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The answer is of course yes. Take any bijection $f:\mathbb{R}^n \to \mathbb{R}$, and define a total ordering on $\mathbb{R}^n$ as follows: for each $x,y \in \mathbb{R}^n$, we define $x<y$ iff $f(x)<f(y)$. This defines a total order on $\mathbb{R}^n$, and now $f$ plus this total order meets your condition.
(See als... |
Consider the following algorithm:
```
Input: a natural number n
Output: the list of prime numbers ≤ n
lst ← []
for x from 2 to n do
if x is not a factor of any number in lst then
append x to lst
return lst
```
Using the [prime number theorem](https://en.wikipedia.org/wiki/Prime_number_theorem), i... |
Below is the language, determine (R), (RE), (coRE). and prove your answer.
$L=\{\langle M_1,M_2\rangle|M_1,M_2$ are Turing-machines and $L(M_1)\cap L(M_2)\neq \emptyset \}$
**Attempt:** I Think the language $L\in RE$ Because we need to use $U-MT$ such that we check all the $L(M_1)\cap L(M_2)\neq \emptyset$ to se... |
Is $L=\{\langle M_1,M_2\rangle|L(M_1)\cap L(M_2)\neq \emptyset \}$ R, RE or coRE? |
Is there a computationally efficient algorithm which can map back and forth a multi-dimensional real number (R^n) to a single dimensional real (R)? |
null |
null |
null |
null |
You can solve it with satisfiability. Here i have created the SAT formula for you. Sorry i deleted this answer earlier because i forgot to add a formula, i fixed it, it should be correct now.
[![enter image description here][1]][1]
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/iBvhk.... |
You can solve it with satisfiability. Here i have created the SAT formula for you. Sorry i deleted this answer earlier because i forgot to add a formula, i fixed it, it should be correct now.
[![enter image description here][1]][1]
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/gpQsn.... |
We use CYK algorithm to check whether the give string is present in the given context free language or not.
By using the CYK algorithm we can check the string belongs to given CFL or not.And also we can find the substrings of that string present in that language.
The process used in the CYK algorithm is
1.Co... |
We use CYK algorithm to check whether the give string is present in the given context free language or not.
By using the CYK algorithm we can check the string belongs to given CFL or not.And also we can find the substrings of that string present in that language.
The process used in the CYK algorithm is
... |
We use CYK algorithm to check whether the give string is present in the given context free language or not.
By using the CYK algorithm we can check the string belongs to given CFL or not.And also we can find the substrings of that string present in that language.
The process used in the CYK algorithm is
... |
Add p and q and make them as a single state.
Combine s and u because they Are final states.
We cannot combine t and r because on the same input they go to different states.
you will understand good when u see the minimized DFA given below
Now we obtain 4terms {(p,q),r,t,(s,u)}[![you can check the minimized DFA in t... |
Given,
M(Q,T,B,£,¥,q0,F)
T = Tape Alphabet = (a,b,X,B)
B = blank symbol
¥ = transition function
( To simplify I just wrote imps)
You can see the below link of image right!(reference image)
Click to see the solution.
PS: HOPE YOU LIKED THE ANSWER....
Don't forget to upvote;)[![reference image][1]][1]
... |
Initially build a finite automata of the language which generates only strings 11 and 111
And then by complementing the finite automata we will obtain the language which accepts all strings except 11 and 111
As we know that L(M)'=U-L(M)[![you can check the finite automata here][1]][1]
[1]: https://i.stack.imgu... |
We know that the post correspondence problem and the modified post correspondence problem are undecidable.
It means for both of the problems, we may or may not get the solution depending on the question.
There is no certain procedure or algorithm for the pcp and mpcp problem. So we may not solve the problem in s... |
I'm doing some Automata Theory exercises through a book and I'm trying to solve the exercise below but I can't figure out how to solve it.
Construct a regular expression that describe the following language given the alphabet $\Sigma = \{0, 1\}$,
$$A = \{w \in \Sigma^ * : |w|_0 = |w|_1 \}\cup \Sigma^*\{00,11\}\Sigm... |
Regular expression for words that have the same number of 0s and 1s or contain 00 or 11 |
Here we can combine p and q to form a single state because p on input 0 goes to q and q on input 0 goes to p and both p and q go to state s on input 1.In similar way we can combine states s and u because s goes to u on both inputs 0 and 1 and u stays in the same state on inputs 0 and 1.We can't combine r and t as they ... |
Here we can combine p and q to form a single state because p on input 0 goes to state q and q on input 0 goes to state p and both p and q go to state s on input 1.Similarly we can combine states s and u because state s goes to u on both inputs 0 and 1 and state u stays in the same state on inputs 0 and 1.We can't combi... |
Here we can combine p and q to form a single state because p on 0 as input goes to state q and q on 0 as input goes to state p and both p and q go to state s on 1 as input.Similarly we can combine states s and u because state s goes to u on both inputs 0 and 1 and state u stays in the same state on inputs 0 and 1.We ca... |
Yes...your intuition to the problem approach is correct.Here are the properties from which you can gain much more clarity:
[Unsolvable Problems Stanford-Stanford University][1]
[1]: https://www.google.com/url?sa=t&source=web&rct=j&url=https://web.stanford.edu/class/archive/cs/cs103/cs103.1142/lectures/21/Small2... |
Your approach to the answer is absolutely correct.
But you can make it in linear run time.
steps:
1.First check for the unreachable final states and remove them.
2.from now you can solve from your approach. |
I am a newbie to algorithm designing. I am working on user association and load balancing algorithm designing in millimeter wave wireless communications.
In this topic I am able to formulate the problem, but my query is how does one decide the algorithm after completing problem formulation.
Any help in this regar... |
How to design algorithm |
|algorithms|algorithm-design|combinatorial-design| |
I beleive modern cpus require a click source signal which is usually generated by a capacitor circuitry rather then a crystal. I Beleive crystals and other ics are used more for arm mcu. For CPU’s an llc circuit is used to deliver the clock source. |
If we take NFA with n states and convert into DFA with 2^n states then there are unreachable states in resulting DFA which gives the result as not a minimal DFA.so the NFA to DFA conversion thus not give the result ad minimal DFA. |
Matching is the same activity for regular expressions (and variants of them) as parsing is for grammars.
Top-down matching/parsing tries to expand parse trees for the input from the root downwards. Standard examples in compiler construction courses are *recursive descent* and *LL(k) parsing*.
With top-down parsin... |
To prove R is a equivalance relation then R must be reflexive,symmetric,transitive
1. Reflexive: for all q ∈ Q then there exists a ∈ Σ then δ(q,a)=δ(q,a).
2. Symmetric: for all q1,q2 ∈ Q, if then there exists a ∈ then δ(q1,a)=δ(q2,a) then there exists b∈Σ then δ(q2,b)=δ(q1,b).
3.Transitive: for all q1,q2,q3 ∈ Q, if... |
Consider the following algorithm:
```
Input: a natural number n
Output: the list of prime numbers ≤ n
lst ← []
for x from 2 to n do
if x is not a multiple of any number in lst then
append x to lst
return lst
```
Using the [prime number theorem](https://en.wikipedia.org/wiki/Prime_number_theorem),... |
null |
You start with simple algorithms, implement them successfully, and then you learn how to do more and more complex things. You need experience, and some built-in cleverness. Without both you will fail.
This is like asking a baker "how do you come up with new cake recipes" or an inventor "how do you invent new things... |
The language of strings containing either 00 or 11 is clearly regular; it's easy to recognise.
The language of strings containing the same number of 0's and 1's is not even regular. But the whole problem is a trick question: The trick is that you don't need the strings containing the same number of 0's and 1's, but... |
I am looking for an algorithm for the following problem:
I have a set of users and a set of books.
Every user has their own set of favorite books, which may be empty, and is a subset the set of books.
We can assume that every book is a favorite book of at least one user.
**Ideally I want to distribute ONE ... |
Algorithm for allocating resources; one resource per one user who accepts it |
|algorithms|sets|probabilistic-algorithms|protocols|optimal-strategy| |
I am looking for an algorithm for the following problem:
I have a set of users and a set of books.
Every user has their own set of favorite books, which may be empty, and is a subset the set of books.
We can assume that every book is a favorite book of at least one user.
**Ideally I want to distribute ONE ... |
I am looking for an algorithm for the following problem:
I have a set of users and a set of books.
Every user has their own set of favorite books, which may be empty, and is a subset the set of books.
We can assume that every book is a favorite book of at least one user.
**Ideally I want to distribute ONE ... |
I am looking for an algorithm for the following problem:
I have a set of users and a set of books.
Every user has their own set of favorite books, which may be empty, and is a subset the set of books.
We can assume that every book is a favorite book of at least one user.
**Ideally I want to distribute ONE ... |
Create a directed bipartite graph $G = (B+U, E)$, where $B$ is the set of books and $U$ is the set of users. The set $E$ contains an edge $(b,u)$ for every pair $(b,u) \in B \times U$ such that $b$ is one of the favorite books of $u$.
To find an optimal solution, you just need to find a [maximum cardinality matchin... |
Create a bipartite graph $G = (B+U, E)$, where $B$ is the set of books and $U$ is the set of users. The set $E$ contains an edge $(b,u)$ for every pair $(b,u) \in B \times U$ such that $b$ is one of the favorite books of $u$.
To find an optimal solution, you just need to find a [maximum cardinality matching](https:... |
I'm not sure if there's a name for this problem, but I'm hoping to validate my proposed solution or hear other ideas for solving it. This stems from a real engineering problem I've encountered at work, so a practical solution is preferred over a perfect one.
## The Problem
I have an infinite **queue** of items $I... |
Stateless item distribution |
|algorithms|greedy-algorithms| |
I'm not sure if there's a name for this problem, but I'm hoping to validate my proposed solution or hear other ideas for solving it. This stems from a real engineering problem I've encountered at work, so a practical solution is preferred over a perfect one.
## The Problem
I have an infinite **queue** of items $I... |
I'm not sure if there's a name for this problem, but I'm hoping to validate my proposed solution or hear other ideas for solving it. This stems from a real engineering problem I've encountered at work, so a practical solution is preferred over a perfect one.
## The Problem
I have an infinite **queue** of items $I... |
null |
Assume we have a balanced binary tree.
On _each_ node, we call:
```python
def node_depths(root, depth):
if not root:
return 0
return depth + node_depths(root.left, depth + 1) + \
node_depths(root.right, depth + 1)
```
`node_depths` itself is O(n) time. But calling it on _each_ node ... |
Why is calling an O(n) time algorithm on each node of a tree O(nlogn) time? |
|time-complexity|trees| |
I need to solve an optimization problem for a given set of [polyominoes][1], for example the five Tetrominoes known from Tetris. The goal is to place each one of the figures on the 2d grid, so the area they enclose is maximal.
Staying with our Tetromino example, the optimal solution would be:
[![Maximal area encl... |
Maximize enclosed area of given figures on 2d grid |
|algorithms|optimization|linear-programming|square-grid|tiling| |
It is not false that `node_depths` is $\mathcal{O}(n)$ time for a tree $T$ of size $n$, but it is imprecise.
To be more precise, `node_depths(x, d)` has a complexity $\Theta(|T_x|)$ where $T_x$ is the subtree rooted in $x$ (can you see why?).
To prove the $\mathcal{O}(n\log n)$ bound, I will consider $T$ to be a ... |
The alternative is to *not* rely on computers. While there are some instances where humans clearly outperform computers (looking at you, Tesla Autopilot), in the vast majority of cases where computers do something safety-critical, they also outperform humans by a significant margin (c.f., anti-lock brakes, traction con... |
**Claim One**: Let $2\le k\le n$. Then $s_n$ starts with $s_k$. In particular, $s_n$ for $n\ge3$ starts with $ab$.
Proof: straightforward induction on $n$ with fixed $k$.
------
**Claim Two**: Let $k\ge3$. $s_k$ ends with $ab$ if $k$ is odd. $s_k$ ends with $ba$ if $k$ is even.
Proof: straightforward indu... |
Extending the @D.W. answer, the approach to solve the imitation game will enlighten us on what intelligence is, or isn't. The experiment proposes a clear and objective way to progressing in solving intelligence, many previous researchers proposed tests to intelligence but when we solved those test we had no meaningful ... |
Here is one possible approach. Introduce boolean variables $x_\ell,y_\ell,z_{\ell,p,r}$, where $\ell$ ranges over locations in the grid, $p$ ranges over polyominoes, and $r$ over rotation amounts (0, 90, 180, 270), with the following meaning:
- $x_\ell$ is true means that $\ell$ is part of the interior region that ... |
null |
**Introduction**
Hi everyone,
I am currently pondering how to efficiently design an algorithm for a sorting problem and hope that someone here may help me out a bit.
**The problem**
I have two lists of integers, let's call them the 'input' list and the 'position' list, and want to sort the 'input' list base... |
Efficiently sorting a list of integers depending on the index of said integers in a second list |
|algorithms|sorting|efficiency|lists| |
Your problem can be solved in $O(\log k)$ time per query by augmenting layered range trees. Let me start with a slower solution first, and then I discuss how the time complexity can be improved.
Consider each segment $[s_i, e_i]$ as a $2$-dimensional point $(s_i, e_i)$ with an associated weight $w_i$. In your case $... |
Your problem can be solved in $O(\log k)$ time per query by augmenting layered range trees. Let me start with a slower solution first, and then I discuss how the time complexity can be improved.
Consider each segment $[s_i, e_i]$ as a $2$-dimensional point $(s_i, e_i)$ with an associated weight $w_i$. In your case $... |
Yes, it is a valid solution. Note that it is sufficient to check whether $L(D)\subseteq L(A)$. Also, note that since you can take D to be an accepting-sink, $q_{acc}$, then checking language containment by checking whether the product $\{q_{acc}\}\times A$ has a reachable state of the form $(q_{acc}, s)$, where $s$ is ... |
null |
null |
It is easy to use straightforward induction to prove the following two claims.
**Claim One**: $s_n$ starts with $s_k$ for $2\le k\le n$. In particular, $s_n$ starts with $ab$ for $n\ge3$.
**Claim Two**: $s_k$ ends with either $ab$ when $k$ is odd or $ba$ when $k$ is even for $k\ge3$.
------
**Claim Thre... |
|compilers|strings|exact-string-matching| |
I am using a Genetic Algorithm framework to solve a path-finding problem. Specifically, given the following 32x32 maze:
```
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
X | X
X | X
X | X
X | G X
X ... |
EvoPathfinding - Stuck in local optimal |
|optimization|genetic-algorithms|path| |
In Dual port RAM write and read operation can be performed independently from both ports from any location as long as there is no collision
Where as in Two port RAM port A is dedicated for read and port B is dedicated for write operation |
let's assume my Turing machine have infinite states.
in this post the claim is that every language is accepted:
https://cs.stackexchange.com/questions/49163/can-a-turing-machine-have-infinite-states
i understand that we can scan over the input from left to right once, and end up at a unique state for each possible... |
Turing machine with infinite states |
|turing-machines|computability|computation-models| |
Let's assume my Turing machine have infinite states.
in this post the claim is that every language is accepted:
https://cs.stackexchange.com/questions/49163/can-a-turing-machine-have-infinite-states
I understand that we can scan over the input from left to right once, and end up at a unique state for each possible... |
We say that dfs runs in $O(n+m)$ . For any connected graph $m \geq n-1 $.
Therefore :
$$m \geq n-1 \implies O(n+m) = O(m)$$
Do you agree ? Because, I have seen in many algorithms proofs this bound $O(n+m)$ but most of the times we are dealing with connected graphs . |
Complexity of DFS : O(m) |
|algorithms|complexity-theory|graphs| |
That's true for an undirected connected graph, but:
- sometimes, the undirected graph isn't connected;
- sometimes, it's a directed graph, and not all vertices are reachable from the starting vertex.
So $\mathcal{O}(n + m)$ (assuming $n = |V|$ and $m = |E|$ for a graph $G = (V, E)$) is always right, whereas $\... |
I want to ask about a turing-machine-like construct with an infinite number of states.
in this post the claim is that every language is accepted:
https://cs.stackexchange.com/questions/49163/can-a-turing-machine-have-infinite-states
I understand that we can scan over the input from left to right once, and end up a... |
I want to ask about a turing-machine-like construct with an infinite number of states.
in this post the claim is that every language is accepted:
https://cs.stackexchange.com/questions/49163/can-a-turing-machine-have-infinite-states
I understand that we can scan over the input from left to right once, and end up a... |
You can decide any language in linear time. Given a language $L \subseteq \Sigma^*$, consider the following Turing machine:
- Create a state for every word $w \in \Sigma^*$ and pick the tape alphabet $\Gamma$ as $\Sigma \cup \{ \bot \}$, where $\bot$ denotes the blank symbol. The initial state is $\varepsilon$.
- ... |
**Small input_list:**
h = {}
for x in input_list:
h[x] = 1
for x in position_list:
if h[x] == 1:
yield x
h[x]=2
Note I'm not using exact python syntax, "if h[x]==1" should handle the case where "x not in h" but you get the idea. Now, let's do elements pres... |
In essence, is kernel basically the code that implements a set of system calls? |
Is kernel essentially an implementation of system calls? |
|computer-architecture|operating-systems|os-kernel| |
null |
It's probably more accurate to think of an operating system is an implementation of a virtual machine.
System calls are part of that; they are operations of the virtual machine callable by programs. |
null |
null |
I'm building an implementation of the dynamo paper, [yottastore][1]. Given a key, I need to find which NVMe block stores the data. To do that I hash the key to find the shard where I have an in memory array in which at position [hash] I can find a struct with:
- 32 bit hash (needed for resizing)
- 64 bit pointer to... |
How to deal with a very big hash table? |
|trees|hash-tables| |
Another fitness function you could try is the following:
Consider the final point of the agent (if it crashes into a wall or outer bound, this the point where it crashed). Then use `(100 - manhattan_distance_from(G)) * penalty` as the fitness function, where penalty is 1.1 if it crashed into a wall or outer bound, ... |
|graphs| |
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