# Questions tagged [space-bounded]

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

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### 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) ...
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 ...
2answers
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### 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 ...
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. ...
0answers
85 views

### The number of words of length $n$ in a context-sensitive language

Let $L$ be a context-sensitive language, $s_{L}(n)$ is denoted by the number of words of length $n$ in $L$. What is known about $s_{L}(n)$? Note that it is known that $s_{L}(n)$ is either polynomial,...
2answers
293 views

1answer
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### $BPL$ 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 ...
0answers
107 views

### Reduction between functions that preserves time and space-complexity

Under which reduction(s) is the class $\mathsf{FTISP}(t(n), s(n))$ closed? Let $\mathsf{FTISP}(t(n), s(n))$ the class of functions from $\{0,1\}^*$ to itself that are computable by a Turing machine ...
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 ...
0answers
313 views

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 ...
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 ...
1answer
117 views

### Base extension in residue number systems with low space

Suppose I have a number $x$ represented in a residue number system, so $x = (x_1, \ldots, x_m)$, where $x_i \equiv x \pmod{p_i}$, and the $p_i$'s are all relatively prime (they can be distinct primes ...
0answers
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### Circuit games in extensive form with imperfect information

Consider $l,m,n,N \in \mathbb{N}$ and circuits $C: \{0,1\}^{l+m} \rightarrow \{0,1\}^l$, $D: \{0,1\}^{l+n} \rightarrow \{0,1\}^l$. Consider the following zero-sum two-layer extensive-form game with ...
0answers
131 views

### Is there a certificate definition for polyL?

Arora explained in his book a certificate definition for $\mathsf{NL}$ using a read-once tape. Can we apply a similar definition for the class $\mathsf{polyL}$?
1answer
362 views

### Can every distribution producible by a probabilistic PSpace machine be produced by a PSpace machine with only polynomially many random bits?

Let $M$ be a probabilistic Turing machine with a unary input $n$ whose space is bounded by a polynomial in $n$ and its output is a distribution $D$ over binary strings. Note that the number of ...
3answers
744 views

2answers
240 views

### Minimal encoding of a set (unordered collection of elements)?

Assume you have universe $\mathcal{U}=\{e_1,e_2,\ldots e_N\}$. If we like to encode an ordered sequence of $k$ elements from $\mathcal{U}$, it's not hard to argue that $k\log |\mathcal{U}|$ bits are ...
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 $NL$,...
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 ...
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 ...
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) (...