Questions tagged [space-complexity]

Asymptotic analyses of the space needed to run algorithms.

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1answer
205 views

Non-existence of PSPACE-hard unary language

I'm trying to prove that unless $\mathsf{P}=\mathsf{PSPACE}$, there is no unary language which is $\mathsf{PSPACE}$-hard. Assuming there is an unary language $A$ which is $\mathsf{PSPACE}$-hard, it ...
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39 views

Time and space complexity of a recursive problem (code included)

I am having trouble finding out the time and space complexity for this recursive solution. I have to create a list of words in order of word length. Each word, must be one character insertion off from ...
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1answer
860 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....
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1answer
101 views

What is space complexity of shrinking one array to increase another?

Say I have an array and I want to add those values to something else. What is the space complexity if I incrementally take one of those values off the first list and add it to the second? For example, ...
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52 views

Relationship between SPACE(t(n)) and DTIME(t(n))

I'm new to complexity theory and am analyzing inclusions between complexity classes. Suppose we are given the following seven complexity classes $DTIME(n)$ $DTIME(n^2)$ $DTIME(2^n)$ $DTIME(2^{2^n})$ $...
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1answer
56 views

How do I prove that SPACE($n^{555}$) $\neq$ NP?

I thought about finding a language with a polynomial verificator "larger" than $n^{555}$, but then I realized it would not imply the space needed for computation is the size of the verificator.
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2answers
113 views

Reachibility between first and last layer in grid graph in logspace

I am trying to prove that there exists logspace deterministic Turing machine that check if exists path between first row and last row in grid graph. Grid graph is matrix of $0s$ and $1s$, the ...
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0answers
46 views

Is there a difference in space complexity between inner product of matrices to multiple of inner products where each containing one matrix at a time?

The book I am reading is suggesting the following: Suppose I have two vectors $v, w$ and $P(n)$ matrices $U_1, U_2, \ldots, U_{P(n)}$. Then performing an inner product of $v$ with $U_1U_2\ldots U_{P(...
4
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1answer
335 views

What's after EXPSPACE?

As far as I'm aware, EXPSPACE is the most inclusive computational complexity class. I was wondering if/how people conceptualize supersets of EXPSPACE. Thinking about this question, I came up with a ...
2
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1answer
597 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 ...
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4answers
419 views

A language in NSPACE(O(n)) and very likely not in DSPACE(O(n))

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

Logic behind O(n) solution for 'Maximum length sub-array having given sum'

I am unable to understand the logic behind O(n) solution of this classical problem- Maximum length sub-array having given sum (...
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2answers
206 views

Return copy of array, each element product of all others, constant additional space, no division

Question: I am trying to solve question 6.10.1 from Elements of Programming Interviews. The task is as follows: Given an array $<a_1, \ldots, a_n>$ of fixed-...
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2answers
97 views

Quick and space-efficient way to find whether two sets intersect

I hope you can help me - Given a lot of sets containing integers, I'd like for any two sets, to quickly (i.e. O(1)) ask whether they intersect. Note that I don'...
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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, ....
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1answer
40 views

Better implementation to find the root of an element in QuickUnion implementation of UnionFind problem

I wanted to know which implementation is better to find the root of the element in the Quick Union implementation of the UnionFind problem. The professor has used a while loop to find the root of the ...
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0answers
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Naming for (memory near optimal) datastructres

Assume that we wish to solve some problem that has a information theoretic memory lower bound of $\mathcal B$ bits. In computer science, there are a few classes for data structures which are close to ...
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50 views

Space complexity for accepting word decision problem of DFAs

It is well known that the the decision problem $w \in \mathcal{L}(M)$ for a DFA $M=(Q,\Sigma, \delta, q_0,F)$ is in $\mathcal{O}(|w|)$. To proof this we assume that the successor state computation can ...
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0answers
22 views

Without using the Space Hierarchy Theorm is there any other way to prove that NL is not equal to PSPACE

From what I know there is no alternate way that NL is not equal to PSPACE. If possible can you link a paper or some book recommendations to show that this is the case. Thank You Akash
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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 ...
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1answer
203 views

Show that $A_\mathrm{LBA}$ is PSPACE-Complete?

I want to show that $A_\mathrm{LBA}$ is PSPACE-Compelte. Say we proved it is in PSPACE. Now for PSPACE-HARD: I had an idea, which was very similar to some solution i found on the web- say we have a ...
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1answer
73 views

Graphs in space efficient representation

Let $G$ be a graph such that $V$ denotes a vertex set and $E$ is an edge set of the graph $G$. Let us consider that for the input graph $G$ it is the case that $|E| \le O(|V| \log |V|)$. Given a graph ...
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0answers
63 views

If problem A is logspace reducible to 2-SAT, is A in NL?

I'm trying to prove that some problem, A, is in NL. I have found a logspace reduction from A to 2-SAT - am right in thinking that this is not sufficient to prove that A is in NL? If so, how does one ...
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2answers
470 views

Time-space tradeoff for missing element problem

Here is a well-known problem. Given an array $A[1\dots n]$ of positive integers, output the smallest positive integer not in the array. The problem can be solved in $O(n)$ space and time: read the ...
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27 views

How to check a graph diameter in LOGSPACE?

Given a graph G, how can I check that its diameter doesn't exceed log(n) (n is the number of vertices)- by using only O(log(n)) space? (adjacency matrix doesn't seems to help...)
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1answer
29 views

What if an $L$-complete problem has $NC^1$ circuits? More generally, what evidence is there against $NC^1=L$?

What if an $L$-complete problem has $NC^1$ circuits? More generally, what evidence is there against $NC^1=L$?
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1answer
151 views

Does Multitape reduction to a one tape machine preserve space complexity?

Suppose a Turing machine $M$ has a read-only input-tape and $k$ read-write work-tapes whose non-blank cells are each bounded by $f(|x|)$ where $|x|$ is the length of the input. Is there some constant ...
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1answer
78 views

Show that an NP-Complete problem exists that is contained in SPACE(n)

The problem I am working on is the following: Show that there exists an $NP$-Complete problem in $SPACE(n)$. Does the existence of such a problem imply that $NP$ is contained in $SPACE(n)$. Why ...
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2answers
225 views

Data structure for finding max, inserting and deleting in O(1) and O(n) space

This is an interview question. I need to implement a data structure that supports the following operations: Insertion of an integer in $O(1)$ Deletion of an integer (for example, if we call delete(7),...
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2answers
17k views

Time and space complexity of Radix sort

I had previously asked a question on space complexity of radix sort here. I have also read this question. However, I still get confused about it which means that the concept is not clear. I have the ...
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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 ...
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41 views

What does actually happens on tile hashing

I am going through Richard Sutton book about Reinforcement Learning and I just encountered the tile coding method. I understood pretty well the principles, however, at the very end of the section, ...
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1answer
344 views

$MIN_{NFA}$ is PSPACE complete

The problem is taken out of Theory of Computational Complexity: Now, I think I've successfully proven that $ALL_{NFA} = \{(A) : A$ is an NFA and $ L(A) = \Sigma^*\} \leq_p MIN_{NFA}$. Which implies ...
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1answer
64 views

Understanding $DSPACE(s(n)) \subseteq DTIME(2^{O(s(n))})$

I'm having trouble understanding this statement: $DSPACE(s(n)) \subseteq DTIME(2^{O(s(n))})$. The logic is that $2^{O(s(n))}$ is the total number of different configurations a Turing Machine M that ...
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0answers
276 views

Space complexity of Coin change with memoization

I've read conflicting answers for the space complexity of the top down implementation w/ memoization for the classic coin change problem. Would this be O(N * M) space as Interview Cake says https://...
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2answers
519 views

Is every regular/context free langauge decidable in LogSpace?

I know all the regular languages are decidable but not sure whether it can be done in LogSpace.
5
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1answer
284 views

Why not polynomial-space reductions for $PSPACE$-hardness?

A language $L'$ is $PSPACE$-hard if for every $L \in PSPACE$ we have $L \le_p L'$. Here $L \le_p L'$ means that $L$ is polynomial-time reducible to $L'$. Why does we use time reductions instead of ...
2
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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 ...
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1answer
305 views

Problem with understanding proof of the Space Hierarchy Theorem

The Space Hierarchy Theorem states that If $f(n)$ is space contructible, then for any $g(n) \in o(f(n))$ we have $SPACE(f(n)) \neq SPACE(g(n))$ An example of a SHT proof can be found here or here ...
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1answer
54 views

Connecting strings in a graph is a PSPACE problem

We define the following problem as: Let $M$ be a TM with alphabet $\Gamma$, with $\{a,b,$ #$\} \subset \Gamma$. We define, for every natural number $n$ the graph $G_{M,n}$ by: $V_{M,n} = \{a,b\}^n$,...
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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
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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)\...
3
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1answer
882 views

What algorithm to use for this subset sum problem

I have a typical subset sum problem and I'm looking to choose the proper algorithm to solve it, the set contains (around) 1000 elements, and elements are constrained to max 22 bits for now. I have ...
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2answers
55 views

STCON in L using matrix multipication algorithm?

I'm trying to understand why the following is incorrect. Given a $STCON$ problem, specifically a graph and nodes $(G, s, t)$, we can assume we are given it's adjacency matrix, $A$. By adding self-...
3
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1answer
126 views

Is $\mathsf{DSPACE}(n)=\mathsf{DSPACE}(n/\log\log n)$?

We know that $\mathsf{DSPACE}(\log\log n) = \mathsf{DSPACE}(1)$ according to this proof. Can we claim that $\mathsf{DSPACE}(n)=\mathsf{DSPACE}(n/\log\log n)$ or something like $\mathsf{DSPACE}(n^3)=\...
3
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1answer
371 views

Does the space complexity of a recursive algorithm depend on the total no of recursive calls?

I am confused whether space complexity of a recursive algorithm depends on the total number of recursive calls or not. Say I have an algorithm which has exponential function calls, but stack size is ...
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2answers
65 views

Why it is not $O(m)$ but $O(\log m)$?

I am reading the lecture notes and have a question. I am trying to understand the beginning of Section 3 on page 2. Problem: Given an input stream $\sigma$, compute (or approximate) its length $m$. ...
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1answer
45 views

A Language Belonges to PSPACE

Let $A,B$ be two languages, for which we know: $A \in PSPACE$ $A\le_LB$ Can we conclude from the above that $B \in PSPACE$ ? I think the answer is no, however I don't know how to prove it. I guess ...
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2answers
802 views

Show that NP is not equal to SPACE(n)

I want to show that $\text{NP} \neq \text{SPACE}(n)$ and tried it like this: Let $L$ be in $\text{SPACE}(n)$ so there is a deterministic $k$-tape TM which decides $L$ in polynomial time. Let's ...
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1answer
391 views

What is a space constructible function?

I've searched this and was not able to understand the answer. What's the idea/intuition behind this? What's the purpose and why is it important? And considering the wikipedia explaination, ...

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