instruction stringlengths 0 29.8k ⌀ |
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I was told that quantum computers are not computationally more powerful than Turing machines.
Could someone kindly help in giving some literature references explaining that fact? |
References on comparison between quantum computers and Turing machines |
|quantum-computing| |
I haven't watched them myself, but you could probably do worse than Michael Nielsen's videos, [Quantum Computing for the Determined](http://michaelnielsen.org/blog/quantum-computing-for-the-determined/).
I'd recommend also that you look online for written lecture notes; there is a very strong culture in the field of... |
What is actually the case is that anything a quantum computer can compute, a Turing machine can also compute. (This is without commenting at all on how *long* it takes the Turing machine to compute the function compared to a quantum computer.)
This is actually not difficult to see, provided you understand quantum co... |
Why is the minimum height of a binary tree $\log_2(n+1) - 1$? |
Given $a,b,c,d \in \mathbb N$ and $b,d \notin \{0\}$,
$$
\begin{eqnarray*}
\frac a b < \frac c d &\iff& ad < cb
\end{eqnarray*}
$$
My questions are:
Given $a,b,c,d$
1. Assuming we can decide $x < y \in \mathbb Z$ in $\mathcal{O}(|x| +|y|)$, is there any way of deciding $ad<cb$ without having to preform ... |
Comparing rational numbers |
|algorithms|integers| |
I have to be honest this is a homework problem, but I just need to discuss this with some one. The problem is there is a row of n houses, with different profit e.g profit1 for house 1, it can be either positive or negative value. But the aim is to maximize the profit by buying a subset of these houses. So infact, you s... |
Adjacent house , dynamic programming problem |
|dynamic-programming|knapsack-problems| |
I'll just leave a broad hint. Suppose $H(1:n)=[h_1,h_2\ldots,h_n]$ is the list of houses. Suppose that we already know the set of houses to buy for the subarray $H(1:n-1)$. This solution either includes $h_{n-1}$ in its optimal list, or it doesn't. If it does include $h_{n-1}$, then we can't add $h_n$ to the list and ... |
My metric of choice would be the communication by **direct message**. However, this metric is hard to come by since measuring it requires the user's consent.
I do not think that using similar hashtags or terms has any say over friendship. People tend to adjust to the language they read and do not require personal bo... |
Consider a quantum gate. Smoothing out all the technical details, it can be represented as a matrix $G$. An input to the gate, say $\vert \phi \rangle$ is just a vector $v$, and the output of the gate is the vector $Gv$.
Now, consider a circuit. A circuit is just a bunch of gates $\{G_1, G_2, ... \}$, and the circui... |
Questions involving rational numbers. Rational numbers are essentially fractions, and consist of two integers, called the numerator and denominator. A rational number is equal to the numerator divided by the denominator. |
null |
Uniform proof complexity is a field devoted (among else) to the study of constructive notions of proofs, and their relation to complexity classes. For each of the popular (uniform) circuit complexity classes, one can define a theory in which everything which is provable has a "backing" in an algorithm in this complexit... |
Three of the options use the same trick: if $C1,C2$ are two classes of languages and $C1 \subseteq C2$; then $L1 \in C2$ doesn't imply that $L1 \notin C1$.
The only choice that cannot be made true is 3: if there is a polynomial time reduction from an undecidable language $L_1$ to a decidable language $L_2$ then $L_1... |
null |
|compilers|artificial-intelligence| |
|algorithm-analysis|proof-techniques|recurrence-relation| |
|automata|regular-languages|finite-automata|check-my-proof| |
Your idea is correct.
You should take the following path to formally write it up/prove it. First assume that $L$ is accepted by some DEA $M=(Q,\Sigma,\delta,q_0,F)$ then define a new NEA $M'=(Q',\Sigma,\delta',q^'_0,F')$ based on $M$. Just express you ideas formally, for example start with
* $Q'=Q\cup\{q_\text{... |
You can do it in $O(N \log N)$ time and $O(N)$ space.
Instead of finding the billboards that you want to keep, find the billboards that you want to get rid of. You need to find the minimum cost set of billboards such that there is no gap of more than $K$ between two billboards. Do this by constructing a table wher... |
What is the time complexity of computing $\frac{1}{2^n} {{n}\choose{(n+2)/2}}$?
$$\frac{1}{2^n} {{n}\choose{(n+2)/2}} = \frac{1}{2^n} \frac{n(n-1)\cdots ((n-2)/2)}{((n+2)/2) (n/2) \cdots 1}$$
The numerator and denominator in $\frac{n(n-1)\cdots ((n-2)/2)}{((n+2)/2) (n/2) \cdots 1}$ will take $O(n)$ multiplication... |
|algorithms|time-complexity|discrete-mathematics| |
|computability|reference-request|turing-machines|quantum-computing| |
|algorithms|terminology|randomized-algorithms| |
null |
I know that the disjoint set datastructure is used to keep track of the connected components of an undirected graph when the edges are added to the graph dynamically . I also know that is is used in [Kruskal's algorithm for minimum spanning trees][1] . What are the other possible applications of this datastructure ?... |
Practical applications of disjoint set datastructure |
|algorithms|graphs|data-structures| |
{"Voters":[{"Id":39,"DisplayName":"Gilles"}]} |
null |
> $L_0$ = { $<M, w, 0>$ | M halts on w} <br>
> $L_1$ = { $<M, w, 1>$ | M does not halt on w}
>
> $L = L_0 \cup L_1$
I need to determine where in the hierarchy of languages(recursive, recursively enumerable, not recursively enumerable) $L$ and $\overline L$ belong. I reasoned as follows
$L$ = { $<M, w, x>$ | M... |
Determining the classification of languages |
|turing-machines| |
> $L_0 = \{ \langle M, w, 0 \rangle \mid \text{$M$ halts on $w$}\}$ <br>
> $L_1 = \{ \langle M, w, 1 \rangle \mid \text{$M$ does not halt on $w$}\}$
>
> $L = L_0 \cup L_1$
I need to determine where in the hierarchy of languages (recursive, recursively enumerable, not recursively enumerable) $L$ and its compleme... |
|turing-machines|undecidability|halting-problem| |
null |
{"Voters":[{"Id":-1,"DisplayName":"Community"}]} |
Here are two hints
1. If $\bar L$ would be decidable (recursive), then so would be $L$, and you just argued that $L$ is not even recursively enumerable. Maybe you have a closer look on what $\bar L_0 \cap \bar L_1$ really is.
2. If you want to argue formally correct that a language is not recursively enumerable, o... |
I'm in a first year discrete math course and we started algorithms. I had to create an algorithm to multiply two numbers together recursively, which resulted in this:
function multiply($n, $r) {
if($n == 1)
return $r;
else if($r == 1)
return $n;
else
return $r + multiply($n - 1, $r);
}
It w... |
How to “prove my algorithm correct” |
|algorithms|correctness-proof|induction| |
The language states that each word of the language consists of two parts `w`, which a condition on the `w`.
As a result you always need a even number of characters in each word of this language. In consequence `a` is not a member of L!
The words `w` created by `{a,b}*` are like you stated: A arbitrary sequence ... |
Given the following regular language `L`:
L={ww : w in {a,b}*}
Why is `a` not a member of this language?
so what is `{a,b}*` exactly? i thought it means `(A|B)*` ? |
Regular language member explanation |
|finite-automata| |
No quick algorithm for this problem is known.
It is a big open problem (for at least 50 years) if there exists a graph of degree 57 order 3250 girth 5 and diameter 2. See http://en.wikipedia.org/wiki/Moore_graph
However, it is known that if such a graph exists its characteristic polynomial has to be $p(x) = (x-5... |
You need to do a proof by induction on $n + r$. Let $P(m)$ be the statement "for any $n,r \geq 1$ satisfying $n+r = m$, $\mathrm{multiply}(n,r) = nr$. The proof by induction goes along the following lines.
Base case: $m=2$, so $n=r=1$. By inspection, $\mathrm{multiply}(1,1)=1$.
Induction step: suppose $P(m)$ is t... |
The notation $\{a,b\}^*$ stands for all strings over the alphabet $\{a,b\}$. A word is in the language $L$ if it is of the form $ww$, where $w$ is an arbitrary word over the alphabet $\{a,b\}$. For example, taking $w = aba$, we see that $abaaba \in L$. If $w$ has length $n$ then $ww$ has length $2n$, and in particular ... |
No quick algorithm for this problem is known.
It is a big open problem (for at least 50 years) if there exists a regular graph of degree 57 order 3250 girth 5 and diameter 2. This is known as a [Moore graph](http://en.wikipedia.org/wiki/Moore_graph).
However, it is known that if such a graph exists its character... |
Given the following regular language $L$:
$$ L=\{ww \mid w\in\{a,b\}^*\}$$
Why is $a$ not a member of this language?
So what is $\{a,b\}^*$ exactly? I thought it means $(A|B)^*$? |
|regular-languages|finite-automata| |
The short answer is **no**. There is no quick algorithm for this problem is known.
There is a big open problem (for at least 50 years) if there exists a regular graph of degree 57 order 3250 girth 5 and diameter 2. This is known as a [Moore graph](http://en.wikipedia.org/wiki/Moore_graph).
However, it is known t... |
Given an adjacency matrix $A_G$ of an undirected graph $G$, it is easy and straightforward to compute the characteristic polynomial $\chi_G(\lambda)$. What about the other way around? The problem can be formulated as follows.
> **Problem** Given a polynomial $P$, decide whether there is a graph $G$ with the correspo... |
null |
The short answer is **no**. No quick algorithm for this problem is known.
A big open problem (for at least 50 years) in algebraic graph theory asks about the existence a regular graph of degree 57 order 3250 girth 5 and diameter 2. This is known as a [Moore graph](http://en.wikipedia.org/wiki/Moore_graph).
Howe... |
Given an adjacency matrix $A_G$ of an undirected graph $G$, it is easy and straightforward to compute the characteristic polynomial $\chi_G(\lambda)$. What about the other way around? The problem can be formulated as follows.
> **Problem** Given a polynomial $P$, decide whether there is a graph $G$ with the correspo... |
null |
The short answer is **no**. No quick algorithm for this problem is known.
A big open problem (for at least 50 years) in algebraic graph theory asks about the existence a regular graph of degree 57 order 3250 girth 5 and diameter 2. This is known as a [Moore graph](http://en.wikipedia.org/wiki/Moore_graph).
Howe... |
null |
The short answer is **no**. No quick algorithm for this problem is known.
A big open problem (for at least 50 years) in algebraic graph theory asks about the existence a regular graph of degree 57 order 3250 girth 5 and diameter 2. This is known as a [Moore graph](http://en.wikipedia.org/wiki/Moore_graph).
Howe... |
* Maze generation (using a modified Kruskal's algorithm)
* [Tarjan's off-line least common ancestors algorithm][1]
* Connected component labeling
* Online maintenance of biconnected components
[1]: http://en.wikipedia.org/wiki/Tarjan%27s_off-line_least_common_ancestors_algorithm
|
* Maze generation (using a modified Kruskal's algorithm)
* [Tarjan's off-line least common ancestors algorithm][1]
* Connected component labeling
* Online maintenance of biconnected components
* [Validation Hindley–Milner rules][2]
* Computing the winner of [Havannah][3] board game (see [Efficient Playouts for th... |
* Maze generation (using a modified Kruskal's algorithm)
* [Tarjan's off-line least common ancestors algorithm][1]
* Connected component labeling
* Online maintenance of biconnected components
* [Validation Hindley–Milner rules][2]
* Computing the winner of [Havannah][3] board game (see [Efficient Playouts for th... |
* Maze generation (using a modified Kruskal's algorithm)
* [Tarjan's off-line least common ancestors algorithm][1]
* Connected component labeling
* Online maintenance of biconnected components
* [Validation Hindley–Milner rules][2]
* Computing the winner of [Havannah][3] board game (see [Efficient Playouts for th... |
Your idea is correct.
You should take the following path to formally write it up/prove it. First assume that $L$ is accepted by some DEA $M=(Q,\Sigma,\delta,q_0,F)$ then define a new NEA $M'=(Q',\Sigma,\delta',q'_0,F')$ based on $M$. Just express you ideas formally, for example start with
* $Q'=Q\cup\{q_\text{n... |
I'm looking for a $\Theta$ approximation of
$$T(n) = T(n-1) + cn^{2}$$
This is what I have so far:
$$
\begin{align*}
T(n-1)& = T(n-2) + c(n-1)^2\\
T(n) &= T(n-2) + c(n-1) + cn^2\\[1ex]
T(n-2) &= T(n-3) + c(n-2)^2\\
T(n) & = T(n-3) + c(n-2)^2 + c(n-1)^2 + cn^2 \\[1ex]
T(n-3) &= T(n-4) + c(n-3)^2 \\
T(n) &=... |
Recurrence relation for time complexity: T(n) = T(n-1) + n^2 |
|time-complexity|algorithm-analysis|proof-techniques|recurrence-relation| |
Recurrence relation for time complexity $T(n) = T(n-1) + n^2$ |
null |
I'll just leave a broad hint. Suppose $H(1:n)=[h_1,h_2\ldots,h_n]$ is the list of houses. Suppose that we already know the set of houses to buy for the subarray $H(1:n-1)$. This solution either includes $h_{n-1}$ in its optimal list, or it doesn't. If it does include $h_{n-1}$, then we can't add $h_n$ to the list and ... |
Consider a quantum gate. Smoothing out all the technical details, it can be represented as a matrix $G$. An input to the gate, say $\vert \phi \rangle$ is just a vector $v$, and the output of the gate is the vector $Gv$.
Now, consider a circuit. A circuit is just a bunch of gates $\{G_1, G_2, ... \}$, and the circui... |
While reading the wikipedia article about the subset sum problem (http://en.wikipedia.org/wiki/Subset_sum_problem) I came across this example: "is there a non-empty subset whose sum is zero? For example, given the set { −7, −3, −2, 5, 8}, the answer is yes because the subset { −3, −2, 5} sums to zero". I have noticed t... |
Shifting subset sum solution by constant positive integer |
|np-complete|p-vs-np| |
Your error is that you add your constant to the left-hand side of the equality multiple times, and only once to the right side. For example, $1+1=2$ clearly holds. If you add $8$ to both sides of it, you get $1+1+8=10$ which also holds. Erroneously you might add it twice to the left hand side and get $9+9=10$, which do... |
null |
Consider the following search tree. Your heuristic search algorithm starts the search at the root node. Unexpanded nodes are marked gray.
![Search tree][1]
For some reason, the heuristic is being guided towards the left-most subtree, when in fact the goal node is in the right-most subtree, marked with green.
!... |
While reading the [Wikipedia article about the subset sum problem](http://en.wikipedia.org/wiki/Subset_sum_problem) I came across this example: "is there a non-empty subset whose sum is zero? For example, given the set $\{ −7, −3, −2, 5, 8 \}$, the answer is yes because the subset $\{ −3, −2, 5 \}$ sums to zero".
I... |
Consider a quantum gate. Smoothing out all the technical details, it can be represented as a matrix $G$. An input to the gate, say $\vert \phi \rangle$ is just a vector $v$, and the output of the gate is the vector $Gv$.
Now, consider a circuit. A circuit is just a bunch of gates $\{G_1, G_2, ... \}$, and the circui... |
Given the following formal language $L$:
$$ L=\{ww \mid w\in\{a,b\}^*\}$$
Why is $a$ not a member of this language?
So what is $\{a,b\}^*$ exactly? I thought it means $(a+b)^*$? |
|formal-languages|finite-automata| |
Intro:
I'm working with huge data set that i need to plot in browser, and since there may be up to 1M points my idea was to create different representations for different zoom levels
lets say i have 100k points, i would average two-by-two until i get 50k, then i would repeat that until i get below 500 points (my ... |
Line smoothing algorithm that perserve data uniformity |
|computational-geometry|approximation| |
null |
Here are two hints
1. If $\bar L$ would be decidable (recursive), then so would be $L$, and you just argued that $L$ is not even recursively enumerable. Maybe you have a closer look on what $\bar L_0 \cap \bar L_1$ really is. For example
$\bar L_0 = \{ \langle M, w, 0 \rangle \mid \text{$M$ does halts on $w$}\} \c... |
null |
When one shuffles playing cards, the goal is evidently to achieve a possibly big derangement
of a given deck. For manual shuffling there are terms like inshuffle, outshuffle etc. I like
to know whether there is a sensible general measure of derangements of n objects and
efficient algorithm to compute that measure an... |
Maximal derangements |
|permutations| |
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