Questions tagged [sorting]
Given a sequence of elements, find a permutation such that the elements are in a certain order.
120
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Algorithm for comparing list elements backwards
Input: A list of real numbers $[x_0, ..., x_n]$
Output: A list of integers $[d_0, ... , d_n]$ where $d_i$ is the largest $d\in\{1,\dots,i\}$ such that
$$
x_i\geq x_{i-1}, x_{i-2},\dots,x_{i-d}
$$
In ...
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1
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Find odd-ranked numbers from a list
From a list of $n$ distinct numbers, I want to find the set consisting of all odd-ranked numbers (1st, 3rd, 5th, ...). How many comparison queries do I need?
I could sort the whole list using $O(n\log ...
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votes
2
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178
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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 ...
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2
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1
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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 ...
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483
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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
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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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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, ...
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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?
3
votes
1
answer
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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{...
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77
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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 ...
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2
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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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117
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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
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135
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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 ...
- 103
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0
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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 ...
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134
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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 $\...
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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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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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104
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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 ...
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1
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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 ...
user15587
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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
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2
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237
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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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Formally prove that the loops of this sorting algorithm will terminate [closed]
Given is the sorting algorithm Bubblesort
...
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195
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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 ...
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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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1
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554
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 (...
- 2,537
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votes
0
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95
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.
...
- 2,537
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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 ...
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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 ...
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480
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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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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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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 ...
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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?
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votes
2
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"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 ...
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votes
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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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votes
2
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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 ...
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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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votes
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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
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0
answers
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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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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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2
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237
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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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votes
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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 ...
user34945
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votes
0
answers
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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 ...
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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 restricted sorting problem:
If we restrict sorting algorithms to use ...
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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 ...
- 13.9k
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votes
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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 ...
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