Questions tagged [space-complexity]

Asymptotic analyses of the space needed to run algorithms.

Filter by
Sorted by
Tagged with
2
votes
3answers
3k views

What happened if we implement quicksort without tail recursion?

On Wikipedia, it said that The in-place version of quicksort has a space complexity of $\mathcal{O}(\log n)$, even in the worst case, when it is carefully implemented using the following strategies:...
2
votes
1answer
97 views

PSPACE-completeness of DFA intersection problem

Let some deterministic finite automata be given. There is a problem of determining whether the intersection of these DFA is empty, and I want to show its PSPACE-completeness. It seems to me that I ...
2
votes
1answer
594 views

Max number of configurations of a Turing Machine

I was wondering about a result in the Sipser book which states that any $f(n)$ space bounded Turing machine also runs in time $2^{O(f(n))}$. Is this because a configuration consists of a state, a ...
2
votes
1answer
47 views

In certificate view of NL can we force the guesses to be in some format like $a^n b^n c^n d^n$?

In certificate view of NL the size of our guess can be polynomial.Can this guesses be like $a^n b^n c^n d^n$. Can we force the guesses to be in some format? I think it(the format) can be in regex ...
2
votes
1answer
747 views

Prove that $L$ is closed under Kleene star iff $L=NL$

Prove that $L$ is closed under Kleene star iff $L=NL$ Hi, I am trying to solve this exercise, but it is quiet difficult. Of course first part is very easy: Let assume that $L=NL$. Lets consider ...
2
votes
1answer
291 views

Does graph connectivity being NP-complete imply NL=P?

I asked this question on cstheory.se before, where someone pointed out that it is equivalent to asking whether P=NP implies NL=P (thus I edited the question accordingly). However, my supervisor ...
2
votes
1answer
460 views

Prove the following language is in L (LogSpace)

I'm trying to prove the following language is in $L$ (decided by a TM in $O(\log n)$ space): $$A = \{ (C,x) \mid C(x)=1\text{ and $C$ is a Boolean formula with depth }\log(n)\}\,,$$ where $n = |(C,x)...
2
votes
1answer
52 views

On graph isomorphism over exponential word sizes

Is it known Graph isomorphism can be done in poly time if we allow exponential word sizes? (Shamir's poly time Integer Factoring algorithm is over exponential word sizes).
2
votes
1answer
130 views

Examples of languages not decidable by a TM using certain upper bounds on space/time

I'm learning about time and space complexity involving Turing Machines at the moment, and would really like some concrete examples of specific languages that belong (or don't belong) to certain ...
2
votes
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 ...
2
votes
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 ...
2
votes
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 ...
2
votes
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(...
2
votes
1answer
92 views

Concluding $SPACE(n^2) \neq SPACE(n^7)$ from universal turing machine running time

Let $M_U$ be an universal Turing machine which fulfills the following condition: If $M$ running $x$ takes $f(x)$ space, then $M_U$ running on $\langle \langle M\rangle,x\rangle$ takes $(f(|x|))^3+2\...
2
votes
1answer
47 views

Uniform Hashing. Understanding space occupancy and choice of functions

I'm having troubles understanding two things from some notes about Uniform Hashing. Here's the copy-pasted part of the notes: Let us first argue by a counting argument why the uniformity property, we ...
2
votes
2answers
43 views

Logarithmic space verifier with unbounded witness

this is a HW question, but its considered a bonus question so I'd appreciate a direction. Definitions: The actual question: **Images taken from HW in TAU Complexity course by Amnon Ta-Shma. My ...
2
votes
1answer
505 views

Sorting an array of strings by length in linear complexity

I am trying to find an algorithm to sort an array of strings by length in O(n) time complexity, and O(1) space complexity. The max length of the strings is known. Because of that, I tried using ...
2
votes
1answer
47 views

Exponential Space Complexity equality

Consider $$ \bigcup_{c \in \mathbb{N}} \mathsf{DSPACE}(2^{c (\log{n})^2}) \quad \overset{?}{=} \quad \bigcup_{c \in \mathbb{N}} \mathsf{DSPACE} ( n^{c \log{n}})$$ My lecture notes say that this is ...
2
votes
1answer
91 views

What is tight NSPACE complexity of $ALT\text{-}SPACE(a(n),s(n))$?

According to Ryan Williams's answer $ALT\text{-}SPACE(a(n),\log n)\subseteq NSPACE(a(n)\log n)$. Does there exist any better bound (for example something like $ALT\text{-}SPACE(a(n),\log n)\...
2
votes
1answer
528 views

How to prove that $n\log n$ is space constructible?

I'm trying to prove that $n\log n$ is space constructible. I've already managed to prove that $\log n$ is space constructible, but I cannot figure out how to prove the same about $n$. I assume, that ...
2
votes
1answer
60 views

What is the complexity to show this theorem?

Given a sum of regular expressions, where each regular expression in the sum is n-1 concatenations of 0, 1 and (0+1). There is need to show that the sum of all regular expressions is either equal to ...
2
votes
2answers
598 views

non deterministic space hierarchy

I want to prove the non deterministic space hierarchy theorem. Let $f(n),g(n)\geq\log n$ be space constructible functions such that $f(n)=o(g(n))$, Prove: $$NSPACE(f(n))\subsetneq NSPACE(g(n))$$ I ...
2
votes
1answer
62 views

Eliminating ambiguity when referring to complexity in cases of multiple parameters and/or multiple results

When looking through a few questions at StackOverflow¹ that all ask for algorithms to select k distinct random numbers out of N, I've become confused about how to compare the answers in terms of time ...
2
votes
1answer
799 views

Why isn't TQBF part of the polynomial hierarchy?

TQBF consists of alternating quantifiers, so does $\Sigma^2_n$ for fixed $n$. So given a formula in TQBF, shouldn't there be a level of the polynomial hierarchy that solves it? I think this is ...
2
votes
1answer
34 views

L contains the concatenations of all k-bit long strings. Why is it decided in PSPACE(loglogn)?

(This exercise is from Computation Complexity: A Conceptual Perspective by Oded Goldreich): For any k $\in \mathbb{N}$, let $w_k$ denote the concatenation of all k-bit long strings (in lexicographic ...
2
votes
1answer
276 views

Prove or disprove that DTIME(n^2)=NL

I need to prove or disprove $DTIME(n^2)=NL$. It kind of feel obvious that I need to disprove it, because if I have non-deterministic machine $M$ that uses $\log n$ space, then it meets at most $|Q| n\...
2
votes
1answer
99 views

Why do we set conditions f(n) ≥ n resp. f(n) ≥ log(n) the Time resp. Space Hierarchy?

In the Space (Time) Hierarchy Theorem and also fully space (time) constructibility of two function we have the condition: being greater than $log(n)$ (being greater than $n$). Why do we have these ...
2
votes
1answer
443 views

Certificate Definition of NL

As per the Sanjeev Arora book, for a certificate based definition of $NL$, the machine is allowed a "read-once" certificate tape to store the certificate along with $O(log n)$ read/write work tape for ...
2
votes
1answer
166 views

How to denote the space complexity in terms of output

Normally the space complexity of an Algorithm $A$ is denoted $\textrm{SPACE}(A)$, which means how much space is needed by the computation itself. I would however like to also describe how much storage ...
2
votes
1answer
481 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 ...
2
votes
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 ...
2
votes
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 ...
2
votes
1answer
158 views

Power of Double - Logarithmic Space

I try to solve exercise "on the power of double - logarithmic space" from the great textbook Computational Complexity by Oded Goldreich. The goal is to show that the given set $S=\left \{ w_k \mid k \...
2
votes
1answer
75 views

Time complexity for the 'Restore IP Addresses' problem

There's a programming problem 'Restore IP Addresses' where given a string containing only digits, restore it by returning all possible valid IP address combinations. Example, "25525511135" returns ["...
2
votes
1answer
38 views

NL-Hardness of Target

When revising for an upcoming exam in complexity theory, I came across this problem on the final part of a question, which I was unable to solve: $ TARGET = \{<G, t> : t\ is\ reachable\ from\ ...
2
votes
1answer
859 views

Is Breadth First Search Space Complexity on a Grid different?

Is the Space Complexity O(number_rows + number_cols) for Breadth First Search on a Grid. This is an attempt to show my reasoning: For example, the flood fill question is described here: https://www....
2
votes
1answer
640 views

Time complexity to find Median of Medians

I recently wrote my Grad school Admission test few days back and the following question appeared in the test. There are 'n' unsorted Arrays : A1, A2, ...., An. Assume that 'n' is odd. Each of A1, A2, ....
2
votes
1answer
309 views

Relation beetween log-space reduction and polynomial time reduction

I read somewhere that given two languages A and B, if A <=(log) B, then A <=(P) B (with <=(log) the log-space reduction and <=(P) the polynomial time reduction), but I'm not sure about the ...
2
votes
1answer
145 views

Space complexity of boolean circuit evaluation

I am given a boolean circuit of depth $D \ge \log n$ where $n$ is the input size. Given an input, I need to find an algorithm that evaluates the circuit in space $O(D)$. Now, assuming the fan in of ...
2
votes
1answer
956 views

STCON is NL complete - but why is the reduction in L?

I saw the proof for STCON being NL complete here : https://en.wikipedia.org/wiki/St-connectivity I understand the reduction, but how is it logspace? I understand each state is of $O(\log(n))$ space....
2
votes
1answer
50 views

Polynomially many times launch $NL$ machine - is it in $NL$? On example of ACYCLIC

Lets consider $$ACYCLIC = \{\langle G \rangle | G \text{ is acyclic}\}$$ We are going to prove that $ACYCLIC\in NL$. I know that the easiest approach for this task is to use the fact that $coNL=NL$. ...
2
votes
1answer
415 views

Space requirement of a universal Turing machine

Given a representation $g$ (e.g. the Gödel number) of a Turing machine $B$, a universal Turing machine $A$ can simulate $B$. If $B$ is restricted to using at most $n$ memory cells of its tape and the ...
2
votes
1answer
53 views

Replacing n with 2n in asymptotic bounds

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. In the proof of the theorem $6$ of the paper on page 632, the authors go on ...
2
votes
1answer
169 views

Time complexity of minimizing Boolean expression

Given any arbitrary boolean expression using AND, OR and NOT gates what is the time complexity of minimizing the expression such that minimum number of gates are used. The following Wikipedia article ...
2
votes
1answer
90 views

Can a cellular automata structure simulates another cellular automata structure?

In Elementary Cellular Automata, rules can show one pattern, but i am wondering if there is something where a cellular automata structure can simulate another structure? Is there a category for this ...
2
votes
1answer
83 views

Looking for an example of proving space upper bounds for computing functions on a DTM

Like think of the function $f\colon \{ 0,1\}^* \rightarrow \{0,1\}^*$ which maps a binary string string $x$ to say a string of $0$s of length $\vert x \vert ^2$ whre $\vert x \vert$ is the length of ...
2
votes
1answer
656 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 $O(1)$ ...
2
votes
1answer
33 views

Algorithmic problem with many different time/space complexity solutions

I am preparing a lesson about algorithmic thinking for beginner programmers. I would like to show them an easy to understand problem which has as many solutions as possible with different time or ...
2
votes
0answers
36 views

Proving inexistence of a PCOMPLETE language in log logarithmic space cannot exist

Hello and thank you for helping me understand the following: I am trying to understand why the following cannot exist: A P-Complete language in regards to a log-logarithmic space. context: Defining ...
2
votes
0answers
24 views

Existence of a P-Complete language with space($\log\log n$) reduction

I have been reading and searching and I still cannot understand if there exists a language as following: Can a language be P-complete with respect to $\mathsf{Space}(\log \log n)$ reductions? ...

1 2 3
4
5
8