Reverse Search Shellability checker ver 0.52

• How to compile.
  Untar the downloaded file and just type make, then you will have the binary rsshell.
• Usage.
  % ./rsshell <datafile>
  Current implementation contains only pure cases. So you can try this checker only with datafiles of pure simplicial complexes.
• Data file.
  In the data file, we express the simplicial complex by a list of simplices. The vertices are numbered by integers. For example, a face with four vertices 1, 3, 4, 5 is expressed as follows.

  1 3 4 5
  

  An example of a simplicial complex is as follows.

  1 2 3 4
  1 3 5 4
  1 4 5 8
  2 3 5 4
  3 5 4 9
  

  The example above shows a 3-dimensional simplicial complex with 5 facets. (This example is shellable.)

The program rsshell is a program (written in C) to check shellability of a given (pure) simplicial complex.
The method is to search a ``shellable h-assignment'' (discussed by Sonoko Moriyama) by using a kind of reverse search. (This idea is essentially suggested by Jörg Rambau in a short conversation during ICM2002. I thank him very much.)

To work within a constant size of memory, our method uses a reverse search. It uses very small memory. It computes exact answer for our question of asking the shellability of simplicial complexes.

Basically, we search an h-assignment (which is equivalent to a shelling) by a depth-first-search like:

• Start from an empty assignment ``* * * ... *''.
• If there is a facet F with k ridges adjacent to the boundary and current h_d+1-k is larger than 0, then
  • assign d+1-k  to F,
  • remove F,
  • h_d+1-k := h_d+1-k-1.
• If all facets are assigned, the assignment we get is a shellable h-assignment. This mean the simplicial complex is shellable.
  If we get stuck before it (i.e., if current assignment is inappropriate), backtrack one step.
  If no shellable assignment is found, the complex is nonshellable.

• Set some ordering of facets: F₁, F₂, ...,F_t.
• Start from an empty assignment ``* * * ... *''.
• Look for a facet F with k ridges adjacent to the boundary and current h_d+1-k is larger than 0. If there are more than one candidates, we check them from the smallest one. Then
  • assign d+1-k  to F,
  • remove F,
  • h_d+1-k := h_d+1-k-1.
  After this modification, we check which way we can go backwards, that is, look for facets G with
  • G is already removed and assigned a label k,
  • G intersects with current complex with k ridges,
  • current h_d+1-k is smaller than the original h_d+1-k. (This last condition is always satisfied.)
  If there is a facet G satisfying these condition and larger than F, we conclude that we already check this situation, so cancell the removal of F and check the next candidate.
  (In short, we check the old assignment is the lexicographically smallest parent of the new or not.)
• If all facets are assigned, the assignment we get is a shellable h-assignment. This mean the simplicial complex is shellable.
  If we get stuck before it (i.e., if current assignment is inappropriate), backtrack one step. For this we cancell the largest facet among candidates to be cancelled.
  If no shellable assignment is found, the complex is nonshellable.

In this searching method, what we need to keep is only the current assignment, thus we need no hash table nor search path to the current status.

In the implementation of rsshell, we keep a list of cancellable facets and the maximum one among them in each step in order to make the computation efficient.