instruction stringlengths 0 29.8k ⌀ |
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Here is a linear-time algorithm that decides whether a DAG contains at least one incomparable pair of nodes.
1. Do a topological sorting with a linear algorithm. (Yes, topological sorting is very helpful!).
2. For each pair of adjacent nodes in the topological order obtained, check if there is an edge between them.... |
null |
**Two nodes $u$, $v$ are said to be incomparable if there is neither a path from $u$ to $v$ nor a path from $v$ to $u$.**
I can't see a linear solution to this. We can do a DFS for each pair $(u,v)$ and $(v,u)$ as in [here][1].
The fact that it is a DAG hints topological order but is this useful?
Can this be s... |
|algorithms|dag|directed-graphs|topological-ordering| |
I need to make a container in which I can store (x,y) as pairs, and for a given number 'a', I have to find a pair (p, q) such that p<=a and q is maximum possible.
Note the constraints: x>=1 and x<= 10^7; y >=1 and y<=10^5.
I am solving this problem in C++.
I tried using a set that stores pairs of integers. I was... |
Constructing a Container for the Given Situation |
|algorithms|data-structures|sets|c++| |
null |
null |
null |
"A window" will also mean "an axis-aligned unit square".
As observed in the question, there is an optimal window such that
- either the bottom left corner is a given point.
- or the left side contains a given point not on the bottom side and the bottom side contains a given point not on the left side.
Hence w... |
As far as I am aware, most functional programming languages today use a call-by-value eager evaluation strategy with some exceptions like Haskell. I am curious if it is possible for a language to have a mixed evaluation strategy: for example, a language where everything is evaluated lazily by default but some functions... |
Is it possible for a language to have mixed evaluation strategies? |
|compilers|type-theory|functional-programming| |
null |
Let $D \subseteq \{0,1\}^*$ be the class of all dyadic languages. Then by $P^D$, we denote the class of languages decidable in polynomial time by a Turing machine with an oracle for a Dyadic language.
Given a dyadic string $x \in \{0,1\}^m$, denote by $B(x)$ the string $y$ such that $y_i = x_{2^i}$. Intuitively, yo... |
I have a problem that I'm struggling to solve or even name, I'd really appreciate any help or pointer to potential existing solutions.
Suppose there is a connected graph $G$ and we are trying to find a tree $M^*$ with the following property. Let $D_M(v)$ be the degree of node $v$ in an arbitrary tree $M$. The soluti... |
Minimum spanning tree with dynamic edge cost based on degrees |
|algorithms|graphs|trees|computer-networks|minimum-spanning-tree| |
null |
In many references I find that the simply typed lambda calculus was introduced because the Kleene–Rosser paradox showed that pure/untyped lambda calculus was inconsistent. However, since it is untyped, the usual definition of inconsistency (i.e., that every type is inhabited) does not apply. Then, in what sense is pure... |
The Kleene–Rosser paradox and the inconsistency of lambda calculus |
|lambda-calculus| |
On most modern systems, peripheral devices do not speak or control DMA directly. Rather, they communicate over a bus protocol (e.g. SCSI or ATA-over-PCI in the case of storage devices), and a special piece of hardware manages both the bus and the mechanics of DMA.
On PC hardware, this is known as the HBA (Host Bus A... |
On most modern systems, peripheral devices do not speak or control DMA directly. Rather, they communicate over a bus protocol (e.g. SCSI or ATA-over-PCI in the case of storage devices), and a special piece of hardware manages both the bus and the mechanics of DMA.
On PC hardware, this is known as the HBA (Host Bus A... |
I need to make a container in which I can store (x,y) as pairs, and for a given number 'a', I have to find a pair (p, q) such that p<=a and q is maximum possible.
Note the constraints: x>=1 and x<= 10^7; y >=1 and y<=10^5.
I am solving this problem in C++.
I tried using a set that stores pairs of integers. I was... |
null |
The [Hamiltonian Path problem][1] on undirected graphs can be reduced to your problem for $k = 2$ and $\beta \gg |V|$. Since the Hamiltonian Path problem is $\mathsf{NP}$-hard, your problem is $\mathsf{NP}$-hard as well.
For the proof of the above statement, you just need to observe the fact that if any *spanning tr... |
I am trying to write down the generalized form of the finite automata which accept strings which contain as a substring an arbitrary string. Here is what I have come up with — I was hoping someone could tell me if I've made any mistakes (i.e., if there are any "bugs" in my "implementation").
Let $\Sigma$ be an alpha... |
Can someone check if the finite state automata I have designed does what I expect it to do? |
|finite-automata| |
null |
Consider the staircase formed by the "left-dominant" points, i.e. those such that they have the maximum ordinate among the points to their left. (If I am right, this staircase is the analogue of the convex hull but for the Manhattan metric.)
[![enter image description here][1]][1]
A dynamic solution to this prob... |
null |
A lot of great answers have already been given, but I would like to name some additional aspects I find quite important as well: not having to define a main function as the entry point encourages less encapsulation and thus less clean, in general uglier (and sometimes also even unusable) code.
This is because high ... |
Almost every mainstream programming language mixes strict and eager evaluation. In particular, in almost all mainstream programming languages, subroutine calls, function applications, and operator applications are evaluated eagerly, *except* conditional operations.
For example, in C, `foo ? x : y` will only evaluate... |
I am seeking a linear algorithm to determine whether a directed acyclic graph (DAG) contains at least one pair of incomparable nodes.
**Two nodes $u$, $v$ are said to be incomparable if there is neither a path from $u$ to $v$ nor a path from $v$ to $u$.**
I can't see a linear solution to this. We can do a DFS for... |
Linear algorithm for determining the presence of incomparable pairs in a directed acyclic graph (DAG) |
I'am trying to prove the following integer programming problem NP hard.
$$
\min \frac{\sum\limits_{i=1}^m(u_i-\bar u)^2}{m}\text{ , where }u_i=\sum\limits_{j=1}^{n}\lambda_{j}x_j\\
s.t. \begin{cases}
Ax\leq b,
\\Cx=d, \\
\forall j: x_{j}\in\{0,1\}, \lambda_{j}\in \mathbb{R}^+\end{cases}
$$
- x is a n-di... |
Is the following integer programming NP-Hard？ |
|computability|np-complete|np-hard| |
null |
I assume by "*equivalent*" you were talking about the language a TM accepts. Then the proof of the two definitions' being equivalent can be made.
On one hand, simply adding a rejecting state to any TM without one does not change what it accepts. On the other hand, removing the rejecting state from a TM with it does ... |
Let $A = \{a_1, \ldots, a_n\}$ be a set of numbers and $w$ - also some number. Define $p_w: \mathbb{R} \rightarrow \mathbb{N}$ as $p_w(r) = |\{i \in \{1, \ldots, n\} | r \leq a_i \leq r + w\}|$. Note that $p_w(\mathbb{R})$ - an image of the whole $\mathbb{R}$ - is finite. Define $MRT_w = \max p_w(\mathbb{R})$.
Let's... |
Data structure for querying maximum of a rolling total |
I have a problem that I'm struggling to solve or even name, I'd really appreciate any help or pointer to potential existing solutions.
Suppose there is a connected graph $G$ and we are trying to find a tree $M^*$ with the following property. Let $D_M(v)$ be the degree of node $v$ in an arbitrary tree $M$. The soluti... |
I assume by "*equivalent*" you were talking about the language a TM accepts. Then the proof of the two definitions' being equivalent can be made.
On one hand, simply adding a rejecting state to any TM without one does not change what it accepts. On the other hand, removing the rejecting state from a TM with it does ... |
null |
I have a problem that I'm struggling to solve or even name, I'd really appreciate any help or pointer to potential existing solutions.
Suppose there is a connected graph $G$ and we are trying to find a spanning tree $M^*$ with the following property. Let $D_M(v)$ be the degree of node $v$ in an arbitrary tree $M$. T... |
An application or program has a start. That's the "main" function. Many languages do not call it "main", and in many it is implicit (e.g. the function defined by the script as a whole).
On the other hand, libraries are collection of functions (and code, data, resources, etc.), and do not have a starting point. You c... |
null |
I'am trying to prove the following integer programming problem NP hard.
$$
\min \frac{\sum\limits_{i=1}^m(u_i-\bar u)^2}{m}\text{ , where }u_i=\sum\limits_{j=1}^{n}\lambda_{j}x_j\\
s.t. \begin{cases}
Ax\leq b,
\\Cx=d, \\
\forall j: x_{j}\in\{0,1\}, \lambda_{j}\in \mathbb{R}^+\end{cases}
$$
- x is a n-di... |
Your specification of the FSA is
- not finished. The transitions where the symbol read is not in $S$ are not defined.
- ambiguous. What happens if $s_4=s_6$? Are $q_4$ and $q_6$ the same state?
With the best intention to interpret your specification, consider the following example.
Let $S=aab$. The FSA wil... |
I'am trying to prove the following integer programming problem NP hard.
$$
\min \frac{\sum\limits_{i=1}^m(u_i-\bar u)^2}{m}\text{ , where }u=Bx,B\in\mathbb{R}^{m\times n}\\
s.t. \begin{cases}
Ax\leq b, \ A\in\mathbb{R}^{m\times n},b\in\mathbb{R}^m
\\Cx=d, \ C\in \mathbb{R}^{m\times n},d\in\mathbb{R}^m\\
\for... |
I'am trying to prove the following integer programming problem NP hard.
$$
\min \frac{\sum\limits_{i=1}^m(u_i-\bar u)^2}{m}\text{ , where }u=Q x,Q\in\mathbb{R}^{m\times n}\\
s.t. \begin{cases}
Ax\leq b, \ A\in\mathbb{R}^{m\times n},b\in\mathbb{R}^m
\\Cx=d, \ C\in \mathbb{R}^{m\times n},d\in\mathbb{R}^m\\
\fo... |
I'am trying to prove the following integer programming problem NP hard.
$$
\min \frac{\sum\limits_{i=1}^m(u_i-\bar u)^2}{m}\text{ , where }u=Q x,Q\in\mathbb{R}^{m\times n}\\
s.t. \begin{cases}
Ax\leq b, \ A\in\mathbb{R}^{m\times n},b\in\mathbb{R}^m
\\Cx=d, \ C\in \mathbb{R}^{m\times n},d\in\mathbb{R}^m\\
\fo... |
> Why do we need the main function and can't we execute the code without it?
It depends entirely on how the language is defined. Programming languages are man-made constructs anyway, so we can make them as we like. With that in mind, we don't "need" a main function, and we can well have a language that can read fro... |
An important philosophical point you might want to consider: Is there really a fundamental difference between _declaring_ a place where the program starts, and simply having the program start at the top of a specific file? `main`, in all the languages that have such a thing, is really just a marker that says "start h... |
I have a problem that I'm struggling to solve or even name, I'd really appreciate any help or pointer to potential existing solutions.
Suppose there is a connected graph $G$ and we are trying to find a spanning tree $M^*$ with the following property. Let $D_M(v)$ be the degree of node $v$ in an arbitrary tree $M$. T... |
Let $A = \{a_1, \ldots, a_n\}$ be a set of numbers and $w$ - also some number. Define $p_w^{A}: \mathbb{R} \rightarrow \mathbb{N}$ as $p_w^{A}(r) = |\{i \in \{1, \ldots, n\} | r \leq a_i \leq r + w\}|$. Note that $p_w^{A}(\mathbb{R})$ - an image of the whole $\mathbb{R}$ - is finite. Define $MRT_w(A) = \max p_w^{A}(\ma... |
I am seeking a linear-time algorithm to determine whether a directed acyclic graph (DAG) contains at least one pair of incomparable nodes.
Two nodes $u$, $v$ are said to be incomparable if there is neither a path from $u$ to $v$ nor a path from $v$ to $u$.
I can't see a linear solution to this. We can do a DFS fo... |
Linear-time algorithm for determining the presence of incomparable pairs in a directed acyclic graph (DAG) |
null |
I have a problem that I'm struggling to solve or even name, I'd really appreciate any help or pointer to potential existing solutions.
Suppose there is a connected graph $G$ and we are trying to find a spanning tree $M^*$ with the following property. Let $D_M(v)$ be the degree of node $v$ in an arbitrary tree $M$. T... |
null |
null |
Here is my code, which passes all the tests. It's based on Matthew C's comment.
def graham_scan(points: list) -> list:
p0 = find_leftmost(points)
hull = []
points.sort(key=cmp_to_key(lambda p, q: orientation_cmp(p0, p, q)))
# handle colinear points at the beginning
... |
An important philosophical point you might want to consider: Is there really a fundamental difference between _declaring_ a place where the program starts, and simply having the program start at the top of a specific file? `main`, in all the languages that have such a thing, is really just a marker that says "start h... |
### Remarks on your FSA
Your specification of the FSA is
- not finished. The transitions from states other than $q_n$ where the symbol read is not in $S$ are not defined.
- ambiguous.
- What happens if $s_1=s_3$? Are $q_1$ and $q_3$ the same state?
- Is the FSA deterministic or nondeterministic? It looks n... |
Show that $ALL-NFA$ = {$\langle M\rangle : M$ is $NFA$ and $L(M) = \Sigma^*$} is $PSPACE-complete$.
I've manged o prove that it is $PSPACE$.
It is easy to see that $\overline{ALL-NFA}\in NPSPACE$ (TM just nondeterministically guess the word and this process is bounded by an appropriate time). $NPSPACE = PSPACE$, w... |
ALL-NFA is PSPACE-complete |
|turing-machines|finite-automata|space-complexity| |
We understand the exponential function to be constantly growing, which we consider bad for a problem. When we say that a problem takes exponentially long time, what we mean is that as we tweak the input parameters a little then the problem takes more time by a 'constant factor'. Then why did we choose polynomials as a ... |
Why are polynomials a natural measure for easiness of computational problems? |
|complexity-theory|polynomial-time| |
null |
**Preliminaries**. Let $n,m,i,j,p,c \in \mathbb{N}$ with $n,m,i,j,p,c \geq 1$.
Let our alphabet be $\{0,1\}$, with
non-empty languages $ L_i \subseteq \Sigma^n$ and $ L'_i \subseteq \Sigma^m$.
The other preliminaries are the same as a [previous question](https://cs.stackexchange.com/questions/159828/counting-stat... |
Counting States in the trim automaton for $\cup_{i=1}^{p} L_i \circ L'_i$ |
|regular-languages|finite-automata| |
null |
{"OriginalQuestionIds":[210],"Voters":[{"Id":755,"DisplayName":"D.W.","BindingReason":{"GoldTagBadge":"complexity-theory"}}]} |
We want to implement a data structure, which can store pairs of $(key, value)$ (where $key$ and $value$ are some numbers) and has the following interface:
1. Add $(key, value)$ pair to a data structure (if such key is already in a data structure then ignore this operation).
2. Remove $key$ from a data structure.
3... |
Segment queries over a key-value store |
|algorithms|data-structures| |
I have a problem that I'm struggling to solve or even name, I'd really appreciate any help or pointer to potential existing solutions.
Suppose there is a connected graph $G$ and we are trying to find a spanning tree $M^*$ with the following property. Let $D_M(v)$ be the degree of node $v$ in an arbitrary tree $M$. T... |
### Remarks on your FSA
Your specification of the FSA is
- not finished. The transitions from states other than $q_n$ where the symbol read is not in $S$ are not defined.
- ambiguous.
- What happens if $s_1=s_3$? Are $q_1$ and $q_3$ the same state?
- Is the FSA deterministic or nondeterministic? It looks n... |
Does my finite state automaton accept a string iff it contains the given string as a substring? |
|regular-languages|finite-automata|substrings| |
I am trying to write down the generalized form of the finite automata which accept strings which contain as a substring an arbitrary string. Here is what I have come up with — I was hoping someone could tell me if I've made any mistakes (i.e., if there are any "bugs" in my "design").
Let $\Sigma$ be an alphabet.
... |
We want to implement a data structure, which can store pairs of $(key, value)$ (where $key$ and $value$ are some numbers) and has the following interface:
1. Add $(key, value)$ pair to a data structure (if such key is already in a data structure then ignore this operation).
2. Remove $key$ from a data structure.
3... |
If you subdivide your two must-include edges creating a node right in the middle of the edge, you have an instance of requiring a simple path with two must-include nodes. The general problem (requiring $k$ must-include nodes) is studied here:
https://utd-ir.tdl.org/bitstream/handle/10735.1/2637/ECS-TR-EE-Vardhan-310... |
I am currently working through Sipser's Introduction to the Theory of Computation. In chapter 5, he defines that a Language $A$ is mapping reducible to language $B$, written $A\leq_m B$ if there is a computable function $f:\Sigma^*\rightarrow\Sigma^*$ where for every $w$, $w\in A \iff f(w)\in B$, where $f$ is called th... |
Understanding reductions and notatoin |
|complexity-theory|computability|reductions|notation| |
We want to implement a data structure, which can store pairs of $(key, value)$ (where $key$ and $value$ are some numbers) and has the following interface:
1. Add $(key, value)$ pair to a data structure (if such key is already in a data structure then ignore this operation).
2. Remove $key$ from a data structure.
3... |
null |
I'am trying to prove the following integer programming problem NP hard.
$$
\min \frac{\sum\limits_{i=1}^m(u_i-\bar u)^2}{m}\text{ , where }u=Q x,Q\in\mathbb{R}^{m\times n}\\
s.t. \begin{cases}
Ax\leq b, \ A\in\mathbb{R}^{m\times n},b\in\mathbb{R}^m
\\Cx=d, \ C\in \mathbb{R}^{m\times n},d\in\mathbb{R}^m\\
\fo... |
|computability|np-complete|np-hard|integer-programming| |
null |
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