# Questions tagged [space-complexity]

Asymptotic analyses of the space needed to run algorithms.

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### 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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### 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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### Complete Problems for $DSPACE(\log(n)^k)$

We know that the $polyL$-hierarchy doesn't have complete problems, as it would conflict with the space hierarchy theorem. But: Are there complete problems for each level of this hierarchy? To be ...
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### Which CNF boolean formulas blow up exponentially at conversion to DNF?

If I'm correct, some boolean formulas in CNF require exponential size when being converted to an equivalent DNF version (and vice versa). But what is an example of such a formula (and is there a ...
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### Complexity of space density and sequentiality

I'm looking for some standard terminology, metrics and/or applications of the consideration of density and sequentiality of algorithms. When we measure algorithms we tend to give the big-Oh notation ...
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### 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 ...
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### 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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### 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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### Are there strongly-polynomial algorithms that take more than polynomial time?

In  strongly-polynomial is defined as either: The algorithm runs in strongly polynomial time if the algorithm is a polynmomial space algorithm and performs a number of elementary arithmetic ...
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### Show Recognizing two Regular Expressions as equal is in PSPACE

If I have $EQ_{REX} = \{\langle R,S \rangle|\text{$R$and$S$are equivalent regular expressions}\}$, how do I show that $EQ_{REX}\in PSPACE$ ? What I know so far is that there are decidable ...
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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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### 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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### What is the big-O (worst-case upper bound) for time and space requirement of the different Chomsky classes?

Everybody knows the Chomsky-hierarchy for describing formal languages and big-O notation for describing time and space complexity of a function. We know, that each class in the Chomsky-hierarchy ...
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### Verifiers equivalent classes

This is a HW question, so Im not expecting full solutions or anything, but would love some direction. Also English is not my first language, so I apologize in advance. We define a new class of ...
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### Show that the multiplication lies in FL

I don't know exactly how to solve the exercise below. Show that the multiplication lies in $\text{FL}$. Hint: A useful approach to a solution is to split the exercise into two parts and to ...
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### 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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### 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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### Acceptance behavior of L and NL with and without cycling

The complexity class NL seems to allow cycling, otherwise we wouldn't have SL $\subset$ NL. What about L? If an algorithm from L cycles for a given input, it certainly cannot accept (because it won't ...
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. ...