instruction stringlengths 0 29.8k ⌀ |
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The fast Fourier transform (FFT) is a popular, efficient algorithm for computing the discrete Fourier transform and its inverse. |
null |
Questions concerning the Halting problem. That is, given a program specification, decide whether it halts. |
null |
Questions concerning the Halting problem which is to decide whether a given a program halts on a given input. |
Questions about graphs in which every edge is associated with a weight. |
null |
John Harrison's book is an exception in going all the way from theory to practice and making all the source code available. I think you will find it difficult to find such a book for model checking.
- [Principles of Model Checking][1] by Baier and Katoen contains a lot of examples and pretty detailed algorithmic de... |
null |
For more details on Teaching-learning-based optimization (TLBO) method and its code, refer to the following paper:
^ Venkata Rao, R.; Patel, V. (2012). [http://www.growingscience.com/ijiec/Vol3/IJIEC_2012_37.pdf "An elitist teaching-learning-based optimization algorithm for solving complex constrained optimization... |
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The time to compute $f(n) = n^{n^2}$ can be decomposed into the time required to compute $f_1(n) = n^2$ and that required to perform $n ^ {f_1(n)}$. Note that $f_1$ requires $O(1)$ time to compute; the exponentiation takes $O(\log(n^2)) = O(\log(n))$ steps by <a href="http://primes.utm.edu/glossary/xpage/BinaryExponent... |
By using the fast Fourier transform, multiplications on $k$-bit numbers can be done in time $\tilde{O}(k)$ (where the tilde [signifies][1] that we're ignoring polylogarithmic factors). By repeated squaring, we can compute $n^{n^2}$ with $O(\log n)$ multiplications, and each multiplication involves no number larger than... |
What is the complexity of computing $n^{n^2},\;n \in \mathbb{N}$? |
Complexity of computing $n^{n^2}$ |
|untagged| |
Last month, I heard a new linear algorithm to determine the longest palindrome named after Jeuring's algorithm. It seemed interesting, therefore I had a try to analyze the algorithm. It's not too difficult to show that it's an $\Theta(n)$-time-algorithm, but I want to observe more closely.
I wrote a C++ implement, a... |
Analysis of an linear algorithm about palindrome |
|algorithms|algorithm-analysis|runtime-analysis| |
From a less formal point of view, I think it may help if you detach yourself from the word "random," as you're correct that a set of truly random bits don't store any information in a practical sense. (If I encrypt a set of names and send the encrypted values to you, they may have very high Kolmogorov complexity but i... |
null |
|complexity-theory|number-theory| |
The [subsumption architecture](http://en.wikipedia.org/wiki/Subsumption_architecture), proposed by Rodney Brooks in 1986, is a "bottom-up" approach, in which robots are designed using simple hierarchical models. These models build upon and subsume the lower modules to form a final product. For example, a robot can be... |
Can bottom-up architectures be effectively programmed in top-down paradigms? |
|artificial-intelligence|algorithm-design| |
Last month, I heard about a new linear algorithm to determine the longest palindrome named Jeuring's algorithm. It seemed interesting, therefore I had a try to analyze the algorithm. It's not too difficult to show that it's an $\Theta(n)$-time-algorithm, but I want to observe more closely.
I wrote a C++ implementati... |
Last month, I heard about a new linear algorithm to determine the longest palindrome called Jeuring's algorithm. It seemed interesting, therefore I had a try to analyze the algorithm. It's not too difficult to show that it's an $\Theta(n)$-time-algorithm, but I want to observe more closely.
I wrote a C++ implementat... |
**Background**
Last month, I heard about a new linear algorithm to determine the longest palindrome called Jeuring's algorithm. It seemed interesting, therefore I had a try to analyze the algorithm. It's not too difficult to show that it's an $\Theta(n)$-time-algorithm, but I want to observe more closely.
I wrote... |
**Background**
Last month, I heard about a new linear algorithm to determine the longest palindrome called Jeuring's algorithm. It seemed interesting, therefore I had a try to analyze the algorithm. It's not too difficult to show that it's an $\Theta(n)$-time-algorithm, but I want to observe more closely.
I wrote... |
A soft heap is a subtle modification of a binomial queue. The data structure is approximate with an error parameter $\epsilon$. It supports insert, delete, meld and findmin. The *amortized* complexity of each operation is $O(1)$, except for insert which takes $\log (1/\epsilon)$ time. The novelty of the soft heap is in... |
For more details on Teaching-learning-based optimization (TLBO) method and its code, refer to the following paper:
> [An elitist teaching-learning-based optimization algorithm for solving complex constrained optimization problems](http://dx.doi.org/10.5267/j.ijiec.2012.03.007) by R. Venkata Rao and V. Patel; Inter... |
|complexity-theory|integers|number-theory| |
|programming-languages|artificial-intelligence|algorithm-design| |
### Background
Last month, I heard about a new linear-time algorithm to determine the [longest palindromic substring](https://en.wikipedia.org/wiki/Longest_palindromic_substring) called Jeuring's algorithm. It seemed interesting, therefore I had a try to analyze the algorithm. It's not too difficult to show that it'... |
Analysis of a linear-time algorithm for longest palindromic substring |
I read about busy beaver numbers and how they grow asymptotically larger than any computable function. Why is this so? Is it because of the busy beaver function's non-computability? If so, then do all non-computable functions grow asymptotically larger than computable ones? |
Do non-computable functions grow asymptotically larger? |
|computability|asymptotics| |
$\newcommand\ldotd{\mathinner{..}}$
### Background
Last month, I heard about a new linear-time algorithm to determine the [longest palindromic substring](https://en.wikipedia.org/wiki/Longest_palindromic_substring) called Jeuring's algorithm. It seemed interesting, therefore I had a try to analyze the algorithm. It... |
If you take any noncomputable set of natural numbers, the characteristic function of the set takes only the values $\{0,1\}$ and is noncomputable. So it is not the case that every noncomputable function grows very quickly, they can even be bounded. |
If you take any noncomputable set of natural numbers, the characteristic function of the set takes only the values $\{0,1\}$ and is noncomputable. So it is not the case that every noncomputable function grows very quickly, they can even be bounded.
The Busy Beaver function grows more quickly than every computable f... |
If you take any noncomputable set of natural numbers, the characteristic function of the set takes only the values $\{0,1\}$ and is noncomputable. So it is not the case that every noncomputable function grows very quickly, they can even be bounded.
The Busy Beaver function grows more quickly than every computable f... |
If a function $f$ grows faster (or slower) than any function in a set $F$ of functions, that is $f \in \omega(g)$ (or $o(g)$) for all functions $g \in F$, then clearly $f \notin F$. This is what is used to show that the busy-beaver function is not computable. Another example is the proof that the -- computable and tota... |
null |
From a less formal point of view, I think it may help if you detach yourself from the word "random," as you're correct that a set of truly random bits don't store any information in a practical sense. (If I encrypt a set of names and send the encrypted values to you, they may have very high Kolmogorov complexity but i... |
I am teaching myself program verification and am currently learning [proof assistants][1]. I have the book [Handbook of Practical Logic and Automated Reasoning][2] which gives the proofs necessary for the understanding of such a system, but more importantly for me it also gives an implementation of the necessary algori... |
> Dijkstra's algorithm is used only when you have a single source and
> you want to know the smallest path from one node to another, but fails
> [in graphs with negative edges]
Dijkstra's algorithm is one example of a _single-source shortest path_ or _SSSP_ algorithm. Every SSSP algorithm computes the shortest-pa... |
I've been studying the three and I'm stating my inferences from them below. Could someone tell me if I have understood them accurately enough or not? Thank you.
1. Dijkstra's algorithm is used only when you have a single source and you want to know the smallest path from one node to another, but fails in cases like ... |
Am I right about the differences between Floyd-Warshall, Dijkstra's and Bellman-Ford algorithms? |
|algorithms|graphs| |
Model checking refers to the following problem: given a model of a system, test automatically whether this model meets a given specification.
Typically, the systems one has in mind are hardware or software systems, and the specification contains safety requirements such as the absence of deadlocks and similar critic... |
null |
null |
John Harrison's book is an exception in going all the way from theory to practice and making all the source code available. I think you will find it difficult to find an equivalent book for model checking, but there are a few that achieve a close approximation.
- [Principles of Model Checking][1] by Baier and Katoe... |
null |
I would like to generalize this into a more common scenario. This I believe makes sense since the particulars of subsumption architecture are perhaps not a primary aspect in the question (and the same problems occur in non-robotic fields as well).
In general software archtitecture there is a bi-directional approach,... |
null |
I've been studying the three and I'm stating my inferences from them below. Could someone tell me if I have understood them accurately enough or not? Thank you.
1. [Dijkstra algorithm](https://en.wikipedia.org/wiki/Dijkstra_algorithm) is used only when you have a single source and you want to know the smallest path... |
Am I right about the differences between Floyd-Warshall, Dijkstra and Bellman-Ford algorithms? |
The middle transition ``fertig`` is the shared action so this must be ran parallel, but after this there are 2 transitions which dont have to run parallel, one is ``kochen`` and the other is ``essen``.
Assume, we are now in the point where ``fertig`` is done by both processes and now we have 2 options, ``kochen`` a... |
I would like to generalize this into a more common scenario. This I believe makes sense since the particulars of subsumption architecture are perhaps not a primary aspect in the question (and the same problems occur in non-robotic fields as well).
In general software archtitecture there is a bi-directional approach,... |
Given a TSP instance $T$, decide whether changing the city coordinates by adding a vector of coordinates $v$ will change the optimal TSP objective by atleast $x$. The city coordinates are integers.
The problem is in PSPACE but even the verification problem seems to be NP-hard. Is that true?
If the verification pr... |
I'm trying to convert a context free grammar to a pushdown automaton (PDA); I'm not sure how I'm gonna get an answer or show you my progress as it's a diagram... Anyway this is the last problem I have on a homework that's due later today, so I'd appreciate some kind of help, even if it's just an explanation of the corr... |
Context-free grammar to a pushdown automaton |
From the information you give in your question, I can not see how to apply standard optimisation methods (that I know of). Your objects are not that complicated (more on that later) but your target function is a nasty one: its values are defined by an external system out of your control, it is unlikely to have any nice... |
I would like to generalize this into a more common scenario. This I believe makes sense since the particulars of subsumption architecture are perhaps not a primary aspect in the question (and the same problems occur in non-robotic fields as well).
In general software archtitecture there is a bi-directional approach,... |
The problem is:
> You have a stream of incoming numbers in range 0 to 60000 and you have
> a function which will take a number from that range and return the
> count of occurrence of that number till that moment. Give a suitable
> Data structure/algorithm to implement this system.
I would like to prove that it... |
Counting with constant space bounded TMs |
|regular-languages|turing-machines|finite-automata|pumping-lemma|space-complexity| |
The problem is:
> You have a stream of incoming numbers in range 0 to 60000 and you have
> a function which will take a number from that range and return the
> count of occurrence of that number till that moment. Give a suitable
> Data structure/algorithm to implement this system.
The stream is infinite, so if... |
The problem is:
> You have a stream of incoming numbers in range 0 to 60000 and you have
> a function which will take a number from that range and return the
> count of occurrence of that number till that moment. Give a suitable
> Data structure/algorithm to implement this system.
The stream is infinite, so if... |
I read about busy beaver numbers and how they grow asymptotically larger than any computable function. Why is this so? Is it because of the busy beaver function's non-computability? If so, then do all non-computable functions grow asymptotically larger than computable ones?
**Edit:**
Great answers below but I wou... |
For simplicity assume, you don't have 600.000 numbers but only 1. At some time $i$ a certain number of `1`s have been sent. We call this number the *prefix* of the stream. There are infinitely many possible prefixes.
Since there is only a constant size work tape, the TM can have at most a constant number of config... |
|regular-languages|turing-machines|finite-automata|space-complexity|streaming-algorithm| |
> Dijkstra's algorithm is used only when you have a single source and
> you want to know the smallest path from one node to another, but fails
> [in graphs with negative edges]
Dijkstra's algorithm is one example of a _single-source shortest path_ or _SSSP_ algorithm. Every SSSP algorithm computes the shortest-pa... |
I'm hard to understand how to calculate the depth of sorting network on n inputs.
In case of Selection sorter,
$D(n)=D(n-1)+2\\D(2)=1\\D(n)=2n-3=\Theta(n)$
How should I approach to understand $D(n)=D(n-1)+2$?
[1]: https://i.stack.imgur.com/hUoqy.png |
How to calculate the depth of sorting network? |
|algorithms| |
Fundamental misunderstanding:
> Every property of a computer program is non-computable
That is not what Rice's theorem talks about. It talks about properties of *functions* and that the set of *programs computing this function* is not decidable. Formally, given $\emptyset \subset P \subset \mathsf{RE}$ the set
... |
From the information you give in your question, I can not see how to apply standard optimisation methods (that I know of). Your objects are not that complicated (more on that later) but your target function is a nasty one: its values are defined by an external system out of your control, it is unlikely to have any nice... |
I have trouble understanding how to calculate the depth of a sorting network on $n$ inputs.
For example, in case of selection sort, we have:
$\qquad \displaystyle D(n)=D(n-1)+2\\\qquad D(2)=1$
which leads to
$\qquad \displaystyle D(n)=2n-3=\Theta(n)$
How should I approach to understand $D(n)=D(n-1)+2$?
... |
How to calculate the depth of sorting networks? |
|algorithms|algorithm-analysis|recurrence-relation|sorting| |
The problem is:
> You have a stream of incoming numbers in range 0 to 60000 and you have
> a function which will take a number from that range and return the
> count of occurrence of that number till that moment. Give a suitable
> Data structure/algorithm to implement this system.
The stream is infinite, so if... |
null |
Let me give an explanation which does not differ in content from the accepted answer, but brings the question back to the realm of regular languages.
The language you are dealing with can be defined as follows. Let $s \in \Sigma^{\mathbb{N}}$ be a (countably) infinite string of symbols from some finite alphabet $\S... |
I have trouble understanding how to calculate the depth of a sorting network on $n$ inputs.
For example, in case of selection sort, we have:
$\qquad \displaystyle D(n)=D(n-1)+2\\\qquad D(2)=1$
which leads to
$\qquad \displaystyle D(n)=2n-3=\Theta(n)$
How should I approach to understand $D(n)=D(n-1)+2$?
... |
null |
Note that there are more than 2 degrees of supervision. For example, see the pages 24-25 (6-7) in the PhD thesis of [Christian Biemann, Unsupervised and Knowledge-free Natural Language Processing in the Structure Discovery Paradigm, 2007.](http://wortschatz.uni-leipzig.de/~cbiemann/pub/2007/Biemann07diss_Structure-Disc... |
Let me give an explanation which does not differ in content from the accepted answer, but brings the question back to the realm of regular languages.
The language you are dealing with can be defined as follows. Let $s \in \Sigma^{\mathbb{N}}$ be a (countably) infinite string of symbols from some finite alphabet $\S... |
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