Questions tagged [sets]
Questions about finite and infinite sets and multisets, related data structures and concepts.
284
questions
1
vote
1answer
14 views
Given a set of sets, what is the largest common intersection between them?
Given a set of sets: $S = \{~\{1, 2, 3\}, \{2, 3, 4\}, \{1, 3, 4\}~\}$, I would like to find the largest common subset of $S$. If $S$ does not have a subset across all elements of $S$, I would like to ...
1
vote
1answer
42 views
Largest number of unique values in sets
Let's say I have 100 sets of values, and each set contains roughly 100,000 values. From these sets, I want to find the 10 sets that collectively have the largest number of unique values.
The brute ...
1
vote
1answer
19 views
Proving set of finite languages vs all languages over finite alphabet to be countable / uncountable
I came across following facts:
Set of finite languages over a finite alphabet is countable.
Set of languages over finite alphabet is uncountable.
I believe proof of this will be similar to ...
3
votes
1answer
36 views
Is it possible to determine if 2 arrays contain the same elements (ignoring duplicates) in faster than O(n log n) time?
So given 2 arrays of equal length, is it possible to determine whether the 2 arrays contain the same elements (ignoring duplicates and where those elements have a total order relation) with time ...
2
votes
1answer
14 views
How to find number of combinations of choosing one from k subsets
Consider we have set S:
S = {1,2,3,4,5,6}
and 3 (say k) subsets of S:
S_1 = {1,2,3}
S_2 = {2,3,4,5}
S_3 = {1,3,6}
What is the total number of cases choosing one element from each subsets?
Same ...
2
votes
1answer
30 views
largest bitwise-or subset with a maximum number of on bits?
Suppose I have the following set of bit sets:
...
2
votes
1answer
50 views
Hitting Set Problem with non-minimal Greedy Algorithm
The Hitting Set Problem is defined as having a universal set $\mathfrak{U}$, and nonempty sets $S_i \subseteq \mathfrak{U}$ for $1 \leq i \leq n$, and finding a set $\mathcal{H} \subset \mathfrak{U}$ ...
3
votes
1answer
29 views
A problem on sets
Given a collection C of sets, with union U, find a choice function which chooses a distinct element from each of the sets such that the union of the singleton distinct elements is U.
As an example, ...
2
votes
1answer
39 views
Efficient data structure for matching 3D lines
I'd like to
Store a set of many infinite undirected 3D lines.
Make lookups against this set - i.e. given an arbitrary line, ask "Does the set contain a line coincident with this one?"
The incidence-...
2
votes
0answers
71 views
Set of maximum overlaps
Assume I have a list of $N$ surfaces $\{S_i\}, i \in [1,N]$ which may or may not overlap.
I also have a boolean function $F(S_{i_1},\dots,S_{i_k})$ (with $1 \le k \le N$) which tests whether all ...
0
votes
2answers
48 views
How to partition disagreeable people into compatible groups
We have a number of people that must be partitioned into groups, but there may be people that dislike other individuals. Partition the people into the minimum number of groups such that no person is ...
1
vote
1answer
53 views
How can $A \cup B$ be decidable if $B$ is undecidable?
My assignment says:
"Determine if the following statement is correct:
If $A$ and $A \cup B$ are decidable, then $B$ is decidable."
The solution says:
"Incorrect. If $B = H_0 \subseteq \{0,1\}^*$ ...
3
votes
1answer
38 views
Proof of sparsification lemma: What exactly does the $\pi$ operator do?
In their proof of the sparsification lemma, Impagliazzo et. al describe the following operator $\pi$:
For a familiy of sets $\mathcal{F}$, let $\pi(\mathcal{F}) \subseteq \mathcal{F}$ be the family ...
0
votes
1answer
39 views
Disjoint Set Connected Components With Weighted Graph
I have been trying to solve this HackerRank problem (link).
The basic premise of this problem is that there is a tree with undirected, but weighted, edges. The cost of a path in this tree is taken ...
3
votes
1answer
77 views
Lambda Calculus as a branch of set theory
This answer to a question about whether C is the mother of all languages contained an interesting tidbit that I am curious about:
The functional paradigm, for example, was developed mathematically (...
0
votes
0answers
23 views
For a collection $S$ of weighted sets $S_i$, find those $k$ elements that maximise the sum of weights of all sets $S_i$ covered by them
I have a collection $S$ of sets $S_i$. Each $S_i$ has a weight given by how many times this set was observed in some data. I now want to find the $k$ elements that maximize the cumulative weight of ...
2
votes
1answer
109 views
Suggest a Data Structure To Manage 2 Sets
I was given the following problem which really baffled me, and I would like some guidance in solving it.
I want to make a data-structure which represents two sets $A,B\subseteq \mathbb{R}$, so that I ...
3
votes
1answer
65 views
Fast hash function for set equality
I'm searching an hash function for integer set equality that must be fast.
It must support update (adding an element already in the set must not change the hash) and union.
MinHash has these 2 ...
1
vote
1answer
120 views
Finding longest subset arithmetic progression with given difference
Given a list of distinct positive integers, I am trying to find the largest subset that forms an arithmetic sequence with a given difference D.
For example, given D = 5, with the set of numbers 1, 5, ...
0
votes
1answer
50 views
How do you compute the Pareto Front of a set?
I need to decide which solution is the best design, in order to do that I need to compare them. Lower energy used and lower weight is better. My initial idea was to order both the fields best to worst ...
1
vote
1answer
36 views
Complexity of set partition generation while equivalence relation is given
Given a binary equivalence relation, R on a set A, Let P be the resulting partition. I want to generate the partition means each subset in the partition. What would be the fastest algorithm for this ...
0
votes
1answer
73 views
Find all subsets with a given sum
How to choose from a set of positive numbers all the subsets that sum to some number x?
For example if the set $S=[1,1,2,3,4,5,6,7]$ and I'm searching for all the subsets that sum to $7$ I would have $...
1
vote
1answer
21 views
Name for multiset with fractional amounts?
What do you call a set which accepts multiples of the same element, even in fractional amount? Is there even an established for this?
Example from a video game about production chains:
For a ...
0
votes
1answer
66 views
Finding combinations of variables that can take value of -1/0/1 that produce sum of 0 with added constraint
I have 64 variables that can either take a value of -1, 0, or 1 and I am interested in finding all possible combinations of variables such that I have n variables in each the positive and negative ...
3
votes
1answer
105 views
Distinct elements count of huge multiset
I know that HyperLogLog can approximate the distinct elements count of a huge multiset but I was wondering if it was possible, using a method I saw mentioned on an IRC channel, to get an exact answer ...
1
vote
0answers
46 views
The maximum number of uniquely intersected elements from the all possible intersection scenarios among the sets in a two-column matrix
Let us define a $n \times 2$ matrix M consisting of integer sets, such
that the first column consists of the so-called intersecting sets,
and the second column ...
2
votes
2answers
100 views
What does the intersection symbol (∩) mean when applied to two non-set elements?
I came across a piece of literature in which I saw the intersection symbol (∩) being used on two non-set elements in the definition of an equivalence relation; I have posted it below for reference. ...
2
votes
1answer
55 views
Basic Set Theory problem
So i'm relatively new to computer science and have been learning set theory and am stumped on a question in it.
The question specifies that we're only looking at subsets of universe U = {0,...,n-1}. ...
-2
votes
1answer
258 views
Count the number of subsets with product less than or equal to k [closed]
You are given an array $A$ of $n$ positive integers, you have to find the number of subsets the product of whose elements is less than or equal to a given integer $k$.
Is there an efficient algorithm ...
1
vote
1answer
32 views
Identifying the Equivalence Classes of a Language with equal number of 10 and 01 strings
I'm doing a problem where I need to find the equivalence classes of the language below:
Let A = {x ∈ {0, 1}* | #(01, x) = #(10, x)}, where, for a, b ∈ {0, 1}*, #(ab, x) is the number of places in x ...
2
votes
1answer
135 views
Union of infinitely many regular languages [duplicate]
I need to prove or disprove the following statement.
If $A_n ⊆ \Sigma^*$ is regular for each $n \in \mathbb{N}$ then $\bigcup\limits_{n=0}^{\infty} A_n$ is regular.
I know that if two languages ...
0
votes
1answer
99 views
Proving the singleton language {x} is regular for all x ∈ Σ*
So I'm aware that the singleton language is in fact regular for all x ∈ Σ*, but I do not understand why it is. A formal proof would help, but also getting some intuition as to why it is regular would ...
2
votes
2answers
112 views
Proving that Every Full Prefix-Free Language is Maximal
I'm practicing a problem where I need to prove that every full prefix-free language is maximal.
I know a prefix-free language A is maximal if it is not a proper subset of any prefix-free language, ...
1
vote
1answer
31 views
Proving existence results
I'm doing a problem where I need to prove that there is a language A ⊆ {0, 1}* with both of the following properties:
(i) For all x ∈ A, |x| ≤ 5.
(ii) Every DFA that decides A has more than 8 states....
1
vote
1answer
31 views
$A^* = B^*$ with $\{0,1\}$ contained in $A$ but not in $B$
I'm trying to exhibit two formal languages $A,B ⊆ \{0,1\}^*$ such that $A^* = B^*$ and $\{0,1\}$ is contained in $A$ but not in $B$.
Finding a language for $A$ is very easy, but I get stuck on $B$, ...
1
vote
1answer
87 views
Proof of an Infinite Binary Sequence
I have a problem where given an infinite binary sequence S ∈ {0, 1}∞ to be "prefix-repetitive" if there are infinitely many strings w ∈ {0, 1}*
such that ww is a prefix of S.
I need to prove that if ...
1
vote
4answers
85 views
Proving B* = B on a given set
I have the set:
B = {x ∈ {0,1}* | there is an equal number of 0's and 1's in x}
and therefore,
B* = {e,01,10,0011,0101,0110,1100,1010,1001,....etc}
I need to either prove or disprove that B*=B
I ...
0
votes
1answer
37 views
Proving $A^\ast = A$ on a given set
I am working on some set theory and am trying to prove how a set can have the property $A^* = A$.
For set $A=\{0^n1^n \mid n \ge0\}$, I still do not understand exactly what $A^*$ is. For example, I ...
0
votes
1answer
94 views
Variant of the “Stable Roommates Problem” when room has not 2 but “n” mates
I'm looking at the name of a variant of the Stable Roommates Problem, when the rooms have more than 2 mates, ie for example 6 to 8. Does this problem has a specific name? A well-known algorithm?
To ...
2
votes
3answers
382 views
N subsets with a given sum?
How to efficiently ¹⁾ choose from a set of numbers $S$, a given number $n$ of disjoint subsets, each with a given sum $K$ of chosen elements?
¹⁾ Not as in $P$, I just want something smarter than $O(n^...
1
vote
0answers
51 views
Find a partition of multiset of binomial coefficients with constriants
Given the multiset $S$ where the elements are defined by the binomial coefficient ${n \choose k}$ where $n \in \mathbb{N}$ and $
0\leq k \leq n$ find the partition $P$ of $S$ such that the sum of ...
1
vote
1answer
43 views
Can You List the Names of Some Algorithms For Determining the Intersection of Two Context Free Grammars?
Suppose we have two sets of strings XS and YS such that set XS is described by grammar GX and YS is described by grammar GY.
We want an algorithm which accepts GX and Gy as inputs. The algorithm will ...
2
votes
1answer
45 views
Find four sets where each element from those four appears in at least two of those four sets
I have a list of sorted arrays ("sets") of integers $A_1..A_n$ where each element is unique w.r.t. the other elements in the same array:
$A_i = \{x_{i,1}..x_{i,c_i}\}$
$x_{i,p} < x_{i,p+1}$
$A_i$ ...
3
votes
2answers
73 views
Given a set $A$ of sets find a minimal set $B$ of pair-wise disjoint sets such that each set in $A$ can be expressed as a union of sets in $B$
I recently thought of the following problem:
Given a set $A$ of sets find a minimal set $B$ of pair-wise disjoint sets such that each set in $A$ can be expressed as a union of sets in $B$.
For ...
1
vote
2answers
373 views
set cover where only certain special subsets are allowed
I am trying to solve a problem which turns out to be a form of the set cover problem. I've implemented the greedy Set cover approximation algorithm for set cover, but it turns out to not be accurate ...
1
vote
1answer
29 views
Is a set $B = \{y, \exists x \in A, f(x)=y\}$ recursive if A is a recursive set and f is a $N->N$ total computable function?
Obviously, B would be recursive if for every TCF f, there was an inverse fuction that would return all possible values, as we could just take these and then check if any of them is in A. However I ...
2
votes
1answer
81 views
Minimal set of subintervals that 'covers' any subinterval in K subintervals
I have a big interval $I = [a, b]$ of size n.
I want an asymptotically minimal set of subintervals of $I$ (let's call it $S$) one can use to construct any subinterval of $I$, by concatenating at most ...
3
votes
1answer
61 views
Algorithm / data structure to filter documents by number of missing words
Is there a data structure or an algorithm or a combination of both to allow me to filter a set of documents based on the number of missing words (compared to another list)?
Problem Definition
We ...
3
votes
1answer
85 views
Parser theory: How to systematically compute FOLLOW sets?
Forgive me for my ignorance as I am self-teaching myself some of this theory... I am having some trouble understanding how to systematically/algorithmically compute FOLLOW sets, given that I have ...
1
vote
1answer
49 views
PL: What solves the type isomorphism $X \cong (X \rightarrow \mathbf{2})$?
In Practical Foundations for Programming
Languages, on page 138 (page 156 of the pdf), it says:
Requiring solutions to all type equations may seem suspicious, because
we know by Cantor’s Theorem ...
