# Questions tagged [sorting]

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

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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 ...
266 views

### Hardness of optimal sorting

For comparison-based sorting algorithms, asymptotically optimal algorithms in worst-case $\Theta(n\log n)$ comparisons are well known. From a purely theoretical perspective, however, exactly optimal ...
890 views

### Enumerating topological sorts of a vertex-labeled DAG

Let $G = (V, E)$ be a directed acyclic graph, and let $\lambda$ be a labeling function mapping each vertex $v \in V$ to a label $\lambda(v)$ in some finite alphabet $L$. Writing $n := |V|$, a ...
3k views

### 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 restricted sorting problem: If we restrict sorting algorithms to use ...
254 views

### For median is it optimal to compare in pairs first?

Median can be done in linear time and is now down to (I think) $2.97n$. The lower bounds is (I think) $(2+\epsilon)n$ where $\epsilon$ is very small. The following theorem, if true, may help improve ...
141 views

### Computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$

This question is about computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$. The computation model we are considering is the unit-cost RAM machine with linear ...
258 views

### 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 ...
103 views

### Height of AVL tree with random elements

I know that for an AVL tree of N nodes, the depth of the tree is bounded by $$\log_2(N + 1) -1 \leq height \leq c \log_2(N + 2) + b$$ where $c,b$ are taken from the golden ratio linked to the worst ...
1k views

### Minimum Number of Adjacent Swaps to Sort Numbers on a 2D Grid

Assume that we have $N$ numbers (labeled from $1$ to $N$) that are placed on a 1D (linear) array. For example, for $N=5$: If we want to sort these numbers with the minimum number of adjacent swaps (i....
160 views

### What is the best upper bound known for the complexity of computing an optimal prefix free code in the RAM model?

In the algebraic decision tree, the result is clear: Elmasry and Belal (Verification Of Minimum Redundancy Prefix Codes)'s lower bound shows that the worst case complexity of computing an optimal ...
142 views

### Is there any efficient Network stable sort (not bubble sort)?

Ok, I realize Bitonic sort is not stable and any attempt to make it stable is inefficient, or is there some efficient way? But is there some other network sort which is indeed stable beside bubble ...
125 views

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### Lower bound for reversing a list using queues

How do you prove (or disprove) that a list of length $n$ cannot be reversed in time $o(n \log n)$ using $O(1)$ queues? Each queue is FIFO. Time refers to the number of operations on the queues. ...
206 views

### 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 ...
101 views

### Can you partially sort using $O(\log n)$ comparisons per element?

Input is a list of $n$ integers in an array A. Desired output is stored in Array B, such that $|rank(B[i])- i | \leq \sqrt{n}$. Can this be done using $O(\log n)$ comparisons per element? Just looking ...
86 views

### Cost of in-place partitioning integer arrays

Suppose we are given an array $a\colon[n]\to[m]$ of length $n$ (and each entry is between 1 and m). We will denote the $i$th entry of the array as $a[i]$. Task: Permute the array $a$ in-place so that ...
192 views

### 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 ...
78 views

### 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 ...
477 views

### estimating the number of comparisons of Shell Sort

I would like to estimate the number of comparisons in ShellSort. I'm using $h_s = 2^s-1$, where $s=\left \lfloor{\log(n)}\right \rfloor, \left \lfloor{\log(n)}\right \rfloor -1, \dots, 1$ ; I know ...