Shortest Path in a Graph: Difference between revisions

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=Internal=
=Internal=
* [[Graphs#Subjects|Graphs]]
* [[Graphs#Graph_Algorithms|Graphs]]
* [[Graph_Search#Breadth-First_Search_.28BFS.29|Graph Search | Breadth-First Search]]
* [[Graph_Search#Breadth-First_Search_.28BFS.29|Graph Search | Breadth-First Search]]
=Overview=
=Overview=
There are several algorithms that compute the shortest path between two vertices in a graph, and they can be used or not depending on the characteristics of the graph, such as whether is directed or undirected, the edges have weights, the weights are. negative or not.  
There are several algorithms that compute the shortest path between two vertices in a graph, and they can be used or not depending on the characteristics of the graph, such as whether is directed or undirected, the edges have weights, the weights are negative or not.
=The Problem=
 
=Shortest Path Algorithms=
=Shortest Path Algorithms=
* [[Breadth-First Search-based Shortest Path Algorithm]]
* [[Breadth-First Search-based Shortest Path Algorithm]]
 
* [[Dijkstra's Shortest-Path Algorithm]]
=TODO=
* [[Bellman-Ford Shortest-Path Algorithm]]
<font color=darkgray>Reshape this page to accommodate [[Dijkstra's Algorithm]].</font>
* [[Shortest Path with Bidirectional Search]]
 
=<span id='All-Pairs_Shortest_Path_Algorithms'></span>All-Pairs Shortest Path Algorithms (APSP)=
=Algorithm=
* https://www.coursera.org/learn/algorithms-npcomplete/lecture/VQStd/problem-definition
 
* [[Floyd-Warshall Algorithm]]
The algorithm is (differences to the [[Graph_Search#BFS_Algorithm_Pseudocode|canonical BFS]] algorithm are emphasized):
* [[Johnson's Algorithm]]
<font size=-1>
BFS_with_Shortest_Path(graph G, start vertex s)
  <font color=teal># All nodes are assumed unexplored</font>
  <font color=SlateGray>initialize a Queue Q (FIFO)
  mark s as explored</font>
  annotate s with distance 0
  <font color=SlateGray>place s in Q
  while Q has elements
    remove the head of the queue v
    for each edge (v, w):
      if w unexplored:
        mark w as explored</font>
        annotate w with a distance dist(w) = dist(v) + 1
        <font color=SlateGray>add w to Q</font>
</font>
 
The distance computed on reachable node gives the "layer" and the distance from the start node s.
 
:::[[File:BFS_Layers.png|160px]]

Latest revision as of 23:16, 24 November 2021

External

Internal

Overview

There are several algorithms that compute the shortest path between two vertices in a graph, and they can be used or not depending on the characteristics of the graph, such as whether is directed or undirected, the edges have weights, the weights are negative or not.

Shortest Path Algorithms

All-Pairs Shortest Path Algorithms (APSP)

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