Questions tagged [dag]

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

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397 views

Find all rooted subgraphs of a DAG

I searched the exchange and couldn't seem to find an answer to this. I am trying to find an algorithm that, given a directed acyclic graph (DAG) $G = (N,E)$ with a single root node $r\in N$, finds ...
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100 views

How do I merge these lists?

I ave a a collection of lists of objects: $a_1$ $a_2$ $a_3$... $b_1$ $b_2$ $b_3$... $c_1$ $c_2$ $c_3$.. I need to merge them into a minimal possible number of lists, each list must be as long as ...
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116 views

Max Flow on low depth DAGs

I have a family of acyclic networks in which every path from the source of a given network to its target has length exactly $3$. I'm aware of a publication that in general, finding a max flow in a DAG ...
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1answer
49 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: ...
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2answers
35 views

Why is there so little literature on partial order production?

Please excuse or improve the poor title of this question. My question is rather undirected, but I guess I am trying to find out if I might be missing a keyword for my problem. So there is plenty of ...
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1answer
38 views

How to remove 'skip' edges from a DAG? (How to find only the longest path from each node to each of its sinks?)

In two separate projects, I have come across this problem and I still don't have a good solution for it, so I thought it was worth describing here. Consider the following problem: I have a set of ...
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1answer
70 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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1answer
93 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
25 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 ...
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1answer
170 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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1answer
74 views

If there is no Hamiltonian path in a DAG then there are at least two different Topological sorts

I understand the concept that if there is no Hamiltonian path so there will be 2 smaller paths and with them I can build more then one topological sort but I am not sure how make it formal. Can you ...
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1answer
104 views

Reference for counting the number of paths in a DAG

Given a connected DAG I know how to compute the number of paths between two nodes. See e.g. Counting number of paths between two vertices in a DAG . Is there a reference or name for the algorithm? ...
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1answer
50 views

Best case “skew height” of an arbitrary tree

Given an arbitrary binary tree on $n$ nodes, choose an assignment $A$ from each parent to one of its children (the "favored child" as it were). We define the skew height of the tree as $H_A(\...
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1answer
231 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
81 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 ...
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1answer
42 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
1k views

SCC of a Directed Acyclic Graph

My instructor asked me to prove that strongly connected component of a DAG is also a DAG. I don't get it. How can a graph be DAG and strongly connected at the same time? Consider a graph on two ...
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47 views

Is there a data structure to represent changes to a DAG over time, with fast reproduction of instances of the DAG?

Consider a directed acyclic graph (DAG) of topics and links. Apart from the root topic, topics are situated under one or more parent topics, and links, as well, are situated under one or more parent ...
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21 views

Proof that “the last vertex in any postordering (in a DFS) of G lies in a source component of G”

From the book Algorithms (Jeff Erickson), there's a lemma that states: The last vertex in any postordering of G lies in a source component of G My initial reaction to this was that the proof would ...
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1answer
27 views

How can follow this this guide to construct a graph with matrix/reachability

Let's we have k matrices. For example we have 3 now, where first one is 8x5 ($a_1$ x $b_1$), second one is 5 x 6 ($a_2$ x $b_2$) and last one is 6 x 8 ($a_3$ x $b_3$). And our goal is to figure out ...
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34 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 ...
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14 views

Shortest sequence of jobs, with dependencies, subject to capacity constraints

Suppose I have $n$ courses, some with some prerequisites, and I can take up to $k$ courses in a semester. I want to compute the least number of semesters needed to complete all courses, while ...
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135 views

Obtaining a graph with no cycles after removing k edges

I am looking for an algorithm that upon an input of a directed graph G and a natural number k,outputs a set of k edges, that upon removing them, the graph will have no cycles. If there are no such k ...
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66 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 ...
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31 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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129 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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1answer
123 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
51 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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0answers
50 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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350 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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312 views

Compression of a complete Directed Acylcic Graph

Consider a DAG $g$ as a label $l$ with a list of sub-nodes $\bar{g}$: $g ::= l \enspace \bar{g}$ This is an "unfolded" representation of the DAG, i.e. it contains double entries, when two paths ...
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185 views

Maximum weighted antichain over a DAG with cardinality constraint

Let $G=(V,E)$ be a vertex weighted DAG (Directed Acyclic Graph), with positive real valued weights. Let also $k\leq \left\vert V\right\vert$, is there any way to find a maximum weighted antichain ...
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83 views

Problem with update in Dynamic Bayesian Networks

Consider the following Bayesian network: I want to impose constraints that state that a node can only be true (1) if at least one of its parents are true (1). So, for node $C$, the constraint takes ...
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0answers
105 views

Fully dynamic k-shortest-path

I have a directed acyclic graph with positive edge weights. It is constantly changing in that nodes are deleted and added. For each change, I need to find the $k$ shortest paths. My current approach ...
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3answers
575 views

Topological Sort without modifying the graph or marking edges

I have a DAG which I want to traverse in a topological order. Wikipedia describes two algorithms for topological sorting, which both work in theory but seem impractical to me from a design point of ...
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1answer
343 views

In a DAG, longest path where every vertex has degree at most 2

I have a directed acyclic graph (DAG). How I can find the longest path using only vertices with degree at most 2? Currently, I try this ...
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2answers
53 views

Finding connected components without building the graph first

What are good algorithms for finding connected components in a graph defined by a set of elements X, where each x in ...
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1answer
458 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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1answer
501 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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2answers
3k views

Longest path in DAG or finding DAG diameter

A directed acyclic graph (DAG), is a directed graph with no directed cycles. That is, it consists of vertices and edges, with each edge directed from one vertex to another, such that there is no way ...
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1answer
21 views

Alternatives for finding sources in a DAG

I have a hard time seeing what the alternative approach is in linearizing a directed acyclic graph (DAG). Chapter 3 of Algorithms by Dasgupta et al. states: Property Every dag has at least one source ...
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1answer
184 views

Given a tournament with $2^n$ vertices, show that there is a sub-tournament with at least $n + 1$ vertices that is acyclic

So a tournament is just a complete directed graph, I believe. I'm having trouble proving this problem. I know it is induction however. I was thinking the base case is $2^1$ vertices, and therefore ...
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1answer
235 views

Why do we do topological sorting to find shortest or longest path in weighted DAG?

I was wondering why do we need to do the topological sort before performing relaxing of edges. Wouldn't it'd be better if we do : if starting vertex is "s" ...
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1answer
379 views

How to remove cycles from a directed graph by edge contraction?

We have cyclic directed graph (possibly disconnected). For cycles consisting of two vertices A->B and B->A, we replace them by a single vertex. In case of A->B, B->C, C->A we also replace them by one ...
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1answer
34 views

Topological Ordering

I have learnt to solve topological ordering using $in-degree$ method where we have to take the vertices having in-degree $0$ at an instance and arrange them in that order. For example consider ...
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1answer
33 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
580 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 ...
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1answer
386 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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0answers
23 views

Iterative Deepening DFS for DAGs

Does Iterative Deepening DFS only guarantee the shortest path for DAGs, assuming a visited list is used to prevent infinite cycles? I can't seem to find any answers online specifically related to ...
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1answer
31 views

Vertices reachable from negative-weight cycles in Bellman-Ford

TLDR: I want to know if there's a simple way to fill in distances for all vertices reachable from negative weight cycles (not just ones on the cycle itself) once Bellman-Ford has found a negative-...