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We can do convolution in $O(n\log n)$ for plus/multiply polynomials with FFT. However, the approach doesn't seem very generalisable to rings in general. Has there been any progress over the naive $O(n^2)$ convolution for the max/plus ring?

I should note that one can transform soft-max/plus into plus/product by doing exponentiation. Here $\text{soft-max}(x,y)=\log(e^x+e^y) = \max(x,y) + \log(1+e^{\min(x,y)-\max(x,y)})$.

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There is a more general question on mathoverflow, and I asked a similar question on CS.SE. jbapple provided a good answer. To quote:

"Necklaces, Convolutions, and X+Y", By Bremner et al. shows a $O\left(\frac{n^2(\lg \lg n)^3}{\lg^2 n}\right)$ algorithm for this problem on the real RAM and a $O(n \sqrt{n})$ algorithm in the nonuniform linear decision tree model.

Any improvement to this bound will shed light to a few tough open problems like sorting $X+Y$ and all pair shortest path.

If one of the functions is convex/concave, we can solve the problem in $O(n\log n)$ time. See "Speeding up Dynamic Programming", By Eppstein et al..

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  • $\begingroup$ I'm wondering if "monotonically increasing" might be a useful property. $\endgroup$
    – Thomas Ahle
    Nov 26 '13 at 17:16
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    $\begingroup$ The first problem the authors tries to solve in the Necklaces paper is monotonically increasing. It's likely there is no known algorithm that performs better than the general case. $\endgroup$
    – Chao Xu
    Nov 26 '13 at 17:41

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