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The task of *deciding* membership is: given any input $x$, decide wether $x \in L$, i.e. compute the following function:
$\qquad \displaystyle \chi_L(x) = \begin{cases}1 &x \in L \\ 0 & x\notin L\end{cases}$
On the other hand, the task of *verifying* membership is: given any input $x$ and a (proposed) *proof* (or... |
Suppose I am given a finite set of points $p_1,p_2,..p_n$ in the plane, and asked to draw a twice-differentiable curve $C(P)$ through the $p_i$'s, such that its perimeter is as small as possible. Assuming $p_i=(x_i,y_i)$ and $x_i<x_{i+1}$, I can formalize this problem as:
<i> Problem 1 </i>Determine a twice-differe... |
A continuous optimization problem that reduces to TSP |
|np-hard|optimization| |
**I want to establish that this is part of my homework for a course I am currently taking. I am looking for some assistance in proceeding, NOT AN ANSWER.**
This is the question in question:
> A 5-pointed-star in an undirected graph is a 5-clique. Show that
> 5-POINTED-STAR $\in$ P, where 5-POINTED-STAR = { <G> ... |
HOMEWORK Question Finding a 5-Poninted Star |
|complexity-theory|time-complexity| |
If there are $V$ nodes, how many subsets of $V$ of size $5$ exist?
If there is a 5-clique, one of this subsets is a clique.
Spoilers below:
>! There are ${V \choose 5}$ possible subsets to check, that is, at most $V^5$ options, which is polynomial in the input. **This is NOT the case** for an arbitrary $k$, sinc... |
>For every computable function $f$ does there exist a problem that can be solved at best in $\Theta(f(n))$ time or is there a computable function $f$ such that every problem that can be solved in $O(f(n))$ can also be solved in $o(f(n))$ time?
If $f$ is [a time-constructible function](https://en.wikipedia.org/wiki/C... |
If $G=(V,E)$ is a graph, how many subsets of $V$ of size $5$ exist?
If there is a 5-clique, one of this subsets is a clique.
Spoilers below:
>! There are ${|V| \choose 5}$ possible subsets to check, that is, at most $|V|^5$ options, which is polynomial in the input. **This is NOT the case** for an arbitrary $k$,... |
If we ignore efficiency issues, there is another example that illustrates the difference by analogy. We know that the halting problem is not decidable: given a code $e$ for a Turing machine, there is no effective way to determine whether the machine stops if it is run with no input.
But if a machine *does* halt, it ... |
**I want to establish that this is part of my homework for a course I am currently taking. I am looking for some assistance in proceeding, NOT AN ANSWER.**
This is the question in question:
> A 5-pointed-star in an undirected graph is a 5-clique. Show that
> 5-POINTED-STAR $\in P$, where 5-POINTED-STAR = $\{ <G... |
Finding a 5-Pointed Star in polynomial time |
null |
On Alpha-Beta pruning, [NegaScout](http://en.wikipedia.org/wiki/Negascout) claims that it can accelerate the process by setting [Alpha,Beta] to [Alpha,Alpha-1].
I do not understand the whole process of NegaScout.
How does it work? What is its recovery mechanism when its guessing failed? |
How does the NegaScout algorithm work? |
null |
null |
By the [Gap theorem][1] (also mentioned [here][2], search for 'gap'), for any computable unbounded function $g : \mathbb{N} \rightarrow \mathbb{N}$, there exists some computable function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $DTIME(f(n)) = DTIME(g(f(n))$.
This answers your question in that there exists s... |
Suppose I am given a finite set of points $p_1,p_2,..p_n$ in the plane, and asked to draw a twice-differentiable curve $C(P)$ through the $p_i$'s, such that its perimeter is as small as possible. Assuming $p_i=(x_i,y_i)$ and $x_i<x_{i+1}$, I can formalize this problem as:
<i> Problem 1 (edited in response to Suresh... |
I am interested in calculating the $n$'th power of a $n\times n$ matrix $A$. Suppose we have an algorithm for matrix which runs in $O(M(n))$ time. Then, one can easily calculate $A^n$ in $O(M(n)\log(n))$ time. Is it possible to solve this problem in lesser time complexity? |
Complexity of computing matrix powers |
|algorithms|time-complexity| |
By the [Gap theorem][1] (using the formulation from [here][2], search for 'gap'), for any computable unbounded function $g : \mathbb{N} \rightarrow \mathbb{N}$, there exists some computable function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $DTIME(f(n)) = DTIME(g(f(n))$.
This answers your question in that th... |
One good way out is Singular Value Decomposition [SVD](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCYQFjAA&url=http%3A%2F%2Fcampar.in.tum.de%2Ftwiki%2Fpub%2FChair%2FTeachingWs05ComputerVision%2F3DCV_svd_000.pdf&ei=ay6CT9LrN6PL2QWl57yLBw&usg=AFQjCNEDIj5efuy3TPm7_jH7SesYcvJsOg&sig2=8_yz1UzNKltvtS... |
If the matrix is [diagonalizable](https://en.wikipedia.org/wiki/Diagonalizable_matrix) then taking the $n$th power will take
$$O(D(n)+ n)$$
where $D(n)$ is the time to diagonalize $A$.
Just to complete the details, if $A=P^{-1}DP$ with a diagonal $D$, then
$$A^n = (P^{-1}DP)^n = P^{-1}D^nP$$
and $D^n$ is just ... |
{"Voters":[{"Id":898,"DisplayName":"PKG"}]} |
{"Voters":[{"Id":898,"DisplayName":"PKG"}]} |
If the matrix is [diagonalizable](https://en.wikipedia.org/wiki/Diagonalizable_matrix) then taking the $n$th power can be done in time
$$O(D(n)+ n\log n)$$
where $D(n)$ is the time to diagonalize $A$.
Just to complete the details, if $A=P^{-1}DP$ with a diagonal $D$, then
$$A^n = (P^{-1}DP)^n = P^{-1}D^nP$$
an... |
I am wondering if there are any experiments that show the existence or the non-existence of a correlation between usage of a dynamic language (such as Python, Ruby, or even Java) over a static language (such as C/C++), and the difference in the productivity. |
Is there a correlation between using a dynamic language (as opposed to a static language) and productivity? |
|software-engineering| |
null |
One good way out is Singular Value Decomposition [SVD](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCYQFjAA&url=http%3A%2F%2Fcampar.in.tum.de%2Ftwiki%2Fpub%2FChair%2FTeachingWs05ComputerVision%2F3DCV_svd_000.pdf&ei=ay6CT9LrN6PL2QWl57yLBw&usg=AFQjCNEDIj5efuy3TPm7_jH7SesYcvJsOg&sig2=8_yz1UzNKltvtS... |
You can get arbitrary difficult problems depending on your $f$.
Let $A$ be a language. Define $f$ as follows:
$$f(x)= \begin{cases} 0 & x\in A \\ 1 & o.w. \\\end{cases}$$
Consider set $S=\{x\}$. There is a non-empty subset $X\subseteq S$ s.t. $f(\Sigma_{x\in X} x) = 0$ iff $x \in A$.
Without putting a comple... |
|algorithms|complexity-theory|time-complexity|linear-algebra| |
|programming-languages|software-engineering| |
|complexity-theory|np-hard|optimization| |
Suppose I am given a finite set of points $p_1,p_2,..p_n$ in the plane, and asked to draw a twice-differentiable curve $C(P)$ through the $p_i$'s, such that its perimeter is as small as possible. Assuming $p_i=(x_i,y_i)$ and $x_i<x_{i+1}$, I can formalize this problem as:
<i> Problem 1 (edited in response to Suresh... |
Suppose I am given a finite set of points $p_1,p_2,..p_n$ in the plane, and asked to draw a twice-differentiable curve $C(P)$ through the $p_i$'s, such that its perimeter is as small as possible. Assuming $p_i=(x_i,y_i)$ and $x_i<x_{i+1}$, I can formalize this problem as:
<i> Problem 1 (edited in response to Suresh... |
|complexity-theory|np-hard|optimization|computable-analysis| |
I am implementing a set of path finding algorithms such as Dijkstra's, Depth First, etc.
At first I used a couple of self made graphs, but now I'd like to take the challenge a bit further and thus I'm looking for either
1. graphs used in benchmarks;
2. graphs of real world cities (or a way to download that kin... |
Where to get graphs to test my search algorithms against? |
|graphs| |
null |
I am implementing a set of path finding algorithms such as Dijkstra's, Depth First, etc.
At first I used a couple of self made graphs, but now I'd like to take the challenge a bit further and thus I'm looking for either
1. graphs used in benchmarks;
2. graphs of real world cities (or a way to download that kin... |
|algorithms|graphs|data-sets| |
null |
This question might seems vague but heres the context:
When we are focusing on HCI we would most likely be interested on knowing first how the user usually deals with a certain object. We then try to see how our system could take away one of the tasks he would do himself and try to do it itself.
- The object ... |
How can I measure the usability of a catalogue? |
|model-checking| |
|hci| |
While I'm not aware of research on productivity, there's been research on comprehension. Philip Wadler has been collecting links to papers on this topic: see [this post][1] and [this one][2].
[1]: http://wadler.blogspot.com/2012/03/empirical-investigation-of-influence-of.html
[2]: http://wadler.blogspot.com/... |
One good way out is Singular Value Decomposition [SVD](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCYQFjAA&url=http%3A%2F%2Fcampar.in.tum.de%2Ftwiki%2Fpub%2FChair%2FTeachingWs05ComputerVision%2F3DCV_svd_000.pdf&ei=ay6CT9LrN6PL2QWl57yLBw&usg=AFQjCNEDIj5efuy3TPm7_jH7SesYcvJsOg&sig2=8_yz1UzNKltvtS... |
null |
My question goes to those who are concerned with computational biology algorithmics. I'm going to take a course on bioinformatics this fall; the problem, however, is that I have too little background in biology and chemistry to feel prepared for that cycle of lections (I was rather weak at these subjects at school).
... |
Introductory books on nature sciences behind bioinformatics |
|algorithms| |
|algorithms|education|computational-biology| |
i have developed the following pseudocode for the following problem -> Sum of pairs:
return YES if for a given array A and an integer b there exist two positions i, j in A such that A[i] + A[j] = b, else NO.
Now i should state a loop invariant, which shows me that my algorithm is correct:
Yes, there are two values... |
Loop invariant algorithm |
|algorithms|invariants| |
This question might seems vague but heres the context:
When we are focusing on HCI we would most likely be interested on knowing first how the user usually deals with a certain object. We then try to see how our system could take away one of the tasks he would do himself and try to do it itself.
- The object ... |
null |
Questions about human computer interaction. |
I have developed the following pseudocode for the sum of pairs problem:
> Given an array $A$ of integers and an integer $b$, return YES if there are positions $i,j$ in $A$ with $A[i] + A[j] = b$, NO otherwise.
Now I should state a loop invariant that shows that my algorithm is correct. Can someone give me a hint ... |
|algorithms|proof-techniques|loop-invariants| |
I am implementing a set of path finding algorithms such as Dijkstra's, Depth First, etc.
At first I used a couple of self made graphs, but now I'd like to take the challenge a bit further and thus I'm looking for either
1. graphs used in benchmarks;
2. graphs of real world cities (or a way to download that kin... |
|algorithms|graphs|data-sets|benchmarking| |
Questions about methods for empirically investigating cost measures of algorithms and data structures. |
Questions asking for real-life or benchmarking data sets. |
|empirical-research|hci| |
Questions about methods that empirically measure the quality of computer science models and techniques when applied to real-world scenarios. |
|programming-languages|empirical-research|software-engineering| |
Research about methods and techniques to ensure the quality of software systems by design and development process. |
I am interested in calculating the $n$'th power of a $n\times n$ matrix $A$. Suppose we have an algorithm for matrix which runs in $O(M(n))$ time. Then, one can easily calculate $A^n$ in $O(M(n)\log(n))$ time. Is it possible to solve this problem in lesser time complexity?
The entries can, in general, be from a semi... |
Is there evidence that using dynamic languages has an impact on productivity? |
I am interested in calculating the $n$'th power of a $n\times n$ matrix $A$. Suppose we have an algorithm for matrix multiplication which runs in $\mathcal{O}(M(n))$ time. Then, one can easily calculate $A^n$ in $\mathcal{O}(M(n)\log(n))$ time. Is it possible to solve this problem in lesser time complexity?
Matrix e... |
|algorithms|complexity-theory|time-complexity|computer-algebra| |
|algorithms|education| |
Properties that hold before and after every execution of a loop. Used to prove correctness of algorithms. |
Questions about computer science expertise applied to computational problems in biology, often in an abstract way. |
Mathematical models of randomness in data and processes. |
By the [Gap theorem][1] (using the formulation from [here][2], search for 'gap'), for any computable unbounded function $g : \mathbb{N} \rightarrow \mathbb{N}$, there exists some increasing (in fact, arbitrarily large) computable function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $DTIME(f(n)) = DTIME(g(f(n))$.
... |
null |
Perhaps something like this?
> After `k` iterations of the loop, `A[i...j]` contains a sorted subarray of size `n - k`, with `k` of the original elements, none of which summing to `b` with any other element of `A`, having been removed.
This loop invariant is useful because, if the loop finishes (i.e., reaches the... |
In the past I worked on usability of interfaces in the information retrieval area, so I give you some **practical** ideas on how to measure the usability.
You didn't give details of what a "catalogue" is, so I will assume that a *user has a problem (or need) and he is searching your catalogue (in paper or electronic... |
One good way out is Singular Value Decomposition [SVD](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCYQFjAA&url=http%3A%2F%2Fcampar.in.tum.de%2Ftwiki%2Fpub%2FChair%2FTeachingWs05ComputerVision%2F3DCV_svd_000.pdf&ei=ay6CT9LrN6PL2QWl57yLBw&usg=AFQjCNEDIj5efuy3TPm7_jH7SesYcvJsOg&sig2=8_yz1UzNKltvtS... |
Here is what I think works:
$\qquad \begin{align}
I = &\phantom{\land}\ A \text{ is sorted} \\
&\land\ j \geq i \\
&\land\ \forall i' \lt i.\ \forall j' \gt j.\ A[i'] + A[j'] \neq b \\
&\land\ \forall i' \lt i.\ b > A[i'] + A[j] \\
&\land\ \forall j' \gt j.\ b < A[i] + A[j']
\end{alig... |
null |
Perhaps something like this?
> After `k` iterations of the loop, `A[i...j]` contains a sorted subarray of size `n - k`, with `k` of the original elements, none of which summing to `b` with any other element of `A`, having been removed.
This loop invariant is useful because, if the loop finishes (i.e., reaches the... |
[Bob Warfield][1] wrote a blog post ([A Picture of the MultiCore Crisis)][2] that has the [kind of graph][3] you're looking for.
**Update**
It [sounds like][4] the Stanford [CPU DB][5] has what you're looking for. You can browse visualizations and download the raw data.
[1]: http://www.smoothspan.com/ma... |
As a previous response stated, it is not an absolute requirement that you study biology in order to do bioinformatics. However, if you wish to have a background understanding of the foundational concepts such as genes, RNA, DNA, etc. then a good introductory biology book should provide everything that you need.
... |
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