instruction stringlengths 0 29.8k ⌀ |
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|formal-languages|formal-grammars| |
Try and build it step by step:
- a language with an odd number of $a$'s: $a(aa)^*$;
- between which there is an even number of $b$'s: $b^*a((bb)^*a(bb)^*a)^*b^*$;
- and an arbitrary number of $x$'s: $(b+x)^*a(x^*(bx^*bx^*)^*ax^*(bx^*bx^*)^*a)^*(b+x)^*$. |
I wanted to ask if someone can help me to construct a regular expression over the alphabet $\{a,b,x\}$ for the language $L$ which is constituted by all strings containing an odd number of $a$'s, and in which between each pair of consecutive $a$'s there is an even number of $b$'s (and an arbirtary number of $x$'s).
F... |
A regular expressions for a given formal language |
|formal-languages|regular-languages|regular-expressions| |
In "Computers and Intractability A Guide to the Theory of NP-Completeness" textbook pp 236, "Sequencing to minimize tardy tasks" is NP-complete.
To be specific the problem is as follows:
For each task $t \in T$, partial order $<$ on $T$, a length $l(t)$, and a deadline $d(t)$ and a positive integer $K \leq |T|$.... |
NP-completeness of scheduling to minimise tardy tasks with release times |
|complexity-theory|scheduling| |
Assume the new problem was polynomially solvable. Take any instance of the old problem and add release dates $r(t)=0$ for all $t \in T$. Then the new problem has a "yes" answer for the modified instance if and only if the old problem had a "yes" answer for the original instance. We have reduced the old problem to the n... |
|algorithms|algorithm-analysis|sorting|average-case| |
I have asked this question at [math.stackexchange](http://math.stackexchange.com/questions/167462/historical-relation-between-computer-science-and-the-theory-of-dynamical-systems), but I have been redirected here.
My question is, if there is any historical relation between the fields of Dynamical systems (and relate... |
|computability|lambda-calculus|type-theory| |
I wanted ask if you know an algorithm to find the witness for $EU(\phi_1,\phi_2)$ using BDDs. In pratice you should use the fixed point for calculating $EU(\phi_1,\phi_2)$, that is:
$\qquad \displaystyle EU(\phi_1,\phi_2)=\mu.Q (\phi_2 \vee (\phi_1 \wedge EX Q)) $
Unwinding the recursion, we get:
$\qquad \disp... |
Is it possible to always construct a hamiltonian path on a [tournament graph](http://en.wikipedia.org/wiki/Tournament_(graph_theory)#Paths_and_cycles) $G=(V,E)$ by sorting (using any sorting algorithm) with the following total order:
$\qquad \displaystyle a \leq b \iff (a,b) \in E \lor \left(\exists\, c \in V. a \le... |
> **Possible Duplicate:**
> [NP-completeness or not](http://cs.stackexchange.com/questions/2634/np-completeness-or-not)
<!-- End of automatically inserted text -->
I know that graph contractability is $NP$-complete. To be specific given $G=(V_1,E_1)$ and $H=(V_2,E_2)$, can a graph isomorphic to H be obtained from... |
{"OriginalQuestionIds":[2634],"Voters":[{"Id":39,"DisplayName":"Gilles"}]} |
NP-completeness of graph isomorphism through edge contractions with an edge validity condition |
|complexity-theory|computability|graphs|np-complete|np-hard| |
> **Possible Duplicate:**
> [NP-completeness or not](http://cs.stackexchange.com/questions/2634/np-completeness-or-not)
<!-- End of automatically inserted text -->
I know that graph contractability is $NP$-complete. To be specific given $G=(V_1,E_1)$ and $H=(V_2,E_2)$, can a graph isomorphic to H be obtained... |
Proof of NP-completeness of graph isomorphism through edge contractions that reduce a metric |
I assume that you intend $\leq$ to be the reflexive and transitive closure of $E$, or directed reachability, that is $\leq$ is really the smallest fixpoint of the equivalence you give:
$\qquad \displaystyle a \leq b \iff \exists\, v_0 \dots v_n. a \to v_o \to \dots \to v_n \to b$
with $n \geq 0$ and $v \to u \iff... |
I want to specify, what it means to give an algebra as input to an algorithm and didn't find very much literature about it. So first I want to ask if you can recommend a book or paper that deals with the topic of complexity analysis of algebras over fields *and clearly define the decision problem*.
After some diggin... |
|terminology|decision-problem| |
A regular expression for a given formal language |
I wanted ask if you know an algorithm to find the witness for $EU(\phi_1,\phi_2)$ ( CTL formula "Exist Until") using BDDs. In pratice you should use the fixed point for calculating $EU(\phi_1,\phi_2)$, that is:
$\qquad \displaystyle EU(\phi_1,\phi_2)=\mu.Q (\phi_2 \vee (\phi_1 \wedge EX Q)) $
Unwinding the recur... |
|algorithms|optimization| |
I wanted ask if you know an algorithm to find the witness for $EU(\phi_1,\phi_2)$ ( CTL formula "Exist Until") using BDDs ([Binary Decision Diagram](http://en.wikipedia.org/wiki/Binary_decision_diagram)). In pratice you should use the fixed point for calculating $EU(\phi_1,\phi_2)$, that is:
$\qquad \displaystyle E... |
There are $n$ people at a table. The $i$th person has to pay $p_i$ dollars.
Some people don't have the right bills to pay exactly $p_i$, so they come up with the following algorithm.
> First, everyone puts all their money on the table. Then each individual takes back the money they overpaid.
An example:
Su... |
Suppose that someone found a polynomial algorithm for a NP-complete decision problem. Would this mean that we can modify the algorithm a bit and use it for solving the problems that are in NP, but not in NP-complete? Or would this just shows the availability of a polynomial algorithm for each NP problem indirectly? |
NP-completeness and NP problems |
|complexity-theory| |
Recall a few definitions. A language $L \subseteq \{0,1\}^*$ is polynomial-time reducible to a language $L' \subseteq \{0,1\}^*$, often denoted as $L \leq_p L'$, if there is a *polynomial-time computable* function $f : \{0,1\}^* \to \{0,1\}^*$ such that for every $x \in \{0,1\}^*$, $x \in L$ iff $f(x) \in L'$. Now, $L'... |
Suppose that someone found a polynomial algorithm for a NP-complete decision problem. Would this mean that we can modify the algorithm a bit and use it for solving the problems that are in NP, but not in NP-complete? Or would this just shows the availability of a polynomial algorithm for each NP problem indirectly?
... |
I just learned that when we have a polynomial algorithm for NP-complete problems, it is possible to use that algorithm to solve all NP problems.
So, the question is how we then distinguish non-NP-complete NP problems from NP-complete problems? It seems that all these problems will have a polynomial algorithm to con... |
How do we distinguish NP-complete problems from other NP problems? |
|complexity-theory| |
Not really. For instance, the [Graph isomorphism problem](http://en.wikipedia.org/wiki/Graph_isomorphism_problem) is believed not to be in P, but on the other hand not being NP-complete (and is obviously in NP). Moreover, also all problems in P (that are also in NP) are not NP-complete, if $P \neq NP$. However, to my k... |
null |
The <a href="http://en.wikipedia.org/wiki/Discrete_logarithm">discrete logarithm</a> is the same as finding $b$ in $a^b=c \bmod N$, given $a$, $c$, and $N$.
I wonder what complexity groups (e.g. for classical and quantum computers) this is in, and what approaches (i.e. algorithms) are the best for accomplishing this... |
How hard is finding the discrete logarithm? |
|algorithms|complexity-theory|time-complexity|discrete-mathematics| |
To provide the flip side of 042's answer: it's important to understand that if $P=NP$ then _every_ problem in $NP$ is $NP$-complete (caveat: under the usual poly-time reductions): $NP$-complete simply means 'every problem within $NP$ can be solved in polynomial time with an oracle for this problem'. But if every probl... |
|formal-methods|model-checking| |
I wanted ask if you know an algorithm to find the witness for $EU(\phi_1,\phi_2)$ (CTL formula "Exist Until") using BDDs ([Binary Decision Diagram](http://en.wikipedia.org/wiki/Binary_decision_diagram)). In pratice you should use the fixed point for calculating $EU(\phi_1,\phi_2)$, that is:
$\qquad \displaystyle EU... |
Is finding the largest subgraph (of a graph) that has hamiltonian path NP-complete?
This may be restated as the following: is finding the largest line path of a graph NP-complete?
Also, is finding the k-length(edge) line path of a graph NP-complete? |
Finding the largest subgraph that has hamiltonian path |
|complexity-theory|np-complete| |
Is finding the largest subgraph (of a graph) that has hamiltonian path NP-complete?
This may be restated as the following: is finding the largest line path of a graph NP-complete?
Also, is finding the k-length(edge) line path of a graph NP-complete?
Add: case is the same except that this time it is to find th... |
First, it is easy to see that the problem is in $\text{NP}$. The longest path is a Hamiltonian one since it visits all vertices. Indeed, there is a straightforward reduction from $\text{HAM-PATH}$ to it. See for example [here](http://en.wikipedia.org/wiki/Longest_path_problem). Likewise, it is $\text{NP}$-complete to d... |
First, it is easy to see that the problem is in $\text{NP}$. The longest path is a Hamiltonian one since it visits all vertices. Indeed, there is a straightforward reduction from $\text{HAM-PATH}$ to it. For details and some special cases, see for example [here](http://en.wikipedia.org/wiki/Longest_path_problem). Likew... |
The problem of finding the largest subgraph of a graph that has a Hamiltonian path can be restated as finding the longest path of a graph. Is this NP-complete? Also, is finding the $k$-length path of a graph NP-complete? Is it still NP-complete if we require the path to visit a given vertex? |
Is finding the longest path of a graph NP-complete? |
To provide the flip side of 042's answer: it's important to understand that if $P=NP$ then _every_ problem in $NP$ is $NP$-complete (caveat: under the usual poly-time reductions): $NP$-complete simply means 'every problem within $NP$ can be solved in polynomial time with an oracle for this problem'. But if every probl... |
The <a href="http://en.wikipedia.org/wiki/Discrete_logarithm">discrete logarithm</a> is the same as finding $b$ in $a^b=c \bmod N$, given $a$, $c$, and $N$.
I wonder what complexity groups (e.g. for classical and quantum computers) this is in, and what approaches (i.e. algorithms) are the best for accomplishing this... |
null |
The question is from Arora-Barak's book Computational Complexity - A Modern Approach:
---
Define a RAM Turing machine to be a Turing machine that has random access memory. We formalize this as follows: The machine has an infinite array A that is initialized to all blanks. It accesses this array as follows. One of t... |
Computational Complexity(Arora-Barak), Exercise 1.9 |
|complexity-theory|turing-machines| |
null |
The question is exercise 1.9 from Arora-Barak's book *Computational Complexity — A Modern Approach*:
---
Define a RAM Turing machine to be a Turing machine that has random access memory. We formalize this as follows: The machine has an infinite array A that is initialized to all blanks. It accesses this array as fo... |
Prove that a boolean function computable in T(n) by a RAM machine is in DTIME(T(n)^2) |
Turns out this is not an easy problem, but it has been solved. A similar case is treated in [Bounded discrete walks](http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAM0103) by Banderier and Nicodème (2010) and they refer to the exact result you need in [Combinatoire analytique des chemins et des car... |
The language is:
L = { (a^n) (b^m) : n + m = 3k, k>=0 }
My attempt at an answer:
(a U b) ^ (3k)
This will work if the a OR b can change for each instance in the string that is (3k) long. If not, what can I do to fix this? |
regular expression given the language |
|regular-languages|regular-expressions| |
First off, putting $k$ in the regular expression is not allowed by typical regular expression syntax, unless I'm mistaken. Even adding constant powers is a shorthand notation, not a part of the regular expression formalism. Even if it were possible, using that notation, you could mix $a$'s with $b$'s, so that not all $... |
Expand the recurrence and sum it up.
Example 1:
T(n) = T(n-5) + O(n^2)
= T(n-10) + O(n^2)
= ....
= T(0) {n/5 terms}
Now let's say T(0) = c. This would be generally given in the question.
so this would sum to (n/5).(n^2) = O(n^3)
Example 2:
T(n) = T(n-2)+log... |
For simplicity, lets assume that 5 divides n
and that n/2=n-5k for some integer k>0.
T(n) = T(n-5) + cn^2
T(n) = cn^2 + c(n - 5)^2 + c(n - 10)^2 + c(n - 15)^2 + ... + c5^2 + c0^2
= c(n^2 + (n - 5)^2 + (n - 10)^2 + (n - 15)^2 + ... + 5^2 + 0^2)
>= c(n^2 + (n - 5)^2 + (n - 10)^2 + (n / 2... |
I've been tasked with solving some recurrence relations, and I've been running into trouble with so called 'chip & conquer' relations.
Here are some example problems:
T(n) = T(n-5) + cn^2
and
T(n) = T(n-2) + log n
I'm supposed to be giving an answer in Θ notation. How do I go around and solving r... |
Solving Recurrence Relations 'Chip & Conquer' |
|recurrence-relation| |
I've been tasked with solving some recurrence relations, and I've been running into trouble with so called 'chip & conquer' relations.
Here are some example problems:
$$T(n) = T(n-5) + cn^2$$
and
$$T(n) = T(n-2) + \log{n}$$
I'm supposed to be giving an answer in $\Theta$ notation. How do I go around and ... |
null |
I am getting started with Graph theory in my algorithms course and trying to understand the approach for this problem:
> "If all streets are one way, there is still a legal way to drive from
> one intersection to another"
The question is to prove that it can be done in liner time. I am not looking for direct ans... |
How to approach this graph theory problem? |
|graphs|algorithm-analysis| |
You can use a segment tree to perform dynamic selection in $O(\log S)$ per oper; where $S$ is the size of the universe. To do this, you need to maintain the total number of elements in the segment represented by a node.
In fact, a binary indexed tree can also be used. Or you could also use a balanced binary search t... |
Given an Undirected graph, I define a structure called $k-key$ as a path contains $k$ vertices which connected to a simple path which contains a $k$ vertices as well.
Here's a problem:
**The '$k-key'$ problem**: Input: an undirected graph $G$ and a number $k$.
question: Does $G$ contains a $k-key$?
I need ... |
The $'k-key'$ problem |
|np-complete|reductions|np-hard| |
I'm working on an expression typing system and looking for insights on what algorithms may be available which solve my problem -- or a proof that its complexity is too high to be reasonable to implement. The problem is defined below.
I have a set of types which form a directed graph $T = (V,E)$ (assume no cycles). T... |
Algorithm for type conversion / signature matching |
|algorithms|typing| |
You want to find a way of creating $G'$ from $G$ such that $G'$ has a $k-key$ iff $G$ has a Hamiltonian path.
If a graph $G$ has $k$ vertices and a simple path of length $k$, then what can we conclude about it having a Hamiltonian path? |
By Lagrange's theorem, $b=O(N)$, since $|a|=O(N)$. (Meaning brute-force is in EXP.)
For a non-deterministic machine, there is a polynomial witness since we can do [modular exponentiation](http://en.wikipedia.org/wiki/Modular_exponentiation) in P. (I.e. the problem is in $NP$.)
The theory that discrete logarithms ... |
I am getting started with graph theory in my algorithms course and trying to understand the approach for this problem:
> "If all streets are one way, there is still a legal way to drive from
> one intersection to another"
The question is to prove that it can be done in linear time. I am not looking for direct an... |
Existence of a route following one-way streets |
Given an Undirected graph, I define a structure called $k-key$ as a path contains $k$ vertices which connected to a simple cycle which contains a $k$ vertices as well.
Here's a problem:
**The '$k-key'$ problem**: Input: an undirected graph $G$ and a number $k$.
question: Does $G$ contains a $k-key$?
I need... |
since $|a|=O(N)$, then $b=O(N)$. (Meaning brute-force is in EXP.)
For a non-deterministic machine, there is a polynomial witness since we can do [modular exponentiation](http://en.wikipedia.org/wiki/Modular_exponentiation) in P. (I.e. the problem is in $NP$.)
The theory that discrete logarithms are in $NP$ but no... |
I've been reading a bit of the literature lately, and have found some rather interesting data-structures.
I have researched various different methods of getting update times down to $\mathcal{O}(1)$ worst-case update time[1][2][3][4][5][6].
Recently I begun looking into lock-free data-structures, to support effic... |
Lock-free, constant update-time concurrent tree data-structures? |
|data-structures|time-complexity|distributed-systems|concurrency|search-trees| |
Expand the recurrence and sum it up.
Example 1:
$$ \begin{align*}
T(n) &= T(n-5) + O(n^2)
= T(n-10) + O(n^2)
= \dots \\
&= T(0) + \{\text{\(n/5\) terms, each \(O(n^2)\)}\}
\end{align*} $$
Now let's say $T(0) = c$. This would be generally given in the question.
so this would su... |
For simplicity, let's assume that $5$ divides $n$
and that $n/2=n-5k$ for some integer $k>0$.
$$ \begin{align*}
T(n) &= T(n-5) + cn^2 \\
T(n) &= cn^2 + c(n - 5)^2 + c(n - 10)^2 + c(n - 15)^2 + ... + c5^2 + c0^2 \\
&= c(n^2 + (n - 5)^2 + (n - 10)^2 + (n - 15)^2 + ... + 5^2 + 0^2) \\
&\ge c(n^... |
The language is:
$$
L = \{ (a^n) (b^m) \mid n + m = 3k, k \ge 0 \}
$$
My attempt at an answer:
$$
(a \cup b)^{3k}
$$
This will work if the a OR b can change for each instance in the string that is (3k) long. If not, what can I do to fix this? |
You want to find a way of creating $G'$ from $G$ such that $G'$ has a $k-key$ iff $G$ has a Hamiltonian path.
If a graph $G$ has $k$ vertices and a simple path of length $k$, then what can we conclude about it having a Hamiltonian path?
EDIT: I'm guessing the intention is that the vertices in the path must be dif... |
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