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Questions tagged [sets]

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

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3k views

What exactly is the semantic difference between set and type?

EDIT: I've now asked a similar question about the difference between categories and sets. Every time I read about type theory (which admittedly is rather informal), I can't really understand how it ...
5k views

Boolean search explained

My mother is taking some online course in order to be a librarian of sorts, in this course they cover boolean searches, so they can search databases efficiently, however, she got a question sounding ...
5k views

in O(n) time: Find greatest element in set where comparison is not transitive

Title states the question. We have as inputs a list of elements, that we can compare (determine which is greatest). No element can be equal. Key points: Comparison is not transitive (think rock ...
376 views

Problems for which algorithms based on partition refinement run faster than in loglinear time

Partition refinement is a technique in which you start with a finite set of objects and progressively split the set. Some problems, like DFA minimization, can be solved using partition refinement ...
7k views

Data Structure for Set Intersection?

Is there any data structure that maintain a collection of set (of finite ground set) supporting the following operations? Any sublinear running time will be appreciated? Init an empty set. Add an ...
6k views

Given a set of sets, find the smallest set(s) containing at least one element from each set

Given a set $\mathbf{S}$ of sets, I’d like to find a set $M$ such that every set $S$ in $\mathbf{S}$ contains at least one element of $M$. I’d also like $M$ to contain as few elements as possible ...
992 views

How to find the maximal set of elements $S$ of an array such that every element in $S$ is greater than or equal to the cardinality of $S$?

I have an algorithmic problem. Given an array (or a set) $T$ of $n$ nonnegative integers. Find the maximal set $S$ of $T$ such that for all $a\in S$, $a\geqslant |S|$. For example: If $T$=[1, 3, 4, ...
10k views

626 views

Existence of Efficient Set Difference Algorithm

As a foreword, I'm not asking what the algorithm is, just whether one can possibly exist (though, if it does already exist and someone knows what it is, that'd be great). Basically, given two sets $S$...
57 views

Small world theorem for set constraints

Let $S_1,\dots,S_n$ be variables representing unknown sets. A set expression has the form $S_i$, $\overline{E}$ (the complement of $E$), or $E \cap E'$, where $E,E'$ are set expressions. A ...
143 views

Test if there are two subsets which cover a set

Given a set $S$ of $n$ elements, and a set $\mathcal{X}$ of $m$ subsets of $S$, decide if there exist $U,V \in \mathcal{X}$, s.t. $U \cup V = S$. Brute force would take time $O(nm^2)$ but is there ...
12k views

Finding the minimum subset of intervals covering the whole set

Suppose we have a set $A$ of pairs $(a,b)$ such that $a$ and $b$ are real numbers and $a < b$. What is the most efficient algorithm to find the smallest subset $B \subseteq A$ such that, for any ...
109 views

Is there a formal difference between $f:X \to X$ and $f\in X \to X$?

We can denote by $X\to X$ the set of all functions from $X$ to $X$. Therefore, we can use the following statement to say that $f$ is a function from $X$ to $X$: $$f\in X\to X$$ But we usually state ...
612 views

Application of set theory subjects as ordinals, forcing, generic filters in software engineering

I am going to teach a course in set theory for software engineering students. I am going to talk in this course about: ordinal numbers, partial orders, well ordering, generic filters and maybe some ...
504 views

2k views

Data structure for partition of a set

A partition of a set S is a separation of the set into an arbitrary number of non-empty, pairwise disjoint subsets whose union is exactly S. What manner of a data structure should be used to represent ...
187 views

How to find a minimum set of axioms within a set of propositions?

I have a set of propositions, for example $\{a_1,a_2,\dots,a_n\}$. Some propositions depend on others (for example, $a_1,a_2\Rightarrow a_3$, means if $a_1,a_2$ are true, then $a_3$ is true). I want ...
113 views

Introduction to type theory for a beginner?

I'm interested to read about type theory, but I'm quite a beginner. I know what sets are and how to work with them, but I don't have a deep understanding of set theory. I don't really understand the ...
134 views

What is the name of this positive integer set data structure?

Google is failing me, so here goes: The data structure is used to describe a set of positive integers. It works conceptually, by keeping track of disjoint ranges [a,b) on the number line. These ...
581 views

Finding set of disjoint sets with additional value optimization

I've got a set $Q$ of pairs $[S, v]$ where $S$ is a nonempty set and $v$ is a value ($v \in \mathbb{N}_{+}$). I need to find a subset $R$ of $Q$ with following properties: Sum of all $v$'s is maximum ...
173 views

225 views

Computing the rank of a multiset after inserting another element

What is the procedure for computing the rank of a multiset after inserting an element? For instance, lets say we have a set $S = (0,1)$ containing $n = 2$ distinct elements. The multiset $M = (1,1)$ ...
96 views

Datastructure for managing (abstract) sets

I am looking for a datastructure to represent the complex relationships between a bunch of abstract sets. The "abstract" means that these sets are not defined by their elements, but by their ...
158 views

Given an amount of sets with numbers, find a set of numbers not including any of the given

Given an amount of sets with numbers (0-20 e.g) , we are asked to find the maximum set of numbers from 0-20 that doesn't include any of the given sets(it can include numbers from a set,but not the ...
Most common subset of size $k$
I'm trying to write an algorithm that detects the most common subset of at least size $k$, from a collection of sets. If there are ties for the most common subset, I want the one of them whose size ...