# Questions tagged [partial-order]

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### Efficiently computing minimal elements over partially ordered sets

I have a list of sets that I would like to sort into a partial order based on the subset relation. In fact, I do not require the complete ordering, only the minimal elements. If I am not mistaken, ...
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### Computing minimum partition of poset of $N$ intervals into chains in $o(N^{2.5})$ time?

Consider a set $P$ of $N$ intervals $\{I_i = (l_i, r_i)\}$ partially ordered according the standard interval order: $I_i < I_j$ iff $r_i \le l_j$. I want to find a minimum cardinality partition of ...
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### Construct neighbourhood relation graph for n sequences

Given $n$ sequences with length $m$, $s_i=\langle c_1^ic_2^i\dots c_m^i\rangle, i = 1,\dots, n$, where $c^i_j\in D$ is a partial ordered set and the partial order relation $\sqsubseteq$ on $D$ answers ...
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### Poset data structure to find least element, greater or equal to given

Let $A$ be a finite set, and $S \subset \mathcal{P}(A)$. Is there a data structure for $S$ that would allow to quickly retrieve an element $q \in S$, given a key $p \in \mathcal{P}(A)$, such that $q$ ...
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### Measuring the Union of Products of Intervals

Verbose Motivation for this Question Inspired by this paper about how the problem of counting unlabelled subtrees that are unique up to isomorphism is #P-complete, I was thinking about the problem ...
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### Does every Partially Ordered relation and its dual have the same number of topological orderings?

Given the Hasse Diagram of a Partially Ordered Relation, is it the case that both the POSET itself and its dual POSET have the same number of topological orderings? I have tried a few examples, and ...
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### The sorting problem for partially ordered sets

I have two questions about sorting for posets, one easy and one hard: Easy: Suppose we have a set of objects and a partial order. Given any two objects such that $a \leq b$, we want to delete $b$ ...
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### Transform a DAG to fork-join format

I have a directed acyclic graph where the nodes are tasks and the edges are dependency relations between tasks - the edges go from the dependency to the task that depends on it. It is possible that ...
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### Retroactively ordering actors' concurrent activity on disjoint sections of a data-structure

Hi! I'm new here, a terrible computer-scientist, and have no idea what I'm doing; I'm more expecting / hoping for links to research on algorithms or data-structures that contribute to problems like ...
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### Efficiently determine relative ordering between two elements in a PO-set

What algorithms/heuristics exist for efficiently determining the relative order between two elements in a partially ordered set? In my case, the PO-set is stored as a directed acyclic graph where an ...
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### How to solve this partial order reduction in $O(n^2)$?

There are two orderings of numbers from the same set. Number $a$ is "immediately before" $b$ iff $a$ appears before $b$ in both sequences and there is no other number that appears between them in both ...
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### Longest chain of pair of points

For chaining two points A and B ...
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### Disproving well-quasi-order by providing an infinite anti-chain

I am currently studying the theory behind Well-Quasi-Orders. However I am having some issues in understanding how an infinite anti-chain can be produced to disprove the claim that a partial order $P$...
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### Generalized sorting algorithm on partially ordered set generated by a relation

Assume we have a finite set $X$ of elements and any relation $\preceq$ on $X$. Such a relation may or may not generate a reflexive transitive anti-symmetric relation $\leq$ on $X$ (a partial order). ...
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### Is < binary relation a strict partial order on IEEE doubles?

To me it looks that it is: irreflexivity: NaN < NaN == false transitivity: if a < b and b < c then a < c (the antecedent is never true for NaNs) asymmetry: if a < b then not b < a (...
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### What is an efficient algorithm to see if a set of nodes ultimately depend on a certain node in a DAG?

Hopefully this question makes sense. Basically, given a DAG, a set of nodes A, and another node b, I'd like to know if node b is an ancestor of any of the nodes in A in that graph. This is my current ...
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### Selection algorithm variant for an array

Have a problem that's a variant of the linear time selection algorithm of a randomized array that I'm struggling with. Let $A = A[1], ..., A[n]$ be an array of $n \ge 4$ distinct keys. ...
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### Prove that any directed cycle in the graph of a partial order must only involve one node

So I have the question: Prove that any directed cycle in the graph of a partial order must only involve one node. So I know that a partial order must be transitive, antisymmetric, and reflective, ...
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### How to prove substring is a partial order

u is defined to be a substring of a string v if v = xuy for some string x and y. Either or both possibly empty. How to you prove that a substring relation on any set of strings is a partial order?
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### trouble with bijection definition [closed]

I have a bijection problem that I cannot get my head around. It goes like this: let f: A -> B and g: B -> C be functions such that g o f is a bijection. Prove that f must be one-to-one and that g ...
Imagine I have an ordering on a bunch of elements like so: Where an arrow $X \leftarrow Y$ means $X < Y$. It is also transitive: $\left(X < Y\right) \wedge \left(Y < Z\right) \implies \left(... 1answer 104 views ### Extracting the set of chains from a partial order Given a partial ordered set (poset)$S$, is there a known procedure or algorithm to find the set of chains (i.e. subsets of$S$where every two elements are comparable)? Note: I am asking here ... 1answer 167 views ### Is there an efficient method to store large DAGs? I have a DAG representing strict partial order where each node is an assignment of variables$V$to their values$v$. Each arc$(u,w)$represents a change in one variable value such that$u\succ w$. ... 2answers 193 views ### Are these two relations on integers partial orders? Are the following relations$R_1$and$R_2$defined on the set$\mathbb{Z}$of integers partial orders? (A partial order is reflexive, antisymmetric and transitive.)$aR_1b$if and only if$a = ...
Inputs. I am given a finite set $S$ of symbols. I know there should exist some total order $<$ on $S$, but I'm not given this ordering and it could be anything. I am also given a collection of ...