[Merged by Bors] - feat: Asymptotic order of divide-and-conquer recurrences (Akra-Bazzi theorem) #6583
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Another question I forgot to mention is if it should be in the |
I agree that filing this under "computability" is a bit of a stretch but yes, let's wait until there is a bit more material about complexity and algorithms! |
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Thanks again for the review -- it took me more time than I thought to get around to implementing the changes, but this is now ready for another round of reviews! |
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…theorem) (#6583) This PR proves the [Akra-Bazzi theorem](https://en.wikipedia.org/wiki/Akra%E2%80%93Bazzi_method), which gives the asymptotic order of divide-and-conquer recurrences of the form ``` T n = (∑ i in Fin k, a i * T (r i n)) + g n ``` where `T : ℕ → ℝ`, and where `r i n` is close to `b i * n` for a set of coefficients `b : Fin k → ℝ`. These recurrences arise mainly in the analysis of divide-and-conquer algorithms such as mergesort or Strassen's algorithm for matrix multiplication. Note that the traditional proof first proves a continuous version (i.e. for `T : ℝ → ℝ`) and then discretizes it to get a version that is relevant for algorithms. Here we prove the discrete version directly, which shortens the proof considerably.
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…theorem) (#6583) This PR proves the [Akra-Bazzi theorem](https://en.wikipedia.org/wiki/Akra%E2%80%93Bazzi_method), which gives the asymptotic order of divide-and-conquer recurrences of the form ``` T n = (∑ i in Fin k, a i * T (r i n)) + g n ``` where `T : ℕ → ℝ`, and where `r i n` is close to `b i * n` for a set of coefficients `b : Fin k → ℝ`. These recurrences arise mainly in the analysis of divide-and-conquer algorithms such as mergesort or Strassen's algorithm for matrix multiplication. Note that the traditional proof first proves a continuous version (i.e. for `T : ℝ → ℝ`) and then discretizes it to get a version that is relevant for algorithms. Here we prove the discrete version directly, which shortens the proof considerably.
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…theorem) (#6583) This PR proves the [Akra-Bazzi theorem](https://en.wikipedia.org/wiki/Akra%E2%80%93Bazzi_method), which gives the asymptotic order of divide-and-conquer recurrences of the form ``` T n = (∑ i in Fin k, a i * T (r i n)) + g n ``` where `T : ℕ → ℝ`, and where `r i n` is close to `b i * n` for a set of coefficients `b : Fin k → ℝ`. These recurrences arise mainly in the analysis of divide-and-conquer algorithms such as mergesort or Strassen's algorithm for matrix multiplication. Note that the traditional proof first proves a continuous version (i.e. for `T : ℝ → ℝ`) and then discretizes it to get a version that is relevant for algorithms. Here we prove the discrete version directly, which shortens the proof considerably.
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This PR proves the Akra-Bazzi theorem, which gives the asymptotic order of divide-and-conquer recurrences of the form
where
T : ℕ → ℝ, and wherer i nis close tob i * nfor a set of coefficientsb : Fin k → ℝ. These recurrences arise mainly in the analysis of divide-and-conquer algorithms such as mergesort or Strassen's algorithm for matrix multiplication.Note that the traditional proof first proves a continuous version (i.e. for
T : ℝ → ℝ) and then discretizes it to get a version that is relevant for algorithms. Here we prove the discrete version directly, which shortens the proof considerably.