13 votes

Problems in NC not known to lie in NC2

Disclaimer: I'm not an expert in fast parallel algorithms, hence the probability that I missed more recent results that put the problems I mention in lower levels of the $\mathsf{NC}$ hierarchy is non-...
user avatar
10 votes

Deterministic Parallel algorithm for perfect matching in general graphs?

A few years later :) and Perfect Matching is now known to be in Quasi-NC (that is, you need quasi-polynomially many processors). See the paper of Fenner, Gurjar and Thierauf (for bipartite graphs) ...
user avatar
10 votes
Accepted

$AC^0$[subexp] vs. NC

No, $\mathrm{AC}^0[2^{n^\delta}]$ is not included in NC; it is not even included in $\mathrm{SIZE}[2^{n^\epsilon}]$ for $\epsilon<\delta$. Indeed, any Boolean function on $n^\delta$ inputs, padded ...
user avatar
8 votes
Accepted

Qubit gates in google supremacy

Based on a fairly cursory inspection of their paper, IBM is clearly aware of the spatial locality. They seem to in fact have taken spatial locality into account when designing their simulation. They ...
user avatar
7 votes

Is there a mathematical analysis/proof available for correctness of solutions to inter process communication problems?

There has been various brand of work for formalizing such tricky code. The one I know about are (but I'm no expert on the topic): using "temporal logic" to study distributed systems; one important ...
user avatar
  • 1,882
5 votes

Cases of Linear programming known to be in $NC$?

Fixed dimensional linear programming (for any constant dimension $d$) is known to be in $\mathsf{NC}$; in fact, it can be done work-efficiently (in the same amount of work as the fastest sequential ...
user avatar
  • 429
4 votes
Accepted

Rank-robustness of the parallel complexity of linear algebra problems

I think the following procedure computes a basis of the column span of an $n \times n$-matrix of rank at most $\log n$ in $\mathrm{AC}_1$. If you have a matrix of size $n \times 2 \log n$, you can ...
user avatar
4 votes

Does such model exists?

The model studied in the following work should be a fairly close match with the model that you described (see in particular graph problems "without edge duplication"): Woodruff & Zhang: "When ...
user avatar
4 votes

To what extent, computational ability for hard tasks helps in solving easy tasks

By counting you should be able to compute about $2^{2^m \cdot s}$ functions with such circuits of size $s$ so I'd guess $s=2^{n-m}$ should be enough to compute all functions.
user avatar
3 votes

Parallel algorithms for reachability in directed planar graphs

See Kao, Ming-Yang; Klein, Philip N. (1993), Towards overcoming the transitive-closure bottleneck: efficient parallel algorithms for planar digraphs, J. Comput. System Sci. 47 (1993), no. 3, 459–500. ...
user avatar
3 votes
Accepted

Is multiprocessing possible on a Turing Machine?

This is a subtle question. TMs are very much a sequential model of computation. So in some sense, TMs cannot (directly) model multiprocessing. However, TMs can do step-by-step simulations of the ...
user avatar
2 votes
Accepted

Embarrassingly Parallel: Formal Definition & Citation

There is no formal answer for that question, as the notion of "embarrassingly parallel" is not a formal one; it is an informal and imprecise notion. I understand it to basically mean that if you do ...
user avatar
  • 10.4k
2 votes

Confusion about a formal definition of PRAM consistency

When they define PRAM (page 11 of the arxiv preprint) they actually state that vis is a partial order (in particular, transitive): We define PRAM consistency by requiring the visibility partial ...
user avatar
2 votes

Is there a mathematical analysis/proof available for correctness of solutions to inter process communication problems?

there is a classic reference/framework increasingly standardized: C.A.R Hoare, Communicating Sequential Processes a 1985 book of same title is now available as ebook with open copyright. wikipedia ...
user avatar
  • 10.8k
1 vote

Modern distributed computing book

You can refer Roger Wattenhofer's Lecture note on PODC. Here I mentioned some other courses on distributed algorithms.
user avatar
1 vote

Multiple independent random number streams

It was already mentioned that it is important to use independent streams. Which is not guaranteed if you, e.g., use the naive approach of just seeding each of your PRNG with a different number. ...
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible