Dijkstra's in Disguise

Covering:

  1 : Bellman-Ford
  2 : Currency Arbitrage
  3 : Directional Shortest-Path
  4 : Q-Learning
  5 : Handling Randomness in Shortest-Path Algorithms
  6 : Modern Q-Learning
  7 : Physically-Based Rendering
  8 : Summary and Questions
  9 : Acknowledgements

on similar lines, we also have multiplicative-weight-update [https://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf], which has been characterized as :

"a basic tool [that should be] taught to all algorithms students together with divide-and-conquer, dynamic programming, and random sampling.” - sanjeev-arora

“so hard to believe that it has been discovered five times and forgotten.” - christos-papadimitrou [https://www.youtube.com/watch?v=KP0WFbdHhJM]

it has formed basis of algorithms in fields as diverse as machine-learning, optimization from e.g. tcp ^^), game-theory, economics, biology etc. etc.

This is one of the tightest, best things I've read on HN in a long time.

The information density in this post is insane! This is easily the best things I have read on hacker news this year.

This was such a pleasure to read, holy cow. I absolutely love the interconnection between different fields bound together by graph theory.

This is a fantastic post! Thank you so much for sharing it.

Perhaps another post (which maybe you should write! Or I may (though I'm certainly nowhere as good), or why not both!) might the unifying principle of finding solutions to fixed-point equations. In general, having a Bellman-style update equation, you can usually construct either (a) DP solutions to this problem or (b) contraction mappings whose fixed point solves this problem.

Perhaps the most general form I know of the above principles is the Hamilton-Jacobi-Bellman equation [0], which is used absolutely everywhere in control theory and has deep connections to several fields (including the theory of PDEs, such as heat equations on manifolds along with several other cases). Many, many problems (including all of the ones mentioned above) reduce immediately to an appropriate instance of the HJB equation (either in the finite or the infinite case), which admits a simple DP solution that can be very quickly computed.

Anyways, again, awesome post. I'll be following your blog quite closely :)

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[0] https://en.wikipedia.org/wiki/Hamilton–Jacobi–Bellman_equati...

> Path tracing integrals are fairly expensive because states and actions are continuous and each bounce requires ray-intersecting a geometric data structure. What if we do light transport simulations on a point cloud with a precomputed visibility matrix between all points, and use that as an approximation for irradiance caching / final-gather?

A crude variation on this is often used in games, known as "light probes". You precompute the light transport simulation on a subset of points in the scene (manually placed by artists), and lighting for dynamic objects simply lerps between the nearest light probes.

What a beautiful write-up. Some colleagues and I used to joke that everything was Dijkstra. It's pretty awesome to see that elucidated so well.

There seems to be a mistake about whether the currency arbitrage cost is -log(B/A) or log(B/A)

Reading this spiked my interest in graphs especially considering I never learned anything more than the basics taught in algorithms and data structures courses in college.

Any resources (i.e. textbooks) recommended for learning more about graph theory as well as currency arbitrage?

That’s a really neat article!

Wonderful article! I absolutely adore graphs just for this reason - you can solve a massive number of problems with a few tools.

And of-fucking-course OP is at Google Brain.