Questions tagged [sorting]

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

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5 votes
2 answers
158 views

Lower bound for sorting without using a decision tree model

Can we prove the lower bound for the sorting problem just by Turing machine model? It seems that available proof of sorting is based on the assumption that the algorithm only uses comparison so we can ...
2 votes
1 answer
53 views

Sampling strategies for Quicksort

I'm studying a variation of Quicksort in which the algorithm samples a subarray of size $f(n)< n$ ($n$ is the size of the input array) and then chooses the pivot from this subarray. The pivot is ...
6 votes
1 answer
469 views

Number of permutations that satisfy a given set of comparisons

We are given a set of comparisons of the form z[i] < z[j] for various i and j and an ...
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13 votes
3 answers
2k views

Expected number of random comparisons needed to sort a list

Consider the task of sorting a list x of size n by repeatedly querying an oracle. The oracle draws, without replacement, a random pair of indices (i, j), with i != j, and returns (i, j) if x[i] < x[...
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3 votes
0 answers
87 views

Linear time in-place stable sort

Surprisingly, linear time in-place stable sort is possible with integer keys of $O(\log n)$ bit length. An algorithm appeared in Radix Sorting With No Extra Space (Franceschini, Muthukrishnan, ...
0 votes
1 answer
53 views

Selecting unique records from a large dataframe with many duplicate records

Suppose we have a dataframe with ~10M rows with ~9M duplicate records. What is the most time efficient way of selecting the unique records from this dataframe? Some sort of sampling algorithm?
2 votes
0 answers
47 views

Sorting multiple columns of a matrix

Let $A \in \mathbb{R}^{n \times k}$ be a matrix where each column contains all of the numbers from $\{1,\dots,n\}$ in some arbitrary order. For example, if $n=3, k=2$, we could have $$ A = \begin{...
0 votes
0 answers
76 views

Low-Treewidth Sorting Networks

It was previously asked if there exist Boolean circuits of treewidth $O(\log n)$ that compute the majority function $\text{MAJ}_n$ on $n$ inputs. While a construction using online algorithms and the ...
0 votes
2 answers
253 views

Finding top-K items in a sliding window

Imagine we have a stream of bank transactions. Each transaction has a target account and some amount of money. I'd like to find top K accounts over some period of time (e.g. last 7 days) which ...
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0 votes
1 answer
113 views

What would be the performance properties of a comb or shell sort where the gap sequence is the prime numbers?

In particular, the genesis of this idea was the notion that you could minimize the number of passes through the list in a comb sort by using the primes as a way to guarantee that every element gets ...
0 votes
1 answer
118 views

A sorting algorithm that uses the minimum comparasions possible

I'm looking for a sorting algorithm that minimizes comparisons instead of time complexity. The algorithm shouldn't compare any two elements for which the relation between them can be derived from ...
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6 votes
0 answers
161 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 ...
  • 775
3 votes
0 answers
126 views

Minimum feedback arc set for dense directed graph

This is really a matrix problem, but the theory I believe lies in graphs. Consider some matrix $A$ and permutation matrix $P$, where we define $\tilde{A}:= PAP^T$. I want to pick $P$ such that if $\...
5 votes
0 answers
152 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 ...
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5 votes
0 answers
258 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 ...
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2 votes
0 answers
104 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 ...
  • 137
2 votes
0 answers
99 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 ...
6 votes
1 answer
249 views

Is sorting pairwise distances as hard as sorting arbitrary points?

If we have $n$ points in $\mathbb{R^d}$, what is the complexity of sorting the $O(n^2)$ pairwise distances? Clearly the complexity is $\Omega(n^2)$ but is there a reduction to show it is as hard as ...
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-2 votes
1 answer
171 views

Is it possible to sort by only knowing the sign of pairwise sums?

I am currently thinking of how much structure one actually needs in order to be able to sort things at all. All comparison-based algorithms need a direct comparability, but are we able to remove this ...
2 votes
2 answers
230 views

Under what models do we know linear time sorting?

The best we know for general case sorting is $O(n\log n)$ (which is also $\theta(n\log n)$ is decision tree model) and the problem of $O(n)$ sorting is open for turing machine models. Under what ...
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-1 votes
1 answer
69 views

Formally prove that the loops of this sorting algorithm will terminate [closed]

Given is the sorting algorithm Bubblesort ...
  • 117
3 votes
1 answer
183 views

how to achieve a topological sort of an given sequence with minimum swaps

For example, given the constraints {$a<b,c<d$} and a sequence $[b,a,c,d]$. we just need swap $a$ with $b$ to get an topological sort, I want to ask how to find the sort solutions with minimum ...
4 votes
0 answers
148 views

Asymptotic complexity of mass production

For a function $f:\{0,1\}^n \rightarrow \{0,1\}^m$, let $C(f)$ be the circuit complexity (for concreteness, constants and NOT gates are free, while 2-input AND gates cost 1). Let $k{\times}f : \{0,1\}...
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14 votes
1 answer
548 views

Sorting with an average of $\mathrm{lg}(n!)+o(n)$ comparisons

Is there a comparison-based sorting algorithm that uses an average of $\mathrm{lg}(n!)+o(n)$ comparisons? Existence of a worst-case $\mathrm{lg}(n!)+o(n)$ comparison algorithm is an open problem (...
3 votes
0 answers
89 views

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. ...
5 votes
3 answers
468 views

Sorting a programs instructions until it works

Lets say I have a computer program below. (define (factorial x) (if (= x 0) 1 (else (* x (factorial (- x 1))))) I then take each line of the ...
7 votes
1 answer
449 views

Computing topological sort while keeping edges "short"

Motivation: I want to compute a topological sort order in which the connected vertices are close to each other. Problem statement: Given a DAG $G(V,E)$ with $n$ vertices, compute a topological sort ...
-5 votes
1 answer
450 views

Does any DAG can be topologically sorted? [closed]

I am not good enough in computer science. My intention is to solve some programming problem in terms of DAG's. The key point is that before getting them into database, I need run "topological sort" in ...
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12 votes
1 answer
1k views

Is sorting $n$ real numbers in time $O(n \sqrt{\log n})$ and linear space possible?

In the recent preprint https://arxiv.org/abs/1801.00776, it is claimed that $n$ real numbers can be sorted in time $$O(n \sqrt{\log n}), $$ and linear space. The paper seems reasonable, though I am ...
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4 votes
0 answers
293 views

How fast can we sort a list if we know how it was written?

Let $G$ be a linear time (deterministic) turing machine that takes positive integers $n$ in unary to lists of length $n.$ For any fixed such $G$, define sparse-sort(G,n) as the problem of sorting the ...
3 votes
1 answer
129 views

Necessary and sufficient number of comparisons by every element to fully sort a set of n elements? [duplicate]

Given $n$ distinct elements. Is there a sorting algorithm which ensures that every element is compared atmost $\lg n$ time? Or is there a higher lower bound?
  • 137
18 votes
2 answers
2k views

"Almost sorting" integers in linear time

I am interested in sorting an array of positive integer values $L = v_1, \ldots, v_n$ in linear time (in the RAM model with uniform cost measure, i.e., integers can have up to logarithmic size but ...
  • 8,224
8 votes
1 answer
294 views

Original reference for Huffman shaped Merge Sort?

What is the first publication of the concept of optimizing merge sort by identifying sequences of consecutive positions in increasing orders (aka runs) in linear time; then repeatedly merging the ...
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12 votes
2 answers
1k 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 ...
  • 8,224
6 votes
2 answers
184 views

Quick-select contiguous subarray

Motivated by the question from this blog post, the following data structure question seems interesting and fun to me. Preprocess: A list of numbers $A = a_1,...,a_n$ Query(s,t,k): Return the $k$-th ...
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4 votes
1 answer
460 views

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 ...
2 votes
0 answers
208 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 ...
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9 votes
1 answer
378 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 ...
  • 307
0 votes
2 answers
235 views

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 ...
  • 143
11 votes
2 answers
278 views

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 ...
  • 8,224
7 votes
1 answer
239 views

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 ...
user avatar
2 votes
0 answers
104 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 ...
5 votes
1 answer
291 views

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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16 votes
1 answer
262 views

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 ...
  • 13.8k
3 votes
0 answers
223 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 ...
11 votes
0 answers
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 ...
14 votes
0 answers
487 views

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 ...
  • 13.8k
5 votes
1 answer
229 views

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 ...
3 votes
1 answer
114 views

Locally sorted sequences

Let $S=s_1,\ldots,s_n$ be a sequence and $p$ be a permutation on the indices of $S$ such that $p$ sorts $S$. Define a sequence to be locally sorted with degree $k$ if $\forall s_i \in S |p(i) - i | \...
  • 663
6 votes
2 answers
2k views

Is the bitonic sort algorithm stable?

I was wondering, is the bitonic sort algorithm stable? I searched the original paper, wikipedia and some tutorials, could not find it. It seems to me that it should be, as it is composed of merge / ...
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