Questions tagged [sets]
Questions about finite and infinite sets and multisets, related data structures and concepts.
418
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Minimal Hitting Sets Problem
Let $\mathcal{I} = \{I_0, \ldots, I_{m-1}\}$ a collection of subset of some universe $U$.
We want to find a partition $P$ of $\mathcal{I}$ of minimal cardinality such that the intersection of each set ...
- 133
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1
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60
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Constructing a Container for the Given Situation
I need to make a container in which I can store (x,y) as pairs, and for a given number 'a', I have to find a pair (p, q) such that p<=a and q is maximum possible.
Note the constraints: x>=1 and ...
- 13
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1
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129
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Best balanced assignment
I'm at a problem I don't know better to name it... maybe it's already a well known problem?
It seems quite simple:
There are objects and labels in a n:m relation.
(Each of the n objects may be ...
- 1
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30
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Finding Minimum Elements for Longest Path in Disjoint Set
I want to know the minimum number of elements needed to create a tree with the longest path having n edges. How can I approach this problem using the forest implementation of disjoint sets with union ...
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25
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Find a set of elements with minimal costs that contains at least one element of given subsets
I have encountered at my work currently a problem where I want to find an efficient algorithm for, although I suspect this might be hard problem. Maybe you can help me to tell if this is a NP hard ...
3
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118
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Maximum set cover with non-overlap
Let the universe be the set $U$ and a set of subsets $S$ be such that $\cup_{s \in S} s = U$. I am interested in computing the longest sequence of sets $s_1, ..., s_k$ such that:
$s_i \in S$ $\forall ...
- 31
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44
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Assignment problem with maximal partitioning
Recently I came across a problem I don't get may hands on:
Given p binary positions.
Let s be the number of "set-bits" (1 < s < p * 2^(p-1) - 1).
I need the maximal set of assigments ...
- 1
3
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58
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Map-like data structure with subsets as keys
I am looking for a map-like data structure with the following properties:
it uses subsets of some set S as keys. The size of S is potentially unbounded, but does not change during the runtime
the ...
- 131
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15
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Trying to figure out how to model the structure of a multilingual dictionary for several constructed languages (basically Wiktionary for my conlangs)
Okay, so, this will be quite a bit, sorry. I'm working on several constructed languages for a worldbuilding project. Up until this point, I have been using a spreadsheet to store the vocab; each row ...
- 1
1
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1
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21
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Is there a practical algorithm for estimating antichain coverage of a superset?
Suppose I'm given a set $S$ and antichain $A \subset 2^S$ ($\forall a_1,a_2\in A: a_1\neq a_2 \iff a_1 \nsubseteq a_2$).
Let's call subset $b \in 2^S$ covered by $A$ if $\exists a \in A :b \subseteq a ...
- 13
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45
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Sending set of string in HTTP query string
I'd like to implement pagination for an API. The elements are ordered by time but there can be multiple elements with the same timestamp. So there can be some duplication between the last elements of ...
- 101
1
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1
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49
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How is the direct product of the functions (A -> B) * (C -> D) equivalent to the function (A * C) -> (B * D)? Is there an error here?
In the simply typed lambda calculus we have type algebra - types can be added, multiplied and exponentiated, where addition corresponds to the sum type, multiplication to the product type, and ...
2
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1
answer
60
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What is the time complexity of removing among $N$ sets of size at most $n$ the sets which are subsets of another set?
A naïve solution would be to first sort all sets, taking time $O(N n \log n)$. Then, for every possible pair of sets, check if one is a subset of the other, and if applicable remove the subset. This ...
- 675
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5
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354
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What is the difference between the set containing the empty string and the set containing nothing at all?
It's an exercise question from chapter 0 of Michael Sipser's book Introduction to the Theory of Computation.
e. The set containing the empty string
f. The set containing nothing at all
I guess the ...
- 11
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3
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63
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Given two sets of coordinates, find out neighboring ones
I have two sets of 2-dimensional coordinates on an integer grid, $A$ and $B$
$A = \{(x_{A1},y_{A1}), (x_{A2}, y_{A2}), (x_{A3}, y_{A3}), \dots\}$
$B = \{(x_{B1},y_{B1}), (x_{B2}, y_{B2}), (x_{B3}, y_{...
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39
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4
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54
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Finding all sets which are not subsets of other sets
I have a set of sets, for example
{
{1, 2, 3},
{1, 2},
{2},
{2, 4}
}
I want to find all sets which are not subsets of another set. For example, ...
- 141
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19
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Randomly Split a Bar Into Beats
So I'm writing a software that generates random MIDI tracks based on a given mode, tonal etc.
As for now the randomisation works on tones building sequences of equal duration.
What I'd like to do is ...
- 111
-4
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2
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31
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Let F be a function defined for all nonnegative integers by the following recursive definition
Let F be a function defined for all nonnegative integers by the following recursive
definition.
F(0) = 0, F(1)= 1
F(n + 2) = 2F(n) + F(n +1), n>0
Compute the first six values of F; that is, write ...
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1
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1
answer
36
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How to argue that an $A$-covering matching exists in this bipartite graph?
In lecture the following was mentioned in the context of matchings in bipartite graphs:
Let $U$ be a finite set and let $\mathcal{S}$ be a family of subsets of $U$.
For $u \in U$ let $r(u) := \lvert \...
- 457
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1
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63
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Way to call and explain: "potentially infinite set of attributes" in databases
This is a bit of a theoretical question. I would like to know how to call the principle described below, in proper computer science terms, or math terms.
Let's say we have a database in which one ...
0
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1
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55
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Logical Consequence - Equivalent Assertions
I have the following slide in my notes and I'm having trouble understanding how the three assertions are equivalent. I understand to a degree how the 2nd and 3rd assertions are equivalent, but the ...
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31
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Reconstruction of the universal set from disjoint subsets
Before I even attempt coming up with an efficient algorithm, I tried googling for similar problems but didn't get far, most queries mentioning "sets" in them led to some sort of Multiple ...
0
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1
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181
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Suppose we have an empty alphabet Σ=∅, what are the possible languages of this alphabet?
Lets say the alphabet is Σ=∅,what are the possible languages of this alphabet?
According to my definitions:
I know that an alphabet is a finite set of symbols Σ
I know words is a set of all finite ...
- 3
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1
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22
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Algorithm for allocating resources; one resource per one user who accepts it
I am looking for an algorithm for the following problem:
I have a set of users and a set of books.
Every user has their own set of favorite books, which may be empty, and is a subset the set of books.
...
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1
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72
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Is there a computationally efficient algorithm which can map back and forth a multi-dimensional real number (R^n) to a single dimensional real (R)?
I believe its possible to achieve this with natural numbers.
The example below is for 2d to 1d conversions both ways, I do believe this generalizes to n-dimensions.
The mapping should work in a way ...
- 165
2
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2
answers
134
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Find if a given number must be in a set that is closed under gcd and lcm with some given elements
Source: https://oj.vnoi.info/problem/cryptkey (problem statements are in Vietnamese, so here it is translated).
There is a set $S$ of positive integers. If $A$ and $B$ are in $S$, then $\gcd(A, B)$ ...
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2
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65
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Why is $\{ w \in \Sigma^* : M_w[\epsilon]\downarrow \land |w| \leq 7\}$ decidable?
I get that the argument for this set $\{ w \in \Sigma^* : M_w[\epsilon]\downarrow \land |w| \leq 7\}$ to be decidable is that $|w|\leq7$ meaning it is a finite set and therefore it can be decided. ...
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209
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Does order of elements in a set matter in Dijkstra's Algorithm?
When we use a set for doing Dijkstra's Algorithm, we use a pair of {distance,node} which we insert in a set. Most of the articles say that the first element of pair should be the distance , else we ...
2
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43
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Lowest total cardinality mutually exclusive construction of a superset
Let there be $N$ sequences containing at least one set each. Each set has at least one element each.
Select exactly one set from each sequence. The selection within each sequence is mutually exclusive....
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140
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Efficient algorithm to count number of intersections of n sets
I've come across this problem when working on a personal project of mine. I need an efficient algorithm of counting the number of overlaps between all pair combinations of n sets.
Example:
Set a = [...
- 23
2
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2
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152
views
Dictionary with sets as keys where lookup can be set intersection
Normally, when working with dictionaries, we expect around O(1) complexity when we go to retrieve a value given the key (and when we insert). I work in Python, but this might apply to any dynamic ...
- 123
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1
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42
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Find the largest subset of unpaired elements [duplicate]
I have a large list (around 200k) of element pairs (e.g. A-B, A-C, B-C, ...). How can I find the largest subset of elements amongst which none are paired?
Example ...
- 113
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90
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Pair points between to sets minimizing the global distance
I have two set of points in the plane or space, which could be for instance radar contacts over two successive scans. I'd like to pair them so that the sum of squared distances is minimal.
One ...
- 111
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19
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Algorithms for computing "optimal set growth order"
Imagine you have a collection of possible "components" (C) and a set of "recipes" assembled from those components (...
- 101
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1
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44
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If $L$ is finite and $R$ is not regular, then $R\cup L$ is not regular
Prove/Disprove: If $L$ is finite and $R$ is not regular, then $R\cup L$ is not regular.
I think that this one is true, but I am stuck:
Since $R$ is not regular, it is infinite, so $R \cup L$ is also ...
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31
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Prove that a predicate is not computable
Prove that the following predicate is not computable:
$P_e(n) =
\begin{cases}
1 & \textrm{if } \phi_n(n) = e \\
0 & \textrm{otherwise}
\end{cases}$
Could someone explain how to approach ...
6
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0
answers
39
views
How to find the minimum number of elements to distinguish several given sets?
Given $n$ distinct sets $S_1, S_2, \cdots, S_n$, how to find a set $X$ such that $X \cap S_1, X \cap S_2, \cdots, X \cap S_n$ are still distinct, and the size of $X$ is minimum?
For example, given $\{...
Acesrc
2
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answer
28
views
Finding all combinations of length k that has at least one of the pairs of T is in it
Let there be a list of $n$ elements $S$. Let $T$ be a set with $m$ elements ($m \leq nC2$), with each element in $T$ being a pair of distinct elements of $S$. For $k\geq2$, is there a polynomial-time ...
3
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1
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69
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Fast algorithm for computing minimal closure of a set of sets under intersection?
A step of an algorithm I’ve designed requires computing the minimal closure under intersection of a set of sets of arbitrary size. By the "minimal closure (of a set $S$) under intersection", ...
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46
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Prove that a dominating set has minimum cardinality in a "unit interval graph"
I am given the definition of a unit interval graph, e.g. $G = (V, E)$ such that $\forall v \in V$ there is a weight $x_v \in \mathbb{R}$ and nodes $u, w$ has an edge iff $|x_u - x_w| < 1$. I am ...
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37
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Locality Sensitive Hashing for Sets
Are there locality sensitive hashes that work nicely with sets? Each set would get a hash, the order of the elements in the set does not change the hash, and sets that share more elements are closer ...
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0
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62
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Find sets which are subsets of the given search set?
The problem is the following:
You are given a collection( set, list, whatever ) C of sets, and you are given a search set S.
We want to find among all sets in C the ones which are subsets of S.
Hence, ...
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148
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Changing a family of sets to become laminar
A family of sets $F = \{S_1, \dots, S_n\}$ on the ground set $S$ is laminar, if for every $1\leq i < j \leq n$, either $S_i \subsetneq S_j$ or $S_j \subsetneq S_i$ or $S_i \cap S_j = \varnothing$ ...
- 287
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53
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Is the equality of Bloom filters analogous to set equivalence?
I have two multisets $A$, $B$ where $A \subseteq B$.
Using these two sets, we construct two Bloom filters $BF(A), BF(B)$; both using bitsets of size $n$ with the same $k$ hash functions.
What's the ...
- 123
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0
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49
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Is there a distributed streaming algorithm to verify set cover?
I have $k$ sets of similar sizes, that cover a universe $U$.
e.g. for $k=3$ and $U = \{1, 2, 3, 4, 5, 6\}$:
$S_0 = \{1, 2, 4\}$
$S_1 = \{2, 3, 4\}$
$S_2 = \{4, 5, 6\}$
I have another larger set $C$ ...
- 123
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1
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32
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DFA and a Partition of $\Sigma^*$
So I'm learning about Myhill-Nerode relations and as an introduction, the book describes possible partitions for $\Sigma^*$. As an example, given a language $L$, a partition of $\Sigma^*$ would be $\{...
- 29
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0
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33
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How to generate supersets from a finite number of subsets efficiently
Let $F$ be a set, for instance $\{a,b,c,d,e \}$. Suppose I have a set of subsets of cardinality two obtained from $F$:
$ ${ a,b },$\{b,c\},${a,d}
I want to create every possible set of cardinality ...
- 111
2
votes
1
answer
181
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Union of multiple overlapping sets efficiently?
I have $n$ sets, each of which overlaps heavily with the other sets, and I want the union of all of them. The obvious solution is to take the union of each set, one by one, which results in $O(n^2)$ ...
- 120
2
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1
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49
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Is there a simplistic way of describing the proof to the undecidability of David Hilbert's 10th problem?
I recently have been reading a bunch about David Hilbert's famous 10th problem, and trying to understand its proof. I am currently in the process of reading through an explanation of the proof, given ...
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