Update research ideas (#17)

research.MD

@@ -1,13 +1,25 @@
+# Dynamic Algorithms
 [Dynamic Graph Alogrithms](http://cs.ioc.ee/ewscs/2012/italiano/dynamic1.pdf)
 
+# Formalization of Math 
 Introduced to the notion of matrix distance products in [All Pairs Shortest Paths using Bridging Sets
 and Rectangular Matrix Multiplication](https://arxiv.org/pdf/cs/0008011.pdf). Distance product for a rectangular 
 matrix ==> $c_ij = min{a_ik + b_kj} for 1 <= i,j <= n$
 
 Introduced to Strassen's method of matrix multiplication in [Introduction to Algorithms](http://mitpress.mit.edu/algorithms)
 
-Looking into Strassen's method, it seems that avos multiplicaiton is likely a semi-ring [Strassen algorithm](https://en.wikipedia.org/wiki/Strassen_algorithm)
+Looking into Strassen's method, it seems that avos multiplicaiton is likely a semiring [Strassen algorithm](https://en.wikipedia.org/wiki/Strassen_algorithm)
 
+# Formalization of Set for Red Black Graphs
+
+A = -1, 0, 1, 2, 3, ...
 Removed reference to unsigned types in sparsetools... Rather than do this, 
 red 1 should be defined as binary 11111...1 for word width w and then only operate 
-on unsigned types. This makes the min function a bit odd, but works well.
+on unsigned types. This makes the min function a bit odd, but works well.
+
+# Idea: Compare hop cardinality from various sampled roots
+Take a sample of individuals and explore hop cardinality, components identified by hop, etc.
+
+# Observation: Sampled crawls exhibit similar characteristics as the entire tree
+We see a large component that contains that vast majority of individuals, there are a handful of smaller compoennts
+and a lot of islands. The next largest component is much smaller in size than the largest. (Todo: compare diameters)

workingOn.MD

@@ -5,6 +5,12 @@ Remove the concept of pedigree number replace it with relationship function.
 
 Editorial changes that result from the simplification
 
+## numpy-like implementation
+Work on core implementations for:
+
+ * transitive closure
+ * canonicalization
+
 ## Sparse implementation
 Work on sparse implementations
 
@@ -23,4 +29,8 @@ being handled properly in the calling code.
 Do this by combining topological_sort with csgraph.find_components
 
 ## Work in Ahnentafel
-The basic numbering scheme is the same as the Ahnen number (see: [Wikipedia Reference](https://en.wikipedia.org/wiki/Ahnentafel))
+The basic numbering scheme is the same as the Ahnen number (see: [Wikipedia Reference](https://en.wikipedia.org/wiki/Ahnentafel))
+
+## Look at GPU implementation for Floyd-Warshall
+See [Discussion thread on GPU impl of Floyd-Warshall](https://devtalk.nvidia.com/default/topic/534223/simple-cuda-implementation-of-the-floyd-warshall-graph-algorithm/)
+it points out several implementations on github