Questions tagged [shortest-path]
The shortest-path tag has no usage guidance.
85 questions
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tailored case for Dijkstra algorithm can reach O(VElog(V))
I know that this has been debated and discussed millions of times, but I couldn't find anything that explains why the outer while loop in a typical Dijkstra's min-heap implementation is considered 𝑂 (...
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107
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Are the algorithms developed for the 'gas station problem' and some seemingly derivative problems equivalent?
I have recently been examining the following recently published papers related to the 'Gas Station Problem,' a generalization of the shortest path problem that accounts for fuel consumption by the ...
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39
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Two-stage Robust Shortest Path Problem - worst- case second-stage of an optimal solution
in the paper Improved Approximations for Two-stage Min-Cut and
Shortest Path Problems under Uncertainty chapter 4, they are using an algorithm to approximate the two-stage robust shortest path problem....
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138
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Bellman-Ford with infinite weights
I have a graph with weights of the form $a \omega + b$ where $a,b \in \mathbb{Q}$ and $ \omega$ is an infinite value, that is, a value such that for any rational number $q$, $q \le \omega$. The ...
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Polynomial time algorihtms for two variants of the decision version of longest walk problem
I want to know if the following variants of the longest path problem over directed graphs have polynomial time algorithm.
As I understand it, the longest path problem doesn't allow repetition of edges....
- 1,071
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67
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Shorter than target vector path algorithm
Consider a generalisation of the shortest path problem on directed graphs with weights in $\mathbb{Q}^k$. Formally, the input is a graph, a source state $s$, a target state $t$, and an objective ...
- 1,071
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229
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Shortest path with permutations and fixed dimension
I'm thinking of extensions of the shortest path problem which are solvable in polynomial time. One way to do this is to consider the shortest path problem on a weighted directed graph with weights on $...
- 1,071
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138
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Shortest path with affine updates and fixed dimension
One may look at the shortest path problem on a weighted directed graph with weights on $\mathbb{Q}$ as the problem of minimizing a rational value $x$ which is updated at each edge of the graph with ...
- 1,071
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How can one find a r-division of a graph with strongly sublinear separation profile (separable graphs)?
Thanks for reading, let me provide the definitions first.
A separator of a graph $G$ is a set of vertices $C$ such that removing $C$ cuts the graph into two disconnected parts $A, B$ such that they ...
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96
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Algorithm for Shortest Path in a DAG with Multiple Transportation Modes and Associated Setup Costs
I am working on a problem involving finding the shortest path in a Directed Acyclic Graph (DAG), where each edge's cost depends on multiple transportation modes, each with its own setup cost. I am ...
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247
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Shortest path property and monadic second order logic
I know that induced paths and Hamiltonian cycles can be expressed with monadic second-order logic ($MS_2$).
Is it possible to express the shortest path in $MS_2$?
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678
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Does Dijkstra's algorithm run faster on a DAG?
I know that Dijkstra's algorithm generally runs in $O(E \log V)$ using a min-heap. And I know we can use dynamic programming to find the shortest path of a DAG in $O(V+E)$. However, I was wondering ...
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394
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Solving All-Pairs Shortest Paths using a distance matrix in sub-cubic time
I'm working on a project centered around the All-Pairs Shortest Paths (APSP) problem. Common algorithms to APSP (Floyd-Warshall, Bellman-Ford, Johnson's) work with the standard definition of the ...
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Proof of SPFA's worst-case complexity?
I am trying to prove the worst-case asymptotic time complexity of the Shortest Path Faster Algorithm (SPFA). I know the complexity is the same as the "original" Bellman-Ford (BF) algorithm, ...
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How can we prove what the shortest line between two points avoiding convex obstacles is? (visibility graphs)?
I came across the observation in russell & norvig's artificial intelligence book that the shortest path between two points while avoiding convex polygonal obstacles is a sequence of line segments ...
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104
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Multi agent path following with collision avoidance with pre-determined path
I am working on a multi-agent pathfinding algorithm. I am aware of other techniques, but planned on the folowing strategy only.
The problem:
There is 12x12 grid, with a few solid blockades within them....
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633
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Recent progress on the next-to-shortest-path problem for directed graphs?
In the paper "Computing strictly-second shortest paths" (1997), Lalgudi and Papaefthymiou consider the following problem:
Let $G$ be a directed graph with edge-weighting $w$. Let $u,v$ be ...
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Shortest path on a hypergraph with no leftovers
In quantum computing, determining the code distance of a stabilizer code is similar to the shortest path problem on a hypergraph. Each node in the graph would be some sort of parity check performed by ...
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Is APSP verification easier than APSP?
In APSP, the input is an $n$-node directed weighted graph $G$, and the output is an $n \times n$ matrix holding pairwise shortest path distances between nodes in $G$. Define "APSP-Verification" as ...
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Find the shortest s-t trail(edge disjoint path) in a graph with negative weight edges
A walk in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. A trail is a walk in which all edges are distinct. Note that a trial may visit a vertex multiple times ...
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Number of simple paths between two vertices in a DAG
Let $G = (N, A)$ be a connected acyclic digraph (DAG). Furthermore, let $s \in N$ and $t \in N$ be two vertices on this graph, such that $t$ is reachable from $s$.
My problem is: how many simple $s-t$...
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115
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Bellman-Ford with Non-edge-decomposable Path Weights
Consider a directed graph $G(V,E)$ with non-negative edge weights. Also, let us define the weight of a path as non-edge-decomposable, that is, the weight of a path cannot be written as the sum of a ...
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When is extra vertex required in arbitrage detection using Bellman Ford?
I am studying applications of shortest path, in particular arbitrage.
Specifically, I was reading these two resources:
https://stackoverflow.com/questions/2282427/interesting-problem-currency-...
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183
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Finding Cheapest n-Path [closed]
Given a weighted directed acyclic graph, how can I find the cheapest path from an Origin Vertex to a Destination Vertex which ...
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415
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Finding shortest path while maximizing the number of overlapping edges
The shortest path problem between 2 arbitrary nodes is one that has been covered extensively and the solution is well-known. Consider the edge costs to be arbitrary.
Consider the following variant:
...
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800
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What exactly is Lawler's modification to Yen's algorithm and how does it work?
I recently read about Yen's algorithm, I understand the algorithm and it seems correct, however Wikipedia mentions that there exists "Lawler's modification" to the algorithm, which is described as ...
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547
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Minimum Union-Sum Cost Path
I have a minimum cost path selection problem that is different from the usual shortest path in that each type of cost is accounted only once in the total cost of the path if multiple edges on the path ...
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603
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Maximum difference between two shortest paths
My problem is the following maximization problem:
Given: A graph $G=(V,E)$, lower bounds $l \in \{0,1,..,K\}^E$ and upper bound $u \in \{0,1,..,K\}^E$ for the edge weights, a source $s$ and two ...
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How to solve the Shortest Hamiltonian Path problem on Sparse Graphs?
Problem: Given a positive-weighted undirected graph, find the shortest path (in terms of total sum of edges) that visits each node exactly once.
For a subset $S$ of nodes and a node $i\in S$, let $D[...
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119
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multi-agent pickup and delivery algorithm and conflict resolution
I am looking for a pathfinding algorithm handling the following issues:
multiple agents
the computed paths for agents may not lead to collisions or deadlocks in space-time
a stream of activities
...
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1
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121
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Multiple source shortest path with one reversal [closed]
Lets say we have a directed graph G, with vertices V, that have lengths l.
I need to find the shortest path between every ordered pair of vertices in the graph, with the following constraint:
In a ...
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140
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K-fold Traveling salesman problem - A variant of TSP
Consider a weighted graph $K_n$ and where the weights between vertices $i,j$ is $w_{ij}$. Consider a path, $\sigma$, passing through each vertex only once. Here $\sigma_i$ is the vertex in the $(i\%n)^...
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Anyone recognize this as a special type of multi-commodity flow problem?
Consider this problem:
$$
\begin{align}
\min_{y,z,l \geq 0} \quad & g(y,z,l) :=
\sum_{(i,j)\in E} \sum_p (-w_{ijp}) y_{ijp} & \\
\textrm{s.t.} \quad & \left( \sum_{(i,j)\in E} y_{ijp} + ...
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Shortest distance problem with length as functions of time
Motivation
The other day, I was travelling around the city with public transport and I made up an interesting graph problem modelling the problem of finding the shortest-time connection between two ...
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What is the proof that visibility graphs can be used to compute the shortest path?
I am trying to understand what the proof is that constructing a visibility graph and searching on can give you the shortest path between two points, avoiding a set of convex polygons. I am trying ...
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Are there any heuristics that works solely on graphs?
I'm exploring heuristics in A* and apparently all heuristics require coordinates of all the locations to calculate a h-cost. This is fine if you are working on grids, but what if you need to work ...
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649
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Path finding on graph with state dependent edge costs
I'm looking for a version of path planning that is able to find paths in a graph where edge costs depend on the state of the moving entity. In such cases, it is required to also consider trade-offs, i....
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Constant Width Max Sum Product Multi-objective Shortest path problem
This question is a follow-up on the question I asked three days ago here.
For convenience I restate it here.
I am given a graph. Each edge is labelled by a vector of numbers, called weights. They ...
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Max Sum Product Multi-objective Shortest path problem
Is anything known about the following problem:
I am given a graph. Each edge is labelled by a vector of numbers, called weights. They are numbers between 0 and 1.
A path is first assigned a vector, ...
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1
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92
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Cooperative Pathfinding to minimize global costs
There are some algorithms and methods around, that allow cooperative pathfinding. Unfortunately they all seem to aim at avoiding collisions or conflicts between entities.
I'm looking for an algorithm ...
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634
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Shortest non-crossing geometric paths
I have a plane graph $G$ and a set of $k$ vertex pairs $\{s_1,t_1\}, \dots, \{s_k, t_k\}$. The goal is to find $k$ non-crossing paths connecting the pairs of terminals $s_i$ with $t_i$ in the graph so ...
user7329
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Assigning edge weights under shortest path constraints
We are given a graph $G = (V,E)$ and we need to find an assignment of non-negative edge weights (You must give every edge a non-negative weight). We are also given a set $R\subseteq V$ and mapping $c_{...
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In a shortest path between two nodes, find if a certain node is unique
So my exact problem is, I have to find if there is any node which is unique in a shortest path. For example, in a square, any node is in the shortest path between any two adjacent nodes,but it is not ...
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Finding the two shortest paths while minimizing the number of nearby/common edges
The shortest path problem between 2 arbitrary nodes is one that has been covered extensively and the solution is well-known. Consider the edge costs to be arbitrary.
Consider the following variant:
...
- 63
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1k
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Lower bounds on single-source shortest paths in directed graphs
Are there any non-trivial lower bounds on the complexity of single-source shortest paths (SSSP) in a directed graph, where all edges have non-negative edge weights? Can we rule out the possibility of ...
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366
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Ref question: K-nearest neighbours in a graph
Given an undirected graph $G$ with $n$ vertices, $m$ edges, and positive weights on the edges, I am interested in the problem of computing for each vertex the $k$ distinct vertices in $G$ that are ...
- 9,636
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Highway dimension
I'm interested in understanding some recent theoretical results on pathfinding. Specifically this paper:
http://research.microsoft.com/apps/pubs/default.aspx?id=201061
I understand from the paper ...
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Shortest path hitting a given vertex
I believe this problem to be NP-Complete, but I'm unable to find any references on possible reductions. Given a weighted graph (either undirected or directed, I cannot find results for either but am ...
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253
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Efficiently computing propagation values for only a few positions in a grid
Consider a matrix filled with some nodes containing positive integers ("starts"), some nodes marked as a wall, and the rest of the nodes given a value of infinity.
The propogation rule is simple: For ...
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Identifying useless edges for shortest path
Consider a graph $G$ (the problem makes sense both for directed and undirected graphs). Call $M_G$ the matrix of distances of $G$: $M_G[i, j]$ is the shortest path distance from vertex $i$ to vertex $...
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