Description
Dijkstra’s algorithm
Implement the classic Dijkstra’s shortest path algorithm and optimize it for maps. Such
algorithms are widely used in geographic information systems (GIS) including MapQuest
and GPS-based car navigation systems.
In this project you will use Dijkstra’s algorithm to find route information between two cities
chosen by the user. The user will be shown a route that results in the lowest price.
Maps. For this assignment we will be working with maps, or graphs whose vertices are
points in the plane and are connected by edges whose weights are Euclidean distances.
Think of the vertices as cities and the edges as roads connected to them. To represent a
map in a file, we list the number of vertices and edges, then list the vertices (index followed
by its x and y coordinates), then list the edges (pairs of vertices. For example, represents
the map below:
6 9
City1 1000 2400
City2 2800 3000
City3 2400 2500
City4 4000 0
City5 4500 3800
City6 6000 1500
City1 City2
City1 City4
City2 City3
City2 City5
City3 City5
City3 City4
City3 City6
City4 City6
City5 City6
Dijkstra’s algorithm. Dijkstra’s algorithm is a classic solution to the shortest path problem
on a weighted graph. The basic idea is not difficult to understand. We maintain, for every
vertex in the graph, the length of the shortest known path from the source to that vertex,
and we maintain these lengths in a priority queue. Initially, we put all the vertices on the
queue with an artificially high priority and then assign priority 0.0 to the source. The
algorithm proceeds by taking the lowest-priority vertex off the PQ, then checking all the
vertices that can be reached from that vertex by one edge to see whether that edge gives a
shorter path to the vertex from the source than the shortest previously-known path. If so, it
lowers the priority to reflect this new information.
This method computes the length of the shortest path. To keep track of the path, we also
maintain for each vertex, its predecessor on the shortest path from the source to that vertex.
Your goal. Optimize Dijkstra’s algorithm so that it can process thousands of shortest path
queries for a given map. Once you read in (and optionally preprocess) the map, your
program should solve shortest path problems in sublinear time. One method would be to
precompute the shortest path for all pairs of vertices; however, you cannot afford the
quadratic space required to store all of this information. Your goal is to reduce the amount
of work involved per shortest path computation, without using excessive space.
Idea. The naive implementation of Dijkstra’s algorithm examines all V vertices in the
graph. An obvious strategy to reduce the number of vertices examined is to stop the search
as soon as you discover the shortest path to the destination. With this approach, you can
make the running time per shortest path query proportional to E’ log V’ where E’ and V’
are the number of edges and vertices examined by Dijkstra’s algorithm. However, this
requires some care because just re-initializing all of the distances to ∞ would take time
proportional to V. Since you are doing repeated queries, you can speed things up
dramatically by only re-initializing those values that changed in the previous query.
Input : Palestinian Cities
Map: Palestinian Map
Output: Show the route on the map
Example for interface:
Notes:
1. You have to choose the city through mouse or keyboard.
2. The path should appear on the map.
3. Your project should include at least 50 cities.
