1

I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / topology are "discretized" (by which I mean "reformulated for a discrete-mathematical setting") to be used for the service of GT/TCS. Examples include:

  • Discrete Morse Theory, which has been useful in the analysis of Boolean functions,
  • The Laplacian matrix, which carries a lot of worthwhile algebro-graphtheoretical properties,
  • The Cheeger constant for graphs, which is useful in the theory of expander graphs, as well as in the theory of distributed computing.

Of course these are based on, respectively, the usual Morse Theory, the Laplace operator, and the Cheeger constant from Riemannian geometry.

My (soft) question is simple: Are there any other cases in GT, TCS or discrete math in general where this type of discretization has occurred? And why are those notions useful discretized?

For example, the Cheeger constant measures the "bottle-neckedness" of a graph, which when interpreted as describing a network used for distributed computing gives us limits on how fast we can compute things in a distributed network. The Laplacian is used heavily in algebraic GT / combinatorics and carries analogous structure to the ordinary Laplace operator, and Discrete Morse theory proves powerful theorems based on inequalities and critical points…(ok, I haven't delved too deep into any form of Morse theory yet…).

(Sure, this is a list-based question, but I figure there can't be that many examples to list?)

Additional Examples, suggested by comments:

  • Discrete Fourier Transform / Analysis has many applications in TCS and combinatorics too, and are obviously based on ordinary Fourier analysis.
MonadBoy
  • 3,252
  • 2
    this is a great equation! I wish I could think of more to add than to note that the Fast Fourier Transform/Discrete Fourier Transform should be part of your list! But I also wonder too how one sees the connection between "discrete" concepts like the graph Laplacian and continuous concepts like the "classical" Laplacian. – Chill2Macht Jun 12 '16 at 02:38
  • 1
    @William Thanks for the addition of the FFT/DFT to the list. Do you think maybe this question would be better posed on MathOverflow? I'm mainly asking since this question earned me the "tumbleweed" badge a couple weeks ago… :/ – MonadBoy Jun 12 '16 at 08:20
  • Haha yeah I'm not sure. It's probably worth a try though; a lot of these discrete concepts seem to be relatively new historically, so are probably still topics of active research. Worst case scenario it gets migrated back to here, so I would try it. – Chill2Macht Jun 12 '16 at 14:11

0 Answers0