Questions tagged [shortest-path]
The shortest-path tag has no usage guidance.
75
questions
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How do the state-of-the-art pathfinding algorithms for changing graphs (D*, D*-Lite, LPA*, etc) differ?
A lot of pathfinding algorithms have been developed in recent years which can calculate the best path in response to graph changes much faster than A* - what are they, and how do they differ? Are ...
32
votes
0
answers
6k
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Combinatorics of Bellman-Ford or how to make cyclic graphs acyclic?
Roughly speaking, my question is:
How costly is to make a cyclic graph
acyclic while preserving all simple $s$-$t$ paths?
Let $K_n$ be a complete undirected graph on vertices $\{0,1,\ldots,n+1\}$.
(...
- 6,655
19
votes
2
answers
870
views
Axioms for Shortest Paths
Suppose we have an undirected weighted graph $G = (V, E, w)$ (with non-negative weights). Let us assume that all shortest paths in $G$ are unique. Suppose we have these $\binom{n}{2}$ paths (sequences ...
- 1,559
18
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0
answers
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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 ...
- 11.1k
17
votes
2
answers
4k
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Finding k shortest Paths with Eppstein's Algorithm
I'm trying to figure out how the Path Graph $P(G)$ according to Eppstein's Algorithm in this paper works
and how I can reconstruct the $k$ shortest paths from $s$ to $t$ with the corresponding heap ...
- 273
15
votes
1
answer
14k
views
Finding the shortest path in the presence of negative cycles
Given a directed cyclic graph where the weight of each edge may be negative the concept of a "shortest path" only makes sense if there are no negative cycles, and in that case you can apply the ...
- 253
12
votes
2
answers
818
views
Graph classes for which the diameter can be computed in linear time
Recall the diameter of a graph $G$ is the length of a longest shortest path in $G$. Given a graph, an obvious algorithm for computing $\text{diam}(G)$ solves the all-pairs shortest path problem (APSP) ...
- 3,160
12
votes
3
answers
2k
views
Subgraph containing all nodes and edges that are part of length-limited simple s-t paths in an undirected graph
Quite similar to my previously posted question. This time however, the graph is undirected.
Given
An undirected graph $G$ with no multiple-edges or loops,
A source vertex $s$,
A target vertex $t$,
...
- 509
11
votes
1
answer
1k
views
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 $...
- 8,223
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votes
1
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450
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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 vertices in ...
- 211
10
votes
3
answers
1k
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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 ...
- 363
8
votes
2
answers
539
views
Dijkstra parallelization
I'd like to know what is the best method to parallelize the Dijkstra algorithm.
Thanks.
- 181
8
votes
1
answer
405
views
Motivation for Developing Shortest Path Simplex Algorithms
I'm reading Efficient Shortest Path Simplex Algorithms by Donald Goldfarb, Jianxiu Hao and Shen-Roan Kai who considered "the specialization of the primal simplex algorithm to the problem of finding a ...
- 183
8
votes
2
answers
6k
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Finding a minimum "node" weight path
Suppose a graph with node weights only (no edge weights). For a given source-sink pair, how can I find a path with the minimal sum of node weights? Does this problem have a name? Is it possible to ...
- 253
8
votes
0
answers
157
views
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 ...
- 2,333
7
votes
1
answer
617
views
Shortest paths perturbation
I have a graph $G=(V,E)$, with positive weights $w_e, e\in E$ on the edges, and I would like to randomly perturb the weights of the edges so that for each pair of distinct vertices $(u,v)$ such that ...
- 569
7
votes
3
answers
585
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Preprocessing Sparse, Directed Non-Planar Graphs for Faster Shortest Path
I'm trying to preprocess a very large and sparse directed graph in order to do faster shortest path searches. The vertices have no natural distance function, and the edges are unweighted.
One ...
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7
votes
2
answers
347
views
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 ...
- 9,586
6
votes
2
answers
3k
views
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 ...
- 169
6
votes
1
answer
235
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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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votes
3
answers
2k
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Dynamic programming and shortest path problem
Several months back, I asked in math.SE the following question
I wonder if any dynamic programming problem can always be converted to a source-sink shortest path problem in a network with source and ...
- 629
5
votes
2
answers
2k
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In a resistor network, is there any relation between the shortest path and the maximum electric current path?
Consider a shortest path problem between the source $s$ and sink $t$ in an undirected weighted graph. There's a well known algorithm such as Dijkstra's algorithm that solves this problem.
Naturally, ...
5
votes
2
answers
10k
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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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votes
2
answers
2k
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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
5
votes
1
answer
247
views
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, ...
4
votes
1
answer
275
views
For which values of $k$ is minimum length undirected $k$-disjoint-paths in $\mathcal{P}$?
In a related question, Saeed and Super8 have mentioned the Robertson-Seymour theory which enables us to find $k$ disjoint paths between pairs of vertices $\{s_i,t_i\}_{i=1}^k$ in poly time for fixed $...
- 9,378
4
votes
1
answer
259
views
Shortest distance/path between two households
If you wanted to know the shortest distance/path between two household addresses, which data structure(s) would you use to return the answer efficiently?
Say you are considering the set of all ...
- 143
4
votes
1
answer
149
views
Shortest path property and monadic second order logic
I know that induced paths and Hamiltonian cycles can be expressed with monadic second-order logic ($MS_2$).
Is it possible to express the shortest path in $MS_2$?
- 45
4
votes
1
answer
391
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Dynamic all-pairs shortest paths - O(1) query
I'm trying to come up with an algorithm to solve all-pairs shortest paths (APSP) problem in dynamic directed planar graph with nonnegative real weights. Additionally:
My primary focus is to achieve ...
- 141
4
votes
1
answer
336
views
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:
...
- 43
4
votes
1
answer
525
views
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 ...
- 175
4
votes
1
answer
88
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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 ...
- 161
4
votes
0
answers
1k
views
Path finding algorithm to maximise points of interest along the route
I am trying to write an algorithm to find a path (not the shortest one) between a given start and end point.
An user will enter the start location, the end location and the available time to travel. ...
4
votes
0
answers
498
views
Dynamic shortest path data structure for DAG
Let $G$ be a dynamic DAG (directed acyclic graph) where new vertices and new edges can be inserted.
I am looking for an efficient data structure/algorithm to maintain the shortest path from a fixed ...
- 141
4
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0
answers
660
views
What about apply maxplus algebra for all-pairs shortest paths?
I didn't find deep informations on Wikipedia about all-pairs shortest path, in particular I do not know what is the best algorithm to solve this problem beyond Floyd-Warshall's one, then I do not know ...
3
votes
1
answer
218
views
Minimum offset while measuring TSP paths
I have Euclidean graph: each vertex is a point on the 2D plane, so the weight of each edge is the Euclidean distance between the vertices.
I am trying to solve TSP with brute algorithm, and I want to ...
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3
votes
1
answer
540
views
Breadth first search and Eppstein K shortest paths algorithm
I'm trying to understand the algorithm for finding K shortest paths in a graph described by Eppstein in this paper: http://www.ics.uci.edu/~eppstein/pubs/Epp-SJC-98.pdf
I have trouble particularly ...
- 133
3
votes
1
answer
288
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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 ...
- 1,478
3
votes
1
answer
624
views
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 ...
- 33
3
votes
1
answer
167
views
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, ...
- 73
3
votes
1
answer
444
views
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 ...
- 151
3
votes
1
answer
188
views
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} + ...
- 31
3
votes
1
answer
207
views
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_{...
- 131
3
votes
0
answers
161
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Multi-Agent Pathfinding
Quoting from Wang and Botea 2011:
An instance is characterized by a graph representation of a map, and a non-empty collection of mobile units $U$. Units are homogeneous in speed and size. Each unit $...
- 1,082
3
votes
0
answers
643
views
A constrained shortest path problem. What is the complexity?
I've got the following problem:
Consider a graph $G=(V,E)$ with $V=\{v_1,\ldots,v_n\}$, and edge-set $E=\{e_1,\ldots,e_m\}$, with associated costs $c_1,\ldots,c_m$. The problem is to find the ...
- 39
2
votes
1
answer
90
views
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 ...
2
votes
1
answer
766
views
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 ...
- 21
2
votes
1
answer
323
views
Do you know a shortest path algorithm for weighted graphs with hard time windows on the edges and waiting allowed?
Title says it all. I have a weighted Graph G={V,E,ETW} where V is the node set, E the edge set and ETW is a set of edge time windows. A edge time window is a 3-Tuple (edge, starttime, endtime) with ...
- 21
2
votes
1
answer
569
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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 ...
- 31
2
votes
2
answers
437
views
Why is label pruning possible with hub labeling?
Hub labeling (HL) computes superlabels using the vertices visited by the forward and reverse Contraction Hierarchies (CH) search. Those labels are then pruned (see HL, sec. 4.2) to generate strict ...
- 159