Print the adjacency list for the full graph Print the edge list and adjacency list for the MST
The problem to solve is using a priority queue to compute a minimum spanning
tree.
Given a fully connected undirected graph where each edge has a weight, find the
set of edges with the least total sum of weights.
You are a civil engineer and you have been tasked with trying to find out the
lowest cost way to build internet access in CIS-Land. There are X (3 ≤ X ≤ 100,000)
towns in CIS-Land connected by Y (X ≤ Y ≤ 100,000) roads and you can travel
between any two towns in CIS-Land by travelling some sequence of roads. With a
limited budget you also know that the cheapest approach to obtain internet
access is to install fiber-optic cables along existing roadways. Fortunately, you
know the costs of laying fiber-optic cable down along all the roads, and you will
be able to cost out how much money CIS-Land will need to spend to successfully
complete the internet access project – that is, every town will be connected along
some sequence of fiber-optic cables. Good thing, you are also CIS-Land’s brightest
computer science professional, and you remember learning about an
Prim’s specifically, in one of your UM-D programming classes. This algorithm
turns out to be exactly what you need to solve this problem, but to implement
the algorithm you will need to use a priority queue.You would ask, why would I use a priority queue in Prim’s algorithm? The reason
why people use it is because it significantly speeds up the runtime of the
algorithm. It turns from O(V2 + E) to O(E*log(V)).
Basically, at each step in Prim’s algorithm, you're looking for the minimum edge
with one vertex in the partial minimum spanning tree, and one vertex not in the
tree, and you're going to add that edge to the tree. How do you do that
efficiently? If you have a way to efficiently order all the edges connected to a vertex in your partial spanning tree, you can simply iterate through them until you
find an edge with an acceptable vertex.
Without such an ordered data structure, you'd have to iterate through all
candidate edges each time to find the minimum, rather than being able to
efficiently grab the minimum directly.
The input data describing the graph will be the nodes and associated list of edges
(roads and their fiber-optic cost). The program will need to covert that input into
to an adjacency list: for every node in the graph (town in CIS-Land), there will be
list of the nodes (towns) it’s connected to and the weight (cost of building fiber-
optic cable along).
adj[0] → (1, 1) (3, 3)
adj[1] → (0, 1) (2, 6) (3, 5) (4, 1)
. . .
First, the adjacency list could be represented as a list of
node. For the priority queue, you may use any built-in library routines to implement the
Priority Queue or Adaptable Priority Queue.
Input Format
Input and output file names are to be entered by the user.
Line 1: Two space-separated integers: X , the number of nodes in the graph, and Y
, the number of edges.
Lines 2 . . . Y+1 :
Line i contains three space-separated numbers describing an edge:
si and ti , the IDs of the two nodes involved, and wi , the weight of the edge. Sample input (input file name format is CIS-Land#.dat where this is file CIS-Land1.dat)
6 9
0 1 1
1 3 5
3 0 3
3 4 1
1 4 1
1 2 6
5 2 2
2 4 4
5 4 4
Line 1: 6 nodes (0 – 5) and 9 edges to follow line 2 - 10
Line 2: 0 1 1 describes edge between 0 and 1 of weight 1
Line 3: 1 3 5 describes edge between 1 and 3 of weight 5
...
The MST sum is 1+1+1+4+2 = 9
Output to Screen and File (output file name format: MST#.out)
Echo print the input (format appropriately per display messages in program)
Print the adjacency list for the full graph
Print the edge list and adjacency list for the MST
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
The problem to solve is using a priority queue to compute a minimum spanning
tree.
Given a fully connected undirected graph where each edge has a weight, find the
set of edges with the least total sum of weights.
You are a civil engineer and you have been tasked with trying to find out the
lowest cost way to build internet access in CIS-Land. There are X (3 ≤ X ≤ 100,000)
towns in CIS-Land connected by Y (X ≤ Y ≤ 100,000) roads and you can travel
between any two towns in CIS-Land by travelling some sequence of roads. With a
limited budget you also know that the cheapest approach to obtain internet
access is to install fiber-optic cables along existing roadways. Fortunately, you
know the costs of laying fiber-optic cable down along all the roads, and you will
be able to cost out how much money CIS-Land will need to spend to successfully
complete the internet access project – that is, every town will be connected along
some sequence of fiber-optic cables. Good thing, you are also CIS-Land’s brightest
computer science professional, and you remember learning about an algorithm,
Prim’s specifically, in one of your UM-D programming classes. This algorithm
turns out to be exactly what you need to solve this problem, but to implement
the algorithm you will need to use a priority queue.You would ask, why would I use a priority queue in Prim’s algorithm? The reason
why people use it is because it significantly speeds up the runtime of the
algorithm. It turns from O(V2 + E) to O(E*log(V)).
Basically, at each step in Prim’s algorithm, you're looking for the minimum edge
with one vertex in the partial minimum spanning tree, and one vertex not in the
tree, and you're going to add that edge to the tree. How do you do that
efficiently? If you have a way to efficiently order all the edges connected to a vertex in your partial spanning tree, you can simply iterate through them until you
find an edge with an acceptable vertex.
Without such an ordered data structure, you'd have to iterate through all
candidate edges each time to find the minimum, rather than being able to
efficiently grab the minimum directly.
The input data describing the graph will be the nodes and associated list of edges
(roads and their fiber-optic cost). The program will need to covert that input into
to an adjacency list: for every node in the graph (town in CIS-Land), there will be
list of the nodes (towns) it’s connected to and the weight (cost of building fiber-
optic cable along).
adj[0] → (1, 1) (3, 3)
adj[1] → (0, 1) (2, 6) (3, 5) (4, 1)
. . .
First, the adjacency list could be represented as a list of vectors, one for each
node. For the priority queue, you may use any built-in library routines to implement the
Priority Queue or Adaptable Priority Queue.
Input Format
Input and output file names are to be entered by the user.
Line 1: Two space-separated integers: X , the number of nodes in the graph, and Y
, the number of edges.
Lines 2 . . . Y+1 :
Line i contains three space-separated numbers describing an edge:
si and ti , the IDs of the two nodes involved, and wi , the weight of the edge. Sample input (input file name format is CIS-Land#.dat where this is file CIS-Land1.dat)
6 9
0 1 1
1 3 5
3 0 3
3 4 1
1 4 1
1 2 6
5 2 2
2 4 4
5 4 4
Line 1: 6 nodes (0 – 5) and 9 edges to follow line 2 - 10
Line 2: 0 1 1 describes edge between 0 and 1 of weight 1
Line 3: 1 3 5 describes edge between 1 and 3 of weight 5
...
The MST sum is 1+1+1+4+2 = 9
Output to Screen and File (output file name format: MST#.out)
Echo print the input (format appropriately per display messages in program)
Print the adjacency list for the full graph
Print the edge list and adjacency list for the MST
by Bartleby Expert
