Time complexity of decision problems or relations among time-bounded complexity classes. (Use [tag:analysis-of-algorithms] for the time taken by particular algorithms.)

learn more… | top users | synonyms

6
votes
3answers
186 views

Total orders which are the transitive closure of a set in P

I am wondering if there is an example of the following form. It seems highly plausible that there should be but I am struggling to come up with one. Consider $T \subseteq \mathbb{N}^2$, a set ...
-2
votes
0answers
71 views

Proof of computational difficulty of RSA factoring? [on hold]

I've been doing some reading on RSA and factoring algorithms. It seems the current fastest algorithm for factoring large (100+ digit) integers is the Number Field Sieve, but are there any proofs that ...
1
vote
0answers
23 views

Limits of parallel computing with local connections?

There are successes with an increasing numbers of individual computational units in GPUs or as processor cores. Given someone made the effort to build a huge array of processors which - however - can ...
2
votes
1answer
136 views

Max flow: either saturate an edge or avoids

Is there a way to create a max flow graph such that it satisfies the condition that a flow either saturates an edge or completely avoids it. It can't have half its flow through one edge and half ...
-1
votes
1answer
160 views

Is the running time of Boyer-Moore linear?

With pattern length $M$, text length $N$, and alphabet $\Sigma$, is the asymptotic running-time of Boyer-Moore $O(N/|\Sigma|)$ (even when $M$ grows larger than $|\Sigma|$)? Are there any sublinear ...
2
votes
1answer
151 views

Time complexity of d-dimensional convex hull

Consider the convex hull problem in $\Re^d$: Input: a list of $n$ points $S$ in $\Re^d$, Output: the vertices of the convex hull of $S$. What is the best lower bound on the time complexity of ...
1
vote
0answers
120 views

proving speedup phenomenon does not apply to any open complexity class separations

Aaronson recently wrote a blog refuting the idea that there could be some "glitch" in the formulation of the P vs NP conjecture[1] which reminds me of this following question. the Blum speedup ...
11
votes
3answers
1k views

Examples of problems where exponential algorithms run faster than polynomial algorithms for practical sizes?

Do you know of any problems (preferably at least somewhat well known), where, for a practical problem size, an exponential algorithm runs much faster than a best-known polynomial time counterpart. ...
1
vote
2answers
179 views

What is the asymptotic time complexity of the number of steps of “Half Or Triple Plus One” ( HOTPO)?

The "Half Or Triple Plus One" process goes as follows: start with $x=n$ for some value of $n$ if ($x$ is odd) $x = 3x+1$ else $x = \frac{x}{2}$ if ($x$ > 1) goto (2) ...
0
votes
0answers
49 views

Known time complexity advantage of quantum algorithms over classical algorithms [duplicate]

I know that this question may depend on how one formulates each complexity class, but in general, what time complexity advantage does quantum algorithms have over classical algorithms?
2
votes
0answers
39 views

Efficient Shamir secret sharing reconstruction

Shamir's secret sharing scheme is a well known way to convert a secret into a polynomial and distribute points in this polynomial. Some of these points can then be regrouped to reconstruct the ...
-1
votes
1answer
62 views

maximum weighted 2D coverage problem by a rectangle

Let $P=\{p_1,\ldots,p_n\}$ be a set of $n$ points in a 2D plane, that is $p_i\in \mathbb{R}^2$, $\forall i=1,\ldots,n$. Each point, $p_i$, is associated with a weight, $w_i \geq 0$. Imagine a ...
13
votes
4answers
326 views

What notable automaton models have polynomially-decidable containment?

I'm trying to solve a particular problem, and I thought I might be able to solve it using automata theory. I'm wondering, what models of automata have containment decidable in polynomial time? i.e. if ...
2
votes
1answer
115 views

Is there an additive time hierarchy theorem?

I would like something like this to be true: Conjecture: There is a function $g(n)$ such that for all functions $f(n)$ (perhaps satisfying some reasonable properties, like time-constructability), ...
4
votes
3answers
113 views

Remove unneeded atoms in CNF minimalization (SAT preprocessing)

This might be a very basic question. I am interested in all atoms of a propositional formula that can be removed from a particular formula, while the derived formula has the same satisfiability ...
4
votes
3answers
366 views

Is it possible to optimize the calculation of $ax+b$ once I know $a$ and $b$?

An "algorithm" for calculating $ax+b$ would take the steps Calculate $a$ times $x$ Calculate $b$ plus the result of previous line. But if the values of $a$ and $b$ are known, can we create a more ...
8
votes
0answers
202 views

Linear space language that requires exponential time without ETH

The $\mathsf{P} \neq \mathsf{PSpace}$ conjecture means that There is a language $L \in \mathsf{DSpace}(O(n^t))$ for some $t>0$ such that for all positive integers $k$, $L$ requires ...
3
votes
1answer
75 views

Example of context-free grammar that triggers exponential behaviour without memoization in RD parsers

It is often said that memoization brings the complexity of recursive-descent parsers from exponential to polynomial. However, I had a hard time finding an example grammar that triggers the exponential ...
1
vote
1answer
87 views

Why does the construction step of Aho-Corasick take linear time in the number of nodes?

The original paper's analysis of this, as far as I can tell is this: "THEOREM 3. Algorithm 2 requires time linearly proportional to the sum of the lengths of the keywords. PROOF. Straightforward." ...
8
votes
1answer
156 views

Multidimensional arithmetic progression variant

For $\vec{d} \in \mathbb{N}^n$, let $Q(\vec{d}) \subset \mathbb{N}^n$ be the set of vertices of the $n$-dimensional cube scaled in the direction of the $i$-th coordinate by $d_i$, i.e. $Q(\vec{d} = ...
2
votes
2answers
102 views

Number of SAT checks that are needed to find all combinations of subset of boolean variables of a propositional formula

Please mind that I sometimes lack formal mathematical knowledge and English is not my first language, so I might miss the right words. Please change the tile if needed. Also, I have choosen this site ...
6
votes
1answer
203 views

Distinguishing between two coins

It is well known that the complexity of distinguishing an $\epsilon$ biased coin from a fair one is $\theta(\epsilon^{-2})$. Are there results for distinguishing a $p$ coin from a $p+\epsilon$ coin? I ...
1
vote
2answers
250 views

Lower-bound of a decision problem [closed]

What's the lower-bound of the decision problem that decides: Whether there is at least one element A[i] such that A[i] = i in a sorted array A of non-negtive integers? (An example is A = ...
0
votes
0answers
20 views

What's the official name for time-driven parallel informed sorting methods?

If you have a set $\Delta = \{A,B,C,D,E,...\}$ and you want to sort $\Delta$ based on a certain quality of the elements $\in \Delta$, then you can use a certain classification or ranking algorithm $R$ ...
0
votes
0answers
208 views

Most efficient algorithm to search an unsorted array with a very precise data structure

(I apologize in advance if this question sounds a bit practical, but I suspect it might have an interesting theoretical aspect.) I have a (large) array of data, not completely sorted, but with which ...
9
votes
0answers
185 views

Does Kannan's theorem imply that NEXPTIME^NP ⊄ P/poly?

I was reading a paper of Buhrman and Homer “Superpolynomial Circuits, Almost Sparse Oracles and the Exponential Hierarchy”. On the bottom of page 2 they remark that the results of Kannan imply that ...
8
votes
1answer
167 views

Consequences of nondeterminism speeding up deterministic computation

If $\mathsf{NP}$ contains a class of superpolynomial time problems, i.e. for some function $t \in n^{\omega(1)}$, $\mathsf{DTIME}(t) \subseteq \mathsf{NP}$, then if follows from the ...
2
votes
0answers
177 views

Does $P\neq NP$ imply any larger separation?

I've asked a similar question in cs.se, but didn't get a satisfying answer. Assuming $P\neq NP$, what can we say about the runtime of any algorithm for an $NP$-complete problem? Obviously, it means ...
4
votes
0answers
118 views

Deciding transitivity of a directed acyclic graph [duplicate]

Is there any algorithm that decides whether a given directed acyclic graph is transitive or not, in time-complexity asymptotically better than boolean matrix multiplication?
15
votes
3answers
969 views

How much time to recognize palindromes in logarithmic space?

It is well-known that palindromes can be recognized in linear time on $2$-tape Turing machines, but not on single-tape Turing machines (in which case the time needed is quadratic). The linear-time ...
4
votes
1answer
62 views

Complexity of generically inverting a class of near linear monotonic functions

Given a monotonic increasing function $f(\mathbb{N}) \rightarrow \mathbb{N}$ and a slack function $a(\mathbb{N})\rightarrow \mathbb{N}$, where $f(n) = n \pm O(a(n))$; how many calls to $f$ do we ...
2
votes
1answer
165 views

Subset Sum bounds

Are there any bounds for the subset problem with respect to the number of the terms involved in the sum and the range of the possible values?For example looking for some 2 terms whose sum equals 3 you ...
2
votes
1answer
117 views

Complexity of determining unique elements of each cycle in a permutation

It is a well known fact that a permutation is a set of cycles, and that one can find all cycles of a permutation in $O(n)$ time, where $n$ is the length of the permutation. But suppose that we know ...
7
votes
3answers
349 views

Can one automate algorithmic analysis?

Has anyone thought about the possibility of a programming language, and a compiler, such that the compiler can automatically do worst-case asymptotic analysis? The use case I have in mind is a ...
9
votes
0answers
314 views

Fundamental assumptions in complexity analysis

I am a software engineer and I need a bit of clarification. The practical performance of algorithms is usually compared against models where arithmetic and dereferencing are instantaneous, such as ...
7
votes
1answer
134 views

What is the fastest algorithm for Weighted Planar Max Cut

In 1990's, the paper Unifying Maximum Cut and Minimum Cut of a Planar Graph described an $O(n^{3/2}\log n)$-time algorithm for Weighted Planar Max Cut, a landmark in the field. Recently, this paper ...
2
votes
0answers
89 views

How is the complexity of PCA $O(\min(p^3,n^3))$?

I've been reading a paper on Sparse PCA, which is: http://stats.stanford.edu/~imj/WEBLIST/AsYetUnpub/sparse.pdf And it states that, if you have $n$ data points, each represented with $p$ features, ...
5
votes
1answer
108 views

Computational Complexity of the Divisor summatory function

The divisor function $d(n)$, is the number of $(a,b)\in\mathbb {N^+}^2$ such that $a\times b =n$. For example, $d(2)=2$ because $2=1\times 2=2\times 1$ and d(6)=4 because $6=1\times 6=2\times ...
1
vote
1answer
84 views

Calculation of Recursive Spawning

I'm reading the book Introduction to Algorithms (Cormel et al., 2009) on the chapter about multithreaded algorithms, and I'm confused about the following: We must also account for the overhead of ...
1
vote
2answers
229 views

Task Multithreading analysis for a Divide and Conquer algorithm

Suppose an algorithm that receive an input array of $n$ elements and it performs a task over each element. All tasks are independent and take $O(k)$ each (being $k$ a variable). Since all tasks are ...
11
votes
3answers
452 views

Can we compute $n$ from the bits of $3^n$ in $O(n)$ time?

I'm seeking an efficient algorithm for the problem: Input: The positive integer $3^n$ (stored as bits) for some integer $n \geq 0$. Output: The number $n$. Question: Can we compute $n$ from ...
-1
votes
1answer
450 views

Time complexity analysis of random forest and k-means?

I am working with random forest for a supervised classification problem, and I am using the k-means clustering algorithm to split the data at each node, where $n$ is the number of points, $K$ is ...
6
votes
2answers
441 views

Is there any task where classical computers outperform quantum computers?

Everybody knows that there are whole classes of problems which quantum computers are able to solve much faster (i.e. with fewer instructions) than classical computers. Is there any problem for which ...
-2
votes
2answers
676 views

Compute Time Complexity of Neural network, SVM and other classification algorithms

I would like to know what is the asymptotic time complexity analysis for general models of Back-propagation Neural Network, SVM and Maximum Entropy. Does it just depend on number of features included ...
1
vote
0answers
145 views

Time complexity of clustering based on random walk

What is the time complexity of the following algorithm (from this paper suggested by Zhou) to partition directed graph? Can I use the complexity of eigen vector computation for this purpose? The ...
3
votes
0answers
81 views

Constant time search for rod segment

(I tried to ask at SO but maybe this has more to do with the CS theory.) Suppose I have a rod which I cut to pieces. Given a point on the original rod, is there a way to find out which piece it ...
-1
votes
1answer
788 views

Flood fill vs depth first search

Is the flood fill algorithm the same as depth first search? If not, how do they differ in complexity?
20
votes
6answers
825 views

Advanced techniques for determining complexity lower bounds

Some of you may have been following this question, which was closed due to not being research level. So, I'm extracting the part of the question which is at a research level. Beyond the "simpler" ...
4
votes
0answers
271 views

How are lower bounds for computational complexity proved? [closed]

As the title says, how are lower bounds for computational complexity proved? I'm interested why it is possible to say that a certain algorithm cannot run in faster time than some lower bound. ...
1
vote
0answers
90 views

Can emptiness of reversal-bounded counter languages be decided in time polynomial to the number of counters?

I was reading this paper, about the complexity of decision problems for reversal bounded counter machines. I got to Theorem 1 on Page 6. The theorem shows that there's a log-space NTM which can ...