Questions tagged [time-complexity]

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

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15
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442 views

An algebra of complexity classes

A key feature of unrelativized computation is its composability out of smaller fragments, and to partially capture the composability, I came up with an algebra of fine-grained complexity classes. For ...
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523 views

Lock-free, constant update-time concurrent tree data-structures?

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 $\mathcal{O}(1)$ ...
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the largest element of a matrix product

Given two matrices, I'm interested in finding the largest element of their product. I wonder if it's possible to do it significantly faster than the matrix multiplication the solution seems to require?...
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245 views

How would proof of the Lindelöf hypothesis improve our understanding of computational complexity classes?

A recent press release from the Viterbi School of Engineering at USC discussed the proof of the Lindelöf hypothesis by Athanassios Fokas, a visiting professor from the Department of Applied ...
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530 views

Simple path on dag with backward edges

What is the complexity of the following problem ($\in$ P? NP-hard?): Input: a directed acyclic graph $D=(V,E)$, a set of backward edges $E'\subset V\times V$, and two distinct nodes $s$ and $t$. ...
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Is 4-in-a-row PSPACE-complete?

This paper by Laurens Kuiper shows that axis-parallel k-in-a-row is PSPACE-complete in complexity for k ≥ 5, but leaves the question open for k = 4. Has there been any research progress on this ...
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148 views

Time complexity of exponentiating s-sparse matrices

Could someone suggest me a reference which discusses the time complexity of algorithms meant for exponentiating (finding $e^A$ approximately given $A$) s-sparse matrices, along with their error rates? ...
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380 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 RAM....
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4k views

Time complexity of a branching-and-bound algorithm

Theoretical computer scientists usually use branch-and-reduce algorithms to find exact solutions. The time complexity of such a branching algorithm is usually analyzed by the method of branching ...
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327 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 $\Omega(n^k)...
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164 views

Cell probe model vs transdichotomous ram

can someone explain me the difference between those two (cell probe model and transdichotomous ram)? In cpm I'm allowed to do computation for free, and complexity of algorithm is just a number of ...
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1k views

Is there an ambiguity test for CFGs faster than trying all strings?

It is well known that testing whether a grammar is ambiguous is undecidable. It is however trivially decidable for any $G$ whether $L_n(G) := \{ w | w \in L(G) \wedge |w| \leq n \}$ for any $n \in \...
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251 views

For median is it optimal to compare in pairs first?

Median can be done in linear time and is now down to (I think) $2.97n$. The lower bounds is (I think) $(2+\epsilon)n$ where $\epsilon$ is very small. The following theorem, if true, may help improve ...
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243 views

How quickly can we find an arbitrary digit in multiplication?

In considering an answer to this question, I once again wondered how quickly we could find a digit in multiplication. We may first consider previous results. Finding the least significant digits is ...
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258 views

Complexity of reachability in Markov Chains

Is anything known about the complexity of the following problem beyond membership in PTIME: Given a finite Markov chain $M$, an initial state $q_0$ and a set $F$ of (absorbing) states, is the ...
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191 views

Nondeterminstic Linear Time vs Other Complexity Classes

Is it known whether or not nondeterministic linear time contains $P$ and/or smaller classes such as Uniform-$NC^1$?
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127 views

Evaluating addition chains

I hope this is a suitable place to ask this question. An addition chain of size $n$ is a sequence $x_1, \dots, x_n$, where $x_1$ is fixed to 1 and $x_i = x_j + x_k$ for some $j,k < i$. I am ...
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110 views

Small universal monotone Turing machines

This paper surveys small universal Turing machines. What are some examples of small universal monotone Turing machines, as described by Schmidhuber? Which of these are efficient (polynomial time) ...
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Looking for an easy/pedantic exposition of Renegar's famous result on polynomial optimization

In September $1989$, Renegar had this famous sequence of 3 papers titled, "On the Computational Complexity and Geometry of the First-order Theory of the Reals, Part I/II/III". I was wondering if ...
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681 views

Is this permutation-sum problem NP-complete?

A new, tighter tardiness bound has been found for global Earliest-Deadline-First scheduling of jobs on symmetric multiprocessors. But this bound seems to be particularly hard to compute. In particular,...
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103 views

Complexity of solving vs verifying in P

Thinking of (seemingly) very different complexity of finding a solution to a NP problem and verifying it as the basis of practical cryptography, I am wondering if such separation is possible among ...
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241 views

Restricted Reachability Problem

Let $G$ be a directed acyclic graph with $V$ vertices and $E$ edges. Choose some subset of $n\leq V$ "special" vertices $\{v_i\}_{i=1}^n$. How efficiently can we preprocess $(G, \{v_i\})$ so that we ...
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122 views

Program size versus program running time

Short "naive" question: Is it true that faster algorithms require longer programs ? Given a decision problem $A$ and a reasonable model of computation, there can be many ways (algorithms) ...
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166 views

Complexity of extracting a coefficient of a polynomial in multiple variables

I'm looking for efficient algorithms for problems of the following type: Let's say we have the variables $x_1,...,x_n$. Over these variables, we are given a function $p_1\cdot ... \cdot p_m$, ...
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278 views

Does Depth-First-Search admit a quasilinear time algorithm in mutitape Turing Machine model?

Depth-First-Search (DFS) has a quasilinear (i.e.,$\widetilde{O}(m+n)$) time algorithm in random access model (RAM). I am curious about whether DFS still admits a $\widetilde{O}(m+n)$ time algorithm in ...
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367 views

Do we know some quasi-polynomial problem that is known to not be in NP?

The title pretty much says it all, but to explain how I got there: I think, that one of the reasons we are unable to prove or disprove , but mainly disprove $P=NP$ (and yes, I was provoked by the ...
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804 views

Does L=P imply any new complexity class separations?

If L=P then P is not equal to PSPACE. This follows from PSPACE properly containing L. I am wondering if L=P implies any stronger separation between complexity classes? Does it imply P is properly ...
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1k views

Optimality of Greedy algorithm for minimization Knapsack Problem

Given items with weight $w_i$ and profits $p_i$, minimization Knapsack problem is to pick a subset of items $I$, s.t. $\sum_{i\in{I}}{w_i} \geq W$ and $\sum_{i\in{I}}{p_i}$ is minimized. The greedy ...
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76 views

Finding a largest symmetrical subset of a k-CNF propositional formula

I have a k-CNF propositional formula $F$ which do not admit any global symmetry i.e. there is no permutation $\sigma$ of its variables such that $\sigma(F) = F$ . I'm interested in finding the largest ...
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155 views

Is $L\subset NC^1$

Arora and Barak's online book claims in exercise 6.11 that $NC^1=L$. While the $NC^1\subset L$ direction is relatively straightforward and explained in many other texts, I couldn't prove or find the $...
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259 views

The problem of whether or not every function computable in time $T(n)$ is computable in time $T(n)^{O(1)}$ and space $T(n)^{o(1)}$ simultaneously

If a function is computable in time $T(n)$, is it computable in time $T(n)^{O(1)}$ and space $T(n)^{o(1)}$ simultaneously? We won't be able to prove it, because it implies the open problems $\text{P} ...
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452 views

What happens when PSPACE contains NEXP?

The complexity class NEXP is defined as the set of all languages that an arbitrary nondeterministic exponential time Turing Machine accepts (i.e. NTIME($2^{p(n)}$) for $p()$ a polynomial). In the ...
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166 views

Complexity of a naive algorithm for finding the longest Fibonacci substring

I already posted this question here but I didn't receive an answer, so I'm posting it here as well :) Given two symbols $\text{a}$ and $\text{b}$, let's define the $k$-th Fibonacci string as follows: ...
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607 views

Nondeterministic linear time vs. the deterministic time hierarchy

How much is known about nondeterministic linear time? I'm aware that $$ \mathrm{NTIME}(n) \neq \mathrm{DTIME}(n).$$ Is there an $m > 1$ so that $\mathrm{NTIME}(n) \not\subset \mathrm{DTIME}(n^m)$? ...
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127 views

Real-time countable vs fully time-constructible

Real-time countable functions were used in time hierarchy theorem in the papers of Hartmanis and Stearns (Theorem 9, 9.1 ...) and also of Hennie and Stearns (Theorems 3, 5, 7 ...). Now it is a "...
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176 views

Complexity of computing logarithm of a prime power

Suppose $n = p^k$ for some prime number $p$ and some non-negative integer $k$. What is (the best-known upper bound on) the complexity of computing $k$ on input $n$ (given in binary)? It is important ...
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287 views

Time complexity for solving linear congruences?

What is the best known algorithm to solve linear congruences of the form below? $$a x + b \equiv 0 \space (n)$$ And what is the time complexity of it?
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102 views

Graph problems in P with unknown lower bounds

I am looking for references to interesting graph problems, which are known to be in P, but their precise big-O lower bounds are elusive. I would split this into 2 classes: problems, where we know of ...
3
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61 views

Fastest known algorithm to enumerate k-cliques in a graph for fixed k

Is the best known algorithm for finding all $k$-cliques in a graph with $n$ nodes, for a fixed $k$, given by https://theory.stanford.edu/~virgi/combclique-ipl-g.pdf ? The time-complexity of the ...
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41 views

Time complexity of finding a point of infinite order on a rank 1 elliptic curve over Q

As an outsider, it sounds like a lot of progress has been made on understanding rank 1 elliptic curves over Q. Much of the BSD conjecture is known for rank 1, and Heegner points provide a way in ...
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66 views

Lower bound for reversing a list using queues

How do you prove (or disprove) that a list of length $n$ cannot be reversed in time $o(n \log n)$ using $O(1)$ queues? Each queue is FIFO. Time refers to the number of operations on the queues. ...
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270 views

Complexity of the mandelbrot set on rationals

(Also posted on mathoverflow) Given two rationals $a,b \in \mathbb{Q}$, call $c = a + ib$, i.e., the complex number represented by these two rationals. A point $c$ is contained within the Mandelbrot ...
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124 views

How does communication complexity relate to time complexity in distributed algorithms?

Some distributed algorithms (e.g. Bracha broadcast) runs in a constant number of rounds. I'm interested on how you'd analyse the time complexity of such algorithm, especially when the message size ...
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188 views

Implications of an $\tilde{O}(n^{1.5})$ 3XORSUM algorithm

Assume one had a (randomised or deterministic) algorithm with asymptotic complexity $\tilde{O}(n^{1.5})$ for the problem of finding $x,y,z\in L$ where $L$ is a list with $n$ binary vectors of ...
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136 views

Testing for satisfiability of a system of linear equations over GF(2)

Consider a system linear equations in $x$, $Ax =b$, where A is an $n\times n$ matrix, and $b$ is a column vector, and all operations are over $GF(2)$. Is it easier to check satisfiability of the ...
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78 views

First-order methods for solving SDP with geometric convergence or better

Is there any first-order method that can solve general SDP in a geometric (linear) rate? or super-geometric (super-linear) rate?
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162 views

Complexity of eigenvalue problem

Many matrix diagonalization algorithms have time complexity $\mathcal{O}(n^3)$ where $n$ is the number of columns/raws (consider only square matrices). What is the best time lower bound it is known? ...
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56 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 ...
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244 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 $...
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91 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 ...