# Questions tagged [topological-ordering]

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### What don't I understand in topological sort using DFS?

Wikipedia says: The algorithm loops through each node of the graph, in an arbitrary order, initiating a depth-first search that terminates when it hits any node that has already been visited since ...
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### Linear-time algorithm for determining the presence of incomparable pairs in a directed acyclic graph (DAG)

I am seeking a linear-time algorithm to determine whether a directed acyclic graph (DAG) contains at least one pair of incomparable nodes. Two nodes $u$, $v$ are said to be incomparable if there is ...
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### Topological sort of DAG that minimizes maximum number of unique-source-edges crossing through any node when placed in 1-d line

Consider a DAG such as one shown below: How do I find a particular topological order of nodes, such that when the nodes are placed in a straight line, the maximum number of unique-edges that cross ...
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### Why compute finish time in topological sort

In depth first search each vertex can be associated with a discovery time and a finish time. I am reading the following implementation of topological sort in terms of depth first search ...
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### Checking if there exists a 'source' vertex

In a directed graph $G=(V,E)$ we denote a vertex $s\in V$ to be a 'source' if there exists in $G$ a path from $s$ to all other vertices $u \in V$. The problem asks for an efficient algorithm to return ...
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### Getting all vertices with fixed index in their topological ordering of a DAG

During my self study for graphs, I'm currently learning about topological sorting and ran into a question I'm not sure how to solve. There are typically more than one order of a topological ordering ...
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### Is there any Algorithm to check a vertex\node's partial order in terms of other vertices\nodes for a given graph?

For the given figure, let's consider vertex v3. For v3, v0 has a higher partial order,v1 & v2 has the same partial order, and v4 & v5 have lower order than v3, e.g., higher: {v0}, same: {v1,v2}...
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### Genetic algorithms applied to topological orderings of a DAG

I need to solve an optimization problem, whose search space is all possible topological orderings of a DAG. There is a cost function associated with each ordering, which has no simple mathematical ...
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### How to mathematically prove that a topological ordering on a cyclic graph will topologically sort its Strongly Connected Components?

Let's have a standard topological ordering algorithm (from CLRS): ...
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### What is the relation between Topological Sort and Strongly Connected Components?

Both the Topological Sorting algorithm and the algorithm to find Strongly Connected Components build a stack whose top is the last visited vertex. I find difficult to find an explaination because ...
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### Topological sort with minimum maximal distance in array

I have a DAG that admits many possible topological sorts. I want to construct one that has the minimum maximum distance between a node and its neighbours in an array storing the nodes in sorted order. ...
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### Topological sort and finding longest path in DAG to solve a stacking boxes variation (no rotation)

Given n elements (boxes) I have to output the max number of boxes that can fit one into another. Each box has width (x), height (y) and depth (z). One box j can hold another box k if: ...
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### Is doing BFS over transitive reduction of a directed acyclic graph equivalent to topological ordering of that graph?

I have a directed acyclic graph. Where each node is a task and each edge denotes a dependency. I want a method to effectively parallelize these tasks. One way would be to topological sort them based ...
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### Constrained/Optimal Topological Order to enhance/reduce the performance/memory usage of other algorithms

I originally posted this question here Lets assume we have a highly connected directed acyclic graph (DAG, more edges then nodes). Since it is a DAG, we can retrieve a topological order of nodes to ...
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### Prove the following claim on Hamilton Path?

I am trying to prove the following claim: Given DAG graph, there is Hamilton path iff the following algorithm returns true: Do topologic sorting. Move on the graph's vertices one by one (from low to ...
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### DFS produces the correct Topologically ordered sequence

Prove that DFS produces the correct topologically ordered sequence. I am having a hard time understanding the question itself. Should I prove the correctness of DFS? Should I use the pseudocode?
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### Valid orderings for topological sort

I am reading Algorithms by Dasgupta et al and the graph section provides an example graph and mentions that there are 4 orderings with one of them being B, A, D, C, E, F. Are the other 3? B, A, D, C,...
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### Ordering of operations in a DAG of git commits

Context: I'm looking for a better state resolution algorithm for https://github.com/MichaelMure/git-bug Summary of the current algorithm and shortcomings git-bug is a distributed bug-tracker that ...
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### Modified topological sort

I recently asked a related question at the theoretical CS stack exchange, but I have modification to the problem that I think is a bit tougher. This seems like a better place anyways. Let's define a &...
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### 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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### How to edge-color a directed acyclic graph so that every path visits none or all edges of each color?

Given a directed acyclic graph $G$ and a start vertex $s$ and an end vertex $e$, consider a coloring of the edges valid if, for every path from $s$ to $e$ and every color $c$, either $c$ is never ...
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### 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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### Is topological sort of an original graph same as post-ordering dfs of its transpose graph

I have an intuition that topo-sort of an original graph A -> B -> C D -> B topo-sort is [D, A, B, C] or [A, D, B, C] If I transpose the graph <...
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### 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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### 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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### Construct a DAG from given multiple topological orderings

I need to construct a DAG, from its given topological orderings (i.e. the graph $G$ created must have all the orderings given as its topological orderings). For simplicity, the vertices are labeled as ...
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### Is there an algorithm to minimize working set during a topological traversal?

I have a dependency graph of tasks, which forms a DAG. I'm looking for an algorithm to calculate the optimal topological traverse to minimize the "working set". Specifically, I define the current ...
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### Determine whether there exists a path in a directed acyclic graph that reaches all nodes without revisiting a node

For this I came up with a DFS recursion. Do DFS from any node and keep doing it until all nodes are Exhausted. I.E. pick the next unvisited node once you cannot keep recursing. The element with ...
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### Order the vertices to maximize the weights of edges in the induced subgraph

I have a complete directed graph $G:=(V,E)$ with directed edge weights $c_{ij}$ for every distinct nodes $i$ and $j$. Goal: Find the topological order such that the smallest edge weight of the ...
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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 ...