# Tagged Questions

Given a sequence of elements, find a permutation such that the elements are in a certain order.

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### Quicksort: compute the expected number of comparisons as a function of $M$ and $t$

I stumbled upon this problem on a list of open problems in the analysis of algorithms dating back to 1997. Is it still open? Can anyone point to a reference with a full or partial solution, or at ...
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### Quicksort optimal partition

Has the question been studied, how to find the shortest sequence of partition choices so that a quick-sort algorithm can sort a set? To be clear, I'm not interested in quick sort per se, but in ...
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### Most efficient inplace merge algorithms (stable and unstable)

I am currently researching the best algorithms available to achieve an inplace merge operation: consider two consecutive sorted arrays of size n and ...
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### Sorting sequence with $O(n^{\frac{3}{2}})$ inversions

There is given sequence $a_1,...a_n$ such that there are $O(n^{\frac{3}{2}})$ inversions in this sequence. I am thinking about sorting algorithm for that. I know lower bound for number of ...
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### Determining what can be achieved by a permutation of elements of a noncommutative group

Fix a finite group $G$. I am interested in the following decision problem: the input is some elements of $G$ with a partial order on them, and the question is whether there is a permutation of the ...
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### Sorting using ring operations

Sorting is in $\mathsf{NP}$. Given a sorted list, it is trivial to check sortedness in linear time. Is there any evidence sorting of elements from an ordered gcd domain(eg: $\Bbb Z$) cannot be done ...
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### Are there any algorithms that are similar to Fagin's Algorithm, but for unranked lists?

Fagin's Algorithm is a popular algorithm for finding the top-$k$ items from multiple ranked lists of the items (i.e., via different scoring functions), using some monotonic aggregation function for ...
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### Lexicographically minimal topological sort of a labeled DAG

Consider the problem where we are given as input a directed acyclic graph $G = (V, E)$, a labeling function $\lambda$ from $V$ to some set $L$ with a total order $<_L$ (e.g., the integers), and ...
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### Reducing sorting to max-flow

Is there a linear-time reduction from the sorting problem to the max-flow problem? If so, what would such a reduction look like?
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### Is it enough to sort for polynomially many 0-1 sequences for a sorting network?

The 0-1 principle says that if a sorting network works for all 0-1 sequences, then it works for any set of numbers. Is there an $S\subset \{0,1\}^n$ such that if a network sorts every 0-1 sequence ...
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### Probabilistic sorting given pairwise comparison probability

Let $X = \{x_1, \dots, x_n\}$ be a set, and $f:[1..n]^2 \to [0, 1]$ be a function, such that $$f(i, j) \cdot f(j, k) \le f(i, k)$$ For all $1 \le i, j, k \le n$. Does there exist a randomized ...
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### Efficient recognition of sequences sortable by transpositions?

While reading the post, Probability of generating a desired permutation by random swaps, by Aaronson, I got interested in this problem of sorting: Input: a sequence $A$ of $2N$ positive integers. ...
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### Is it possible to find the median with a linear size sorting network?

Is there a sorting network that makes only $O(n)$ comparisons and finds the median? The AKS sorting network sorts with $O(\log n)$ parallel steps, but here I am only interested in the number of ...
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### How to Quantify Entropy in a Data Set

I'm currently creating a program in Java to analysis the pathological cases of Quicksort. Namely, the transition of complexity from O(n^2) to O(nlogn) as a data set gets less ordered. Since Quicksort ...