Questions tagged [sets]
Questions about finite and infinite sets and multisets, related data structures and concepts.
74
questions with no upvoted or accepted answers
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Alternative to Bloom filter for extreme parameters
A Bloom filter is a space-efficient probabilistic data structure to perform membership-tests on a set (see Wikipedia's page for a definition; I use the same notations below).
I am interested in a ...
- 391
7
votes
0
answers
166
views
Overlap Maximization problem
Here's the problem:
I have a collection of collections, $C$, where each $c\in C$ is a collection of sets $X\subset U$. Denote $c_i$ as the i-th $X$ in $c$. Informally, I want to map all the sets in ...
- 378
6
votes
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
6
votes
0
answers
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Time complexity of obtaining the support set of an unsorted sequence?
Consider a sequence $s$ of $n$ integers (let's ignore the specifics of their representation and just suppose we can read, write and compare them in O(1) time with arbitrary positions). What's known ...
- 955
4
votes
0
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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
4
votes
0
answers
114
views
Sequence where every subset exists as some contiguous subsequence
Given a set (i.e., a collection of distinct elements), how would you find a minimal sequence where every subset of that set can be found as the elements in some contiguous subsequences? The order of ...
- 103
3
votes
0
answers
118
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 ...
- 31
3
votes
0
answers
57
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 ...
- 131
3
votes
0
answers
148
views
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
3
votes
0
answers
39
views
Components of subset partial order
Given a collection C of sets, there are a number of proposed algorithms for building the subset partial order, e.g. this paper.
But is there any work on algorithms ...
- 177
3
votes
0
answers
321
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 ...
- 131
3
votes
0
answers
121
views
effective, efficient algorithms on antichains
In a partially ordered set L, an antichain is a subset A of L such that no two elements of A are comparable.
Antichains are commonly used to represent upward-closed subsets of L, that is, sets S such ...
- 181
2
votes
2
answers
125
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 $\gcd(A, B)$ ...
2
votes
0
answers
43
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....
- 237
2
votes
0
answers
149
views
Algorithm to find largest intersection of sets
This is a cross-posting from here, on the mathematics Stack Exchange. I thought this might be a more appropriate venue.
The problem is this:
I have a list of sets $$S_1, S_2,... S_N$$ where each set ...
- 121
2
votes
0
answers
39
views
Online algorithm for compressing multiple SETS?
Algorithms like LZW and others compress data sequences.
What I'm looking is an algorithm that compress multiple Sets. if possible online algorithm.
For example :
...
- 139
2
votes
1
answer
75
views
Efficient Implementation of Boolean Lattice-Esque Operation
Let $X = \{1,2,\dots n\}$, and $Y_i= \{T \in \mathcal{P}(X): |T| \le i\}$. I am interested in "avoidance sets" $A \subset Y_n$. We say a subset $S \subset X$ is valid with respect to an ...
- 385
2
votes
0
answers
139
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 ...
2
votes
0
answers
52
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 ...
- 131
2
votes
0
answers
451
views
hash-table subsets
Having trouble figuring this out.
If I have 2 sets of integers how would I use a hash table to test if set A is a subset of set B (in pseudocode).
I think I understand that basically I would need to ...
- 51
2
votes
0
answers
127
views
Minimum overlap partitioning
We are given $N$ sets of $M$ non-unique elements each.
The amount of overlap (computed as the element count in the set intersection) between the elements of these sets is stored in a $N \times N$ ...
- 121
2
votes
0
answers
59
views
Smallest set non-disjoint with other given sets
Given a number of sets, what is the best algorithm to calculate the smallest set S such that S is not disjoint with any of the given sets?
- 315
2
votes
0
answers
75
views
Finding subsets in a large collection of sets
Given a large collection $\mathcal{X} = \{X_1, X_2, \dots, X_n\}$, where each $X_i$ is a set of integers, what's a fast algorithm to identify all pairs $(i,j)$ with $i \ne j$ such that $X_i \subseteq ...
- 203
2
votes
0
answers
142
views
physical significance of membership function greater than one
In fuzzy logic, when we associate an element with a set, we usually do it in terms of membership grade which suggests the "belonging" of this element to the set. Membership grade value 0 means that ...
- 149
2
votes
0
answers
63
views
Ordered set transformation data structure
Assume an ordered set $M = \{\tau_1, \tau_2, ..., \tau_n\}$ and a subset $S = \{\tau_k,\tau_l,...,\tau_m\}\subset M$ where $1\leq k,l,m \leq n$. All the items of $S$ are randomly ordered.
The task is ...
2
votes
0
answers
714
views
Efficient algorithm to approximate membership in a set of strings
I devised an algorithm / data structure and I would like to ask whether it already exists. The problem statement is: after having added some number of strings to the set, determine whether a given ...
- 121
2
votes
0
answers
77
views
Tradoff between space and false positive rate when using bloom filters
Bloom Filters have false positive rate of $\epsilon = 2^{-k}$ with a data structure of size $m = n\log (\frac{1}{\epsilon})\ln 2$. Suppose you fix the number of hash functions at $k \le 3$. What is ...
- 109
2
votes
0
answers
129
views
Message protocol to probabilistically infer missing object from Union of two subsets of a larger set
This was a challenge problem I read some time ago and just remembered it:
Say you have two people, $A$ and $B$, collect objects distinctly labeled $1,...,n$. They will each separately collect sets ...
- 125
2
votes
1
answer
60
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 ...
- 675
2
votes
1
answer
124
views
How to speed up finding a subset of a given set?
Is there a data indexing technique that speeds up finding subsets of a given set in a collection, or do I always have to scan all of the data?
For example, let's say that I have a collection of sets:
...
- 121
1
vote
0
answers
19
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 ...
- 111
1
vote
0
answers
46
views
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 ...
- 159
1
vote
0
answers
62
views
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, ...
- 121
1
vote
0
answers
49
views
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
1
vote
0
answers
35
views
Selecting sets that maximise the cardinality of the union minus the cardinality of the difference
I have a sparse $60000\times10000$ matrix where each element is either a $1$ or $0$ as follows.
$$M=\begin{bmatrix}1 & 0 & 1 & \cdots & 1 \\1 & 1 & 0 & \cdots & 1 \\0 &...
- 13
1
vote
0
answers
21
views
Streaming maximum pair matching with limited memory
I am trying to find as many pairs of elements as possible from two distinct data streams, while being constrained by the number of elements I can hold in memory at any given time. Once a pair of ...
- 11
1
vote
0
answers
67
views
Disjoint groups using maximum matching
In the 3-Path Packing problem, we are given an undirected graph $G$ and a parameter
$k \in \mathbb{N} \cup \{0\}$. We need to answer Yes/No if there exists a collection of $k$ vertex disjoint paths on ...
- 203
1
vote
0
answers
41
views
Minimal number of unions of sets such that no union has more than N elements
I have some sets, and can combine them by taking their union. I can take unions of the unions, too. I want to take unions until the total number of sets is as small as possible, with one caveat: that ...
- 131
1
vote
0
answers
52
views
Clustering sets by set difference
Suppose you have $n$ nonequal sets $S_1, \ldots, S_n$ and some constant $0 \le k < n$. The goal of set clustering is to find a partition of the set $\mathbf{S} = \{S_1, \ldots, S_n\}$ such that the ...
- 364
1
vote
0
answers
43
views
Set data structure for data too large to fit into memory
I'm trying to solve the following exercise:
Given N data items and memory that can hold M/B blocks of size B.
Describe a data structure that needs at most N/B blocks of external memory and allows ...
- 11
1
vote
0
answers
43
views
Optimally find one of the total orderings for a poset based on some metadata about the elements
Given a finite, partially ordered set with the following two properties:
Every element in the set has one of two types: "A" or "B". The type does not define the total ordering of the set and is ...
- 11
1
vote
0
answers
34
views
Algorithm for finding the set of systems of distinct representatives
Given a collection of finite sets, is there an algorithm for finding the set (unordered) of all systems of distinct representatives for the collection?
Example:
S: {{1, 2}, {1, 2, 3}}
Unordered ...
1
vote
0
answers
255
views
Data structure for overlapping sets
Is there a good data structure for storing overlapping sets? Consider having multiple sets which can overlap in various ways and would like to store them in the memory and access efficient way.
...
1
vote
0
answers
55
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 ...
- 11
1
vote
0
answers
86
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 ...
- 187
1
vote
0
answers
73
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 ...
- 11
1
vote
0
answers
187
views
Representing multisets by a bit vector
What would be the most space-efficient way to represent a multiset (a set that can contain duplicates) using a (static) bit string (bit vector, bit array, etc.)? All of the elements in the multiset ...
- 23
1
vote
0
answers
723
views
Complexity of set oprations in algorithm
I am designing a graph algorithm. Some steps of the algorithm, are set operations (union, difference, intersection, set-membership).
Can I assume them as $~ \mathcal{O}(1)$ operations? Have someone ...
- 133
1
vote
0
answers
82
views
Create the shortest list that contains a set of subsets in block
I have a problem where I have a set of subsets and I want to create the shortest list where I can find all subsets in it. Each subset must be a block of that list.
The input is a set of set of ...
- 11
1
vote
0
answers
408
views
Show that every infinite recursive set has both a nonrecursive r.e. subset and a non-r.e. subset
My attempt to solve this:
If $\mathcal{A}$ is an arbitrary infinite recursive set then the members of $\mathcal{A}$ can be ordered in ascending order. We can do bijection between $\mathcal{N}$ and $\...
- 11