How to resolve the algorithm Maze solving step by step in the Julia programming language
How to resolve the algorithm Maze solving step by step in the Julia programming language
Table of Contents
Problem Statement
For a maze generated by this task, write a function that finds (and displays) the shortest path between two cells.
Note that because these mazes are generated by the Depth-first search algorithm, they contain no circular paths, and a simple depth-first tree search can be used.
Let's start with the solution:
Step by Step solution about How to resolve the algorithm Maze solving step by step in the Julia programming language
This code applies Dijkstra's algorithm to find the shortest path from a given source node to all other nodes in a given graph. Dijkstra's algorithm is a greedy algorithm that iteratively builds a tree of shortest paths from the source to all other nodes.
Here's a detailed explanation of the functions involved in the code:
dijkstra function: This function takes two arguments:
- A graph represented as an adjacency matrix.
- An optional source node (defaulting to 1).
It returns a tuple containing two dictionaries:
- The first dictionary,
dist, maps each node in the graph to its minimum distance from the source node. - The second dictionary,
prev, maps each node in the graph to its previous node in the shortest path from the source node.
Inner function _minimumdist: This function is a helper function that finds the node in the given set of nodes with the smallest distance from the source node. It returns both the node with the smallest distance and the minimum distance itself.
printpath function: This function takes two arguments:
- A dictionary mapping nodes to their previous nodes in the shortest path from the source node.
- A target node.
It prints the shortest path from the source node to the target node by recursively following the prev dictionary.
Example usage:
The code also includes an example of how to use the dijkstra and printpath functions to find and print the shortest path from node 1 to node 6 in a given graph.
The code first defines a graph as an adjacency matrix. Then, it calls the dijkstra function to find the shortest path from node 1 to all other nodes in the graph. The resulting distances and previous nodes are stored in the dist and prev dictionaries, respectively.
Finally, the code calls the printpath function to print the shortest path from node 1 to node 6.
Overall, this code snippet provides a detailed implementation of Dijkstra's algorithm for finding shortest paths in a graph, along with an example of how to use the algorithm in practice.
Source code in the julia programming language
"""
+ +---+---+
| 1 2 3 |
+---+ + +
| 4 5 | 6
+---+---+---+
julia> const graph = [
0 1 0 0 0 0;
1 0 1 0 1 0;
0 1 0 0 0 1;
0 0 0 0 1 0;
0 1 0 1 0 0;
0 0 1 0 0 0]
julia> dist, path = dijkstra(graph, 1)
(Dict(4=>3,2=>1,3=>2,5=>2,6=>3,1=>0), Dict(4=>5,2=>1,3=>2,5=>2,6=>3,1=>0))
julia> printpath(path, 6) # Display solution of the maze
1 -> 2 -> 3 -> 6
"""
function dijkstra(graph, source::Int=1)
# ensure that the adjacency matrix is squared
@assert size(graph, 1) == size(graph, 2)
inf = typemax(Int64)
n = size(graph, 1)
Q = IntSet(1:n) # Set of unvisited nodes
dist = Dict(n => inf for n in Q) # Unknown distance function from source to v
prev = Dict(n => 0 for n in Q) # Previous node in optimal path from source
dist[source] = 0 # Distance from source to source
function _minimumdist(nodes) # Find the less distant node among nodes
kmin, vmin = nothing, inf
for (k, v) in dist
if k ∈ nodes && v ≤ vmin
kmin, vmin = k, v
end
end
return kmin
end
# Until all nodes are visited...
while !isempty(Q)
u = _minimumdist(Q) # Vertex in Q with smallest dist[]
pop!(Q, u)
if dist[u] == inf break end # All remaining vertices are inaccessible from source
for v in 1:n # Each neighbor v of u
if graph[u, v] != 0 && v ∈ Q # where v has not yet been visited
alt = dist[u] + graph[u, v]
if alt < dist[v] # Relax (u, v, a)
dist[v] = alt
prev[v] = u
end
end
end
end
return dist, prev
end
function printpath(prev::Dict, target::Int)
path = "$target"
while prev[target] != 0
target = prev[target]
path = "$target -> " * path
end
println(path)
end
const graph = [
0 1 0 0 0 0;
1 0 1 0 1 0;
0 1 0 0 0 1;
0 0 0 0 1 0;
0 1 0 1 0 0;
0 0 1 0 0 0]
dist, path = dijkstra(graph)
printpath(path, 6)
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