Merge Branches in Computation Graph

[lukem12345]

We note that we can take this idea of contracting chains further by eliminating branches.

In other words, a binary tree of depth 2, (with update steps at the leaves) requires (assuming all functions on edges are unique) 6 matrices, 2 intermediary vectors, and performs 6 matrix-vector multiplications. This could be re-written as an equivalent 1-deep 4-ary tree which requires just 4 matrices, 0 intermediary vectors, and performs 4 matrix-vector multiplications. This result cannot be improved upon, unless edge pruning of duplicate operations is performed.

This speed-up and memory improvement of an entire decapode in this fashion is unlikely to occur, since Op2s are likely to break this downwards tree-like property.

Originally posted by @lukem12345 in AlgebraicJulia/Decapodes.jl#93 (comment)

[lukem12345]

This issue would allow for the most speed-ups in downstream Decapodes simulations ATM, since most bottlenecks relating to operator efficiency are resolved.