New coloring methods

[pkj-m]

• Implemented greedy star-1 coloring (ref: "What Color is your Jacobian?" pg 662 )
• Implemented greedy star-2 coloring (ref: "What Color is your Jacobian?" pg 663)

#29

[pkj-m]

While it doesn't make a lot of sense to use star coloring for simple Jacobian computation using orthogonal column partitioning followed by forward mode AD or finite differencing, it can be used later for easier implementation of substitution-based recovery methods.

I ran the new algorithm on some random and special graphs and this is what I observed.

For a star graph with 200,000 vertices:

• the original greedy algorithm takes 46.170148 seconds (400.01 k allocations: 298.040 GiB, 19.11% gc time)

• the new star coloring algorithm takes 324.176172 seconds (8 allocations: 3.052 MiB) which is probably the worst case for this algorithm.

So evidently the simple coloring algorithm performs better in this case, by an order of magnitude. We should also take note of the huge difference in the number of allocations and memory consumed.

For a cycle graph again with 200,000 vertices,

• the original greedy algorithm takes 44.821616 seconds (400.01 k allocations: 298.040 GiB, 18.88% gc time)

• the star coloring algorithm takes 0.008580 seconds (8 allocations: 3.052 MiB)

Thus, for cycle graphs, our new star coloring algorithm is much, much faster than the original greedy coloring, in terms of both time and memory use. However, note that star coloring would return the coloring vector [1, 2, 3, 1, 2, 3,...] while the original algorithm would return [1, 2, 1, 2...]. Although I think the huge difference in time taken to compute the vector would easily nullify any advantage that one less f call could provide.

Finally, for randomized input graphs with number of vertices varying from 500 to 2500 and edges per graph varying from (1, V^2), where V is the number of vertices, the original algorithm came up with solutions for all 5 test cases in under a second, while the star coloring algorithm failed to come up with a solution even after 2 minutes, so it's safe to say that without further optimizations, there is no reason to use star coloring as a primary coloring mechanism.

EDIT: added latest data

[matbesancon]

Given that you don't use vertex removal, any graph can be passed (not only VSafeGraph). For the tests, generate a bunch of graphs, and compute the coloring using different algorithms. One basic property you can test is that the coloring is a valid one. Another one is that the number of colors is upper-bounded (you have to know the true minimum number of colors for that)

[matbesancon]

In your tests, you could add the two reference graphs from page 664, fig 4.3

[matbesancon]

@pkj-m you are missing random in the package it seems, go to the project, ]add Random