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Questions tagged [space-complexity]

Asymptotic analyses of the space needed to run algorithms.

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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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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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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(...
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Proof of Space Hierarchy Theorem incompatible with Linear Speed Up Theorem for time

In this proof of the Space Hierarchy Theorem the following langugae is defined $$ L = \{ (\langle M \rangle, 10^k) : M \mbox{ does not accept } (\langle M \rangle, 10^k) \mbox{ using space } \le f(|\...
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274 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 ...
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36 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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11 views

Time complexity of well know constraint satisfaction problem algorithms with heuristics

I have know that complexity of csp algorithms as follow: Backtracking algorithm for constraint processing space:O(n) ,Time :O(expn) Backjumping algorithm for constraint satisfaction problem ...
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22 views

Is there a known lower bound on the time/space complexity of DFA minimization?

I've read the Wikipedia page on the topic, but there's no mention of a lower bound on the time complexity, and there's no mention of space complexity at all.
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34 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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480 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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1answer
83 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
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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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28 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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103 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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55 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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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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23 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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Create a binary search tree from a sorted array: Space complexity reasoning

Here is the Python code. The solution is fairly common and is seen in most textbooks like 'Cracking the Coding Interview' and 'Element of Programming Interviews'. ...
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41 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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22 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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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
61 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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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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What is the Space Complexity of Tail Recursive Quicksort?

Looking at the following tail recursive quicksort pseudocode ...
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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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94 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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28 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
44 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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126 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
164 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.
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1answer
68 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 ...
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27 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
55 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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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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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 ...
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1answer
87 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)\...
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2answers
33 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-...
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1answer
84 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)=\...
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1answer
169 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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1answer
50 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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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
44 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 ...
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1answer
83 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
464 views

Fast, stable, almost in-place radix and merge sorts

I've developed LSD radix sort algorithm that is stable, about as fast as the classic LSD radix sort, require only $O(\sqrt{RN})$ extra space when we sort into R buckets. The same technique also ...
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2answers
260 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
107 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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2answers
143 views

Space/Time Hierarchy Theorem - proving the language of the machine is in the larger space class

I am trying to understand the space and time Hierarchy theorems according to Sanjeev Arora, Boaz Barak: Computational Complexity: A Modern Approach but the more general case. What I don't ...
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0answers
25 views

EXPTIME $\neq$ EXPSPACE consequences?

We know that $EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE$, where $EXPSPACE = DSPACE(2^{poly(n)})$. Question: Is known any consequences of $EXPTIME \neq EXPSPACE$? (nothing in complexity zoo or wikipedia)
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Why is a pointer constant space?

If a pointer specifies a point in memory, would the amount of space a pointer takes not be dependent on how much memory it could possibly range over? So for example, if we have 4 locations of memory ...
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1answer
34 views

Is it known that $AC^1 \subseteq L$?

A good exercise is to show $NC^1 \subseteq L$. (According to the complexity zoo page this was first shown by Borodin, 1977.) Although the details must be checked, the proof is simple: take the $NC^1$ ...