This task has been clarified. Its programming examples are in need of review to ensure that they still fit the requirements of the task.

Dijkstra's algorithm, conceived by Dutch computer scientist Edsger Dijkstra in 1956 and published in 1959, is a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge path costs, producing a shortest path tree.

This algorithm is often used in routing and as a subroutine in other graph algorithms.



For a given source vertex (node) in the graph, the algorithm finds the path with lowest cost (i.e. the shortest path) between that vertex and every other vertex.



For instance

If the vertices of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a direct road,   Dijkstra's algorithm can be used to find the shortest route between one city and all other cities.

As a result, the shortest path first is widely used in network routing protocols, most notably:

   IS-IS   (Intermediate System to Intermediate System)   and 
   OSPF   (Open Shortest Path First).


Important note
 
The inputs to Dijkstra's algorithm are a directed and weighted graph consisting of 2 or more nodes, generally represented by:

   an adjacency matrix or list,   and
   a start node. 


A destination node is not specified.

The output is a set of edges depicting the shortest path to each destination node.



An example, starting with

                                         a──►b,  cost=7,   lastNode=a  
                                         a──►c,  cost=9,   lastNode=a 
                                         a──►d,  cost=NA,  lastNode=a
                                         a──►e,  cost=NA,  lastNode=a 
                                         a──►f,  cost=14,  lastNode=a
 
    The lowest cost is    a──►b    so    a──►b    is added to the output. 
 
    There is a connection from   b──►d   so the input is updated to: 
                                         a──►c,  cost=9,   lastNode=a 
                                         a──►d,  cost=22,  lastNode=b 
                                         a──►e,  cost=NA,  lastNode=a 
                                         a──►f,  cost=14,  lastNode=a
 
    The lowest cost is    a──►c    so    a──►c    is added to the output. 
 
    Paths to    d    and    f    are cheaper via    c    so the input is updated to:
                                         a──►d,  cost=20,  lastNode=c 
                                         a──►e,  cost=NA,  lastNode=a 
                                         a──►f,  cost=11,  lastNode=c
 
    The lowest cost is    a──►f    so    c──►f    is added to the output. 
 
    The input is updated to:
                                         a──►d,  cost=20,  lastNode=c 
                                         a──►e,  cost=NA,  lastNode=a
 
    The lowest cost is    a──►d    so    c──►d    is added to the output. 
 
    There is a connection from    d──►e    so the input is updated to:
                                         a──►e,  cost=26,  lastNode=d
 
    Which just leaves adding    d──►e    to the output.
 
    The output should now be:
                                       [ d──►e
                                         c──►d
                                         c──►f
                                         a──►c
                                         a──►b ]   


Task

 Implement a version of Dijkstra's algorithm that outputs a set of edges depicting the shortest path to each reachable node from an origin.
 Run your program with the following directed graph starting at node    a.
 Write a program which interprets the output from the above and use it to output the shortest path from node    a    to nodes    e    and  f. 

 Vertices


 Number 
 Name


 1 
 a


 2 
 b


 3 
 c


 4 
 d


 5 
 e


 6 
 f


 Edges


 Start 
 End 
 Cost


 a 
 b 
 7


 a 
 c 
 9


 a 
 f 
 14


 b 
 c 
 10


 b 
 d 
 15


 c 
 d 
 11


 c 
 f 
 2


 d 
 e 
 6


 e 
 f 
 9



You can use numbers or names to identify vertices in your program.



See also
 Dijkstra's Algorithm vs. A* Search vs. Concurrent Dijkstra's Algorithm (youtube)


private with Ada.Containers.Ordered_Maps;
generic
   type t_Vertex is (<>);
package Dijkstra is
 
   type t_Graph is limited private;
 
   -- Defining a graph (since limited private, only way to do this is to use the Build function)
   type t_Edge is record
      From, To : t_Vertex;
      Weight   : Positive;
   end record;
   type t_Edges is array (Integer range <>) of t_Edge;
   function Build (Edges : in t_Edges; Oriented : in Boolean := True) return t_Graph;
 
   -- Computing path and distance
   type t_Path is array (Integer range <>) of t_Vertex;
   function Shortest_Path (Graph    : in out t_Graph;
                           From, To : in t_Vertex) return t_Path;
   function Distance      (Graph    : in out t_Graph;
                           From, To : in t_Vertex) return Natural;
 
private
   package Neighbor_Lists is new Ada.Containers.Ordered_Maps (Key_Type => t_Vertex, Element_Type => Positive);
   type t_Vertex_Data is record
      Neighbors : Neighbor_Lists.Map; -- won't be affected after build
      -- Updated each time a function is called with a new source
      Previous  : t_Vertex;
      Distance  : Natural;
   end record;
   type t_Graph is array (t_Vertex) of t_Vertex_Data;
end Dijkstra;