instruction stringlengths 0 29.8k ⌀ |
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|algorithms|image-processing| |
null |
|algorithms|optimization| |
|terminology|data-mining| |
{"OriginalQuestionIds":[9063],"Voters":[{"Id":4249,"DisplayName":"Pål GD"},{"Id":1636,"DisplayName":"Luke Mathieson"},{"Id":98,"DisplayName":"Raphael"}]} |
{"OriginalQuestionIds":[9835],"Voters":[{"Id":3011,"DisplayName":"Paresh"},{"Id":6447,"DisplayName":"vonbrand"},{"Id":98,"DisplayName":"Raphael"}]} |
|combinatorics|linear-algebra| |
Does the difficulty of a strongly NP-hard or NP-complete problem (as e.g. defined [here](http://en.wikipedia.org/wiki/Strongly_NP-complete)) change when its input is unary instead of binary encoded?
What difference does it make if the input of a strongly NP-hard problem is unary encoded? I mean, if I take for instan... |
Plot the intervals for all the jobs (sorted by earliest finishing time) on the x axis. Start with y = 1.
If two jobs overlap, then plot the intervals of the later job on a higher y -- i.e., above the already plotted job interval. Stop when all jobs have had their intervals plotted. Sort the levels such that the bot... |
Seeing past the formulation issue pointed out in JeffE's answer, the answer is yes.
Consider the [Knapsack](https://en.wikipedia.org/wiki/Knapsack_problem) problem. It does have a [pseudo-polynomial](https://en.wikipedia.org/wiki/Pseudo-polynomial_time) algorithm, that is one with runtime bounded by a polynomial in ... |
I'm trying to figure out what are currently the two most efficent algorithms that permit, starting from a Left/Right pair of stereo images created using a traditional camera (so affected by some epipolar lines misalignment), to produce a pair of adjusted images plus their depth information by looking at their disparity... |
|algorithms|reference-request|image-processing| |
Consider the [interval scheduling problem](http://en.wikipedia.org/wiki/Interval_scheduling), see also [here](http://www.phailed.me/2012/08/interval-scheduling-problem/).
In order to schedule the $n$ job requests over one resource, you sort the requests in order of finish time, choose the request with earliest fini... |
|algorithms|optimization|scheduling| |
|regular-languages|automata|pumping-lemma| |
I was reading the proof of pumping lemma from Sipser's book. I couldn't understand certain things mentioned there.
![pumping lemma proof][1]
In the second paragraph he has written, "because $r_l$ occurs among first $p+1$ places, we have $l \le p+1$". Here, does $l$ denotes the number of states visited?
Also he... |
|complexity-theory|graphs|np-complete| |
The function is no longer one-way.
we build a counter example in the following way. Assume $g$ is a one-way function that preserve size, and define
$$f(bx_1x_2) = \begin{cases} g(x_1)\,x_2 & b=0 \\ x_1\, g(x_2) & b=1 \end{cases}$$
(assuming $b\in\{0,1\}$ and assuming $|x_1|=|x_2|$.)
It is easy to see that $f$ is ... |
{"Voters":[{"Id":157,"DisplayName":"Ran G."}]} |
I went to listen to a workshop and someone from the audience asked the presenter how the moments can improve the [mutual information](https://en.wikipedia.org/wiki/Mutual_information). I am learning about MI (Mutual Information) so didn't have enough knowledge to understand what it means. Then, I did some research but ... |
null |
> Does the local search algorithm start from one node in state space? Check if constraints are satisfied? If Yes, this is goal? If not, move to neighbours?
A local search would typically start from an arbitrary solution. For example in the case of SAT, each variable could be assigned either a 0 or a 1 with independe... |
Mapreduce does not seem to have received significant academic attention and/or widespread enthusiasm nor does it seem to have yet been integrated into much of the large previously-existing literature and theory on parallelism. This is partly due to it being relatively new, but there may be other reasons.
One possibl... |
The function is no longer one-way.
we build a counter example in the following way. Assume $g$ is a one-way function that preserves size, and define $f$ on input $w=bx_1x_2$ in the following way,
$$f(bx_1x_2) = \begin{cases} g(x_1)\,x_2 & b=0 \\ x_1\, g(x_2) & b=1 \end{cases}$$
(assuming $b\in\{0,1\}$ and $|x_1|=... |
Try [Latent Semantic Analysis](http://en.wikipedia.org/wiki/Latent_semantic_analysis) or the similar [Latent Semantic Indexing](http://en.wikipedia.org/wiki/Latent_semantic_indexing) which reduces documents to vectors, and typically finds principal components via [singular value decomposition, SVD](http://en.wikipedia.... |
I have some questions regarding local search / optimization as explained in chapter 4 of the book : http://aima.cs.berkeley.edu/
In classical search (Chap 3), the search starts from an initial node, then the search continues based on strategies of BFS, DFS, etc. What I am unsure of is the process of local search (Ch... |
I have some questions regarding local search / optimization as explained in chapter 4 of the book : http://aima.cs.berkeley.edu/
In classical search (Chap 3), the search starts from an initial node, then the search continues based on strategies of BFS, DFS, etc. What I am unsure of is the process of local search (Ch... |
null |
This is a Data structures & Algorithms question. For instance I have the following grades of functions: $O(1), O(2^n), O(n \log n), O(e^n), O(n^3), O(n^{1/3})$ and $O(\log \log n)$
I need to show and proof to which of these function grades does the function: $n^7$ belong to.
I didn't get a chance to ask my profe... |
Show that a function belongs to grade of incline |
|algorithms|data-structures| |
|complexity-theory|asymptotics|landau-notation| |
|algorithms|reference-request|parallel-computing| |
|computer-architecture|coding-theory|arithmetic| |
There are not enough details in the question. Yuval's answer gives the basic arithmetic answer. I'll just add that usually modems do not transfer the information "as-is" but encode it (see [Asynchronous serial communication](http://en.wikipedia.org/wiki/Asynchronous_serial_communication)).
The most common encoding ... |
I have a problem that I was able to conceptualize as following:
##Problem##
We have a set of n people. And m subsets representing their ethnicity like White, Hispanic, Asian etc. Given any combination of these people, I want to check if it is a diverse group.
A diverse group is a group that satisfies several req... |
Assignment problem with no cost |
|graphs|bipartite-matching|matching|assignment-problem| |
Consider the [interval scheduling problem](http://en.wikipedia.org/wiki/Interval_scheduling), see also [here](http://www.phailed.me/2012/08/interval-scheduling-problem/).
In order to schedule the $n$ job requests over one resource, you sort the requests in order of finish time, choose the request with earliest fini... |
Let $P$ be the set of people and $\mathcal{S}$ be the modified multi-set of subsets. By checking if a $P$-saturated matching into $\mathcal{S}$ exists (i.e. Hall's condition) you are indeed checking that your groups are diverse.
This can be done quickly via the Hopcraft-Karp algorithm (as you don't have weights ove... |
Let $P$ be the set of people and $\mathcal{S}$ be the modified multi-set of subsets. By checking if a $P$-saturated matching into $\mathcal{S}$ exists (i.e. Hall's condition) you are indeed checking that your groups are diverse. If you are "checking" if some group is a solution, simply make $P$ this group and solve.
... |
There are not enough details in the question. Yuval's answer gives the basic arithmetic answer. I'll just add that usually modems do not transfer the information "as-is" but encode it (see [Asynchronous serial communication](http://en.wikipedia.org/wiki/Asynchronous_serial_communication)).
The most common encoding ... |
null |
Let $P$ be the set of people and $\mathcal{S}$ be the modified multi-set of subsets. By checking if a $P$-saturated matching into $\mathcal{S}$ exists (i.e. Hall's condition) you are indeed checking that your groups are diverse. If you are "checking" if some group is a solution, simply make $P$ this group and solve. If... |
Consider the following problem. Given a $m \times n$ integer matrix $A$ and a $p \times q$ integer matrix $B$, do there exist one-to-one functions
$$r:\{1,2,...,m\} \rightarrow \{1,2,...,p\}$$
$$c:\{1,2,...,n\} \rightarrow \{1,2,...,q\}$$
where for all $1 \leq i \leq m$ and $1 \leq j \leq n$, $A[i,j] \leq B[r(i),c(... |
Prove Matrix Correspondence is NP-complete |
|np-complete| |
As you mentioned, we can reduce maximum clique problem to this. In fact we can reduce the more general [subgraph isomorphism problem](http://en.wikipedia.org/wiki/Subgraph_isomorphism_problem) to this one easily. Given graphs $G$ and $H$, define $A$ and $B$ as their (undirected) incidence matrices. Then we can see that... |
null |
Is induced subgraph isomorphism easy on an infinite subclass? |
|complexity-theory|np-complete|reductions| |
The term of $O(...)$ means representation of the performance of a solution. It is not entirely clear what you see under the expression n ^ 7. And because it is a representation of the complexity of the algorithm is needed to know how this power will count. There can be several calculation algorithms, each of which will... |
{"OriginalQuestionIds":[824],"Voters":[{"Id":157,"DisplayName":"Ran G."},{"Id":6447,"DisplayName":"vonbrand"},{"Id":472,"DisplayName":"Juho"},{"Id":96,"DisplayName":"Mohammad Al-Turkistany"},{"Id":667,"DisplayName":"sdcvvc"}]} |
Suppose, I have an array of length 100.
At alternate positions, I have 0s and 1s.
What is the ***fastest method*** by which I can separate the **0s** and **1s**, so that we get all the 0s at the beginning and all the 1s at the remaining positions / vice-versa? |
Fastest Algo to separate the 0s and 1s |
|program-optimization| |
i=0, j=size(ar)-1;
while(true)
increase i to position of first 1;
decrease j to position of first 0;
if(i>=j) break;
ar[i]=0, ar[j]=1; |
Suppose, I have an array of length 100.
I have 0s at some positions and 1s at some other positions.
What is the ***fastest method*** by which I can separate the **0s** and **1s**, so that we get all the 0s at the beginning and all the 1s at the remaining positions / vice-versa? |
A general trick that works for a lot of DP problems is to look at the parameters in your recursion -- the run time is then $O(n \times max(parameter_1) \times \dots \times max(parameter_k))$ where $n$ is the number of iterations (or, the number of inputs).
Consider for example a DP to solve the subset sum -- the rec... |
By the nondeterministic version of the [time-hierarchy theorem](http://en.wikipedia.org/wiki/Time_hierarchy_theorem), we have $\mathsf{NP} \subsetneq \mathsf{NEXP}$, where [$\mathsf{NEXP}$](http://en.wikipedia.org/wiki/NEXPTIME) is the class of problems solvable in non-deterministic exponential-time. Thus it suffices t... |
null |
Here is vonbrand's solution.
int C = 0, i;
for (i = 0; i < 100; i++)
C += A[i];
C = 100 - C;
for (i = 0; i < C; i++)
A[i] = 0;
for (; i < 100; i++)
A[i] = 1;
This solution has the advantage that we only scan the array forwards. Also, there are no conditionals.
The i... |
null |
I'm working on a type system and hit upon a problem that seems similar to lowest common ancestor. Given two types types, I need to find the smallest sequence of conversions which will result in the same target type. If I had a simple type tree I know how to get the result, but unfortunately I have a slightly more compl... |
Lowest common ancestor similar algorithm for a graph |
|algorithms|graphs| |
null |
A problem is a set of instances. We consider a problem "hard" if there are instances that require lots of resources to solve (time, space, brain cells, whatever). If a subset of the problem is hard, then the problem is hard. If a subset is easy, that doesn't mean the problem is easy (there might be hard instances elsew... |
A problem is a set of instances. We consider a problem "hard" if there are instances that require lots of resources to solve (time, space, brain cells, whatever). If a subset of the problem is hard, then the problem is hard. If a subset is easy, that doesn't mean the problem is easy (there might be hard instances elsew... |
I'm working on a type system and hit upon a problem that seems similar to lowest common ancestor. Given two types, I need to find the smallest sequence of conversions which will result in the same target type. If I had a simple type tree I know how to get the result, but unfortunately I have a slightly more complex gra... |
this is a question from the dragon book: (I apologize for translation mistakes, I don´t have the book in english language at hand.)
> Which languages are produced from this grammar?
> S -> a S b S | b S a S | € (empty String)
I don´t know what I´m supposed to do here. The definition in the book about langu... |
which languages are produced by this grammar? |
|context-free|compilers| |
This is a question from the Dragon book (I apologize for translation mistakes, I don´t have the English version on hand):
> What language is generated by this grammar?
> $S \rightarrow a S b S |b S a S | \epsilon$
I don´t know what I´m supposed to do here. The definition in the book about languages says this (... |
Finding the language generated by a context-free grammar |
Hint: Construct some words that are generated by this grammar. Do you see a pattern? Can you describe some properties of all words generated by the grammar, just by looking at the rules? Once you have a (correct) guess as to the language generated by the grammar, it won't be too difficult to prove it. |
Hint: What can you say about the number of $a$s and $b$s in the produced words?
> It would be very helpful if someone were able to provide some tips on
> solving these sorts of problems.
There is no one-size-fits all recipe here. It is undecidable in general, whether two CFGs produce the same language or two CFL... |
null |
Hint: What can you say about the number of $a$s and $b$s in the produced words?
> It would be very helpful if someone were able to provide some tips on
> solving these sorts of problems.
There is no one-size-fits-all recipe here. It is undecidable in general, whether two CFGs produce the same language or two CFL... |
This [blogpost][1] on lattice theory has a useful reference section, which contains among others "Lattice Theory with Applications" by Vijay K. Garg.
Chapter 2 "Representing Posets" describes some data structures for representing posets, and discusses how to compute join(x,y) using such a data structure.
The first ... |
If we regard a tree as a partial ordered set, it becomes a special case of a join-semilattice. For a join-semilattice, we want to be able to compute the (unique) least upper bound of two elements (more or less) efficiently. In the case of a tree, a data structure which would enable this would be to store for each eleme... |
> Are you able to determine what the algorithm or logic is contained within the neural network? Other than feeding in all possible inputs and studying the outputs it produces.
No, I don't think so, not in a meaningful way. That would be akin to studying the bits in each individual byte of a computer program in an e... |
If the lowest common ancestor of two types always exists and is unique, then your structure is a join-semilattice. Computing the lowest common ancestor is possible, but the worst case runtime complexity is not as good as I would have expected intuitively. I asked a [related question][1] some time ago, but was too lazy ... |
As encouraged in the comments, I'm giving a more general answer regarding constraint violation or constraint propagation in search in the context of CSP solvers. In a sense, the theory of search in CSP/SAT majorizes other approaches. My answer deals with complete search methods, although hybrids exist (combine local se... |
null |
|formal-languages|context-free|formal-grammars| |
Grammar for describing boolean expressions with AND, OR and NOT |
It is an almost copy of the grammar Wirth gives for arithmetic expressions in "Algorithms + data structures = programs" (as far as I can remember it; I might be wrong with the source, though). So I strongly believe it is correct.
Write the corresponding recursive descent parser, and try it on some correct strings, a... |
I'm working on a type system and hit upon a problem that seems similar to lowest common ancestor. Given two types, I need to find the smallest sequence of conversions which will result in the same target type. If I had a simple type tree I know how to get the result, but unfortunately I have a slightly more complex gra... |
Suppose we're receiving numbers in a stream. After each number is received, a weighted sum of the last N numbers needs to be calculated, where the weights are always the same, but arbitrary. How efficiently can this done if we are allowed to keep a data structure to help with the computation? Can we do any better than ... |
Weighed sum of last N numbers |
|algorithms|data-structures| |
There is a sentence with N words.
The words are randomly shuffled.
I have a heuristic algorithm that tries to restore the original order.
I want to evaluate my algorithm on a dataset of several hundred long sentences from the internet.
The main goal of the algorithm is that humans looking at the restored s... |
Evaluation metric for an ordering algorithm |
|algorithm-analysis|sorting|natural-language-processing| |
|algorithms|data-structures|online-algorithms| |
|natural-language-processing|string-metrics| |
null |
This is a question from the Dragon book (I apologize for translation mistakes, I don´t have the English version on hand):
> What language is generated by this grammar?
> $S \rightarrow a S b S \mid b S a S \mid \epsilon$
I don't know what I'm supposed to do here. The definition in the book about languages say... |
Suppose we're receiving numbers in a stream. After each number is received, a weighted sum of the last N numbers needs to be calculated, where the weights are always the same, but arbitrary. How efficiently can this done if we are allowed to keep a data structure to help with the computation? Can we do any better than ... |
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