Questions tagged [space-complexity]

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22
votes
12answers
2k views

What are some algorithms where space complexity tends to be the limiting factor in practice?

Time complexity can't be any lower than space complexity (at least one operation is required to use a unit of memory), so what are some algorithms where space actually tends to be the limiting factor? ...
22
votes
2answers
865 views

Best current space lower bound for SAT?

Following on from a previous question, what are the best current space lower bounds for SAT? With a space lower bound I here mean the number of worktape cells used by a Turing machine which uses a ...
21
votes
3answers
3k 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 ...
19
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2answers
926 views

What is the space complexity of calculating Eigenvalues?

I am looking for a survey paper or a book covering results about the space complexity of common linear algebra operations such as matrix rank, eigenvalues calculation, etc. I stress the "space ...
17
votes
1answer
945 views

Quadratic relationship between nondeterministic and deterministic space?

Savitch's theorem shows that $\mathrm{NSPACE}(f(n)) \subseteq \mathrm{DSPACE}(f(n)^2)$ for all large enough functions $f$, and proving that this is tight has been an open problem for decades. Suppose ...
14
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1answer
741 views

Is Quasi-polynomial time in PSPACE?

I had done some search on this but I was not able to find an answer either way. Huck answered it fully. Thanks :)
14
votes
1answer
340 views

Space-approximation Trade-off

In their paper Approximate Distance Oracles, Thorup and Zwick showed that for any weighted undirected graph, it is possible to construct a data structure of size $O(k n^{1+1/k})$ that can return a $(...
13
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0answers
382 views

Non-deterministic logspace with two-sided error

The class BPL is the set of all problems solvable by a Turing machine running in logarithmic space and polynomial time with two-sided error; that is, if $x\in L$ then the machine accepts with ...
11
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1answer
285 views

Does the space hierarchy theorem generalize to non-uniform computation?

General Question Does the space hierarchy theorem generalize to non-uniform computation? Here are a few more specific questions: Is $L/poly \subsetneq PSPACE/poly$? For all space ...
10
votes
1answer
296 views

Is bounded-width SAT decidable in logspace?

Elberfeld, Jakoby, and Tantau 2010 (ECCC TR10-062) proved a space-efficient version of Bodlaender's theorem. They showed that for graphs with treewidth at most $k$, a tree decomposition of width $k$ ...
8
votes
2answers
345 views

The hardness of generating an instance of a problem that is harder than the complexity of the resulting problem

In the movie Inception Cobb asks a asks Ariadne to design a maze that takes twice as much time to design. This lends itself to a generalized problem where we have an situation where we are resource ...
8
votes
1answer
142 views

Savitch's use of measurability

In Savitch's 1969 paper, "Relationships Between Nondeterministic and Deterministic Tape Complexities", he states that "all common storage functions L(n) >= lg n are measurable. In particular, any ...
8
votes
1answer
120 views

Can Quarter-Subset Membership be decided space-efficiently?

Consider the following decision problem. Let $q = \sum_{i=0}^{n/4} \binom{n}{i}$ and let $(C_0^n, C_1^n,\dots,C_{q-1}^n)$ be a suitable enumeration of those subsets of $\{0,1,\dots,n-1\}$ that have at ...
8
votes
1answer
290 views

Storing a bit vector in uninitialized memory and minimal space

A well-known trick for storing bit vectors using uninitialized memory can allocate a bit vector of size $n$ in which all of the bits are set to $0$ by allocating $(2 n + 1)\lceil \lg n \rceil$ bits of ...
8
votes
0answers
336 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)...
7
votes
2answers
342 views

Are space and time hierarchies even comparable?

I am wondering if there are any results to what extent the space and time hierarchies "disagree" on which problem is harder. For example, is it known whether there are languages $L_1$ and $L_2$ such ...
7
votes
0answers
169 views

Range min-gap query

The min-gap of an array $A[1..n]$ of $n \ge 2$ elements is defined as $\min_{1 \le i < j \le n}{|A_i - A_j|}$. Now, I am considering a query version of it. Given $A$, a query receives two integers $...
6
votes
1answer
2k views

k-Vertex Cover problem is in parameterized Log space

$k$-Vertex Cover: Given a graph $G = (V, E)$ where $V$ is a set of vertices and $E$ a set of edges, and an integer $k$, the $k$-Vertex Cover problem determines if there exists a subset of vertices $...
6
votes
1answer
160 views

Is $L \subset 1NL$ when $L \neq NL$?

A log-space Turing machine has a read-only input tape, a write-only output tape and uses at most $O(\log n)$ space in its read-write work tapes. The classes $L$ and $NL$ contain those languages which ...
6
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0answers
155 views

Immutable Space Model

I have heard it said that time is more precious than space because we can reuse space but not time. What if we treat space with this much reverence? What is generally known about models of ...
6
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0answers
364 views

PSPACE completeness, with different kinds of reductions

PSPACE-complete$_{FP}$ problems are the PSPACE problems such that every other PSPACE problem can be transformed to it with a polynomial time reduction, i.e. the reduction is an algorithm $\in$ FP. ...
5
votes
1answer
265 views

Is the primality problem with unary input NLOGSPACE-Hard?

Consider the language $L=\{1^n : n \text{ is prime}\}$. Is this language NLOGSPACE-Hard? The motivation for this question is that $L$ is a good candidate for reducing to other languages related to my ...
5
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0answers
612 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 ...
5
votes
0answers
185 views

On space complexity of permanent modulo $2^t$?

We know from here that permanent of $0/1$ matrix modulo $2^t$ is in $DTIME(n^{t+3})$ and hence in $P$. My question is whether permanent of $0/1$ matrix modulo $2^t$ is in $L$ as well or is the current ...
5
votes
0answers
845 views

Any known connections between open problems for time and space: P vs L, NP vs NL, BPP vs BPL, ⊕P vs ⊕L

It would be nice to show that $P=L$ implies $NP=NL$. Or, $NP=NL$ implies $UP=UL$. Or maybe, $⊕P = ⊕L$ implies $PP = PL$. Are there any known connections between the problems: P vs L, UP vs UL, NP ...
5
votes
0answers
877 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 ...
4
votes
1answer
178 views

Is $UL\neq PSPACE$ known?

$L\neq DSPACE[\omega(\log n)]$ is known. Is $UL\neq DSPACE[\omega(\log n)]$ and $UL\neq PSPACE$ known?
4
votes
1answer
189 views

On complexity class $\mathsf{\Pi_2 L}$

I suggest the following definition of $\mathsf{\Pi_2 L}$ (similarly to the certificate definition of $\mathsf{NL}$): A language $L$ belongs to $\mathsf{\Pi_2 L}$ iff there exists a deterministic ...
4
votes
1answer
367 views

state-of-the-art bit complexity of the determinant

I'm trying to understand the full bit-complexity of computing the determinant of an $n\times n$ integer matrix, with each entry represented by $M$ bits. I would like to know what is the state-of-the-...
4
votes
1answer
311 views

Satisfiability for various branching programs

Clearly, SAT for CNF, formula, or many classes of boolean circuits have strong significance on the entire theory community. Branching program, or BDD is one of the most basic and popular computational ...
4
votes
2answers
169 views

Function with space-depending computation time

Does a function exist which is easily computable for one space capacity and is hard to compute for another? I am looking for a function which can be computed in polytime when available space is at ...
4
votes
1answer
750 views

Space complexity for multiplying $m$ matrices

Suppose you have $m$ $n$ by $n$ matrices $M_1,M_2,\dotsc,M_m$, and you want to calculate their product $\prod_{i=1}^{m} M_i$. The naive method use $m \cdot poly(n)$ times but needs $poly(n)$ memory. ...
4
votes
1answer
236 views

Low space computation and branching program

One of the most elemental result of relationship between boolean circuit size and polynomial uniform computation is Pippenger and Fishers simulation: $DTIME[T(n)]\subseteq SIZE[T(n)\log T(n)]$. I ...
4
votes
0answers
94 views

Relationship between SC and NL

It is a major open problem whether $NL \subseteq SC$, or equivalently, whether directed reachability can be solved (simultaneously) in poly-logarithmic space and polynomial time. What is known ...
4
votes
0answers
263 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} ...
4
votes
0answers
180 views

Is there a language in NSPACE(O(n)) and (very likely) not in DSPACE(O(n))?

Actually I found that the set of context-sensitive Languages, $\mathbf{CSL}$ ($\mathbf{=NSPACE}(O(n))$ $= \mathbf{LBA}$ accepted languages) are not so widely discussed as $\mathbf{REG}$ (regular ...
3
votes
2answers
190 views

Nonregular Languages in Sublogarithmic Space

Are there any known languages that require $o(\log n)$ space but are also not regular? I'm looking for a problem "just below" undirected connectivity.
3
votes
1answer
2k views

Computational complexity of classifying with an already-trained SVM

If I have a support vector machine which has already been trained, what is the computational complexity of classifying a new example using that machine? I care about both time and space complexity. ...
3
votes
1answer
126 views

Detect if a graph has a $k$ cycle in space complexity $O((\log k)^d)$ for fixed $d \geq1$

For a graph $G$, I want to test if it contains a cycle of length $k$, for some $k$ much smaller than $|G|$. I am interested in particular in an algorithm with low space complexity. The cycle need not ...
3
votes
1answer
123 views

Family of functions with properties similar to k-wise independent hash functions

I am looking for a family of functions that has similar properties to a family of $\ell$-wise independent hash functions. The goal is to hash $\ell$ pairwise different bit strings of length $k$ to a ...
3
votes
1answer
268 views

Space complexity of integer programming

Given $A\in\Bbb Z^{n\times k}$, $v\in\Bbb Z^n$ and variables $x_1,\dots,x_k$ given as a vector $x$ we know that solving $x\in\Bbb Z^k$ in $Ax\leq v$ is fixed parameter tractable. There is a ...
3
votes
0answers
88 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 ...
3
votes
0answers
219 views

What is consequence of $PH\subseteq NSPACE((\log n)^2)$?

What is consequences of $PH\subseteq NSPACE((\log n)^2)$? We don't even know PH is equals to L or not. I am wondering what will be happened when $PH\subseteq NSPACE((\log n)^2)$?
3
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0answers
324 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 ...
2
votes
1answer
322 views

What is the space complexity of CTL model checking?

What is the space complexity of the CTL model checking algorithm via labeling without fairness (see e.g. Model Checking by Clarke at al Section 4.1 or Principles of Model Checking by Baier et al ...
2
votes
1answer
261 views

Space requirements for solving True Quantified Boolean Formulas problem [closed]

I came across this section on the wikipedia page for the TQBF solving problem, and just can't wrap my head about the fact that the space requirement is linear. Moreover, it does not provide any ...
2
votes
1answer
534 views

What is best known space requrement for solving SATISFIABILITY problem in exp time

I searched a lot for finding best space requirement algorithm for SATISFIABILITY problem but I didn't find any thing better than brute force that is in DSPACE(n). is there exists better bound? and ...
2
votes
1answer
101 views

Is sliding blocks linear space complete?

Sliding blocks is PSPACE complete even in its simplest form involving 1x2 and 2x1 blocks (without rotation or fractional positions) in a rectangular area, with goal being to move a designated block to ...
2
votes
0answers
87 views

Simultaneous time-space complexity of directed reachability on bounded diameter graphs

Is directed reachability on a graph with bounded diameter (e.g. diameter $O(n^{1/2})$) known to be solvable simultaneously in polynomial time and sublinear space? Is anything known if the diameter is ...
2
votes
0answers
175 views

Is $SUBLOG\subset DTIME(n)$?

In the course of trying to give a more natural answer to a previous question of mine involving the complexity classes $$SUBLOG=\{L\mid L \text{ is recognizable by a sublogarithmic-space TM} \}$$ and $...