
+ }
+ }
- assert(rbm == bm * tm);
+ // A lowest one remained, recall it
+ if (rbm.get_column(c).size() > 0) {
+ const IT r = rbm.get_column(c).back();
+ assert(lowestones[r] == 0);
+ assert(r < c);
+ lowestones[r] = c;
+ }
+ }
+ }
+
+ /** Get the lowest-one matrix of 'order'. */
+ const lowestones_matrix& get_lowestones_matrix() const {
+ assert(done_matrices);
+ return lowestones;
}
/** Get the boundary matrix of 'order'. */
- const boundary_matrix& get_boundary_matrix() {
- compute_matrices();
+ const boundary_matrix& get_boundary_matrix() const {
+ assert(done_matrices);
return bm;
}
/** Get the reduced boundary matrix of 'order'. */
- const boundary_matrix& get_reduced_boundary_matrix() {
- compute_matrices();
+ const boundary_matrix& get_reduced_boundary_matrix() const {
+ assert(done_matrices);
return rbm;
}
/** Get the transformation matrix of 'order', which transforms the
* boundary matrix to the reduced boundary matrix, when multiplied from
* the right. */
- const transformation_matrix& get_transformation_matrix() {
- compute_matrices();
+ const transformation_matrix& get_transformation_matrix() const {
+ assert(done_matrices);
return tm;
}
-
/** Print the sequence of births and deaths of cycles (homology classes). */
- void interpret_reduction(std::ostream &os) {
- compute_matrices();
- for (unsigned c=0; c < bm.size(); ++c) {
+ void interpret_reduction(std::ostream &os) const {
+ assert(done_matrices);
+
+ for (unsigned c=0; c < bm.width(); ++c) {
os << c << ". inserting ";
switch (bm.get_column(c).size()) {
os << std::endl;
os << " boundary: " << bm.get_column(c) << std::endl;
- typename boolean_colrowmatrix<IT>::colbase::column_type col = rbm.get_column(c);
+ typename boolean_colmatrix<IT>::column_type col = rbm.get_column(c);
if (col.size() == 0) {
os << " \033[1;32mbirth\033[0;m of a cycle: " << tm.get_column(c) << std::endl;
} else {
}
}
+ /** Get the 'dim'-dimensional peristence diagram. */
+ const diagram& get_persistence_diagram(int dim) const{
+ assert(-1 <= dim);
+ assert(dim <= MAXDIM);
+ assert(done_diagrams);
+
+ return diagrams[dim+1];
+ }
+
+ /** Compute persistence diagrams */
+ void compute_diagrams() {
+ if (done_diagrams)
+ return;
+ done_diagrams = true;
+
+ compute_matrices();
+
+#ifndef NDEBUG
+ std::set<IT> deaths, births;
+#endif
+ //Clear all diagrams
+ for (int d=0; d < MAXDIM + 2; ++d)
+ diagrams[d] = diagram();
+
+ for (unsigned c=0; c < lowestones.size(); ++c) {
+ // A cycle was born
+ if (rbm.get_column(c).size() == 0) {
+ const int dim = order.get_simplex(c).dim;
+
+ // Create a diagram point
+ diagram_point p;
+ p.birth = c;
+ p.death = 0;
+ // If the class actually dies, get the index of the killing simplex
+ if (lowestones[c] > 0)
+ p.death = lowestones[c];
+
+ _get_persistence_diagram(dim).births.push_back(p);
+#ifndef NDEBUG
+ assert(births.count(p.birth) == 0);
+ assert(p.death == 0 || deaths.count(p.death) == 0);
+ births.insert(p.birth);
+ deaths.insert(p.death);
+#endif
+ }
+ }
+#ifndef NDEBUG
+ // Shakespeare principle: A simplex either gives birth or kills, be
+ // or not be, tertium non datur.
+ for (unsigned c=0; c < lowestones.size(); ++c) {
+ // A class died with c
+ if (rbm.get_column(c).size() != 0)
+ assert(c == 0 || deaths.count(c) > 0);
+ else
+ assert(births.count(c) > 0);
+ }
+#endif
+ }
+
+ /** Print birth and death information in the persistence diagrams. */
+ void interpret_persistence(std::ostream &os) const {
+ assert(done_diagrams);
+
+ for (int d=-1; d <= MAXDIM; ++d) {
+ os << "Dimension " << d << std::endl;
+ const diagram &dia = get_persistence_diagram(d);
+ for (unsigned i=0; i < dia.births.size(); ++i) {
+ os << " ";
+ if (dia.births[i].death > 0)
+ os << dia.births[i].birth << "\033[1;31m -> " << dia.births[i].death;
+ else
+ os << "\033[1;32m" << dia.births[i].birth;
+ os << "\033[0;m" << std::endl;
+ }
+ }
+ }
+
private:
+ diagram& _get_persistence_diagram(int dim) {
+ assert(-1 <= dim);
+ assert(dim <= MAXDIM);
+ assert(done_diagrams);
+
+ return diagrams[dim+1];
+ }
+
/** The underlying simplex order of this diagram. */
const simplex_order ℴ
bool done_matrices;
+ bool done_diagrams;
+ /** The boundary matrix of 'order' */
boundary_matrix bm;
+ /** The reduced boundary matrix of 'order' */
boundary_matrix rbm;
+ /** Only the lowest ones per column of the reduced boundary matrix.
+ * lowestones[i] contains the column-index at which the lowest one at
+ * row i appears, and zero if no such column exists. */
+ lowestones_matrix lowestones;
+ /** The transformation matrix that transforms bm into rbm, i.e. rbm = bm * tm. */
transformation_matrix tm;
+
+ /** The container for all p-dimensional diagrams. The p-th diagram is
+ * save at index p+1, with -1 <= p <= MAXDIM. */
+ diagram diagrams[MAXDIM+2];
};
* invariant. Hence, the resulting b is equal to the product of the boundary
* matrix times v. */
template<class IT, class D>
-void reduce_boundary_matrix(boolean_colrowmatrix<IT> &b, boolean_colmatrix_base<IT, D> &v) {
- assert(b.size() == v.width());
+void reduce_boundary_matrix(boolean_colmatrix_base<IT, D> &b, boolean_colmatrix_base<IT, D> &v) {
+ assert(b.width() == v.width());
+
+ // lowestones[i] gives the column whose lowest 1 is at row i. If it contains
+ // zero, there is no such column.
+ std::vector<IT> lowestones(b.width());
// Make every column reduced, i.e., it is a zero-column or contains only one 1.
- for (unsigned c=0; c < b.size(); ++c) {
- const typename boolean_colrowmatrix<IT>::colbase::column_type &col = b.get_column(c);
- if (col.size() == 0)
- continue;
-
- // The column is not a zero-column, take the lowest 1, and reduce the other columns at the row
- IT r = col.back();
- assert(b.get(r, c));
-
- // Get a copy of the r-th row
- typename boolean_colrowmatrix<IT>::rowbase::row_type row(b.get_row(r).size());
- copy(b.get_row(r).begin(), b.get_row(r).end(), row.begin());
- assert(row.size() >= 1);
-
- // Get rid of 1s at that row right of column c.
- typename boolean_colrowmatrix<IT>::row_type::const_iterator it = lower_bound(row.begin(), row.end(), c);
- for(; it != row.end(); ++it) {
- if (*it <= c)
- continue;
-
- assert(b.get(r, *it));
- b.add_column(*it, col);
- v.add_column(*it, v.get_column(c));
- assert(!b.get(r, *it));
+ for (unsigned c=0; c < b.width(); ++c) {
+ //if (c % 100 == 0)
+ //std::cout << "c = " << c << " (" << (100.0*float(c)/b.width()) << " %)" << std::endl;
+ // Reduce as long as we need to reduce
+ while (b.get_column(c).size() > 0) {
+ // (r, c) is the lowest one of col
+ const IT r = b.get_column(c).back();
+ // This column contains its lowest one on the same row
+ const IT c2 = lowestones[r];
+ assert(c2 < c);
+
+ // There is no valid c2, hence we have completely reduced
+ if (c2 == 0)
+ break;
+ // Reduce col by column c2
+ else {
+ assert(b.get(r, c));
+ b.add_column(c, b.get_column(c2));
+ v.add_column(c, v.get_column(c2));
+ assert(!b.get(r, c));
+ }
}
- }
-}
-template<class T>
-std::ostream& operator<<(std::ostream& os, const std::vector<T> &vec) {
- os << "[";
-
- typename std::vector<T>::const_iterator it = vec.begin();
- while ( it != vec.end()) {
- os << *it;
- if (++it != vec.end())
- os << " ";
+ // A lowest one remained, recall it
+ if (b.get_column(c).size() > 0) {
+ const IT r = b.get_column(c).back();
+ assert(lowestones[r] == 0);
+ assert(r < c);
+ lowestones[r] = c;
+ }
}
-
- os << "]";
- return os;
}
projects / libstick.git / blobdiff