Questions tagged [sets]

Questions about finite and infinite sets and multisets, related data structures and concepts.

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Is this intersection set problem NP-Hard?

Suppose we have collection of n sets $S_1, S_2, \dots, S_n$. Each set has a size of at least $k$. We know for sure that $\exists k$ sets where all of them contain the same $k$ elements; $|S_1 \cap S_2 ...
Xfae's user avatar
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2 votes
1 answer
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Decide whether this Problem NPC or P?

Input: A finite set A, subsets S1, . . . , Sn ⊆ A, and a number k ∈ N. Question: Does there exist a set R ⊆ A with |R| = k such that |R ∩ Si| = |Si| for all 1 ≤ i ≤ n? I read somewhere (without ...
Moh's user avatar
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2 votes
1 answer
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Given two sets of integers find an integer in the first set furthest away from all members of the second set

If given two sets of integers how can I find an integer in the first set furthest away from all members of the second set. The distance of two integers is the absolute value of difference of the two ...
hovnatan's user avatar
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CSES Sliding Cost - why is this formula for updating the cost correct?

In the CSES problem Sliding Window Cost, given an array and an integer k, we are asked to compute the minimum cost for each sliding window of fixed size ...
Andrei Onoie's user avatar
4 votes
0 answers
37 views

Cardinalities in set coverings

Let $I$ be a set of items; $C \subseteq \mathcal{P}(I)$ be a set of subsets of $I$, where $\mathcal{P}(I)$ stands for the power set of $I$; And $C(i) = \{ c \in C \mid i \in c \}$ be the set of sets, ...
Matheus Diógenes Andrade's user avatar
2 votes
1 answer
59 views

"Consecutive statements" in Static control Part(SCoP)

Context: I was reading a research paper related to polyhedral representation(citation given in last). Got confused while trying to understand the notation by implementing them with example code. Paper ...
F.C. Akhi's user avatar
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2 votes
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114 views

SCoP, Iteration Domain in Polyhedral Optimization and use of Presburger arithmetic

Context: While exploring the fundamentals of polyhedral optimization and attempting to explore a connection from the input Static Control Part (SCoP) to the iteration domain from birds eye view, I am ...
F.C. Akhi's user avatar
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Does adding all of these sequences actually get us the set of all infinite sequences?

Section 1.9 Binary Strings of the textbook Introduction to Lattices and Order, second edition, by Davey and Priestley, says the following: Let $\Sigma^\ast$ be the set of all finite binary strings, ...
The Pointer's user avatar
1 vote
1 answer
49 views

If A U B and A ∩ B are recognizable, then is one of A, A', B, B' also recognizable?

I know that if decidability of $A \cap B$ and $A \cup B$ doesn’t guarantee the decidability of any of $A$ or $B$. We can prove that: ATM is not decidable. Since decidable languages are closed under ...
Luis Ramirez's user avatar
2 votes
0 answers
24 views

Windowed LogLog/HyperLogLog algorithm to get a count of the cardinality of the set of the last $k$ elements?

LogLog/HyperLogLog provides a great way for estimating the cardinality of the set of $n$ objects. At its simplest, you hash all $n$ objects into binary strings, find the largest number of leading 0's $...
chausies's user avatar
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Enumerating proper intersections

Let $U \subset \mathbb{N}$ be a finite universe set; $B$ be a set of nonempty subsets of $U$ such that $B$ covers all elements in $U$, i.e. $\bigcup_{b \in B} b = U$, and if $b \in B$ then $b \...
Matheus Diógenes Andrade's user avatar
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0 answers
52 views

Binary range identification in terms of subset

Given a range of "K" bit numbers, we want to identify the whole range by using a subset within that range, with the following criteria - If there are > "n" consecutive 0s, it ...
Dave's user avatar
  • 23
1 vote
1 answer
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Linked Lists, Ordered Pairs?

I would like to model linked lists using set theory similar to that in Scheme and LISP. There is a set theoretic definition of the ordered pair: $p = \{\{a, 1\}, \{b, 2\}\}$ My question is how does ...
notaorb's user avatar
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Is this variant of multiset covering problem NP-hard?

Consider this variant of multiset covering problem. Input: a collection of sets $S = \{s_1, s_2, \ldots, s_n\}$ and a universal set $U$, in which $s_k \subseteq U$ and $s_k \neq \emptyset$ for all $k$...
Josh's user avatar
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1 vote
1 answer
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Order in a subset

Lets consider a range of "K" binary digit numbers. In that range, we want to take a subset of those values which have (<="n" consecutive 0s) AND (<="n" consecutive ...
Dave's user avatar
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2 votes
1 answer
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Algorithm to identify common subsets

Given a large dataset $D$ and multiple sets of filters that can be applied to $D$, e.g. $setA = \{filterOnColorRed\}$ $setB = \{filterOnAgeGreaterThan20\}$ $setC = \{filterOnColorRed, ...
foo's user avatar
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2 votes
1 answer
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Algorithm to find intersection between collection of sets

I have two dataframes representing products two distributors sell. They look like this: df1 for distributor 1. ...
nepee's user avatar
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1 vote
2 answers
111 views

Can we efficiently find the most common pair in a set of sets?

Given a collection of sets $C=\{S_1, S_2, \ldots, S_n\}$ with each $S\in C$ holding some integers, I would like to find a pair $\{x,y\}, x\ne y$ such that $\{x,y\}\subset S$ for the highest number of $...
fuz's user avatar
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1 answer
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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 ...
matteo_c's user avatar
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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 ...
harshmangalamv's user avatar
0 votes
1 answer
136 views

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 ...
User42's user avatar
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3 votes
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160 views

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 ...
in_question's user avatar
0 votes
1 answer
52 views

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 ...
User42's user avatar
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3 votes
0 answers
101 views

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 ...
Minop's user avatar
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1 vote
1 answer
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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 ...
NooneAtAll3's user avatar
0 votes
0 answers
47 views

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 ...
Karsten's user avatar
  • 101
1 vote
1 answer
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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 ...
Wasabi Kurosawa's user avatar
2 votes
1 answer
82 views

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 ...
J. Schmidt's user avatar
0 votes
5 answers
891 views

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 ...
Gabe Ebag's user avatar
0 votes
3 answers
120 views

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_{...
2080's user avatar
  • 191
1 vote
1 answer
67 views

Binary subsets for a given set

Lets take an example of the range of N=32 bits ...
Dave's user avatar
  • 23
4 votes
0 answers
84 views

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, ...
Daniel M.'s user avatar
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1 vote
0 answers
21 views

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 ...
Carlo Moretti's user avatar
-4 votes
3 answers
97 views

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 ...
Max's user avatar
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1 vote
1 answer
40 views

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 \...
3nondatur's user avatar
  • 457
0 votes
1 answer
92 views

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 ...
Lucas Johnston's user avatar
1 vote
1 answer
78 views

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 ...
A. Boy's user avatar
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0 votes
1 answer
34 views

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 ...
Mantas Kandratavičius's user avatar
0 votes
1 answer
290 views

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 ...
MohG's user avatar
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0 votes
1 answer
29 views

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. ...
rapt's user avatar
  • 113
0 votes
1 answer
75 views

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 ...
newlogic's user avatar
  • 165
3 votes
2 answers
145 views

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 $\gcd(A, B)$ ...
Minh Đức Hoàng's user avatar
0 votes
2 answers
72 views

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. ...
linuxxx's user avatar
0 votes
1 answer
431 views

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 ...
Rasmus IN's user avatar
2 votes
0 answers
45 views

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....
Reinderien's user avatar
2 votes
1 answer
316 views

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 = [...
 snow's user avatar
  • 23
2 votes
2 answers
284 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 ...
mathlete42's user avatar
1 vote
1 answer
75 views

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 ...
cubiii's user avatar
  • 113
1 vote
1 answer
169 views

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 ...
galinette's user avatar
  • 111
0 votes
0 answers
19 views

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 (...
copumpkin's user avatar
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