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# Questions tagged [shortest-path]

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

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### Why doesn't Dijkstra's use a shortest-path first search?

When using BFS search on an unweighted graph to find the single-source shortest paths, no relaxation step is required because a breadth-first search guarantees that when we first make it to a node, we ...
430 views

### Finding Shortest Paths of weighted graph using stacks

I will be given some kind of this graph as in the picture below. I've searched some algorithms but it seams as if it is something impossible for me to figure them out. In fact using Floyd–Warshall ...
209 views
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### How hard is finding the shortest path in a graph matching a given regular language?

Suppose we are given a directed graph $G = (V, E)$ with edge weights $w : E \rightarrow \mathbb{R}$ (we can assume there are no negative cycles) and edge labels $\ell : E \rightarrow \Sigma$ from some ...
82 views

### Dijkstra without decrease key

I was reading though this paper, which suggests using dijkstra without edge relaxation, but to rather to just insert new nodes, cf page 16 for the pseudo code. But to me the code looks wrong. I think ...
129 views

### Why is my algorithm version so slow with this input?

Here I'm trying to do a comparison of two simple as possible algorithms, solving the symmetric travelling salesman problem, finding the optimal solution without the support of heuristics. I'm showing (...
4k views

### Why can't we find shortest paths with negative weights by just adding a constant so that all weights are positive?

I'm currently reading introduction to algorithms and came by Johnson’s algorithm that depends on making sure that all paths are positive. the algo depends on finding a new weight function (w') that ...
36 views

### Time complexity analysis of shortest path algorithm

Below is Dijkstra's algorithm from CLRS: In the time complexity analysis of Dijkstra, CLRS says, RELAX() contains call to DECREASE-KEY(), which is essentially reducing edge weights associated with ...
25 views

### How Dijkstra's algorithm forms shortest path tree when multiple nodes have same shortest path length

I came across following problem: Consider below graph: What will be the shortest path tree starting with node $A$ returned by Dijkstra's algorithm, if we assume priority queue is implemented ...
208 views

### Finding negative cycle using Bellman Ford

Given a graph with |V| vertexes and |E| edges, I have to find a negative cycle, if there is one, in a graph. The wanted complexity is O(|V|*|E|). I was thinking about using Bellman-Ford to solve the ...
28 views

### Dijkstra shortest path yields unintuitive results

Considering the following nodes with edge weights in red, Dijkstra's shortest path algorithm seems to return incorrect results, at least by the definition of the steps on wikipedia. By those rules, ...
23 views

### Dijikstra's algorithm with “hull” value catch

Whilst preparing for the CCC(Canadian Computing Competition), I encountered CCC 2015 Seniors problem 4, linked here. Anyway, the problem describes a set of vertices(points) numbered from $1$ to $N$, ...
65 views

### Conditional lower bounds on the running time of the single source shortest path problem

Just out of curiosity, I was wondering whether there is a conditional lower-bound on the running time of an algorithm for the Single Source Shortest Path Problem (on directed or undirected graphs). I ...
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 ...
386 views

### Shortest paths between given red vertices and arbitrary blue vertices

Given an undirected weighted graph, where each vertex has one of two colors - red or blue. I have to answer queries to find the shortest path between a given red vertex and any blue vertex in the ...
47 views

### Dynamic all pairs shortest path edge removal

I have a planar(|E|=O(V)) undirected graph with positive edge weights. I have already calculated all pairs shortest path with Floyd–Warshall algorithm. Now I want to recalculate APSP with an edge ...
885 views

### Dijkstra with max instead of sum

Is it true that if we replace in the Dijkstra algorithm + with max, then the resulting algorithm correctly solves the problem of ...
256 views

### How to deal with cost variation in a dynamic graph when applying Dijkstra

What are the methods to deal with variations in cost in a dynamic graph when applying Dijkstra? For instance, I select the shortest path in a graph, however, the weight of this path changed after I ...
359 views

### Find a path that contains specific nodes without back and forward edges

I have a directed graph and and a set of nodes(set = [1,2,5,9,24...]). I want to find a path that contains all the set of nodes and this path dont contain back edges(cycles) and forward edges. For ...
58 views

### What's the best algorithm to find the shortest path between 2 vertices in a graph?

Considering an undirected and unweighted graph, what's the best algorithm to find the shortest path between two vertices?
16 views

### Decomposition of graph to subgraphs according to parallel edges

I am supposed to calculate all-pair shortest path lengths of a graph. However, I first need the graph to be decomposed/expanded to a simple one based on the presence of parallel edges. If N parallel ...
45 views

### Maximum value of arbitrage

I'm trying to solve this from The Algorithm Design Manual: 6-23. Arbitrage is the use of discrepancies in currency-exchange rates to make a profit. For example, there may be a small window of time ...
58 views

### Determine shortest path in a 4x5 grid graph

Suppose we had a graph $G = (V,E)$. This graph can also be seen as a $4x5$ grid graph as shown in the image. There is a directed edge from $v_{i,j} \rightarrow v_{i,j+1}$ for $1 \leq i \leq n$ and ...
14 views

### Performant algorithm to find edges without cross overs

I have a series of graphs with points plotted like this: Like in the image, I need to join these points to create a complete edge. I am currently doing this with nearest-neighbour, but because I don'...
45 views

### Optimal root in shortest path tree (SPT)

I would like to find the "optimal" shortest path tree (SPT) in some undirected weighted graph. As "optimal" SPT, I mean so its maximal path from root to leaf is minimal from any other potential SPTs. ...
76 views

### How can I avoid re-computation of Dijkstra algorithm if I add or remove one edge from a graph?

I have a nested graph filtration and each step I have to find the shortest path between two nodes. At each step I just add one edge to the graph so the re-computation of the Dijkstra algorithm is ...
25 views

### Proving inequalities related to Dijkstra's algorithm

Define $spdist(s,t)$ as the distance of the shortest path from vertex $s$ to $t$. Define $IN(v)$ as the set of in-neighbors of $v$. Define $w(u,v)$ as the weight of the edge $(u,v)$. I am asked to ...
92 views

### Remove a vertex from a graph keeping shortest path distance same

How could we delete an arbitrary vertex from a directed weighted graph without changing the shortest-path distance between any other pair of vertices? We are allowed to reweight the edges.
96 views

### Shortest path between any origin to any destination through some way stations

How can one find the shortest path between any one of the origins to any one of the destinations through a number of way stations on the way using Dijkstra algorithm? You can visit those way stations ...
757 views

### Verifying whether a description of a shortest path tree is actually the shortest path tree in O(V+E) time

This is from CLRS problem 24.3-5: Professor Gaedel has written a program that he claims implements Dijkstra’s algorithm. The program produces $v.d$ and $v.\pi$ for each vertex $v \in V$ . Give ...
97 views

### Minimum distance of nodes from a set of two nodes

In an unweighted tree, suppose that we want to delete (or mark) any node which is closer to node $v$ than node $w$ ($dist(x,v) < dist(x,w)$). The solution that comes to my mind is running two BFS, ...
36 views

### Using the decision problem, PATH in order to solve the optimization problem, SHORTEST-PATH in polynomial time

So, if I were using a black-box decision algorithm, PATH in which I could say, "does a path of weight k exist in this graph from ...
669 views

### Questions on shortest path and minimum spanning tree

T/F Questions Adding a constant to every edge weight does not change the solution to the single-source shortest-paths problem. Solution - False I think this should be True, as Dijkstra's Algorithm ...
108 views

### Modifies Dijkstra’s Algorithm to find the maximum cost path

In a DAG and all weights are larger than 0. Is it possible to use a max heap to get the maximum cost?
134 views

### How does the slow All-pairs-shortest-paths algorithm work?

I am trying to fully understand the following algorithm from CLRS book: I like to think that it works similarly to Bellman-Ford algorithm by relaxing all edges once for every vertex in the graph. ...
24 views

### How to compute the predecessor-subgraph in all-pairs-shortest-paths algorithm?

The following slow algorithm (implemented from CLRS book) which runs in $\Theta(V^4)$ works fine for computing shortest paths distances: ...
4k views

### Why do we need to run the bellman-ford algorithm for n-1 times?

I'm a little confused about the concept of the Bellman-Ford(BF) algorithm to compute the shortest path in a general graph with negative weights knowing that there are no negative cycles present. I ...
17 views

### Maintaining SCCs in directed graphs (on-line, under edge deletion) with ES-trees

I'm interested in efficiently maintaining the set of strongly connected components (SCC) in a directed (unweighted) graph under edge deletions only. While searching for ways I came across an article [...
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, ...
106 views

### Algorithm to find longest path in a tree that is smaller than x

Suppose we have a weighted binary tree $G$ where the nodes are towns and edges are streets with edge weights being the travel time and we want to find out whether it is possible to travel from any ...
81 views

### Why haven't I solved the Travelling Salesman problem with the following argument using djikstras algorithm?

I claim to have solved the travelling salesman problem as follows. (You will have to be familiar with djikstra's algorithm for this.) 1) I am about to start using djikstra's algorithm on any given ...
346 views

### Why is Dijkstra's Algorithm more popular compared to Grassfire algorithm?

Consider algorithms to find shortest paths in a graph. The grassfire algorithm has a complexity of O(|V|) where V is the number ...
112 views

### C++ finding the shortest path, reducing time complexity, dijkstra v Floyd Warshall Algorithm?

I have an algorithm that I am performing on a graph and I am looking to do an analysis of how to speed it up and would appreciate any comments. The algorithm iterates over every edge in the graph. ...
70k views

### Minimum spanning tree vs Shortest path

What is the difference between minimum spanning tree algorithm and a shortest path algorithm? In my data structures class we covered two minimum spanning tree algorithms (Prim's and Kruskal's) and ...
15 views

### APSP on GPGPU with paths reconstruction

There is a huge number of works on efficient solution of all-pairs shortest paths (APSP) by using GPGPU. But the main goal of these works is to compute the length of the shortest path. Are there exist ...
47 views

### Similar-path shortest paths

Consider a directed graph with an out-degree of 2 for every vertex, i.e. all vertices have exactly two outgoing edges. This means, considering $n$ as the number of vertices, that the number of edges ...
684 views

### Routing algorithm for train network

I am trying to analyse a weighted multi-graph which represents a snapshot of a rail network for a particular day. As such, the vertices of the graph can be considered stations and the weighted edges ...
183 views

### A* without heuristic more efficient than Dijkstra

I am using the module networkx to operate on graphs made from OpenStreetMap. I wanted to compare the shortest path algorithm to ...
39 views

### Graphs - Shortest Path Algorithms - Summary

Are following statements valid? Shortest Path in an undirected graph can be found using BFS. Is DFS an option here? If DFS is not an option, why. Dijkstra's SPT works if there are no negative ...
While reading Thorup's Algorithm to solve SSSP problem, I have one point that I can't understand: super distance. It says: "For each vertex we have a super distance $D(v)\geq d(v)$" $d(v)$ must ...