A partition or partitioning of a set A is a collection of disjoint sets whose union yields A.
Questions tagged [partitions]
183 questions
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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 quite efficiently. I don't know of any other problems…
Juho
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"partial sorting" algorithms (aka "partitioning")
Context:
When trying to tame real-world datasets that contain outliers and noise, the interquartile mean is a handy tool: you sort the data, throw away the top and bottom 25% of the data and take the mean of what's left. (Of course, you can choose…
fearless_fool
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What is a compact way to represent a partition of a set?
There exist efficient data
structures for representing set
partitions. These data structures have good time complexities for operations
like Union and Find, but they are not particularly space-efficient.
What is a space-efficient way to represent a…
cberzan
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Partition problem with distinct integers
The partition problem is a well-known NP-complete problem. In the definitions I have seen, the input is assumed to be a multiset of integers, and we want to decide the existence of a partition into two sets that have the same sum. My question…
Mohammad Al-Turkistany
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Is the ABC-partition problem NP-hard?
In the ABC-partition problem, there are three sets $A, B, C$ with $m$ positive integers in each. The sum of all integers is $m T$. The goal is to construct $m$ triplets with the same sum $T$, each of which contains exactly one integer from $A, B$…
Erel Segal-Halevi
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What is the trick of "adding a huge number" for in the reduction from 3-Partition?
Problem: To prove the $\textsf{NP-Completeness}$ of the problem of "Packing Squares (with different side length) into A Rectangle", $\textsf{3-Partition}$ is reduced to it, as shown in the following figure.
In the $\textsf{3-Partition}$ instance,…
hengxin
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Fastest known algorithm for $3$-$\mathrm{Partition}$ problem
$3$-$\mathrm{Partition}$ problem is $\mathsf{NP}$-Complete in a strong sense meaning there is no pseudo-polynomial time algorithm for it unless $\mathsf{P=NP}$. I am looking for the fastest known exact algorithm that solves…
user742
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Partitioning an undirected, unweighted, square planar graph paths that join certain pairs of nodes
I am trying to find a way to efficiently solve a puzzle that I play a lot by turning it into a graph partitioning problem (which is basically is in its actual form). I know that generally, graph partitioning problems are considered NP-hard or even…
saltthehash
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Hardness proof of EVEN-ODD PARTITION
The PARTITION problem is NP-complete:
INSTANCE: finite set $A$ and a size $s(a) \in \mathbb{Z}^+$ for each $a \in A$
QUESTION: Is there a subset $A' \subseteq A$ such that $\sum_{a \in A'} s(a) = \sum_{a \in A \setminus A'} s(a)$
The problem…
Vor
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Partitioning bag of sets such that each set in a group has a unique element
Suppose I have a bag (or multiset) of sets $S = \{s_1, s_2, \dots, s_n\}$ and $\emptyset\notin S$. I wish to partition $S$ into groups of sets such that within each group each set has at least one element not found in any other set in that group.…
orlp
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How to distribute items of varying sizes into bins of varying sizes, such that percent utilization across all bins is minimized?
I have a bunch of databases, each having different access patterns, such that each puts a different amount of load on its database cluster. I would like to distribute them around my set of database clusters such that the workload for the clusters is…
spiffytech
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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 a partition of a set if the following methods…
user1448338
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Is the set partitioning problem NP-complete?
I know that the set partitioning problem defined like this:
Given $$S = \left\{ x_1, \ldots x_n \right\}$$ find $S_1$ and $S_2$ such that $S_1 \cap S_2 = \emptyset$, $S_1 \cup S_2 = S$ and $\displaystyle\sum_{x_i \in S_1} x_i=\sum_{x_i \in S_2}…
SashaGreen
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Partitioning a graph into connected pairs and triplets
We need to partition an undirected graph into connected subgraphs of size between $2$ and $k$, where $k$ is an integer.
When $k=2$, the problem is equivalent to the perfect matching problem which is known to be solvable in polynomial time.
When…
Chaya
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Can almost equal partition problem be solved in polynomial time?
Given a list of positive integers with sum $s$, decide if there is a subset with sum $0.5s$. This is the well-known PARTITION problem, which is NP-hard.
What about the following: Given a list of positive integers with sum $s$, decide if there is a…
user355066
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