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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 ...
• 82.1k
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Does an algorithm's space complexity include input?

It depends on the chosen convention. I often prefer the convention that considers that the input is not part of the space complexity, for different reasons: the space complexity of a function answer ...
• 16k

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 ...
• 82.1k
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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 ...
• 29.6k
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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 ...
• 279k

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 ...
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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 ...
• 279k
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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, ...
• 13.4k
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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 ...
• 82.1k
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Is every PSPACE-complete problem complete with respect to logspace reductions?

If every PSPACE complete problem is also complete under logspace reduction, then $\mathsf{P\neq PSPACE}$. To see why, suppose for the purpose of contradiction that the definition of completeness ...
• 13.5k
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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 ...
• 82.1k

• 279k
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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 ...
• 608
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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), (...
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$UCYCLE$ 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 ...
• 279k
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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 $\mathsf{SPACE}(n)$ ...
• 279k
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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 ...
• 461
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Max number of configurations of a Turing Machine

Yes, that's right. If there are $k$ possible configurations, then any such Turing machine either runs in time at most $k$, or it loops forever. That's because if it runs for at least $k+1$ time ...
• 164k
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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 ...
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Time complexity for count-change procedure in SICP

Order of growth of number of steps: $\theta (n^5)$ We can prove that, in general, the order of growth of number of steps is $\theta (n^m)$, where $m$ is the number of types of coin available. Here is ...
• 1,101
Accepted

Understanding why ALL_nfa is in co-nspace

To show that $ALL_{\mathsf{NFA}}$ is in $\mathrm{co-NSPACE}(n)$, we must show that the complement $\overline{ALL_{\mathsf{NFA}}}$ is in $\mathrm{NSPACE}(n)$. The complement is \overline{ALL_{\...
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Every language $X$ in PSPACE would be complete under your proposed definition, except for $\emptyset$ and $\Sigma^*$. You could reduce any PSPACE language $Y$ to $X$ by a reduction that ...