more README

README.md

@@ -5,6 +5,10 @@ and the patterns are centered on composition.
 The code demonstrates how to *compose* either a Dijkstra algorithm,
 either a fast-marching algorithm, with different neighborhoods for each.
 
+
+Avalaible languages
+===================
+
 So far, the demo contains:
 - Python:
     - a purely functional pattern (functions which returns parametrized closures),
@@ -18,3 +22,56 @@ So far, the demo contains:
 The algorithm machinery itself is located within the `code.*` files
 and is not of major interest,
 look for the other files for the architecture.
+
+
+Algorithmics
+============
+
+The Dijkstra algorithm is a well-known way to compute shortest paths on a graph.
+Here, we use a simple regular discretization of a bounded plan as the
+navigation domain (i.e. a grid).
+The Dijkstra algorithm proceed by going on the edge from one node of the graph
+to another *neighbor* node (in a specific order, which does not matter here).
+As such, it needs to know what is the neighborhood of a node.
+Here, we define two choices: either a set of four nodes, or a set of eight
+nodes.
+
+The Fast-Marching algorithm is a generalization of the Dijkstra's algorithm
+for continuous space instead of a graph.
+Instead of going from one node to another, it will traverse a triangle
+(i.e. one of the 2D *simplex* of a partition of the domain space),
+from one node to the opposite edge.
+The difference between the two thus rely on how they define a *transit* on the domain.
+This traversal is called the *Hopf-Lax operator* in the litterature.
+
+Just like Dijkstra needs to know how many nodes are in the neighborhood,
+Fast-Marching needs to know how many triangles are in the neighborhood.
+
+
+Architecture
+============
+
+Hence, we have two *operator slots* for each algorithms:
+- neighborhood (where to go?),
+- Hopf-Lax operator (how to go there?).
+
+For each slot, we have two options:
+- neighborhood:
+    - four neighbors,
+    - eight neighbors;
+- Hopf-Lax:
+    - on the edge of a grpah,
+    - across a triangle.
+
+At the end, by combining those options, we can make up for four different
+algorithms:
+- four neighbors + on edges (4-Dijkstra),
+- eight neighbors + on edges (8-Dijkstra),
+- four neighbors + in triangles (4-Fast-Marching),
+- eight neighbors + in triangles (8-Fast-Marching).
+
+Each of those algorithms may have different behavior/performances and be useful
+in different cases.
+It can however be handy to easily come up with the one you want,
+especially if you have many more operators to manage.
+