Questions about graphs in which every edge is associated with a weight.

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Modification of Dijkstra's algorithm

Let $G=(V,E)$ be a weighted, directed graph with non negative weight function $w:E \to \{1,2,....,W\}$ for some non negative integer $W$. Modify Dijkstra’s algorithm to compute the shortest paths from ...
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0answers
31 views

Applications of min spanning trees

What are the significant applications of minimum spanning trees? After doing some research online and in several textbooks, I have found three real-world applications: Building a connected network. ...
2
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1answer
33 views

Choice of algorithm for hierarchical clustering for minimizing network communication costs

Suppose I have a large distributed task running on a cluster system where part of the workload is compute bound and part depends on network performance. Data transfer is not completely homogeneous ...
2
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0answers
37 views

What is a semantic cognitive map

Based on: J. P. Carvalho, "On the Semantics and the Use of Fuzzy Cognitive Maps in Social Sciences" (WCCI, 2010 -- PDF) and Richard Dagan's web page Cognitive Mapping. A cognitive map consists of ...
5
votes
2answers
156 views

Adding a node between two others, minimizing its maximum distance to any other node

We are given an undirected graph weighted with positive arc lengths and a distinguished edge $(a,b)$ in the graph. The problem is to replace this edge by two edges $(a,c)$ and $(c,b)$ where $c$ is a ...
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2answers
126 views

Maximum length path between any two nodes in a tree with possible negative edge weights

Is there an efficient way to find the longest path between any two nodes in a tree. Given that edges can have negative weights. I know about the diameter problem which finds the longest path in a ...
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0answers
37 views

Proof of shortest-paths optimality conditions

I am struggling with understanding the proof of shortest-paths optimality conditions. Let $G$ be an edge-weighted digraph. Then values in $distTo[]$ are the shortest path distances from $s$ iff: ...
2
votes
1answer
39 views

Is traversing an unconnected graph possible?

I have been assigned a fun project: design and implement a program that maintains the data of a simple social network. Each person in the network should have a profile that contains his name, current ...
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1answer
49 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 ...
1
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1answer
78 views

Shortest directed path connecting given subset of vertices

Given weighted directed graph $G = (V,E,w)$, where $w : E \to \mathbb R^+$ source vertex $v \in V$ vertex subset $U \subset V$ how to find a shortest directed path from $v$ containing all vertices ...
1
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0answers
100 views

Algorithm to determine a minimal cost graph [closed]

I'm trying to solve this problem: Given a collection of cities and the number of commuters between cities, design a network of roads for minimal cost where cost includes the cost of building the ...
14
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1answer
918 views

Why does Dijkstra's algorithm fail on a negative weighted graphs?

I know this is probably very basic, I just can't wrap my head around it. We recently studied about Dijkstra's algorithm for finding the shortest path between two vertices on a weighted graph. My ...
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2answers
360 views

Dijkstra's algorithm for edge weights in range 0, …, W

Suppose I want to run Dijkstra's algorithm on a graph whose edge weights are integers in the range 0, ..., W, where W is a relatively small number. How can I modify that algorithm so that it takes ...
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1answer
253 views

Calculating the number of non-intersecting routes in an Euclidean graph

I have an 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 found a geometric proof that every optimal TSP solution ...
2
votes
0answers
67 views

Is there a relationship between graph entropy and node entropy?

Eagle, et al [1] discuss the notion of node entropy and this is captured in igraph via the diversity metric. I was wondering if there was any relationship between these node entropies and the idea of ...
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1answer
180 views

Efficient way to find intersections

I have an 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 randomly creating a path thru all the vertices and I ...
0
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0answers
67 views

Trim graph to minimum

I have an Euclidean graph: each vertex is a point on the 2D plane, so the weight of each edge is the Euclidean distance between the vertices, also all the vertices are connected with edges. I want to ...
1
vote
1answer
127 views

Converting graphs to sets of paths

I have an Euclidean, undirected graph: each vertex is a point on the 2D plane, so the weight of each edge is the Euclidean distance between the vertices. The number of vertices with no edges is ...
11
votes
2answers
767 views

Shortest non intersecting path for a graph embedded in a euclidean plane (2D)

What algorithm would you use to find the shortest path of a graph, which is embedded in an euclidean plane, such that the path should not contain any self-intersections (in the embedding)? For ...
2
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1answer
2k views

A* to find the longest path in a directed cyclic graph

I have written an A* algorithm to find the shortest path through a directed cyclic graph. I am trying to modify it to find the longest path through the same graph. My attempt was to write it so that ...
1
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2answers
1k 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 ...
5
votes
1answer
196 views

Dividing a weighted planar graph into $k$ subgraphs with balanced weight

I've been looking for an algorithm which divides an undirected, weighted, planar and simple graph into $k$ disjoint subgraphs. Here, the graph is sparse, $k$ is fixed, and there are no negative edge ...
2
votes
0answers
270 views

Effect of increasing the capacity of an edge in a flow network with known max flow

I need your help with an exercise on Ford-Fulkerson. Suppose you are given a flow network with capacities $(G,s,t)$ and you are also given the max flow $|f|$ in advance. Now suppose you are ...
2
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0answers
58 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 ...
5
votes
2answers
1k 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 ...
7
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1answer
605 views

Minimum s-t cut in weighted directed acyclic graphs with possibly negative weights

I ran into the following problem: Given a directed acyclic graph with real-valued edge weights, and two vertices s and t, compute the minimum s-t cut. For general graphs this is NP-hard, since one ...
5
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2answers
116 views

An edge that connects more than two nodes in a graph?

Is there a way to create a single edge on a graph that connects 3 or more nodes? For example, let's say that the probability of Y occurring after X is 0.1, and the probability of Z occurring after Y ...
8
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1answer
1k views

Do the minimum spanning trees of a weighted graph have the same number of edges with a given weight?

If a weighted graph $G$ has two different minimum spanning trees $T_1 = (V_1, E_1)$ and $T_2 = (V_2, E_2)$, then is it true that for any edge $e$ in $E_1$, the number of edges in $E_1$ with the same ...