Questions tagged [space-bounded]

Questions about space resources of computations in computational complexity or algorithms.

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51
votes
4answers
3k views

Why do we consider log-space as a model of efficient computation (instead of polylog-space) ?

This might be a subjective question rather than one with a concrete answer, but anyway. In complexity theory we study the notion of efficient computations. There are classes like $\mathsf{P}$ stands ...
33
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2answers
1k views

Does LOGLOG = NLOGLOG?

Define LOGLOG as the class of languages which can be computed in space O(loglog n) by a deterministic Turing machine (with two-way access to the input). Similarly define NLOGLOG as the class of ...
31
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1answer
875 views

Treewidth and the NL vs L Problem

ST-Connectivity is the problem of determining whether there exists a directed path between two distinguished vertices $s$ and $t$ in a directed graph $G(V,E)$. Whether this problem can be solved in ...
28
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2answers
845 views

Tight Lower bounds on Savitch's theorem

First of all, I apologize in advance for any stupidity. I am by no means an expert on complexity theory (far from it! I am an undergraduate taking my first class in complexity theory) Here's my ...
26
votes
3answers
743 views

Intermediate problems between L and NL

It is well-known that directed st-connectivity is $NL$-complete. Reingold's breakthrough result showed that undirected st-connectivity is in $L$. Planar directed st-connectivity is known to be in $UL \...
25
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4answers
2k views

Separating Logspace from Polynomial time

It is clear that any problem that is decidable in deterministic logspace ($L$) runs in at most polynomial time ($P$). There is a wealth of complexity classes between $L$ and $P$. Examples include $NL$,...
23
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1answer
867 views

Is there any justification to believe that $NL\neq L$?

I wonder if there is any justification to believe that $NL=L$ or to believe that $NL\neq L$? It is known that $NL \subset L^2$. The literature on derandomization of $RL$ is pretty convincing that $RL=...
23
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1answer
864 views

Logspace algorithms on graphs with bounded tree width

Tree width measures how close a graph is to a tree. It is NP-hard to compute tree width. The best known approximation algorithm achieves $O(\sqrt{{\log}n})$ factor. Courcelle's theorem states that ...
22
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2answers
846 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
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1answer
1k views

Alternate proofs of Immerman-Szelepcsenyi theorem

Immerman and Szelepcsenyi independently proved that $NL=coNL$. Using their technique of inductive counting, Borodin et al proved that $SAC^i$ is closed under complementation, for $i > 0$. Prior to ...
21
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1answer
409 views

Can $\{a^nb^n\}$ be recognized in poly-time probabilistic sublogarithmic-space?

Consider language $ \mathtt{EQUALITY} = \{ a^nb^n \mid n \geq 0 \} $. It is known that $ \mathtt{EQUALITY} $ cannot be recognized by any sublogarithmic-space alternating Turing machine (ATM) (...
20
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3answers
490 views

space-bounded TMs and oracles

In general, the query-tape for an oracle counts towards the space-complexity of a TM. However, it seems plausible to allow a write-only oracle-tape (such as is used in L-space reductions). Is such a ...
19
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1answer
693 views

How to prove that USTCONN requires logarithmic space?

USTCONN is the problem that requires deciding whether there is a path from the source vertex $s$ to the target vertex $t$ in a graph $G$, where these are all given as part of the input. Omer Reingold ...
18
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4answers
863 views

If P = BQP, does this imply that PSPACE (= IP) = AM?

Recently, Watrous et al proved that QIP(3) = PSPACE a remarkable result. This was a surprising result to myself to say the least and it set me off thinking... I wondered what if Quantum Computers ...
18
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3answers
677 views

CFG parsing using $o(n^2)$ space

There are a multitude of algorithms that can parse a context-free grammar in $O(n^3)$ time. Using matrix multiplication, one can even go asymptotically faster than that. However, all algorithms for ...
18
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2answers
596 views

Storage requirements for median selection (two passes algorithms)

In a classic paper Munro and Paterson study the problem of how much storage is required for an algorithm to find the median in a randomly sorted array. In particular they focus on the following model:...
18
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1answer
792 views

Context-sensitive grammar for SAT?

By a classic result of Kuroda, the complexity class NSPACE[$n$] (also known as NLIN-SPACE) is precisely the class CSL of context-sensitive languages. The satisfiability problem SAT is in NSPACE[$n$], ...
17
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3answers
2k views

Efficient logspace algorithms

It is easy to see that any problem that is decidable in deterministic logspace ($L$) runs in at most polynomial time ($P$). Many known logspace algorithms (For example : undirected st-connectivity, ...
17
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1answer
515 views

Fooling arbitrary symmetric functions

A distribution $\mathcal{D}$ is said to $\epsilon$-fool a function $f$ if $|E_{x\in U}(f(x)) - E_{x\in \mathcal{D}}(f(x))| \leq \epsilon$. And it is said to fool a class of functions if it fools every ...
17
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1answer
351 views

Which results make quantum space interesting?

Time-bounded quantum computation is obviously very interesting. What about space-bounded quantum computation? I know many interesting results for quantum computation with sublogarithmic space bounds ...
16
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1answer
936 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 ...
16
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1answer
522 views

Does Nisan's pseudo-random generator relativize?

Nisan proved in "Psuedorandom Generators for Space-Bounded Computation", that there exists a pseudo-random generator which "fools" space-bounded computations. Does this construction hold for every ...
16
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1answer
772 views

Completeness and Context-Sensitive Languages.

I'm interested in two questions regarding context-sensitive languages (CSL) and completeness: Is there a notion of completeness for CSL, and which languages are complete? Are there natural CSL that ...
15
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2answers
967 views

Quantum analogues of SPACE complexity classes

We often consider complexity classes where we are bounded in the amount of space our Turing machine can use, for example: $\textbf{DSPACE}(f(n))$ or $\textbf{NSPACE}(f(n))$. It seems that early in ...
15
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2answers
682 views

SC^2 algorithms for st-connectivity

Savitch gave a deterministic algorithm to solve st-connectivity using $O({\log}^2{n})$ space, implying $NL \subseteq DSPACE({\log}^2{n})$. Savitch’s algorithm runs in time $2^{O({\log}^2{n})}$. It is ...
14
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1answer
297 views

Machine characterization of $SAC^i$

$SAC^i$ is the class of decision problems solvable by a family of $O({\log}^i{n})$ depth circuits with unbounded-fanin OR and bounded-fanin AND gates. Negations are only allowed at the input level. It ...
13
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2answers
1k views

Space alternating hierarchy

It is known thanks to Immerman and Szelepcsényi that ${\rm NSPACE}(f)={\rm coNSPACE}(f)$ if $f=\Omega(\log)$ (even for non-space constructible functions). In the same paper, Immerman state that the ...
13
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0answers
313 views

Consequences of bipartite perfect matching not in NL?

Are any significant consequences known of $\text{BPM} \not\in \textsf{NL}$? I'm interested in the status of the following well-studied decision problem, in particular whether it is known to be in $\...
13
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0answers
374 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 ...
13
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0answers
301 views

Is there any known nontrivial result on QIP systems having a space-bounded verifier?

Is there any known nontrivial result on quantum interactive proof (QIP) systems having a space-bounded verifier? The only paper I know is An application of quantum finite automata to interactive ...
12
votes
3answers
468 views

How to iterate over vectors in order of probability in small space

Consider an $n$ dimensional vector $v$ where $v_i \in \{0,1\}$. For each $i$ we know $p_i = P(v_i = 1)$ and let us assume the $v_i$ are independent. Using these probabilities, is there an efficient ...
12
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1answer
432 views

Time Hierarchies in DSPACE(O(s(n)))

The time hierarchy theorem states that turing machines can solve more problems if they have (enough) more time. Does it hold in some way if the space is limited asymptotically? How does $\textrm{DTISP}...
12
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2answers
342 views

Can multipebble automata decide all deterministic context-sensitive languages?

A MPA (multipebble automaton) is a 2DFA (two-way deterministic finite automaton) that can use arbitrary number of pebbles (actually at most $ |w|+2 $ pebbles on a given input $ w $ - the input is ...
12
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0answers
1k views

Is CFL strictly contained in NL?

We know that $\mathsf{REG}=\mathsf{NSPACE}(O(1))$ and $\mathsf{CSL}=\mathsf{NSPACE}(O(n))$. What is the relation of $\mathsf{CFL}$ and $\mathsf{NSPACE}(O(\log n))=\mathsf{NL}$? Is $\mathsf{CFL}$ a ...
11
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1answer
266 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 ...
11
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1answer
531 views

Average-case space complexity

I am trying to find problems whose average-case space complexity has been analyzed. More specifically, I am interested to know if there are any problems with a proven space complexity lower bound ...
10
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3answers
1k views

An explicit separation between time-constructibility and space-constructibility?

Show a function $f(n)$ which is space-constructible but not time-constuctible. Is this problem related to a possible separation between complexity classes DTIME(f(n)) and SPACE(f(n))?
10
votes
2answers
326 views

Are non-deterministic tree-walking automata stronger than deterministic ones?

Update: It seems this problem has been studied and solved recently, see this wiki article: http://en.wikipedia.org/wiki/Tree_walking_automaton And also this survey: http://www.mimuw.edu.pl/~bojan/...
10
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0answers
149 views

Alternative proofs of Savitch's theorem?

Question: Are there any known proofs of Savitch's theorem that $NL \subseteq L^2$ besides the usual one? By the usual one I mean the proof based on recursively querying whether there is a midpoint. ...
10
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0answers
2k views

Which algorithm can do a stable in-place binary partition with only O(N) moves?

I'm trying to understand this paper: Stable minimum space partitioning in linear time. It seems that a critical part of the claim is that Algorithm B sorts stably a bit-array of size n in O(nlog2n) ...
10
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0answers
230 views

Statistical tests between L and LogCFL

Statistical tests are used to check whether a source of randomness is "good". Blum and Goldreich (1992) mentioned two types of statistical tests : (1) deterministic polynomial time statistical tests ...
9
votes
1answer
437 views

Does There exist a particular PSPACE Complete Problem which has a FPTAS algorithm?

It is well known that the NP-Complete Problem called Subset Sum has a FPTAS. I was wondering if there existed an PSPACE Complete problem which also has a FPTAS? Thanks in advance.
9
votes
1answer
119 views

$BPL$ with polylog random bits is in $L$

Consider a $BPL$ machine (namely, a probabilistic algorithm that uses logspace and polynomially many random bits). It is known (Saks-Zhou) that $BPL \subseteq DSPACE(log^{1.5}(n))$. My question is ...
9
votes
1answer
197 views

Do bounded-visit nondeterministic linear bounded automata recognize only regular languages?

Do bounded-visit nondeterministic linear bounded automata recognize only regular languages? By a nondeterministic linear bounded automaton (nLBA) I mean a single-tape nondeterministic Turing machine ...
8
votes
1answer
286 views

Hamming weight of powers

Given positive integers $b$ and $e$, what is known about the space and time complexity of finding the Hamming weight (number of binary 1s) of $b^e$? If $e\log b$ bits are available, the number can ...
8
votes
1answer
119 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
282 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 ...
7
votes
1answer
166 views

Complexity class of efficient streaming algorithms

Consider the class of problems $\mathsf{StreamL}$ which can be solved in logarithmic space reading the input in a single pass from left to right. In other words: $L \in \mathsf{StreamL}$ if there ...
7
votes
0answers
233 views

Integer queue summation

As part of a project I'm working on, we came up with an efficient algorithm for approximating the sum of an integers queue. The setting is as follows: Let $\epsilon>0$. we need to maintain a space-...
7
votes
0answers
483 views

complexity of Sokoban with a small number of boxes

(I asked a very concise version of this one month ago on cs.stackexchange, and although it got edited, it was not (otherwise) responded to.) In this post, for positive integer values $k$, "$k$-...