Questions tagged [shortest-path]

Questions about the algorithmic problems of finding shortest paths between nodes in a graph.

Filter by
Sorted by
Tagged with
7
votes
2answers
11k views

Shortest path that passes through specific node(s)

I am trying to find an efficient solution to my problem. Let's assume that I have positive weighted graph G containing 100 nodes(each node is numbered) and it is an ...
6
votes
1answer
4k views

Running Floyd-Warshall algorithm on graph with negative cost cycle

I am trying to find the answer to the following question for the Floyd-Warshall algorithm. Suppose Floyd-Warshall algorithm is run on a directed graph G in which every edge's length is either -1, 0, ...
4
votes
1answer
489 views

An incrementally-condensed transitive-reduction of a DAG, with efficient reachability queries

Is there an incremental directed graph data structure that has the following properties: Keeps an internal graph structure as a DAG, and the graph is accessible (notwithstanding other helper data-...
3
votes
2answers
2k views

Shortest Minimax Path via Floyd-Warshall

I am trying to modify the Floyd-Warshall algorithm to find all-pairs minimax paths in a graph. (That is, the shortest length paths such that the maximum edge weight along a path is minimized.) Floyd-...
6
votes
1answer
329 views

Approximability of the edge-disjoint shortest paths problem

In the edge-disjoint paths problem (EDP), we are given a (possibly directed) graph $G=(V,E)$, and a set of distinct source-sink pairs $\{ (s_i,t_i) \mid 1 \leq i \leq k \}$, and we want to maximize ...
2
votes
1answer
208 views

Finding path with minimum weight

There is a river which can be considered as an infinitely long straight line with width W. There are A piles on the river, and B types of wooden disks are available. The location of the $i$-th pile ...
4
votes
1answer
1k views

Can the shortest simple cycle between two given nodes be found in polynomial time?

Given an undirected simple graph $G$ and two nodes $s$ and $t$, the question asks for an algorithm to find the shortest simple cycle (no edge or vertex reuse) that contains the two. As far as I know, ...
3
votes
0answers
3k views

Route planning in public transport application [closed]

This is a cross-post of this StackOverflow question, (I'm not aware of linking questions between StackExchange sites). You can ignore the part about programming. I'm making a journey planner (or a ...
2
votes
0answers
145 views

Multicommodity shortest path problem on a directed acyclic graph

I have n commodities with each a unique source and sink node. Each source-sink pair is connected in some manner on a directed acyclic graph. All arc weights are non-negative. The goal is to find the ...
1
vote
1answer
81 views

For Djikstra's algorithm, why are we surely done if we update all edges $|V|-1$ times?

Apparently, if we use Djikstra's algorithm to find the shortest path between the root node and all other nodes in a weighted graph with no negative cycles, we are done after updating the distance of ...
-2
votes
1answer
381 views

Find the weight of the lightest path from u to v

Find the weight of the lightest path from u to v the goes through node a or/and b. Do you have a suggestion on how it can be done?
5
votes
2answers
2k views

Finding all vertices on negative cycles

Given a weighted digraph, I can check whether a given vertex belongs to a negative cycle in $O(|V|\cdot|E|)$ using Bellman-Ford. But what if I need to find all vertices on negative cycles? Is there a ...
2
votes
2answers
1k views

Shortest path with odd weight

Let G be a directed graph with non-negative weights. We call a path between two vertices an "odd path" if its weight is odd. We are looking for an algorithm for finding the weight of the shortest odd ...
4
votes
1answer
9k views

Finding the path of a negative weight cycle using Bellman-Ford

I wrote a program which implements Bellman-Ford, and identifies when negative weight cycles are present in a graph. However what I'm actually interested in, is given some starting vertex and a graph, ...
2
votes
1answer
251 views

Shortest paths candidate

Let $G = (V,E)$ be a directed graph with a weight function $w$ such that there are no negative-weight cycles, and let $v \in V$ be a vertex such that there is a path from $v$ to every other vertex. ...
4
votes
1answer
5k views

Shortest path with exactly $k$ edges

From Skiena's book The Algorithm Design Manual, chapter 6, problem 22: Let $G = (V,E,w)$ be a directed weighted graph such that all the weights are positive. Let $v$ and $u$ be two vertices in $G$ ...
1
vote
1answer
481 views

Bellman-Ford parent pointer (?) negative cycle

First of all, let me preface by saying that this question is not completly new but the original question hasn't been answered. More important, this is only basic question on understanding the proof ...
5
votes
1answer
610 views

Route on a square grid with only (x,y) → (x,x+y) and (x,y) → (x+y,y) moves

This problem is about finding a route on a square grid. The starting point is $(1,1)$ and the target point $(n,m)$. I can move each step from my current point $(x,y)$ either to $(x+y,y)$ or $(x,y+x)$. ...
6
votes
4answers
3k views

Solving system of linear inequalities

I am trying to solve a system of inequalities in the following form: $\ x_i - x_j \leq w $ I know these inequalities can be solved using Bellman-Ford algorithm. ...
13
votes
2answers
22k views

Finding shortest and longest paths between two vertices in a DAG

Given an unweighted DAG (directed acyclic graph) $D = (V,A)$ and two vertices $s$ and $t$, is it possible to find the shortest and longest path from $s$ to $t$ in polynomial time? Path lengths are ...
-1
votes
1answer
252 views

WP pseudocode for Dijkstra does not work

I mean Dijkstra's algorithm for the shortest path. In all descriptions that I saw (including wikipedia), on every step, it always selects the nearest neighbor based on examining their weights. ...
7
votes
1answer
4k views

All paths of less than a given length in a directed graph between couple of nodes

Counting all possible paths, or all possible paths with a given length, between a couple of nodes in a directed or undirected graph is a classical problem. Attention should be given to what all means, ...
4
votes
3answers
588 views

Find a vertex that is equidistant to a set of vertices?

I need help with the following problem: Input: An undirected, unweighted graph $G = (V,E)$ and a set of vertices $F \subseteq V$. Question: Find a vertex $v$ of $V$ such that the distance ...
3
votes
1answer
7k views

Destination-based vs source-based routing

I understand that destination-based routing builds the "route" from the destination backwards to the source (e.g. if using a spanning tree, then the tree is routed at the destination). With source-...
6
votes
1answer
2k views

What is the maximum number of shortest paths between any pair of vertices in a chordal graph?

A graph $G$ is chordal if it doesn't have induced cycles of length 4 or more. Chordal graphs are precisely the class of graphs that admit a clique tree representation. A clique tree $T$ of $G$ is a ...
-1
votes
3answers
841 views

Dijkstra algorithm: equal number of shortest paths

If I had a Dijkstra graph with the number shortest paths from Node A to O being 1, would it be correct to say: the equal number of shortest paths from A to O is 1 and not 0, because that node is ...
19
votes
2answers
25k views

Is Dijkstra's algorithm just BFS with a priority queue?

According to this page, Dijkstra's algorithm is just BFS with a priority queue. Is it really that simple? I think not.
5
votes
2answers
2k views

Why doesn't the Floyd-Warshall algorithm work if I put k in the innermost loop

The Floyd-Warshall algorithm is defined as follows: ...
1
vote
1answer
127 views

Path on an edge-colored DAG using exactly $k$ colors

I have the following problem: Given an edge-colored DAG $G = (V,A)$, vertices $s$ and $t$, a set of colors $C$ and $k \in \mathbb{N}$, does there exist a path from $s$ to $t$ using exactly $k$ ...
4
votes
1answer
5k views

What's the complexity of calculating the shortest path from $u$ to $v$ with Dijkstra's algorithm using binary heap?

Problem: Consider a graph $G = (V, E)$ on $n$ vertices and $m > n$ edges, $u$ and $v$ are two vertices of $G$. What is the asymptotic complexity to calculate the shortest path from $u$ to $v$ ...
-1
votes
3answers
296 views

BFS in K shortest paths

Do we need to use BFS or DFS algorithm to find the k shortest loopless paths in a graph between any two nodes? If so where can it be useful?
5
votes
2answers
497 views

Why not relax only edges in Q in Dijkstra's algorithm?

Can someone tell me why almost in every book/website/paper authors use the following: foreach vertex v in Adjacent(u) relax(u,v) when relaxing the edges, ...
4
votes
3answers
6k views

Dijkstra's algorithm for undirected graphs with negative edges

INPUT: "an undirected, weighted graph (negative weights allowed)" Could someone give an example for an undirected graph with negative edges where Dijkstra's algorithm doesn't work? As far as i ...
23
votes
3answers
24k views

What is the fastest algorithm for finding all shortest paths in a sparse graph?

In an unweighted, undirected graph with $V$ vertices and $E$ edges such that $2V \gt E$, what is the fastest way to find all shortest paths in a graph? Can it be done in faster than Floyd-Warshall ...
2
votes
1answer
508 views

Bellman-Ford: shortest path

my assumption: - we have an undirected graph with only positive edges - the edges are sorted alphabetically:     e.g A-B, A-C, B-D     and e.g not C-A, D-B, A-...
23
votes
3answers
11k views

Retrieving the shortest path of a dynamic graph

I'm studying shortest paths in directed graphs currently. There are many efficient algorithms for finding the shortest path in a network, like dijkstra's or bellman-ford's. But what if the graph is ...
20
votes
2answers
27k views

Getting negative cycle using Bellman Ford

I have to find a negative cycle in a directed weighted graph. I know how the Bellman Ford algorithm works, and that it tells me if there is a reachable negative cycle. But it does not explicitly name ...
14
votes
2answers
7k views

Bellman-Ford algorithm - Why can edges be updated out of order?

The Bellman-Ford algorithm determines the shortest path from a source $s$ to all other vertices. Initially the distance between $s$ and all other vertices is set to $\infty$. Then the shortest path ...
10
votes
3answers
9k views

Modifying Dijkstra's algorithm for edge weights drawn from range $[1,…,K]$

Suppose I have a directed graph with edge weights drawn from range $[1,\dots, K]$ where $K$ is constant. If I'm trying to find the shortest path using Dijkstra's algorithm, how can I modify the ...
8
votes
2answers
5k views

Finding negative cycles for cycle-canceling algorithm

I am implementing the cycle-canceling algorithm to find an optimal solution for the min-cost flow problem. By finding and removing negative cost cycles in the residual network, the total cost is ...
6
votes
1answer
7k views

Formalization of the shortest path algorithm to a linear program

I'm trying to understand a formalization of the shortest path algorithm to a linear programming problem: For a graph $G=(E,V)$, we defined $F(v)=\{e \in E \mid t(e)=v \}$ and $B(v)=\{ e \in E \mid h(...
4
votes
0answers
634 views

What is the complexity of Hoffman and Pavley's Nth best path algorithm?

I am currently working on a project where I'm using an implementation of Hoffman and Pavley's "Method for the Solution of the Nth Best Path Problem" to find n-th best path through a directed graph. ...
6
votes
1answer
12k views

Finding paths with smallest maximum edge weight

I need to find the easiest cost path between two vertices of a graph. Easiest here means the path with the smallest maximum-weigth edge. In the above graph, the easiest path from 1 to 2 is: ...
9
votes
6answers
37k views

Why can't DFS be used to find shortest paths in unweighted graphs?

I understand that using DFS "as is" will not find a shortest path in an unweighted graph. But why is tweaking DFS to allow it to find shortest paths in unweighted graphs such a hopeless prospect? ...
8
votes
1answer
3k views

Dijkstra to favor solution with smallest number of edges if several paths have same weight

You can modify any graph $G$ so that Dijkstra's finds the solution with the minimal number of edges thusly: Multiply every edge weight with a number $a$, then add $1$ to the weight to penalize each ...
5
votes
1answer
6k views

Finding the Shortest path in undirected weighted graph

Is there an algorithm for finding the shortest path in an undirected weighted graph?
6
votes
4answers
16k views

Using Dijkstra's algorithm with negative edges?

Most books explain the reason the algorithm doesn't work with negative edges as nodes are deleted from the priority queue after the node is arrived at since the algorithm assumes the shortest distance ...
6
votes
1answer
2k views

Bellman-Ford variation

I have a "smarter" version of Bellman-Ford here; this version is more clever about choosing the edges to relax. ...
4
votes
1answer
246 views

Modified Djikstra's algorithm

So, I'm trying to conceptualize something: Say we have a weighed graph of size N. A and B are nodes on the graph. You want to find the shortest path from A to B, given a few caveats: movements on ...
18
votes
2answers
17k views

Shortest Path on an Undirected Graph?

So I thought this (though somewhat basic) question belonged here: Say I have a graph of size 100 nodes arrayed in a 10x10 pattern (think chessboard). The graph is undirected, and unweighted. Moving ...