# Tag Info

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### Does space complexity analysis usually include output space?

Typically, we consider space complexity in terms of Turing machines with: one read-only input tape one write-only output tape however many read-write working tapes you want. The space usage is the ...

### How is the problem, {⟨G⟩|G has no triangle} in Logspace?

FOR x := 1 TO n DO FOR y := 1 TO n DO FOR z := 1 TO n DO IF E(x,y) && E(y,z) && E(z,x) THEN REJECT ACCEPT Each of the ...
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### How is the problem, {⟨G⟩|G has no triangle} in Logspace?

You don't need to first write all 3-tuples and then check, for each of them, whether it induces a triangle. You can just enumerate the 3-tuples one at a time and reject as soon as you find one that ...
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### why does LSPACE(log space) complexity class exist but not logtime?

Turing machines operating in logarithmic time cannot even read the entire input. This makes them rather uninteresting. What you have in mind is not Turing machines, but random-access machines, for ...
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### What does sublinear space mean for Turing machines?

When dealing with restricted space, we use the following model. The Turing machine has three tapes: a read-only input tape, a read-write work tape, and a write-only output tape. We only measure space ...

### Algorithms with O(sqrt(N)) SPACE complexity?

$\sqrt{n}$ space is somewhat unusual; the most likely reason for such a complexity to emerge is as a result of a so-called meet in the middle scheme. Two notable examples off the top of my head are ...

### How to check if two strings are permutations of each other using O(1) additional space?

Denote the arrays by $A,B$, and suppose they are of length $n$. Suppose first that the values in each array are distinct. Here is an algorithm that uses $O(1)$ space: Compute the minimum values of ...

### Is there an algorithm for algorithms time/space complexity optimisation?

Look up Blum's speedup theorem (yes, this article is less than informative; look at a book on complexity theory). It essentially says that there are programs for which there is a program doing the ...
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### How can an algorithm have exponential space complexity but polynomial time complexity?

This algorithm doesn't run in polynomial time, it runs with polynomial delay. As the paper notes: Observe that the number of minimal, and even minimum solutions, can be exponential in the size of ...
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### space complexity of DFA intersection problem

Solving intersection Non-Emptiness for 2 DFA's: It essentially just becomes a reachability problem for the product DFA. Roughly, we can solve it deterministically in $O(n^2)$ time using $O(n^2)$ ...

### is Co-NP in PSPACE?

By the same reasoning that NP is in PSPACE, co-NP is in co-PSPACE. But co-PSPACE = PSPACE (you can just flip the answer), so co-NP is in PSPACE, too.
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### Why is PH in PSPACE?

No, it is not necessary to remember all $y$'s tried before. In order to remember that I've tried the numbers $1,2,\ldots,200$, I do not need to remember $3,4,5,6,\ldots,199$. If you try them in order, ...
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### Proving that $\mathrm{SPACE}(o(\log\log n)) = \mathrm{SPACE}(O(1))$?

You're basically there. If the machine uses at most $s(n)$ tape cells for inputs of length $n$, then it can't visit more than $C(n)=|Q||\Sigma|^{s(n)}s(n)$ different configurations (possible ...

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### Can FPSPACE give exponentially long outputs?

Space classes always only include working space: the model is that we have a read-only input tape and write-only output tape, plus a read-write work tape (or multiple such tapes) on which we're only ...
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### How hard are PSPACE-complete problems?

By definition, PSPACE consists of all languages decided by some Turing machine using polynomial space. So every language in PSPACE can be decided by some Turing machine using space $O(n^\ell)$ for ...
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### How to check if two strings are permutations of each other using O(1) additional space?

The naive approach would be building histograms of both strings and checking whether they are the same. Since we are not allowed to store such a data structure (whose size would be linear to the size ...
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### Is there a polynomial time algorithm to determine whether an 'up down' language is 'emptible'?

Reduction from 3-SAT: a variable in 3-SAT becomes a character in your problem and is paired with its negation. Each clause becomes a word. e.g. 3 SAT: (a,b,-c) && (-b,c) => pairs: (a,-a), (...

### $UCYCLE$ is in $L$

First of all, let me give the correct attribution to this algorithm: Cook and McKenzie, Problems complete for deterministic logarithmic space. The setup of Cook and McKenzie is that you are given an ...
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### Show that NP is not equal to SPACE(n)

In order to prove that $\mathsf{SPACE}(n) \not\subseteq \mathsf{NP}$, you need to identify a language in $\mathsf{SPACE}(n)$ which is not in $\mathsf{NP}$. Not every language in $\mathsf{SPACE}(n)$ ...
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### Logic behind O(n) solution for 'Maximum length sub-array having given sum'

This is an example of a dynamic algorithm. I will adapt the algorithm in the link, as I don't think that it is written in the most helpful way. Initialise sum = 0 ...
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### What's after EXPSPACE?

EXPSPACE is not the most inclusive computational complexity class. There's a huge number of complexity classes; see, e.g., the Complexity Zoo for some that have been studied (and in principle there ...