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2
votes
0answers
101 views

Sieve Methods for Twin Primes - How to extract algorithm from formula

I am reading Cojocaru and Murty's Introduction to Sieve Methods and their Applications. They wait until Chapter 5 to discuss the Sieve of Eratosthenes for finding primes - and their version of it is ...
4
votes
1answer
62 views

Complexity of generically inverting a class of near linear monotonic functions

Given a monotonic increasing function $f(\mathbb{N}) \rightarrow \mathbb{N}$ and a slack function $a(\mathbb{N})\rightarrow \mathbb{N}$, where $f(n) = n \pm O(a(n))$; how many calls to $f$ do we ...
11
votes
1answer
79 views

Decidable theory of asymptotic growth

What are the known limits of the decidability of the comparison of growth rate of functions from $\mathbb{N} \to \mathbb{N}$? I am here thinking of the decidability of questions like "Is $x^x \sim ...
3
votes
0answers
66 views

What is the state-of-the-art asymptotics for convex optimization?

I've got a convex program of the form: Choose $v \in \mathbb{R}^n$ to minimize $vAv^T$ subject to $O(n)$ linear contraints (including $v \ge 0$). $A$ is a square binary matrix. What algorithm gives ...
1
vote
0answers
161 views

Big-Theta extension of Brent's Theorem?

Is there an extension or translation of Brent's theorem into asymptotics aside from big-$O$? Brent's Theorem: source Running time of a parallel algorithm with $p$ processors (say, $f(n,p)$), $W(n)$ ...
8
votes
1answer
348 views

Can we get a sorted list from a sorted matrix in $O(n^2)$

I'm confused. I want to prove that that the problem of sorting a $n$ by $n$ matrix i.e. the rows and columns are in ascending order is $\Omega(n^2\log n)$. I proceed by assuming that it can be done ...
12
votes
0answers
126 views

Is the theory of asymptotic bounds finitely axiomatizable?

Let $F$ be the set of functions over real numbers. Consider the structure $M = \langle F, <, \leq, =, \geq, > \rangle$ where the $<, \leq, =, \geq, >$ are defined as asymptotic notions ...
16
votes
2answers
304 views

Under what circumstances do $O(n^{a + \epsilon})$ algorithms imply $O(n^{a+o(1)})$ algorithms?

Suppose that, for each $\epsilon > 0$, there is an Turing machine $M_{\epsilon}$ that decides a language $L$ in time $O(n^{a + \epsilon})$. Is there a single algorithm deciding $L$ in time $O(n^{a ...
5
votes
1answer
362 views

$O(m+n)$ really necessary for graph algorithms?

It is standard to express the running time of linear-time graph algorithms as $O(m+n)$ (such as depth-first-search, etc.). For nearly all such algorithms, vertices of degree zero have no effect on ...