Questions tagged [shortest-path]

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

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181 views

Minimum bottleneck path between two vertices in an undirected graph

I have an undirected graph, where the value of the path is the maximum weight among all weights edges included in it. And I want find the path of minimum value between two given vertices in time $O(n ...
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33 views

Algorithms (optimization problem): find collection of objects whose permutation satisfies criteria

I'm putting together a personal list of recipes that I enjoy, and would like to construct an algorithm that parses this recipe database and automatically builds me a meal plan for the week. For N ...
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39 views

calculating a shortest path in a table structure that changes in real time

I have a table that looks like this In table NPC - are AI like characters that move from one point to another. Player - a character that is controlled by the user. In any moment the player ...
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1answer
456 views

Shortest path in a incomplete graph

I know the Dijkstra algorithm to solve the "single source shortest path" problem in a graph. And I've seen people discuss solutions in a dynamic graph where edge/vertices are subject to change. ...
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1answer
56 views

Is the number of shortest paths between every two vertices at most 4*n^3?

In every weighted graph with $n$-vertices with negative weights, with $n > 10$, a weight can't appear $n$-times in graph, there are between every two vertices at most $4n^3$ shortest paths. I'm ...
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72 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 ...
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1answer
62 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 ...
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117 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 ...
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477 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 ...
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1answer
2k 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 ...
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1answer
187 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 ...
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185 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 ...
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2k 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)\}...
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84 views

How do I solve a maze using velocity and acceleration?

I need to solve a maze. I can move an object around in a 2D maze by giving it 1 acceleration in any 8 of its 8 directions. The object starts at rest at 0 velocity. By only adding an acceleration at ...
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1answer
194 views

Recursive DP vs Graph Traversal solutions to path-based problems

I am studying some algorithms interview questions and I am seeing many path-based questions like "if a robot is at the top left of a grid and can only move down or to the right, how many paths can ...
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1answer
128 views

Shortest path in a graph where edges are forbidden depending on the path taken

I have a problem similar to Shortest path problem where edge weight depends on path taken but not quite the same. In my case each edge has either a fixed finite weight, or an infinite weight, ...
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505 views

Dynamic All Pairs Shortest Paths algorithm

I heard about the following problem in a competitive programming camp: Given an undirected weighted graph $G$ with one vertex initially. Suppose you are given two types of queries: Add a new vertex ...
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1answer
442 views

weight constrained shortest path problem variants

Given a graph $G=(V,E)$, metric spaces $\delta:E\rightarrow \mathbb{Z}^{+}$ and $w:E\rightarrow \mathbb{Z}^{+}$, terminal vertices $s,t\in V$, do there exists $s\rightarrow t$ path $P=(V_{p},E_{p})$ ...
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488 views

How to find the minimal path cost from left edge of a grid to the right edge using dynamic programming?

I need to find the minimal path cost from left edge of a $n\times n$ grid to the right edge where each node has some non-negative weight $cost(i,j)$. $i$ represents horizontal coordinates while $j$ ...
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1answer
388 views

Find minimum time path between two nodes

I am trying to write an algorithm for finding best path for an electric vehicle to navigate through network of chargers. A graph with Vertices representing charges and Edges representing distances ...
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1answer
77 views

Movement on Labyrinth with Best First Search

I have the following labyrinth where R is the robot(the parent node), red blocks are the obstacles where the R cannot move and <...
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1answer
310 views

Shortest path tree from each vertex implies a unique MST?

Given a connected, undirected graph G, edge-weighted (positive), prove that If G has a spanning tree T which, for each vertex r in G, is a shortest path tree from r, then G has a unique MST. I know ...
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2answers
245 views

Sum of all distances in connected DAG in $O(n\log n)$

I have a DAG with $n$ nodes and $n-1$ edges. The edges in DAG (fixed) are defined as follows: For every node $i$, $1 \le i \le n-1$ is connected to node $i+1$. The lengths of the $n-1$ edges are ...
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65 views

Shortest past including an edge from a given set

I am working on the following problem. There are N vertices and M roads connecting them. Some of the roads are broken. I have to go from vertex 1 to vertex N taking at least one good road and find ...
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1answer
881 views

Correctness of Dijkstra's algorithm

This question is about the correctness proof of Dijkstra's algorithm in the third edition of Introduction to Algorithms by Cormen et al. (pages 660–661). The proof makes a case that considering path $...
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2answers
555 views

modify Dijkstra's algorithm to compute shortest path only for the vertex which is no more than three edges away from the start vertex

i want to modify Dijkstra algorithm to compute shortest path only for the vertex which is no more than three edges away from the start vertex I tried it with BFS(breadth first search). Initially ...
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949 views

Algorithm to find a path connecting given nodes in a graph

Suppose I have $n$ nodes in a graph and I identify $x$ nodes in the graph (where $x < n$). I would like to find a path to connect all those $x$ nodes I have identified. Is there any algorithm for ...
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1k views

Shortest-Path for Weighted Directed Bipartite Graphs

I did a research project in which I seek to move a car through zones from origin to destination. This allows for the formulation of a bipartite graph because only adjacent zones can be connected. Each ...
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1answer
890 views

Shortest walk that covers $k$ nodes

I have the following problem, I would like to know an efficient algorithm to solve it. Suppose I have a weighted graph $G$ and a set of vertices $K$, I want to find a walk which starts at a vertex $s$...
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2answers
456 views

Using Dijkstra to find shortest path in relation to two weight functions?

I'm given a graph and two weight functions, $w_1$ and $w_2$, such that there doesn't exist a negative loop in the graph in $w_1$ and $w_2$. I'm also given two vertices, $s$ and $t$, and am asked to ...
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2answers
975 views

Finding all paths with lengths in a fixed interval in sparse graphs

What is the most efficient way to find all paths of length M to N in a large sparse graph? Some general information: Graph has 30,000 to 50,000 nodes Average number of edges per node ~ 10 M=4, N=7 ...
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1answer
601 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 ...
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16 views

Can the Bellman-Ford Algorithm be used to find the longest path in an undirected graph through first negating the weight of all the edges? [duplicate]

I understand that the Bellman-Ford Algorithm can solve the single-source shortest-paths problem. However, can it also be used to determine the longest path in an undirected, graph through first ...
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21 views

How do i make sure i get the correct Bellman Ford path?

I was studying shortest path algorithms and was met with an issue regarding Bellman Ford for the image below. Following the graph, i see that node 3 has a length of 1 while node 2 has a length of 2. ...
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0answers
27 views

Dijkstra's algorithm - additional properties

Say we let $R$ denote the set of currently chosen vertices in Dijkstra's algorithm, $d$ be the currently stored path-length estimates, and $s$ be the source. The standard property that we know is true ...
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45 views

Proving that every graph has an order such that Bellman Ford can run in one iteration

I need to prove that for every given graph, that doesn't contain negative cycles, there is an order of edges so that Bellman-Ford algorithm will finish running after one iteration. I could only solve ...
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0answers
20 views

Solving multiple pathfinding problems efficiently

Let $V$ be a set of nodes, $c : V \times V \rightarrow \mathbb{R} \cup \{\infty\}$ be an edge cost function, and $h : V \times V \rightarrow \mathbb{R}$ be an admissible heuristic. Suppose we want to ...
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28 views

Finding homotopies in a 2-complex

Are there any efficient algorithms to find the shortest homotopy between two paths in a $2$-complex?
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45 views

How to get the shortest simple path in a directed Graph with an additional constraint that it needs to use two arcs in the said path

I have a directed graph that has positive weights (but there are reverse arcs) and I am trying to find the shortest path between a given source, s and a given sink, ...
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235 views

Johnson's vs Floyd-Warshall for dense graphs

I often read that Floyd-Warshall is a good fit for dense graphs and Johnson's for sparse ones. While it's easy to see why Johnson outperforms Floyd-Warshall on sparse graphs, I'm noticing that ...
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16 views

Queries on unbounded knapsack

Given $n$ types of items with integer cost $c_{i}$ (there is an unlimited number of items of each type), such that $c_{i} \leq c$ for all $i = 1, 2, \dots, n$, answer (a lot of) queries of form "is ...
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60 views

A pathfinding algorithm for graphs in which arc weights can change over time

So I'm not really sure even what to be googling for solutions to this. Hence this question, hopefully, someone can point me in the right direction. Here's the situation, I have a weighted undirected ...
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1answer
30 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, ...
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424 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.
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156 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. ...
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94 views

Can the shortest path problem be solved using Monte Carlo Tree Search?

I think Monte Carlo Tree Search could be used to find the shortest path, but it seems that this method is only used considering win/lose outcomes in the simulation step. If we consider the path ...
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290 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, ...
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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 ...
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48 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 ...
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263 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 $...

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