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Symbolic multiplication of degree $<n$ polynomials $A(x)$ and $B(x)$ would take $O(n^2)$ time. That may be fine, but they are giving an algorithm that takes $O(n \log n)$ time, which is better. Inste …

This is a neat question and I've thought about it before. Here's what we came up with: You run your algorithm $n$ times to get outputs $x_1, \cdots, x_n \in \mathbb{R}^d$ and you know what with high …

Here's a generic randomized solution. (Do we even have deterministic solutions in the unweighted case? Don't Space Saving and Batch Decrement both need hash maps?) This is probably not the ideal solut …

Expanding my comment: Computing the permanent of a matrix is #P-hard (Valiant 1979) even if the matrix entries are all either 0 or 1. We can interpret a matrix $M \in \{0,1\}^{n \times n}$ as a bipa …

Elaborating Paul's suggestion for a $O(n \log n)$-time algorithm: Input: Let $u \in [m]^k$ and $v \in [m]^n$ with $k \leq n$, where $U=[m]=\{1,2,\cdots,m\}$. Define polynomials $$p(x,y) = \sum_{i \i …

Finding eigenvalues is inherently an iterative process: Finding eigenvalues is equivalent to finding the roots of a polynomial. Moreover, the Abel–Ruffini theorem states that, in general, you cannot e …