"a basic tool [that should be] taught to all algorithms students together with divide-and-conquer, dynamic programming, and random sampling.” - sanjeev-arora
Glad you like it! Happy to answer any questions. I’ve been trying to make my writing accessible to both the HN and academic research community; let me know if you have feedback on how I can improve here
Thank you very much, great article indeed! The PDF download link to the article at the top of the page requires access approval though, is that the way it is supposed to be?
Hmm, even in PDF form it's not possible to read this without humming the Transformers theme and putting in 'Dijkstra's in Disguise' at the relevant places.
The usage of text color on this image caption is brilliant.
It never occurred to me that you can do this technique to explain a diagram. Definitely love it. Will steal this technique for my own purpose.
Couldn't agree more, it had me going back to the Algorithms book to refresh on most of part VI (Graph Algorithms). For those interested, Bellman-Ford is 24.1, and Dijkstra's algo is 24.3 in CLRS. Reviewing those helped step through the VisuAlgo link in the article: https://visualgo.net/en/sssp
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 :)
Great suggestions, thanks! Indeed, that's a beautiful connection and many proofs of reinforcement learning algorithms with function approximators actually use contraction mappings.
Say hello to Kwabena for me if you get the chance ;)
> 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.
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?
Graph theory and abstract algebra were the two elective math classes I took that almost tempted me to switch from physics to math. But then I took real analysis and was like "Yup, nope, this is not for me"
Covering:
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.
Glad you like it! Happy to answer any questions. I’ve been trying to make my writing accessible to both the HN and academic research community; let me know if you have feedback on how I can improve here
Thank you very much, great article indeed! The PDF download link to the article at the top of the page requires access approval though, is that the way it is supposed to be?
My bad, I updated it to fix some typos and forgot to open up sharing settings. Should be fixed.
Hmm, even in PDF form it's not possible to read this without humming the Transformers theme and putting in 'Dijkstra's in Disguise' at the relevant places.
This is very distracting.
https://i.imgur.com/mlpUy4g.png
The usage of text color on this image caption is brilliant. It never occurred to me that you can do this technique to explain a diagram. Definitely love it. Will steal this technique for my own purpose.
You might enjoy articles from BetterExplained. For example: https://betterexplained.com/articles/q-why-is-e-special-2-71...
There might be an article about this specific technique.
Thanks for the pointer -- I have an article here.
https://betterexplained.com/articles/colorized-math-equation...
If you hover any diagram, there are options for a colorblind-friendly version.
If it's for broad consumption, it's best to consider people with poor colour vision.
Progressive enhancement?
The information density in this post is insane! This is easily the best things I have read on hacker news this year.
Couldn't agree more, it had me going back to the Algorithms book to refresh on most of part VI (Graph Algorithms). For those interested, Bellman-Ford is 24.1, and Dijkstra's algo is 24.3 in CLRS. Reviewing those helped step through the VisuAlgo link in the article: https://visualgo.net/en/sssp
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 :)
---
[0] https://en.wikipedia.org/wiki/Hamilton–Jacobi–Bellman_equati...
Great suggestions, thanks! Indeed, that's a beautiful connection and many proofs of reinforcement learning algorithms with function approximators actually use contraction mappings.
Say hello to Kwabena for me if you get the chance ;)
Hehe, will do!
> 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)
Thanks! There was an error in the embedded graphic, which I've updated. Edge costs E(A,B) should be log (A/B)
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.
Graph theory and abstract algebra were the two elective math classes I took that almost tempted me to switch from physics to math. But then I took real analysis and was like "Yup, nope, this is not for me"