Questions tagged [space-complexity]

Asymptotic analyses of the space needed to run algorithms.

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2
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1answer
480 views

Combined linked/array-like data structures for a set of non-intersecting sub-intervals of integer interval?

This question is related to my previous question: Looking for a set implementation with small memory footprint I'm looking for information about combined data structures, which can efficiently ...
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1answer
87 views

Space complexity problem, relation between $DSPACE(log^kn)$ and $DSPACE(log^{k+1}n)$

I need help with the following: Let $k\in \mathbb{N}$, define: $L^k=DSPACE(O(log^k(n)))$ $NL^k=NSPACE(O(log^k(n)))$ and: $PolyL=\bigcup_{k=1}^{\infty}L^k$ $PolyNL=\bigcup_{k=1}^{\infty}NL^k$ I need ...
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1answer
544 views

Why is it necessary to use binary numbers in logspace?

I have noticed that a lot of problems that are in L and NL use binary numbers. I don't understand why this is the case. Does a TM use less space by storing a binary number, than a "normal" one. In my ...
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1answer
141 views

How to show that FPATH is in NL?

Consider this problem: $\qquad\displaystyle \mathsf{FPATH} = \{\langle G, a_1,\dots,a_n\rangle \mid G \text{ is a digraph with directed path } (a_1,\dots,a_n)\}$ It's allowed to visit nodes outside ...
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1answer
245 views

Finding dynamic programming algorithm

I got a matrix of integers of size $3\times n$. Of each one of the three rows, for each column I got to choose one number, with the restriction that, for each $i$, the numbers chosen in the $i$th and $...
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1answer
71 views

Local and Global storage with multithreading pools + locking threads

I am having difficulty answering the following questions relating to the use of threading. Question 1 is of relating to the possibility of a local storage per thread and a global storage accessible ...
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0answers
120 views

General object recognition versus specific object recognition

I have a question about the difference between general object detectors and specific object detectors. By specific object detectors, I'm referring to classifiers/object recognizers that are built to ...
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1answer
986 views

Why is the set of NFA that accept all words in co-NPSPACE?

In Sipser's book there is a section describing how to decide $\qquad\displaystyle \mathrm{ALL}_\mathrm{NFA} = \{ \langle N \rangle \mid N \text{ is an NFA}, L(N) = \Sigma^*\}$ in polynomial space. ...
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1answer
180 views

Inclusion of complexity classes (Deterministic Turing Machine)

I can't understand what my professor wrote about these inclusions concerning deterministic classes: $$ DTIME(f) \subseteq DSPACE(f) \subseteq \sum_{c\in\Bbb N}DTIME(2^{c(log+f)}) $$ I understood ...
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2answers
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Has there been any more progress on P vs. PSPACE compared to P vs. NP?

I understand this is a slightly vague question, but there are results for P vs. NP, such as the question cannot be easily resolved using oracles. Are there any results like this which have been shown ...
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0answers
430 views

Clique and PSPACE [closed]

I was wondering how I could go about creating an algorithm that gets all the cliques in a graph in PSPACE So far, based on some of the readings I've done, I am considering to use bit-strings (that ...
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1answer
343 views

What is the time/space complexity of $n!$? Can $n!$ has polynomial space complexity?

Given an integer $n$, calculate $n!=n\times(n-1)\times(n-2)\dotsc 3\times2\times1$. What is the best time and space complexity of calculating $n!$? P.S. I do not have any idea about this topic. I ...
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1answer
244 views

Polynomial space complexity with exponential size witnesses

Define the complexity class $C$ to be the class of all languages that can be verified by a TM that has: Input tape: Read only, move in both directions. Witness tape: Read only, move only in one ...
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0answers
19 views

Generalized Geography with repetitions [duplicate]

Consider the "Generalized Geography" game: on directed graph G with selected start node, players take turns moving along edges, without ever going back to previously visited nodes. Last player to move ...
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1answer
342 views

Generalized Geography with repetitions

Consider the "Generalized Geography" game: on directed graph G with selected start node, players take turns moving along edges, without ever going back to previously visited nodes. Last player to ...
3
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1answer
376 views

Log-Space Reduction $CO-2Col \le_L USTCON$

I want to show that $CO-2Col \le_L USTCON$ (Log-Space reduction) $USTCON$ The $s-t$ connectivity problem for undirected graphs is called $USTCON$. [Input]: An undirected graph $G=(V,E)$, $s,...
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1answer
1k views

The crux of Savitch's Theorem

In "Introduction to the Theory of Computation" by Sipser, Savitch's theorem is explained as an improvement to a naive storage scheme for simulating non-deterministic Turing machines (NTM). I am going ...
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1answer
1k views

How to Study Space Complexity

I am working through Sipser, and I am trying to understand some of the algorithms described in Space Complexity, but I am having a hard time understanding the presentation of the material (especially ...
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2answers
1k views

Language with $\log\log n$ space complexity?

We know that every non-regular language can be recognized with $ \Omega (\log\log n) $ space complexity. I'm looking for an example of a language which is $ \Theta (\log\log n) $ space complexity (...
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1answer
144 views

ALL_{REGEX} in PSPACE algorithm

$ALL_{REGEX}$ is the computational problem of determining for regular expression x if $L(x) = \Sigma^*$. In a proof for $ALL_{REGEX} \in PSPACE$, the following non-deterministic turing machine $M(R)$ ...
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1answer
2k views

Relation of Space and Time in Complexity?

I'm looking for some clarification on some concepts/facts I came across while studying for a class. I was reading the following wikipedia article. The below specific section and statement intrigued ...
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1answer
574 views

Logspace Transducer

I know that a logspace transducer is a deterministic Turing machine that enables us to use log-space complexity. I do not understand though why that is correct. Whatever algorithms can be implemented ...
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2answers
2k views

Looking for a set implementation with small memory footprint

I am looking for implementation of the set data type. That is, we have to maintain a dynamic subset $S$ (of size $n$) from the universe $U = \{0, 1, 2, 3, \dots , u – 1\}$ of size $u$ with operations ...
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2answers
729 views

Is DSPACE properly contained in NSPACE?

It may be a dumb question, but is $\mathsf{DSPACE}(f(n)) \subset \mathsf{NSPACE}(f(n))$ or is $\mathsf{DSPACE}(f(n)) \subseteq \mathsf{NSPACE}(f(n))$? In other words, is the containment relation ...
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1answer
314 views

If a problem is PSPACE-complete what do we know about NL-completeness

I have a problem $A$ which was shown to be PSPACE-complete by reduction from planning. However, $A$ can also be transformed into reachability problem which is NL-complete. I know that $NL=NSPACE(...
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1answer
1k views

Relation between space and time complexity for machines with write once read many (WORM) memory

While thinking about different calculi for predicate logic (like natural deduction and sequent calculus), I noticed that these calculi are (often) presented in a form suitable for "human computers". A ...
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1answer
2k views

Prove that $TQBF \notin SPACE(n^{\frac{1}{3}})$

I would like some hints on how to approach this problem, I know for instance that $TQBF$ is $PSPACE$-$Complete$, so it can solved in poly space and any other $PSPACE$-$Complete$ problems can be log ...
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1answer
2k views

Proving that the language SPACE TMSAT is PSPACE-complete? [closed]

I'm trying to prove that the language SPACE TMSAT (where SPACE TMSAT = {⟨$M$, $w$, $1^n$⟩ : DTM $M$ accepts $w$ in space $n$}) is PSPACE-complete. My solution is as follows: SPACE TMSAT $= \{<M,w,...
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1answer
655 views

Known bounds on space complexity of multiplication decision problem

Given three numbers $m$, $n$ and $p$ in interleaved binary encoding1, it's obviously possible to check in $O(1)$ space whether $m+n=p$. It's less obvious2 that it isn't possible to check in $O(1)$ ...
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2answers
13k views

Memory complexity?

I am unclear about finding the memory complexity of an algorithm. Some places refer memory complexity as what container would be carrying for instance: ...
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1answer
1k views

NTIME(f) subset of DSPACE(f)

As the question states, how do we prove that $\textbf{NTIME}(f(n)) \subseteq \textbf{DSPACE}(f(n))$? Can anyone point me to a proof or outline it here? Thanks!
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1answer
886 views

Set combination data structure (And storage complexity)

I have already posted this question on Stackoverflow, but I'm starting to think that this is the right place. I have a problem where I am required to associate unique combinations from a set (unique ...
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1answer
363 views

Is the memory-runtime tradeoff an equivalent of Heisenberg's uncertainty principle?

When I work on an algorithm to solve a computing problem, I often experience that speed can be increased by using more memory, and memory usage can be decreased at the price of increased running time, ...
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1answer
11k views

Time complexity and space complexity in recursive algorithm

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1answer
837 views

Checking whether a digraph on $n$ vertices contains exactly $10\sqrt{n}$ strongly connected components in NL

I am studying now for a test in my complexity course. When I solved previous exams I saw the following question: Prove that the language $L$ of all directed graphs on $n$ vertices that contain exactly ...
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1answer
161 views

'Stones' game complexity

I'm trying to find complexity class of finding winning strategy for first player in following game: Intance of 'Stones' game is: finite set $X$ relation $R \subseteq X^3$ set $Y \subseteq X$ and ...
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2answers
180 views

Proving language in Space Complexity

I'd like to know if I have the right intuition and my answer is headed the correct way. I am given a function $ f = \{0, 1\}^* \rightarrow \{0, 1\}^* $ that is computable in space $O(\log n)$ assume ...
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1answer
625 views

Proving that Turing Machine M runs in time $O(2^{dn})$

I'm trying to solve this question in order to review for my exam, and this one has got me a bit stumped. From the looks of it, it seems like a fairly straight-forward question, but I can't figure out ...
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1answer
76 views

Showing transitivity of PSPACE?

For the following question: If B is an element of PSPACE and A is an element of PSPACE-Complete, and A polynomial reduces to B, then B is an element of PSPACE-Complete. I am trying to prove this, ...
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1answer
807 views

What is a compact way to represent a partition of a set?

There exist efficient data structures for representing set partitions. These data structures have good time complexities for operations like Union and Find, but they are not particularly space-...
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4answers
1k views

Space complexity for finding the minimum number outside the list of numbers

We are given an (unsorted) list $L=(a_1,\dots,a_n)$ of numbers of size $n$, where $a_i\in \{ 1,\dots,B\}$. We want to find the minimum number $x$ from $\{ 1,\dots,B\} \backslash L$. What is the ...
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2answers
349 views

Why is one often requiring space constructibility in Savitch's theorem?

When Savitch's famous theorem is stated, one often sees the requirement that $S(n)$ be space constructible (interestingly, it is omitted in Wikipedia). My simple question is: Why do we need this? I ...
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2answers
417 views

Why does a polynomial-time language have a polynomial-sized circuit?

I wish to understand why P is a subset of PSCPACE, that is why a polynomial-time langauge does have a polynomial-sized circuit. I read many proofs like this one here on page 2-3, but all the proofs ...
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1answer
655 views

Relationship between L and PSPACE

I have a question have to answer, so that, if anyone have the answer, please help me. The problem is: Give a self-contained proof that $\mathsf{L} \neq \mathsf{PSPACE}$ where: $\qquad \mathsf{L} ...
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2answers
108 views

Prove that $0$-$1$ $\mathsf{ Ineq}$ is $\mathsf{NL}$-complete

I need to prove that the following problem $0$-$1$ $\mathsf{ Ineq}$ is $\mathsf{NL}$-complete. Given a finite set of variables $V$, a finite set of inequalities of the form $x \le y$ (where $x, y \in ...
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1answer
129 views

Prove that 2-Colourability is in L from Undir-Reachability is in L

Let Undir-Reachability be the following problem: given an undirected graph G and two specified vertices s and t in G, is there a path from s to t in G? I need to prove that the 2-Colourability is in ...
7
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1answer
151 views

Does $\mathsf{NSPACE}( f (n)) = \mathsf{coNSPACE}( f (n))$ hold for $ f(n) < \log (n) $?

It's known that for $f(n) \geq \log n$, $\mathsf{NSPACE}(f(n)) = \mathsf{coNSPACE}(f(n))$. What if $f(n)<\log n$? Are they also equal?
5
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1answer
159 views

NL- definition and a problem

The question is: What is the smallest complexity class in which the following problem is contained: Given a graph with $n$ nodes, Is there independent set of size of at least $n-10$? I have a little ...
8
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1answer
526 views

Bit complexity of O(1) time range query in a $k$-ary array

Consider the following problem: Let $k$ be a constant. We are given a $k$-ary array $A_{d_1\times\ldots\times d_k}$ of $0$ and $1$'s. Let $N = \prod_{i=1}^k d_i$. We want to create a data structure ...
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1answer
994 views

Bipartite Problem is Log-Space Reducible To $s$-$t$ Undirected Connectivity

Prove that the problem of determining if graph is bipartite is computationally equivalent under log-space reductions to $s$-$t$ undirected connectivity. Problem of $s$-$t$ undirected connectivity is ...

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