While the underlying implementation (whether adjacency matrix or adjacency list) is hidden behind the graph ADT, these two implementations can have an impact on the efficiency of the resulting program. For Dijkstra’s shortest paths algorithm, two different implementations were given in Section 11.4.1 that provide different ways for determining the next closest vertex at each iteration of the algorithm. The relative costs of these two variants depend on who sparse or dense the graph is. They might also depend on whether the graph is implemented using an adjacency list or adjacency matrix.

Design and implement a study to compare the effects on performance for three variables: (i) the two graph representations (adjacency list and adjacency matrix); (ii) the two implementations for Djikstra’s shortest paths algorithm (searching the table of vertex distances or using a priority queue to track the distances), and (iii) sparse versus dense graphs. Be sure to test your implementations on a variety of graphs that are sufficiently large to generate meaningful times.