# Questions tagged [space-complexity]

Asymptotic analyses of the space needed to run algorithms.

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### 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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### 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 $NC^1$ ...
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### Why is DTIME(n) not equal to NP, and consequently, DSPACE(n) not equal to NP?

Intuitively it would seem like these equalities are false since DTIME(n) and DSPACE(N) are in terms of deterministic Turing machines and NP is non-deterministic, but I'm struggling to come up with a ...
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### Does the circuit value problem require only log space in alternating Turing machines

Why does the circuit value problem run in log space on an alternating Turing machine? It is claimed to be so in my university's lecture notes. Also, it is claimed that monotone circuit value problem ...
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### 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 ...
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### Proving that the sum of DTIME and DSPACE are not equal

I have an example question from a textbook where it asks to prove that $\Sigma_k DTIME(2^{n^k}) \neq DSPACE(2^n)$. There isn't a solution provided in the textbook. I've been working with a solution ...
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### show doubly connected graph is NL complete

The question:A directed graph is doubly connected if every two vertices are connected by a directed path in each direction. Let DCG = {| G is a doubly connected graph} Prove that DCG is NL-complete. (...
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### Space complexity for connectivity problem with given graph diameter

Given $(G,D,d)$, a graph, the graph diameter and the maximum outdegree of the graph. Verify that $G$ is strongly connected in $O(D\log d + \log n)$ space complexity. I thought about using the $STCON$ ...
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### Generalized Geography problem time and space complexity

We all know the algorithm that solves the Generalized Geography problem using polynomial space (it's described on wiki). My question is: what is the time complexity of this algorithm? I'd like a more ...
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### Algorithm to generate non-repeating random numbers of O(1) memory?

Is it possible to create $O(1)$ memory consuming algorithm, which is generating non-repeating pseudo random numbers? I can remember numbers in the hash set and it will be $O(1)$ time, but the set ...
527 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 ...
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### Row-Column agnostic Matrix datastructure

Due to the linear nature of computer memory, row-major and column-major matrices have different performance for row- vs. column access. The obvious solution would be to keep two copies of the same ...
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### How to show MULT={a#b#c| a,b,c binary natural numbers and a ✕ b = c} is in Log Space?

Let MULT$=\{a\#b\#c| a,b,c \text{ binary natural numbers and } a\times b=c\}$ Prove that MULT $\in L$ How do I show that this language, MULT, is computable in Logarithmic space? Let us assume a#b#c ...
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### United space-time complexity of finite strings

Let's consider bit string as a program for some computational model. If after $k$ steps program represented by number $n$ halts and outputs bit string $s$, then complexity of s is (n+1)*k. For example ...
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### Best way we know search for an integer

Basically, you have $n$ integers. The data structure is for your choice, it is ok to do polynomial time preprocessing on them. Then you have multiple questions "Is an integer $k$ in the set?" My ...
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### Median-of-medians in O(log n) memory

Is there a way to use median-of-medians to find a median in, simultaneously, ​ ​O(log n) ​ ​memory and O(n) comparisons? The user orlp on this site seems to claim that there is. Getting ​ ​O(log n) ...
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### PSpace-completeness under PSpace reductions

A language $L$ is PSpace-complete, if it meets two conditions: It is in PSpace. Every other PSpace-complete language reduces to it in polynomial time. Question: suppose we change the second ...
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### Prerequisites of computational complexity theory

what's the prerequisite topics needed for understanding computational complexity theory and analysis of algorithm ...including big-O and Big-theta notations and these staff. I want a mathematical ...
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### Space-unconstructable function in the proof of Savitch's theorem

I'm learning about the Savitch's theorem, and while the construction proof is straightforward, I still don't understand one part about it. The proof I'm talking about is the same as is currently on ...
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### Limited oracle TM

Let $M$ be a Turing machine with oracle to $B$ that can decide $B$ in polynomial time. In the general case it means nothing, since we can just pass the input as a query to the oracle of $B$ and accept/...
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### Natural logspace complete problems

After Omer Reingold's famous proof (from 2005?) that SL = L, the distinction between natural L complete problems and natural SL complete problems has been mostly dropped, so that it became difficult ...
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### What is a good example of an NL-complete context free language?

Setting Exactly as the title stated: Give an example of an $\mathsf{NL}$-complete context free language. $\newcommand{\angle}{\langle #1 \rangle}$ Current Solution Recall in the past we ...
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### Is PSPACE closed under the following forall-there-exists construction?

Suppose I have a language $L$ (over alphabet $\Sigma$), such that $$w \in L \iff (\forall x \in \Sigma^*) (\exists y \in \Sigma^*) P(x,y,w).$$ and I can give a turing machine that decides $P(x,y,w)$ ...
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### Does space complexity analysis usually include output space?

Since most examples of complexity analysis I've seen involve functions that return either nothing (e.g. in-place sort) or a single value (e.g. computation, lookup), I haven't been able to figure this ...
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### Proving SAT is in L

How to prove that SAT is in L if and only if NP=L? I know that reducing SAT in cook-levin theorem is computable in deterministic linear space . How to do it in log space? Any reference will also help.
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### 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 ...
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### Confusing method of proving PSPACE-completness

I don't understand a way of proving PSPACE-completness. The way was used by my lecturer. I can use reduction, however following method confuse me: We consider sequence (of polynomial length) of ...
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I'm trying to represent a directed acyclic graph using a structure similar to an adjacency list. The difference is, for a given vertex v, I need to know precisely which nodes are inwardly adjacent to ...
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### Show that problem is PSPACE-complete - path in directed graph

I have a following problem: Given $n$ and graph of size $2^n$, and circuit with $2n$ input gates. Directed edge between $k$ and $l$ exists iff only and only we encode $k$ and $l$ as bits and launch ...