Questions tagged [shortest-path]

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

Filter by
Sorted by
Tagged with
4
votes
0answers
84 views

Can the running time be reduced to something lower than $O(d^4)$?

Imagine I have a weighted complete directed graph $G$ with $d$ vertices(so $d(d-1)$ edges) and I want to do the following: Set $D$ to be a DAG with the same set of vertices but without any edges sort ...
0
votes
1answer
18 views

Term for an A*-like pathfinding strategy where only the heuristic goal distance matters

I am trying to find a proper term for the A*-like best-first pathfinding strategy where the node to expand next is the one with the least estimated distance from the goal, regardless of its distance ...
0
votes
1answer
99 views

Hitting probability of random walk within given number of steps

Given m,n dimensions of a 2D matrix; (i,j) initial co-ordinates; (x,y) final co-ordinates. What is the probability of being at (x,y) after at most k steps if we start from (i,j) initially? We can ...
1
vote
1answer
321 views

Floyd–Warshall algorithm on an undirected graph contains negative weight edges

According to this answer, the Bellman-Ford algorithm doesn't work when an undirected graph contains negative weight edges since any edge with negative weight forms a negative cycle, and the distances ...
6
votes
1answer
158 views

Finding a negative cycle in a bipartite graph

The Bellman-Ford algorithm can be used to find a negative cycle in a general graph, in time $O(|V||E|)$. Is there a faster algorithm for finding a negative cycle in a bipartite directed graph, where ...
0
votes
1answer
63 views

Distance function such that we visit every “color region” once [closed]

Consider the following image: Starting at (0,0) top left, the objective is to find a dijikistra path to the bottom right. We must go through each color exactly once, and once we go outside a color, ...
3
votes
1answer
144 views

Constructing a minimum spanning tree from an all-shortest path graph?

Given an $n \times n$ shortest path distance matrix $D$. And a complete graph $G(D)$ on $n$ nodes, where edge $(i, j)$ has weight $D_{ij}$. Furthermore, the distance matrix $D$ is computed for a ...
1
vote
1answer
63 views

Combinatorial Optimization: Shortest distance given sets of drivers and riders

Problem: I have 2 sets, one of drivers and one of riders. All my participants need to reach one common destination. I wish to calculate the shortest combined distance in order for all participant to ...
2
votes
2answers
391 views

Given all pairs shortest paths matrix, find graph with minimum total sum of edges

I was looking at some problems about graphs, and I got stuck on this one. Namely, we have given matrix of size $N \cdot N$ representing the length of the shortest path in undirected graph between some ...
3
votes
2answers
619 views

How many iterations does the Bellman-Ford algorithm need for directed and undirected graphs

The Bellman-Ford algorithm on a graph with $n$ vertices, normally includes a loop executed $n-1$ times. Each time through the loop we iterate over the list of edges $(u,v)$ and relax $v$. Note that ...
2
votes
1answer
60 views

All pair shortest path in a tripartite graph

I have a tri-partite graph with three sets of vertices source, bridge and destination nodes. I want to find the shortest path between every vertex in the source set to every vertex in the destination ...
3
votes
1answer
30 views

Longest simple walk below a certain weight

Given a directed graph G and a starting vertex $v$ and a cutoff weight $w$, I want to find a simple walk with net weight < $w$ that visits as many nodes as possible. Currently, I have a recursive ...
0
votes
1answer
81 views

Running Dijkstra on particular graph with negative weight

After running Dijkstra on this graph from S, which shortest paths will be incorrect? This graph has a negative weight, so which shortest paths will be incorrect? after my first attempt I got that Y ...
0
votes
1answer
543 views

Dijkstra complexity analysis using adjacency list and priority queue?

I just got to look at the Implementation of Dijkstra using adjacency list and priority queue. The time complexity is $O(E\log V +V)$, I tried looking for the proof but couldn't find one. Any help will ...
0
votes
0answers
45 views

Contraction Hierarchies minimal distance proof

I am trying to implement "Contraction Hierarchies" algorithm and reading the white paper and watching video lectures [6,7]. But still I can't understand proof for the following lemma: Lemma 1. $d(s,...
1
vote
1answer
220 views

Shortest path from source to all vertices, but with some wildcards

Here is problem in Sprinklr Interview Experience | Set 5 (On campus – FTE for Product Engineer). You are given a graph of $n$ nodes with $m$ bidirectional edges. Each edge has some value associated ...
0
votes
0answers
171 views

Successive shortest path without reduced costs

The successive shortest path algorithm, used to solve the minimum-cost flow problem, can be described as follows : Successive shortest path (for minimum-cost flow) : while all flow is not ...
2
votes
1answer
52 views

Djikstra algorithm analysis

My textbook says that the Dijkstra algorithm's runtime is $O(n) + O(m \log(n)) = O((n+m) \log(n))$. How did they come up with that? Dijkstra algorithm pseudocode: ...
1
vote
1answer
32 views

How is Johnson's shortest path weighting function $\hat{w}(u, v) = w(u, v) + h(u) - h(v)$ proven by the triangular inequility?

Recap to the Johnson's shortest path algorithm: By the procedure extending the original graph $G^\prime = (V^\prime, E^\prime), V^\prime = V\ \cup \{s\}, E^\prime = E\ \cup \{(s, v)\ |\ \forall v \in ...
0
votes
2answers
528 views

Understanding connection between minimum spanning tree, shortest path, breadth first and depth first traversal

In CLRS, in the later part of breadth first search topic, for unweighted graphs, it says: At the beginning of this section, we claimed that breadth-first search finds the distance to each reachable ...
0
votes
0answers
177 views

Finding all edges on any shortest path between two nodes using dijkstra

Given a directed weighted graph, we need to mark all edges (represented by an ordered triple of (source,destination,weight) ) which lie on some shortest path from source to destination (there could be ...
3
votes
1answer
190 views

Intersection of two shortest paths in connected weighted graph

Let $G=(V,E)$ be a connected directed weighted graph with non-negative weights on edges. Let $u,v,s,t$ be vertices in the graph $G$. I need to find an algorithm which in $O(|E|\log |V|)$ time checks ...
2
votes
0answers
130 views

Why is it true that given a monotonic heuristic function, A* can be seen as Dijkstra's algorithm where no node needs to be processed more than once?

Maybe I am missing something very easy and obvious. But, I don't understand why estimate cost of source vertex is subtracted from the overall estimate cost, if heuristic function $h$ is monotonic: $$...
1
vote
1answer
37 views

Shortest path in graph by flipping binary colored nodes to one color [closed]

Given a graph consists of two-colored nodes(e.g. white and black) and a starting node, and every time you visit a node, its color is switched(from black to white, or, white to black), how to find the ...
1
vote
2answers
546 views

Time complexity of Dijkstra's algorithm for sparse graph

I'm not sure I understand the answer to this question: Question 9. What is the running time of Dijkstra's algorithm in a graph that is sufficently sparse - in particular, $E=o(V^2/\log V)$, ...
-1
votes
1answer
98 views

Computer Networks, OSI model

What layer of OSI model does define the route of information transmission between sender and receiver computers? A) Session layer B) Physical layer C) Data link layer D) Network layer E) Transport ...
2
votes
1answer
425 views

Confused about the correctness proof of Dijkstra's algorithm

In the proof of the correctness of Dijkstra algorithm, there is a lemma stating as follow: Let u be v's predecessor on a shortest path P:s->...->u->v from s to v. Then, If d(u) = δ(s,u) and edge (...
1
vote
1answer
45 views

What's the usage of $S$ in Dijkstra shortest path algorithm in the book Introduction to Algorithms?

I don't understand how the $S$ is needed in dijkstra shortest path algorithm. For each node $v$ in $G.V$, the $v.\pi = previous\_node$ is used to denote it previous node in the shortest path to the ...
1
vote
1answer
82 views

Will MST find the shortest path for each pair $(r,v)$?

Will local best choice will lead to global best choice? In other words, I'm thinking about whether it's possible that the MST has to put its branch location in the middle of two far nodes ...
1
vote
1answer
351 views

Word ladder problem for words with different length

Is there some one who know any algorithm for word ladder problem with words of different length? Actually we have some strings with same length and some strings with one length longer but not from ...
2
votes
1answer
61 views

Shortest path with nodes containing collectibles of negative cost

Suppose you have a graph with weighted edges and nodes. Edges always have non-negative costs (representing e.g. fuel costs), and nodes always have non-negative benefits (representing e.g. collectible ...
1
vote
2answers
60 views

Two Problems in understanding the algorithm for computing shortest paths in undirected graphs with possibly negative edge weights

Section 2 of this Lecture Note: Shortest Path Algorithms Luis Goddyn, Math 408 describes an algorithm using Edmonds' Minimum Weight Perfect Matching Algorithm to solve the shortest path problem for ...
1
vote
0answers
39 views

Concurrent Shortest Paths with “Congestion Penalty”

Given a graph with positive edge weight representing "time to travel through", and 2 or more pairs of start/end vertices, we can find concurrent paths for the pairs such that the maximum cumulative ...
2
votes
2answers
523 views

Find shortest path that goes through at least 5 red edges

Let $G=(V,E)$ be a directed graph, $\omega : E \rightarrow R$ a weight function, and $s,t \in V$ a pair of different nodes. It's given that $G$ doesn't have a negative cycle. Moreover, 10 of its edges ...
1
vote
1answer
432 views

Dijkstra’s shortest path algorithm

I'm learning about single source shortest path algorithms and need to clarify few doubts- Does Dijkstra’s assumes that the weights of edges in a graph searched by it are positive integers. Does ...
2
votes
1answer
439 views

Shortest path with a start vertex that touches all nodes at least once with repeats allowed

I tried looking this problem up for quite a bit now, but can't seem to find a whole lot of discussion about this. At first it sounded like the TSP to me, but I don't think so (it's much harder to do I ...
-2
votes
1answer
653 views

Please indicate whether each of the following statements is TRUE or FALSE and provide a brief justification

I provided my answers in the "answer your own question" bit. I have applied the same logic for my answers to a&b and c&c which seem to be essentially the same questions. Am I right though? a)...
0
votes
2answers
203 views

Shortest distance from multiple points to one point

I am looking for an algorithm to find the shortest distance from multiple nodes to one end node. For example let $v_1,v_2,\dots,v_r$ be the nodes on a graph with distance $d_1,d_2,\dots,d_r$ to the ...
1
vote
1answer
42 views

What happens if I replace $<$ with $\le$ in Dijkstra's algorithm?

The following is Dijkstra's algorithm for finding the shortest path in a graph. I know something wrong happens if I replace d[u] + weight(u,v) < d[v] with ...
0
votes
0answers
137 views

Intuition behind Floyd-Warshall being faster

I know the Floyd-Warshall, and I also clearly understand the proof of running time of $O(V^3)$ of F-W algorithm. However, consider this algorithm: Let $dp[i][j][n]$ denote the shortest path from $...
0
votes
2answers
64 views

Finding shortest path between two nodes with a set of forbidden nodes

I have undirected and unweighted graph, in which I would like to find the shortest path between two entered nodes. There is also a set of forbidden nodes. How to find the shortest path, if I am ...
1
vote
1answer
588 views

Behaviour of Dijkstra in Case of Negative Edge Weight [duplicate]

I came across a serious doubt regarding the implementation of the Dijkstra algorithm and hence wanted to discuss. For the given below graph What should be the Cost to reach Node G from Source Vertex ...
1
vote
1answer
61 views

Calculate Shortest Path (Shortest Time) Through a Store in a Graph

There is an undirected graph and some of the vertices are said to be stores. Person A wants to reach to person B with a present. That means person A has to stop by one of the vertices marked as stores ...
1
vote
1answer
170 views

Find shortest path for a volatile graph

Let me define a volatile graph first: It is an undirected graph in which the weight of each edge varies every time we query it. That is, when we request the weight w(e) of edge e, we obtain an ...
1
vote
0answers
48 views

Finding simple min-weight path between two vertices in graph with negative edge weights

Given a weighted graph (negative weights are allowed) and two vertices $u$ and $v$, can we find the simple min-weight path between $u$ and $v$? There can be a negative cycle on the path from $u$ to $v$...
1
vote
1answer
733 views

Confirmation of alternative correctness proof for Floyd-Warshall's all-pair shortest-path algorithm

The most common proof for Floyd-Warshall's algorithm is an induction proof on the outer-most loop, which says $\delta^k(i,j)=\begin{cases} \min\{\delta^{k-1}(i,j),\delta^{k-1}(i,k)+\delta^{k-1}(k,j)\}...
1
vote
0answers
122 views

Minimizing cost of shortest paths to a group of vertices by adding minimal edges to an unconnected vertex

Let $G=(V,E)$ be directed graph, where the weights of the edges are non-negative. The graph might have cycles, but without parallel edges. Consider a $T \subset V$, and $u \notin V$. I'm trying to ...
1
vote
2answers
317 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 ...
5
votes
1answer
269 views

Shortest path when allowed to reverse an edge

We're given an unweighted directed graph with vertices $V$ and edges $E$. We're trying to find the shortest path from $s$ to $t$ but we're allowed to travel along up to one edge in the ...
2
votes
0answers
110 views

Shortest route through ordered points

My algorithm-fu is really weak and I do not know how to express following problem in terms of any other problem known to me: Given a small rectilinear grid and coordinates of four cells in this grid (...