Questions tagged [dag]

For questions about the usage and manipulation of Directed Acyclic Graphs (DAG).

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36 views

Is there a term for these “descendancy” subgraphs of directed acyclic graphs?

Consider a directed acyclic graph $G$ with vertex set $V$. Choose a vertex $v$, and let $H$ be the subgraph containing $v$ and all other vertices in $G$ that are reachable from $v$ (along with the ...
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1answer
90 views

Algorithm for finding an irreducible kernel of a DAG in O(V*e) time, where e is number of edges in output

An irreducible kernel is the term used in Handbook of Theoretical Computer Science (HTCS), Volume A "Algorithms and Complexity" in the chapter on graph algorithms. Given a directed graph $G=(V,E)$, ...
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41 views

Time complexity analysis of shortest path algorithm

Below is Dijkstra's algorithm from CLRS: In the time complexity analysis of Dijkstra, CLRS says, RELAX() contains call to DECREASE-KEY(), which is essentially reducing edge weights associated with ...
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29 views

Find a longest path with k vertices in a directed graph

Given a weighted directed acyclic graph $G=(V,E)$, find a path (with $k$ vertices) so that the sum of edge-weights is maximum.
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1answer
176 views

Counting number of paths between two vertices in a DAG

I need an algorithm that computes the number of paths between two nodes in a DAG (Directed acyclic graph) I need a dynamic porgramming solution if possible.
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26 views

Finding all non-comparable nodes in DAG

I have a DAG and I want to list all pairs of vertices that are not comparable (there is no path from the first to the second or the second to the first). In this image (taken from this StackOverflow ...
3
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1answer
26 views

Given an unsorted list of $n$ items, how many random comparisons are needed on average to be able to sort the list?

There is an unsorted list of $n$ items $x_1, \ldots, x_n$. Until you can sort the list, you are given one of the ${n \choose 2}$ possible binary comparisons uniformly at random (with replacement). On ...
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1answer
30 views

Name for Turning DAG into redundant tree

I am looking for a term: How is the tree called that you can obtain from a DAG by going top-down and appending all visited nodes to a tree, thereby copying nodes from the DAG into multiple occurences ...
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1answer
42 views

An algorithm for topological sorting based on depth-first search: why do we need two tags?

Wiki gives an alternative algorithm for topological sorting is based on depth-first search, as follows: ...
2
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1answer
44 views

Time complexity of finding a node with no incoming edges in a DAG: O(n) or O(m+n)

I'm reading Algorithm Design by Jon Kleinberg. In section 3.6, in order to compute the topological ordering of a DAG, one first finds a root node in this DAG, then deletes it from the DAG. The author ...
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1answer
59 views

Does topological sort exist for any complete directed acyclic graph?

Let's assume that DAG is complete: there is directed edge among every to nodes. Does topological sort of vertices exist for any such graph? I.e. is it possible to make linear list of nodes in which ...
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1answer
53 views

Extracting a spanning tree from a directed acyclic graph with minimum total distance between terminal nodes

I have a directed acyclic graph that has uniform edge weights. I would like to extract from this graph a spanning tree (an arborescence) with the property that the total distance between all pairs of ...
3
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1answer
265 views

Merging nodes of a DAG

I would like to merge connected nodes with a specific attribute of a directed acyclic graph. The purpose is to detect max connected clusters of blue nodes and merge them. After each merge operation, ...
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0answers
53 views

Directed Acyclic Graph partition into minimum subgraphs with a constraint

I have this problem, not sure there is a name for it, wherein a Directed Acyclic Graph has different colored nodes. The idea is to partition it into minimum number of subgraphs with the following 2 ...
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1answer
39 views

Path of exact cost k in DAG

struggling with this question from an exam: input:   DAG G=(V,E). each edge $e_i$ has weight $w_i\in \text{{0,1,2,3}} $   Two vertices : s,t   Number: k output: ...
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0answers
82 views

Assign weights to the edges in a DAG so that, for all S and T, all paths from S to T have equal weight

I have a DAG, and on each edge, I have a minimum and maximum weight. I would like to assign (or determine it's impossible to assign) exact weights to each edge so that Each edge's weight is between ...
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0answers
27 views

Routing a DAG through 5 consecutive butterfly networks

I have two questions concerning the paper Nearly linear-size holographic proofs. In the second paragraph of section 6, A Graph Coloring Problem, it is claimed that Using standard packet-routing ...
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0answers
34 views

Are these algorithms for detecting cycles in directional graph correct?

I want to detect whether a subset of a directional graph reachable from a given root has a cycle, and print some useful debug information about the cycle. It's not a problem if there's a cycle not ...
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2answers
97 views

Fewest traversals to visit all vertices of DAG

I want to find the fewest traversals to visit all vertices of a DAG. To take a very simple case: ...
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0answers
121 views

Minimum Number of Edges Added to a DAG to get Unique Topological Order

The question is simple: Given an unweighted directed acyclic graph, $G = (V, E)$, what is the minimum number of directed edges we need to add to $E$ such that the resulting graph $G = (V, E')$ has ...
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81 views

Algorithm for finding single input/output sub graphs

I'm running into an interesting directed acyclic graph (DAG) problem and was wondering if this is a known problem and if it has an efficient algorithm for it. I will use 'graph' and 'DAG' ...
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84 views

Can the running time be reduced to something lower than $O(d^4)$?

Imagine I have a weighted complete directed graph $G$ with $d$ vertices(so $d(d-1)$ edges) and I want to do the following: Set $D$ to be a DAG with the same set of vertices but without any edges sort ...
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1answer
35 views

Set of DAG vertices disconnecting a vertex from forbidden vertices

Let $v$ be a vertex with in-degree 0 in an (acyclic) DAG $G$, and let $F$ be a subset of $G$'s vertices (the "forbidden") vertices. Now suppose $U$ is a set of vertices such that every path from $v$ ...
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1answer
71 views

Generating project network graph

I had a problem of generating project network graph (like there and there) from list of activities and their dependencies. Informal description: Every activity is represented as edge of directed ...
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1answer
311 views

Approximating the biggest acyclic subgraph of a given weighted digraph $G= (V,E)$

The base scenario is this: We're given a weighted, directed graph $G= (V,E)$, and are tasked to find an approximating algorithm which returns a digraph $G' = (V,E')$ (i.e. which only deletes edges) so ...
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1answer
44 views

Let G be a graph directed without circles. Suggest a method to find a minimum set of vertices So that all the vertices in the graph can be reached

Let G be a graph directed without circles. Suggest a method to find a minimum set of vertices So that all the vertices in the graph can be reached. I thought to run an SCC algorithm to find binding ...
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1answer
71 views

Can Dynamic programming be applied to solve problems if and only if the subproblem form a DAG?

I assume Dynamic Programming can be used only when the corresponding subproblems form a Directed Acyclic Graph, otherwise you're stuck in a loop. Is this reasoning correct or is there more to it?
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1answer
23 views

There could not be an edge from u to v in a DAG, if w is before v in a topological order

I am trying to prove that given a DAG. There exists a valid topological ordering that has v in front of u iff there is no path from u to v. The proof is related to the fact that reverse DFS post ...
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1answer
487 views

How to find all topological sortings of a special DAG in O(N^2)

I came across the following question in a hackerrank competition, which is based on topological sorting of a DAG. https://www.hackerrank.com/contests/hourrank-29/challenges/birthday-assignment/forum ...
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2answers
114 views

What is the name of a rooted tree whose nodes may have edges to their descendants?

A tree is a special kind a graph. However, I came across a data structure which is a like a rooted tree, but where nodes are authorized to have direct links to any of their descendants. Shortcuts if ...
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0answers
214 views

Finding unique topological ordering wrt to another vertex ordering

Given a directed acyclic graph with $n$ nodes labeled from $1$ to $n$, what is an efficient way to produce the unique topological ordering with lower labeled nodes prioritized over higher labeled? ...
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1answer
23 views

Extend reachability to total ordering

Given a finite DAG $G = (V, E)$ with $|E| \in \mathcal{O}(|V|)$, I want to compute a total ordering $<$ over $V$ that is compatible to reachability: If there is a transition $(u, v) \in E$, then $u ...
2
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1answer
110 views

Is the following figure DAG?

I have been through multiple definitions of DAG and all of them say that it is a directed graph without cycles. Also, it is said that it has topological ordering. Now the following figure is directed ...
2
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1answer
143 views

Topological sorting and NP-hard proof

I meet a problem. I can find a sub-optimal solution, but cannot find an optimal one and cannot prove its NPC hardness. The problem can also be described as follows. Given a sequence $X=\{x_1,x_2,...,...
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1answer
434 views

DAG Hamiltonian Path NP-complete

The book computers and Intractability mentions that Hamiltonian Path problem is not NP-complete in DAG. But if Hamiltonian Cycle is NP-complete in digraph then I can split a vertex and create two ...
2
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1answer
204 views

Find minimum set of vertices that are reachable from all other vertices

I have a DAG and I wish to find the minimal set of vertices that have a path to all other vertices. My solution was to simply find all the vertices that their incoming degree is 0. However, if there ...
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1answer
305 views

Longest path in a DAG: source to sink?

Is the longest path in a (weighted) DAG always from a source to a sink? This seems correct to me by intuition, but I'm not 100% confident. Like, for example, if I had an array in which each index ...
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0answers
45 views

DAGs and Equivalence Class of DAGs

I am learning DAGs and Equivalence Class of DAGs, I am reading the material by Prof. Campos Ibáñez here: https://www.cs.cmu.edu/afs/cs/project/jair/pub/volume18/acid03a-html/node2.html However, I ...
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0answers
249 views

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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1answer
4k views

maximum weighted path(s) in a DAG

Given a weighted directed acyclic graph (DAG), I need to find all maximum weighted paths between the start node(s), i.e. zero incoming edges, and the end node(s), i.e. zero outgoing edges. My current ...
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0answers
55 views

maximally exclusive paths in a DAG

Suppose we have a DAG $G$ with one source node $s$, ie. it has a path to every node, and target node $t$ which every node has a path to it. For a pair of paths from $s$ to $t$ we can define a distance ...
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1answer
275 views

How to tell if a spanning tree is a shortest-spanning tree of a DAG?

I know how to calculate the shortest paths from source s to all other reachable vertices in a DAG (with no negative weight on the edges) By iterating the ...
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2answers
593 views

Formal definition on graph levels

I'm looking for a formal definition of "graph levels" on a DAG. This example should illustrate what I mean by this. The node 0 has no edges directed towards it, therefore this has level 0. Next is ...
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1answer
351 views

Flipping all incoming/outgoing edges from a vertex in a DAG

I'm working on a problem where I have a directed acyclic graph and I need to repeatedly flip all incoming (or outgoing, or both incoming and outgoing) edges from a single vertex. I think that ...
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0answers
321 views

Multithreaded algorithm to traverse DAG

I have a DAG of tasks designed as: DAG { Dictionary<NodeAddresses,Node> } Node { List<Node> Parents; List<Node> Children; Task T; } ...
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0answers
1k views

Lowest single common ancestor in a Directed Acyclic Graph?

I was reading how to find the Lowest common ancestor in a DAG. A DAG can have scenarios where the LCA yields multiple solutions and I feel the accepted answer explains that pretty well. However, one ...
3
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1answer
292 views

Construct a DAG using Tarjan

This is a question related to a homework assignment so I guess I'm asking for hints, if that's ok? Basically, I'm wondering if there is some way to use Tarjan to compress the nodes in every SCC found ...
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1answer
155 views

Approximating the dominating set on a certain kind of DAG

A dominating set on a directed graph $G=(V,E)$ is a subset $D$ of $V$ such that for all $v\in V$ it holds that $v\in D$ or there is an $u\in D$ such that $(u,v)\in E.$ For a concrete problem (size ...
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227 views

spanning tree of a DAG (directed acyclic graph) with less forward arcs

I am new to this algorithm and graphs. Just started learning. Could someone help me which algorithm is best suited to find the spanning tree of a Directed Acyclic Graph with less forward edges?
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195 views

In a DAG, what is the name for a vertex with only one predecessor and one successor?

In a directed acyclic graph (DAG), what is the name for a vertex with only one predecessor and one successor?