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1500 questions
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Can the "divide" step in a merge sort be avoided?

So merge sort is a divide and conquer algorithm. While I was looking at the above diagram, I was thinking if it was possible to basically bypass all the divide steps. If you iterated over the original array while jumping by two, you could get the…
Jimmy_Rustle
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Easy proof for context-free languages being closed under cyclic shift

The cyclic shift (also called rotation or conjugation) of a language $L$ is defined as $\{ yx \mid xy \in L \}$. According to wikipedia (and here) the context-free languages are closed under this operation, with references to papers from Oshiba and…
Hendrik Jan
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Importance of recursion in computability theory

It is said that computability theory is also called recursion theory. Why is it called like that? Why recursion has this much importance?
user5507
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Proving P = NP without mathematical statements / computer program

This is my first post after being a passive user for some time now. I wish to ask some questions if I may. I am not a mathematician but my question relates to the field of maths/computer science. In particular, the P vs NP problem. I am aware…
user5287
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1 answer

Random Access Machines with only addition, multiplication, equality

The literature is fairly clear that unit-cost RAMs with primitive multiplication are unreasonable, in that they cannot be simulated by Turing machines in polynomial time can solve PSPACE-complete problems in polynomial time However, all of the…
Mike Battaglia
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7 answers

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

Given two strings how can you check if they are a permutation of each other using O(1) space? Modifying the strings is not allowed in any way. Note: O(1) space in relation to both the string length AND the size of the alphabet.
Teodor Dyakov
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Discrepancy between heads and tails

Consider a sequence of $n$ flips of an unbiased coin. Let $H_i$ denote the absolute value of the excess of the number of heads over tails seen in the first $i$ flips. Define $H=\text{max}_i H_i$. Show that $E[H_i]=\Theta ( \sqrt{i} )$ and…
Plummer
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Optimal strategy for an abstract game

I've been given the following problem in an interview (that I've already failed to solve, not trying to cheat my way past): The game starts with a positive integer number $A_0$. (E.g. $A_0 = 1234$.) This number is converted to binary representation,…
millimoose
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Counting permutations whose elements are not exactly their index ± M

I was recently asked this problem in an algorithmic interview and failed to solve it. Given two values N and M, you have to count the number of permutations of length N (using numbers from 1 to N) such that the absolute difference between any…
Gena
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How does 0 have two values in one's complement?

It is said that in 2's complement 0 has only one value, while in 1's complement both +0 and -0 have separate values. What are they?
user136782
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Finding optimal sequence of questions to minimize total student time

Suppose there is a tutorial session at a university. We have a set of $k$ questions $Q = \{ q_1 \ldots q_k \}$ and a set of $n$ students $S = \{ s_1 \ldots s_n \}$. Each student has a doubt in a certain subset of questions, i.e. for each student…
skankhunt42
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Decidable restrictions of the Post Correspondence Problem

The Post Correspondence Problem (PCP) is undecidable. The bounded version of the PCP is $\mathrm{NP}$-complete and the marked version of the PCP (the words of one of the two lists are required to differ in the first letter) is in $\mathrm{PSPACE}$…
Vor
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Matrix chain multiplication and exponentiation

If I have two matrices $A$ and $B$, of dimensions $1000\times2$ and $2\times1000$, respectively, and want to compute $(AB)^{5000}$, it's more efficient to first rewrite the expression as $A(BA)^{4999}B$ and only then evaluate numerically, because…
isekaijin
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Gödelization in Turing Machine

I was looking at Gödelization in Theory of Computation course. I could understand the Gödel numbering concepts, but couldn't understand its importance in Theory of Computation. Could anyone please point to some good materials or point out its…
user5507
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Algorithms with O(sqrt(N)) SPACE complexity?

Are there any known algorithms for formulated problems that require a SPACE complexity of O(sqrt(N))? I know that algorithms with that big-O time complexity exist.
vawd_gandi
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