Computer Science

Questions tagged [sets]

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

270 questions
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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 ...
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1answer
94 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
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1answer
55 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 ...
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1answer
68 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, ...
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1answer
40 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 ...
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1answer
24 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 ...
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1answer
50 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 $...
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1answer
19 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 ...
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1answer
41 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 ...
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1answer
54 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 ...
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0answers
45 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 ...
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2answers
93 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. ...
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1answer
54 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}. ...
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1answer
118 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 ...
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1answer
24 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 ...
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1answer
74 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 ...
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1answer
87 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 ...
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2answers
88 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, ...
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1answer
29 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....
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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 $B$, ...
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1answer
62 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 ...
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4answers
82 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 ...
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1answer
35 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 ...
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1answer
36 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 ...
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3answers
178 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^...
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0answers
43 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 ...
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1answer
40 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 ...
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1answer
41 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
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2answers
60 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 ...
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2answers
155 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 ...
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1answer
28 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
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1answer
64 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
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1answer
53 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
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1answer
59 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 ...
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1answer
45 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 ...
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1answer
111 views

Prove/disprove that the class of decidable (resp. partially decidable) languages is closed under symmetric difference

Prove/disprove that the class of decidable (resp. partially decidable) languages is closed under symmetric difference. A symmetric difference of sets A and B is the set (A \ B) ∪ (B \ A). I know that ...
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0answers
35 views

Algorithm for Minimum Subset Needed to Satisfy all Constraints

I was wondering what is the most efficient algorithm to solve something like the following: You have $P$ people. You have $T$ tasks, each of which is a set of sets that represent all of the possible ...
3
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0answers
288 views

Algorithm for minimum number of partitions to transform list of sets into Laminar Set Family

I have a list of sets $L$. I want to partition the sets in $L$ to produce a new list $L'$ that is a Laminar Set Family Concretely: For any $L'_i, L'_j \in L'$ if $L'_i \not\subseteq L'_j$ and $L'_j ...
2
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1answer
118 views

What is the algorithm for a decider to get the language accepted by a DFA?

I am trying to understand the larger problem of the decidability of the equality of two DFAs. I understand that this problem can be solved using minimizing DFAs, but my textbook states this can be ...
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1answer
45 views

How to find sets of polynomially bounded numbers whose subset sums are different?

Let $n$ be any positibe integer and set $N=\{1,\dots,n\}$. Now select two arbitrary but different subsets of $N$, say $S$ and $S'$. We are interested in finding a function $\pi(A)=\sum_{i\in A}a_i$ ...
2
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2answers
311 views

Prove: Every decidable set is Turing reducible to the empty set

Question- Prove: Every decidable set is Turing reducible to the empty set. Can anyone help me with this please? All reductions tutorials I've seen use practical examples of reduction such as sipser'...
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1answer
80 views

Is it possible to add every word in a file to a set in $\mathrm{O}(n)$ time?

The Problem: I am currently analyzing a simple program that takes a file of length $n$, splits it into its individual words (seperated by white space) and adds those words to a set: ...
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1answer
51 views

For each given set choosing either it or its complement such that their union exactly has a given size

Given an integer $k$ and $n$ sets $A_1,\ldots,A_n$, denote $U=A_1\cup A_2\cup\cdots\cup A_n$, $A_i^0=A_i$ and $A_i^1=U\backslash A_i$. The problem asks whether there exists $(b_1,\ldots, b_n)\in\{0,1\}...
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0answers
39 views

What is the most efficient algorithm for creating a list of unique values from a list of pairs of value?

Background I have a list of 50 million $A-A_i$ pairs, where $i>1$, and $A$ and $A_i$ are some text. I need to extract the $A$ values from the list, so I get a new list of unique $A$ values.: $$ \...
2
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1answer
93 views

Find 2 sets with an empty intersection

I have the following problem. The problem can be formulated in three different ways Given sets $B_{-n},\ldots,B_n \subset \{1,\ldots,m\}$. Find $i,j \in \{-n,\ldots,n\}$ with $|i| \neq |j|$ and $i,...
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1answer
44 views

What is the optimal algorithm for finding all sets of overlapping ranges?

I have a set of (integer) ranges and want to compute the (possibly non-disjoint) set of all subsets of overlapping ranges. The data structure used for the output is not of particular importance to me; ...
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0answers
39 views

Abstract Data Type

I have been studying data structures. In that I have come across topics like Array being defined as Power set of cross product of set of objects and set of natural number and list being defined as ...
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1answer
45 views

number of random sets needed to generate subset

Let $A\subseteq \{1\ldots n\}$ with $|A|=\alpha n, 0<\alpha\leq1$. Now we start generating random sets $B_i \subseteq \{1\ldots n\}$ with $|B_i|=\beta n$ where $0<\beta\leq\alpha$. How many $...
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2answers
939 views

Given n numbers How to find out a set of numbers whose sum equal to a certain given number

I am given an list of numbers and A number-s. I need to find out the collection(s) of numbers from the list of numbers whose sum corresponds to the given number s. ...
0
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0answers
19 views

What does it mean for a set to be NP-complete? [duplicate]

For homework I have the task Assuming P ≠ NP, is the following set NP-complete: {(G,w) | G is a Graph and w is a Hamilton cycle in G} and I don't understand how to show that a set is NP-complete. I ...

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