# Questions tagged [dc.parallel-comp]

Theoretical questions in Parallel Computing

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### Linear modular equalities with $0/1$ solution

Let $Ax\equiv b\bmod q$ be a $n\times n$ modular linear system known to have $0/1$ solution where $q$ is a large prime. We can solve in $NC^2$ for general linear systems using determinant and matrix ...
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### Is modular square roots modulo primes in $NC$?

Assume modulus is prime. Is modular square roots then in $NC$? If not then, are there special primes or prime powers or numbers related to these (such as $2^k\pm i$ where $i\in\{-1,0,+1\}$) where it ...
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### Parallel complexity of fixed dimension fixed constraints integer programming

Papadimitriou in https://lara.epfl.ch/w/_media/papadimitriou81complexityintegerprogramming.pdf shows ILP is fixed parameter tractable in number of constraints and Lenstra in https://people.csail.mit....
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### Boolean vs algebraic circuits difference

Valiant, Skyum, Berkowitz and Rackoff in https://epubs.siam.org/doi/10.1137/0212043 showed that $VP=VNC^2$, namely, that arithmetic circuits can be parallelized. What is the central reason such a ...
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### Some examples of tools to demonstrate problem is in $NC$ [closed]

Unlike the class $P$ or $NP$ the class $NC$ does not have any complete problems. To show a problem is in $NC$ one needs to marshal efforts to directly show the problem is in $NC$ since there are no ...
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### Is black box parallel quantum speedup ever nontrivial?

Grover's algorithm is not parallelizable, in that $p$ quantum processors searching over $n$ elements can't do better than $O(\sqrt{n/p})$ queries. Are there any oracle problems where quantum ...
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### Embarrassingly Parallel: Formal Definition & Citation

I've been unable to find a good answer for this question: Formally, what makes a problem embarrassingly parallel? Intuitively, it would seem to me that an embarrassingly parallel problem is one where: ...
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### Fast algorithms for evaluating functions with high Kolmogorov complexity

Motivation: I am motivated by a concrete example that occurs in neuroscience, dendritic computation, which may be approximated by functions computable on binary trees [1]. To be more precise, I ...
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### Qubit gates in google supremacy

The gates in quantum supremacy experiment are nearest-neighbor and have spatial locality. Would this additional information help bolster IBM's argument to perhaps simulate quantum supremacy experiment ...
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### Space complexity of global minimum cut

Are there any non-trivial bounds on the space complexity of global minimum cut? The problem is known to be in $\mathsf{RNC}$. Is anything known about containment in either $\mathsf{L}$ or $\mathsf{NL}$...
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### Parallel building time of a k-d tree on n points with n processors

Given a point set with $n$ points to build a k-d tree on. We have $n$ processors available. What is the time-optimal building time for the k-d tree? A straight forward parallelization would be as ...
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### Problems which will be in $NC$ if fixed dimension Linear Integer Programming in $NC$

We know if fixed dimension linear integer programming is in $NC$ then integer $GCD$ is in $NC$. Is this the only non-trivial implication of fixed dimension linear integer programming in $NC$?
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### Complexity class of approximating perfect match count

We know we can approximate perfect matching count of bipartite and approximate volume of convex bodies in randomized polynomial time. Is there any evidence these approximations could be in Nick's ...
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### Confusion about a formal definition of PRAM consistency

I am reading the paper "Consistency in Non-Transactional Distributed Storage Systems" by Paolo Viotti and Marko Vukolić. The authors provide a comprehensive survey of various consistency semantics ...
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### Fixed dimension Integer programming minus LLL in fixed parameter $NC$?

If you remove LLL part then is remaining part of a. Lenstra algorithm b. Barvinok algorithm in $O(f(n)(\log(mL))^c)$ time on $O(g(n)(mL)^c)$ processors with fixed $c>0$ in fixed $n$ dimension, $m$ ...
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### Problems in $\mathsf{NC^{2}}$ that are not known to be in $\mathsf{AC^{1}}$ or $\mathsf{DET}$

Do we know of any problems in $\mathsf{NC^{2}}$ that are not known to be in $\mathsf{AC^{1}}$ or $\mathsf{DET}$? Context: based on Josh's answer to this question, it could be possible that all ...
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### Cases of Linear programming known to be in $NC$?

Linear programming is $P$-complete. However are there special situations where we know an $NC$ algorithm?
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### Rank-robustness of the parallel complexity of linear algebra problems

It is known that most computational problems related to linear algebra can be computed in $NC^2$ - i.e. for an $n\times n$ matrix $A$, over the reals or a finite field, we can compute the rank of $A$, ...
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### On analogies between parallel complexity and polynomial time hierarchy structure?

Is it known $\mathsf{RNC=NC\iff P=RP}$ or $\mathsf{BPNC=NC\iff P=BPP}$? Are there any analogies (such as collapse results, problems which suggest analogies such as gcd(in NC) and factoring (in P), ...
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### Problems in NC not known to lie in NC2

Are there interesting problems that are in $\mathsf{NC}$ but not known to be in $\mathsf{NC^{2}}$? In the paper 'A Taxonomy of Problems With Fast Parallel Algorithms', Cook mentions that MIS was known ...
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### Does such model exists?

I have a problem on distributed graph, with the following model: 1. There is a Global Graph $G=(V,E)$ 2. There are $k$ computers. 3. Each computer $1 \leq i \leq k$ knows ALL the nodes of the graph,...
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### Modern distributed computing book

Lynch's Distributed Algorithms book is a classic but it is from 1996 and rather out of date. Are there any recent distributed computing books that can be used as textbooks for a graduate distributed ...
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### To what extent, computational ability for hard tasks helps in solving easy tasks

In short, the question is: to what extent, computational ability for hard tasks really helps you in solving easy tasks. (There could be various ways to make this question interesting and non-trivial, ...
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### Is multiprocessing possible on a Turing Machine? [closed]

I recently created a parallel implementation of the Merge Sort, in which the sorting of several groups was accomplished by different processes, and was wondering if this was theoretically possible on ...
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### What are some natural problems that we can quickly find a solution to using massive parallelism but not a canonical solution?

For many problems, more than one output is acceptable. For instance, the problem of finding an assignment that satisfies a boolean formula. If randomness buys us something then it could be that it ...
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### NC algorithm for rank of skinny matrix

Suppose I want to find the rank of an $m \times n$ matrix $A$ over $GF(2)$, where $m \ll n$. The algorithms for rank in the literature seem to be focused on the case when $m = n$, giving a time ...
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### Limits of parallel computing with local connections?

There are successes with an increasing numbers of individual computational units in GPUs or as processor cores. Given someone made the effort to build a huge array of processors which - however - can ...
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### Algorithm (parallel and serial) for Gram-Schmidt

Suppose we are given $m$ vectors $v_1, \dots, v_m$ in $n$-dimensional space $\mathbf R^n$ (or perhaps they are specified up to $b$ bits of precision). I would like to find an orthonormal basis for the ...
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### Lower bounds in PRAM model for evaluation of straight-line code

Miller, Ramachandran and Kaltofen showed that any straight line program can be executed in parallel time O(log n) using M(n) processors where M(n) is the number of processors for multiplying nxn ...
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### Is there a mathematical analysis/proof available for correctness of solutions to inter process communication problems?

I've been going over some material related to IPC recently from Tanenbaum's "Modern Operating Systems" and revisited semaphore after many years. There is a lot of code and pseudo code based ...
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### Evidence of non P-hard problems that require polynomial space?

It is admitted that a $\mathsf{P}$-complete problem requires polynomial space and thus cannot be efficiently parallelized. One purpose of these problems is that they can be used to 'defeat' an (...
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### Parallel (NC) replacements for Gaussian elimination?

Suppose one has an matrix $A$ with $c$ columns and $r$ rows with entries in the binary field $GF(2)$. One wants to determine its rank, and a basis for its null-space. These can be computed easily ...
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### Are the sets of executions of data-race free programs equal, when run on causal memory and on sequentially consistent memory respectively?

In the paper "Causal Memory: Definitions, Implementations, and Programming (Distributed Computing [DC] 1995)", the authors present a formal definition of causal memory, an abstraction of distributed ...
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### Best complexity bound for parallel matrix-vector product?

I'm looking for the best known complexity (and a bound for the number of processors invoved) to do the calculation of a $(n,n)$ matrix-vector product. Thank you
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Has any research been done on paradox-driven computers? An example of what I mean by "paradox-driven": Given a computer which can send information back through time, an algorithm to instantly break ...
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### Randomized Parallel Algorithm for Maximal Independent Set

There are a couple of randomized parallel algorithms for the maximal independent set problem, e.g. A Simple Parallel Algorithm for the Maximal Independent Set Problem, A fast and simple randomized ...
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### Differences between Quantum Computing and Parallelism [closed]

What are the differences between Quantum Computing and Parallelism? thanks in advance
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### DAG partitioning for parallel computing

Consider a set of processes ($P=\{p_1, p_2,\dots, p_n \}$) and their data dependencies. Each process $p_i$ has an execution runtime which is denoted by $d_i$. We are interested to parallelize these ...
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### Multiple independent random number streams

Having multiple streams of pseudo-random numbers known to be independent and with a uniform distribution I want to do Monte Carlo simulations in parallel. In other words, one thread will have a full-...
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### Calculation of Recursive Spawning

I'm reading the book Introduction to Algorithms (Cormel et al., 2009) on the chapter about multithreaded algorithms, and I'm confused about the following: We must also account for the overhead of ...
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Suppose an algorithm that receive an input array of $n$ elements and it performs a task over each element. All tasks are independent and take $O(k)$ each (being $k$ a variable). Since all tasks are ...
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### Should the Schedule of High-level Operations'' Respect the Linearizability of Low-level Operations'' in Proof of Simulation Algorithm?

Backgroud I am reading Chapter 10 Fault-Tolerant Simulations of Read/Write Objects'' of the Book Distributed Computing (by Hagit Attiya & Jennifer Welch). Specifically, in section 10.2.3, it ...
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### Multiple k-selection using GPU

What I am trying to achieve is multiple k-selections (different but small datasets) running in parallel on a GPU. Basically, my aim is to select kth smallest element from an array of floats such that ...
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### Is there a quantum NC algorithm for computing GCD?

From the comments on one of my questions on MathOverflow I get the feeling that the question regarding GCD being in $\mathsf{NC}$ vs. $\mathsf{P}$ is akin to the question regarding Integer ...
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### Status on circuit lower bounds for polylog-bounded depth circuits

Bounded depth circuit complexity is one of the main areas of research within circuit complexity theory. This topic has origins in results like "the parity function is not in $AC^{0}$" and "the mod $p$ ...
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