I have interval data:
A = (0,50)
B = (20,500)
C = (80,420)
....
And realized that there's an associated graph with this data, the interval graph
I'd like to find the most efficient path to go from A to G (assume I know all of the positive vertex weights, wa, wb, wc...). I need to start at A and go to G, so the minimum spanning tree must be bound between these points. One of the constraints in our application is that the interval starting at A and ending at G must be covered in full (no gaps). I'm looking at networkX's minspanning tree method, and don't understand how to specify that A and G must be the start and endpoints.
Some other questions that come to mind are:
Since this problem is NP-hard, should I even bother looking for a min-spanning tree if the number of nodes is high? How many nodes would be too many?
Notice that interval F has a unique region. In other words, to completely cover the interval A-G, one HAS to go through F. Therefore, my minimum spanning tree probably should only connect A-F, not A-G. Is there a standard way, given a larger graph, to find all of the subgraphs whose intervals contain no unique patches? In other words, since all paths have to pass through F to get to G, A-F is the min spanning path of interest, not A-G. How does one reduce a graph in such a way without inspecting it manually?
Becasue I have to go from A-G, I would never go backwards or take a cyclic path. For example, I'd never go A-B-A. Do spanning trees incorporate this? And does this make my graph directed? Consider point C: from C one could go to D, E, or F, but never back to A (for our use case). What does this mean in regard to directionality of the graph?
Sorry for novice Q's, new to most of this.
If you must go from A to G in an efficient way, you aren't looking for a minimum spanning tree algorithm. A simple shortest path algorithm is enough. You just have to adapt you graph to put the weights in the edges instead of the nodes. But it's just a matter of setting the node's weight to the incoming edge.
Also, both shortest path and minimum spanning tree problems aren't NP-hard. There are known polynomial algorithms for all these problems. In special, shortest path can be solved by Dijkstra's algorithm (if your graph doesn't have negative edges, which seems to be true) and minimum spanning tree can be solved by Prim's or Kruskal's algorithm.
And finally, any tree, by definition doesn't have cycles.
As mentioned in another answer, Dijkstra's algorithm is the solution. What wasn't mentioned is how to implement that solution in networkx. Here it is. Simple as this:
import networkx as nx
my_graph = nx.Graph()
my_graph.add_edges_from([('A','B'),('B','C'),('A','C'),('C','D'),('A','D'),('C','E'),('D','E'),('D','F'),('F','G')])
#graph is now defined.
shortestpath = nx.dijkstra_path(my_graph, 'A', 'G') #optional weight argument here.
shortestpath
> ['A', 'D', 'F', 'G']
In general, more documentation on how to do the shortest path algorithms (and there are many variations thereof) in networkx is here.
Note if you have weights on nodes and you want to minimize the sum of the nodes in the path, what you do is place weights on the edges so that the weight of (u, v) is (w[u]+w[v])/2.
Then run nx.dijkstra_path with the optional argument telling networkx where to find the weight of the edges. The weight of the entire path will equal the sum of the intermediate weights, plus half the values of the end nodes. You can then correct for the end node weights.
Related
It is easy to prove that if P is a shortest path between u and v, then every subpath is also a shortest path.
Given a connected Graph, I want to preprocess a the shortest path between every pair of nodes in a Matrix, such that:
Path[u,v] = Path[v,u]
If x,y in Path[u,v] then Path[x,y] is a subpath of Path[u,v].
I can not figure out an algorithm or a prove and actually I do not know if this is posible.
Any idea is welcome.Thank you.
You can only get (1) if you are working with undirected graphs OR if it is guaranteed that the weight of the arc (a, b) is equal to the weight of the arc (b, a) for all arcs in your graph.
The problem you describe sounds like the all-pairs shortest path problem: for each pair of nodes in a connected graph, find the shortest paths between nodes in the pair. The Floyd-Warshall algorithm can be used to find the lengths of paths and it is straightforward to reconstruct the shortest paths from there.
This algorithm further requires that there are no negative cycles (otherwise a shorter path could always be obtained by running through that cycle again) but that requirement seems reasonable.
To guarantee property (2), you need to make sure when reconstructing paths you are reconstructing "canonical" paths whenever more than one shortest path may be possible. To do this, impose an ordering on the vertices and always test candidate nodes in ascending order, always preferring the lowest-ordered node which maintains the shortest-path property.
Wikipedia has a fairly good write-up.
What is the exact difference between Dijkstra's and Prim's algorithms? I know Prim's will give a MST but the tree generated by Dijkstra will also be a MST. Then what is the exact difference?
Prim's algorithm constructs a minimum spanning tree for the graph, which is a tree that connects all nodes in the graph and has the least total cost among all trees that connect all the nodes. However, the length of a path between any two nodes in the MST might not be the shortest path between those two nodes in the original graph. MSTs are useful, for example, if you wanted to physically wire up the nodes in the graph to provide electricity to them at the least total cost. It doesn't matter that the path length between two nodes might not be optimal, since all you care about is the fact that they're connected.
Dijkstra's algorithm constructs a shortest path tree starting from some source node. A shortest path tree is a tree that connects all nodes in the graph back to the source node and has the property that the length of any path from the source node to any other node in the graph is minimized. This is useful, for example, if you wanted to build a road network that made it as efficient as possible for everyone to get to some major important landmark. However, the shortest path tree is not guaranteed to be a minimum spanning tree, and the sum of the costs on the edges of a shortest-path tree can be much larger than the cost of an MST.
Another important difference concerns what types of graphs the algorithms work on. Prim's algorithm works on undirected graphs only, since the concept of an MST assumes that graphs are inherently undirected. (There is something called a "minimum spanning arborescence" for directed graphs, but algorithms to find them are much more complicated). Dijkstra's algorithm will work fine on directed graphs, since shortest path trees can indeed be directed. Additionally, Dijkstra's algorithm does not necessarily yield the correct solution in graphs containing negative edge weights, while Prim's algorithm can handle this.
Dijkstra's algorithm doesn't create a MST, it finds the shortest path.
Consider this graph
5 5
s *-----*-----* t
\ /
-------
9
The shortest path is 9, while the MST is a different 'path' at 10.
Prim and Dijkstra algorithms are almost the same, except for the "relax function".
Prim:
MST-PRIM (G, w, r) {
for each key ∈ G.V
u.key = ∞
u.parent = NIL
r.key = 0
Q = G.V
while (Q ≠ ø)
u = Extract-Min(Q)
for each v ∈ G.Adj[u]
if (v ∈ Q)
alt = w(u,v) <== relax function, Pay attention here
if alt < v.key
v.parent = u
v.key = alt
}
Dijkstra:
Dijkstra (G, w, r) {
for each key ∈ G.V
u.key = ∞
u.parent = NIL
r.key = 0
Q = G.V
while (Q ≠ ø)
u = Extract-Min(Q)
for each v ∈ G.Adj[u]
if (v ∈ Q)
alt = w(u,v) + u.key <== relax function, Pay attention here
if alt < v.key
v.parent = u
v.key = alt
}
The only difference is pointed out by the arrow, which is the relax function.
The Prim, which searches for the minimum spanning tree, only cares about the minimum of the total edges cover all the vertices. The relax function is alt = w(u,v)
The Dijkstra, which searches for the minimum path length, so it cares about the edge accumulation. The relax function is alt = w(u,v) + u.key
Dijsktra's algorithm finds the minimum distance from node i to all nodes (you specify i). So in return you get the minimum distance tree from node i.
Prims algorithm gets you the minimum spaning tree for a given graph. A tree that connects all nodes while the sum of all costs is the minimum possible.
So with Dijkstra you can go from the selected node to any other with the minimum cost, you don't get this with Prim's
The only difference I see is that Prim's algorithm stores a minimum cost edge whereas Dijkstra's algorithm stores the total cost from a source vertex to the current vertex.
Dijkstra gives you a way from the source node to the destination node such that the cost is minimum. However Prim's algorithm gives you a minimum spanning tree such that all nodes are connected and the total cost is minimum.
In simple words:
So, if you want to deploy a train to connecte several cities, you would use Prim's algo. But if you want to go from one city to other saving as much time as possible, you'd use Dijkstra's algo.
Both can be implemented using exactly same generic algorithm as follows:
Inputs:
G: Graph
s: Starting vertex (any for Prim, source for Dijkstra)
f: a function that takes vertices u and v, returns a number
Generic(G, s, f)
Q = Enqueue all V with key = infinity, parent = null
s.key = 0
While Q is not empty
u = dequeue Q
For each v in adj(u)
if v is in Q and v.key > f(u,v)
v.key = f(u,v)
v.parent = u
For Prim, pass f = w(u, v) and for Dijkstra pass f = u.key + w(u, v).
Another interesting thing is that above Generic can also implement Breadth First Search (BFS) although it would be overkill because expensive priority queue is not really required. To turn above Generic algorithm in to BFS, pass f = u.key + 1 which is same as enforcing all weights to 1 (i.e. BFS gives minimum number of edges required to traverse from point A to B).
Intuition
Here's one good way to think about above generic algorithm: We start with two buckets A and B. Initially, put all your vertices in B so the bucket A is empty. Then we move one vertex from B to A. Now look at all the edges from vertices in A that crosses over to the vertices in B. We chose the one edge using some criteria from these cross-over edges and move corresponding vertex from B to A. Repeat this process until B is empty.
A brute force way to implement this idea would be to maintain a priority queue of the edges for the vertices in A that crosses over to B. Obviously that would be troublesome if graph was not sparse. So question would be can we instead maintain priority queue of vertices? This in fact we can as our decision finally is which vertex to pick from B.
Historical Context
It's interesting that the generic version of the technique behind both algorithms is conceptually as old as 1930 even when electronic computers weren't around.
The story starts with Otakar Borůvka who needed an algorithm for a family friend trying to figure out how to connect cities in the country of Moravia (now part of the Czech Republic) with minimal cost electric lines. He published his algorithm in 1926 in a mathematics related journal, as Computer Science didn't existed then. This came to the attention to Vojtěch Jarník who thought of an improvement on Borůvka's algorithm and published it in 1930. He in fact discovered the same algorithm that we now know as Prim's algorithm who re-discovered it in 1957.
Independent of all these, in 1956 Dijkstra needed to write a program to demonstrate the capabilities of a new computer his institute had developed. He thought it would be cool to have computer find connections to travel between two cities of the Netherlands. He designed the algorithm in 20 minutes. He created a graph of 64 cities with some simplifications (because his computer was 6-bit) and wrote code for this 1956 computer. However he didn't published his algorithm because primarily there were no computer science journals and he thought this may not be very important. The next year he learned about the problem of connecting terminals of new computers such that the length of wires was minimized. He thought about this problem and re-discovered Jarník/Prim's algorithm which again uses the same technique as the shortest path algorithm he had discovered a year before. He mentioned that both of his algorithms were designed without using pen or paper. In 1959 he published both algorithms in a paper that is just 2 and a half page long.
Dijkstra finds the shortest path between it's beginning node
and every other node. So in return you get the minimum distance tree from beginning node i.e. you can reach every other node as efficiently as possible.
Prims algorithm gets you the MST for a given graph i.e. a tree that connects all nodes while the sum of all costs is the minimum possible.
To make a story short with a realistic example:
Dijkstra wants to know the shortest path to each destination point by saving traveling time and fuel.
Prim wants to know how to efficiently deploy a train rail system i.e. saving material costs.
Directly from Dijkstra's Algorithm's wikipedia article:
The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim's algorithm. Prim's purpose is to find a minimum spanning tree that connects all nodes in the graph; Dijkstra is concerned with only two nodes. Prim's does not evaluate the total weight of the path from the starting node, only the individual path.
Here's what clicked for me: think about which vertex the algorithm takes next:
Prim's algorithm takes next the vertex that's closest to the tree, i.e. closest to some vertex anywhere on the tree.
Dijkstra's algorithm takes next the vertex that is closest to the source.
Source: R. Sedgewick's lecture on Dijkstra's algorithm, Algorithms, Part II: https://coursera.org/share/a551af98e24292b6445c82a2a5f16b18
I was bothered with the same question lately, and I think I might share my understanding...
I think the key difference between these two algorithms (Dijkstra and Prim) roots in the problem they are designed to solve, namely, shortest path between two nodes and minimal spanning tree (MST). The formal is to find the shortest path between say, node s and t, and a rational requirement is to visit each edge of the graph at most once. However, it does NOT require us to visit all the node. The latter (MST) is to get us visit ALL the node (at most once), and with the same rational requirement of visiting each edge at most once too.
That being said, Dijkstra allows us to "take shortcut" so long I can get from s to t, without worrying the consequence - once I get to t, I am done! Although there is also a path from s to t in the MST, but this s-t path is created with considerations of all the rest nodes, therefore, this path can be longer than the s-t path found by the Dijstra's algorithm. Below is a quick example with 3 nodes:
2 2
(s) o ----- o ----- o (t)
| |
-----------------
3
Let's say each of the top edges has the cost of 2, and the bottom edge has cost of 3, then Dijktra will tell us to the take the bottom path, since we don't care about the middle node. On the other hand, Prim will return us a MST with the top 2 edges, discarding the bottom edge.
Such difference is also reflected from the subtle difference in the implementations: in Dijkstra's algorithm, one needs to have a book keeping step (for every node) to update the shortest path from s, after absorbing a new node, whereas in Prim's algorithm, there is no such need.
The simplest explanation is in Prims you don't specify the Starting Node, but in dijsktra you (Need to have a starting node) have to find shortest path from the given node to all other nodes.
The key difference between the basic algorithms lies in their different edge-selection criteria. Generally, they both use a priority queue for selecting next nodes, but have different criteria to select the adjacent nodes of current processing nodes: Prim's Algorithm requires the next adjacent nodes must be also kept in the queue, while Dijkstra's Algorithm does not:
def dijkstra(g, s):
q <- make_priority_queue(VERTEX.distance)
for each vertex v in g.vertex:
v.distance <- infinite
v.predecessor ~> nil
q.add(v)
s.distance <- 0
while not q.is_empty:
u <- q.extract_min()
for each adjacent vertex v of u:
...
def prim(g, s):
q <- make_priority_queue(VERTEX.distance)
for each vertex v in g.vertex:
v.distance <- infinite
v.predecessor ~> nil
q.add(v)
s.distance <- 0
while not q.is_empty:
u <- q.extract_min()
for each adjacent vertex v of u:
if v in q and weight(u, v) < v.distance:// <-------selection--------
...
The calculations of vertex.distance are the second different point.
Dijkstras algorithm is used only to find shortest path.
In Minimum Spanning tree(Prim's or Kruskal's algorithm) you get minimum egdes with minimum edge value.
For example:- Consider a situation where you wan't to create a huge network for which u will be requiring a large number of wires so these counting of wire can be done using Minimum Spanning Tree(Prim's or Kruskal's algorithm) (i.e it will give you minimum number of wires to create huge wired network connection with minimum cost).
Whereas "Dijkstras algorithm" will be used to get the shortest path between two nodes while connecting any nodes with each other.
Dijkstra's algorithm is a single source shortest path problem between node i and j, but Prim's algorithm a minimal spanning tree problem. These algorithm use programming concept named 'greedy algorithm'
If you check these notion, please visit
Greedy algorithm lecture note : http://jeffe.cs.illinois.edu/teaching/algorithms/notes/07-greedy.pdf
Minimum spanning tree : http://jeffe.cs.illinois.edu/teaching/algorithms/notes/20-mst.pdf
Single source shortest path : http://jeffe.cs.illinois.edu/teaching/algorithms/notes/21-sssp.pdf
#templatetypedef has covered difference between MST and shortest path. I've covered the algorithm difference in another So answer by demonstrating that both can be implemented using same generic algorithm that takes one more parameter as input: function f(u,v). The difference between Prim and Dijkstra's algorithm is simply which f(u,v) you use.
At the code level, the other difference is the API.
You initialize Prim with a source vertex, s, i.e., Prim.new(s); s can be any vertex, and regardless of s, the end result, which are the edges of the minimum spanning tree (MST) are the same. To get the MST edges, we call the method edges().
You initialize Dijkstra with a source vertex, s, i.e., Dijkstra.new(s) that you want to get shortest path/distance to all other vertices. The end results, which are the shortest path/distance from s to all other vertices; are different depending on the s. To get the shortest paths/distances from s to any vertex, v, we call the methods distanceTo(v) and pathTo(v) respectively.
They both create trees with the greedy method.
With Prim's algorithm we find minimum cost spanning tree. The goal is to find minimum cost to cover all nodes.
with Dijkstra we find Single Source Shortest Path. The goal is find the shortest path from the source to every other node
Prim’s algorithm works exactly as Dijkstra’s, except
It does not keep track of the distance from the source.
Storing the edge that connected the front of the visited vertices to the next closest vertex.
The vertex used as “source” for Prim’s algorithm is
going to be the root of the MST.
What is the exact difference between Dijkstra's and Prim's algorithms? I know Prim's will give a MST but the tree generated by Dijkstra will also be a MST. Then what is the exact difference?
Prim's algorithm constructs a minimum spanning tree for the graph, which is a tree that connects all nodes in the graph and has the least total cost among all trees that connect all the nodes. However, the length of a path between any two nodes in the MST might not be the shortest path between those two nodes in the original graph. MSTs are useful, for example, if you wanted to physically wire up the nodes in the graph to provide electricity to them at the least total cost. It doesn't matter that the path length between two nodes might not be optimal, since all you care about is the fact that they're connected.
Dijkstra's algorithm constructs a shortest path tree starting from some source node. A shortest path tree is a tree that connects all nodes in the graph back to the source node and has the property that the length of any path from the source node to any other node in the graph is minimized. This is useful, for example, if you wanted to build a road network that made it as efficient as possible for everyone to get to some major important landmark. However, the shortest path tree is not guaranteed to be a minimum spanning tree, and the sum of the costs on the edges of a shortest-path tree can be much larger than the cost of an MST.
Another important difference concerns what types of graphs the algorithms work on. Prim's algorithm works on undirected graphs only, since the concept of an MST assumes that graphs are inherently undirected. (There is something called a "minimum spanning arborescence" for directed graphs, but algorithms to find them are much more complicated). Dijkstra's algorithm will work fine on directed graphs, since shortest path trees can indeed be directed. Additionally, Dijkstra's algorithm does not necessarily yield the correct solution in graphs containing negative edge weights, while Prim's algorithm can handle this.
Dijkstra's algorithm doesn't create a MST, it finds the shortest path.
Consider this graph
5 5
s *-----*-----* t
\ /
-------
9
The shortest path is 9, while the MST is a different 'path' at 10.
Prim and Dijkstra algorithms are almost the same, except for the "relax function".
Prim:
MST-PRIM (G, w, r) {
for each key ∈ G.V
u.key = ∞
u.parent = NIL
r.key = 0
Q = G.V
while (Q ≠ ø)
u = Extract-Min(Q)
for each v ∈ G.Adj[u]
if (v ∈ Q)
alt = w(u,v) <== relax function, Pay attention here
if alt < v.key
v.parent = u
v.key = alt
}
Dijkstra:
Dijkstra (G, w, r) {
for each key ∈ G.V
u.key = ∞
u.parent = NIL
r.key = 0
Q = G.V
while (Q ≠ ø)
u = Extract-Min(Q)
for each v ∈ G.Adj[u]
if (v ∈ Q)
alt = w(u,v) + u.key <== relax function, Pay attention here
if alt < v.key
v.parent = u
v.key = alt
}
The only difference is pointed out by the arrow, which is the relax function.
The Prim, which searches for the minimum spanning tree, only cares about the minimum of the total edges cover all the vertices. The relax function is alt = w(u,v)
The Dijkstra, which searches for the minimum path length, so it cares about the edge accumulation. The relax function is alt = w(u,v) + u.key
Dijsktra's algorithm finds the minimum distance from node i to all nodes (you specify i). So in return you get the minimum distance tree from node i.
Prims algorithm gets you the minimum spaning tree for a given graph. A tree that connects all nodes while the sum of all costs is the minimum possible.
So with Dijkstra you can go from the selected node to any other with the minimum cost, you don't get this with Prim's
The only difference I see is that Prim's algorithm stores a minimum cost edge whereas Dijkstra's algorithm stores the total cost from a source vertex to the current vertex.
Dijkstra gives you a way from the source node to the destination node such that the cost is minimum. However Prim's algorithm gives you a minimum spanning tree such that all nodes are connected and the total cost is minimum.
In simple words:
So, if you want to deploy a train to connecte several cities, you would use Prim's algo. But if you want to go from one city to other saving as much time as possible, you'd use Dijkstra's algo.
Both can be implemented using exactly same generic algorithm as follows:
Inputs:
G: Graph
s: Starting vertex (any for Prim, source for Dijkstra)
f: a function that takes vertices u and v, returns a number
Generic(G, s, f)
Q = Enqueue all V with key = infinity, parent = null
s.key = 0
While Q is not empty
u = dequeue Q
For each v in adj(u)
if v is in Q and v.key > f(u,v)
v.key = f(u,v)
v.parent = u
For Prim, pass f = w(u, v) and for Dijkstra pass f = u.key + w(u, v).
Another interesting thing is that above Generic can also implement Breadth First Search (BFS) although it would be overkill because expensive priority queue is not really required. To turn above Generic algorithm in to BFS, pass f = u.key + 1 which is same as enforcing all weights to 1 (i.e. BFS gives minimum number of edges required to traverse from point A to B).
Intuition
Here's one good way to think about above generic algorithm: We start with two buckets A and B. Initially, put all your vertices in B so the bucket A is empty. Then we move one vertex from B to A. Now look at all the edges from vertices in A that crosses over to the vertices in B. We chose the one edge using some criteria from these cross-over edges and move corresponding vertex from B to A. Repeat this process until B is empty.
A brute force way to implement this idea would be to maintain a priority queue of the edges for the vertices in A that crosses over to B. Obviously that would be troublesome if graph was not sparse. So question would be can we instead maintain priority queue of vertices? This in fact we can as our decision finally is which vertex to pick from B.
Historical Context
It's interesting that the generic version of the technique behind both algorithms is conceptually as old as 1930 even when electronic computers weren't around.
The story starts with Otakar Borůvka who needed an algorithm for a family friend trying to figure out how to connect cities in the country of Moravia (now part of the Czech Republic) with minimal cost electric lines. He published his algorithm in 1926 in a mathematics related journal, as Computer Science didn't existed then. This came to the attention to Vojtěch Jarník who thought of an improvement on Borůvka's algorithm and published it in 1930. He in fact discovered the same algorithm that we now know as Prim's algorithm who re-discovered it in 1957.
Independent of all these, in 1956 Dijkstra needed to write a program to demonstrate the capabilities of a new computer his institute had developed. He thought it would be cool to have computer find connections to travel between two cities of the Netherlands. He designed the algorithm in 20 minutes. He created a graph of 64 cities with some simplifications (because his computer was 6-bit) and wrote code for this 1956 computer. However he didn't published his algorithm because primarily there were no computer science journals and he thought this may not be very important. The next year he learned about the problem of connecting terminals of new computers such that the length of wires was minimized. He thought about this problem and re-discovered Jarník/Prim's algorithm which again uses the same technique as the shortest path algorithm he had discovered a year before. He mentioned that both of his algorithms were designed without using pen or paper. In 1959 he published both algorithms in a paper that is just 2 and a half page long.
Dijkstra finds the shortest path between it's beginning node
and every other node. So in return you get the minimum distance tree from beginning node i.e. you can reach every other node as efficiently as possible.
Prims algorithm gets you the MST for a given graph i.e. a tree that connects all nodes while the sum of all costs is the minimum possible.
To make a story short with a realistic example:
Dijkstra wants to know the shortest path to each destination point by saving traveling time and fuel.
Prim wants to know how to efficiently deploy a train rail system i.e. saving material costs.
Directly from Dijkstra's Algorithm's wikipedia article:
The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim's algorithm. Prim's purpose is to find a minimum spanning tree that connects all nodes in the graph; Dijkstra is concerned with only two nodes. Prim's does not evaluate the total weight of the path from the starting node, only the individual path.
Here's what clicked for me: think about which vertex the algorithm takes next:
Prim's algorithm takes next the vertex that's closest to the tree, i.e. closest to some vertex anywhere on the tree.
Dijkstra's algorithm takes next the vertex that is closest to the source.
Source: R. Sedgewick's lecture on Dijkstra's algorithm, Algorithms, Part II: https://coursera.org/share/a551af98e24292b6445c82a2a5f16b18
I was bothered with the same question lately, and I think I might share my understanding...
I think the key difference between these two algorithms (Dijkstra and Prim) roots in the problem they are designed to solve, namely, shortest path between two nodes and minimal spanning tree (MST). The formal is to find the shortest path between say, node s and t, and a rational requirement is to visit each edge of the graph at most once. However, it does NOT require us to visit all the node. The latter (MST) is to get us visit ALL the node (at most once), and with the same rational requirement of visiting each edge at most once too.
That being said, Dijkstra allows us to "take shortcut" so long I can get from s to t, without worrying the consequence - once I get to t, I am done! Although there is also a path from s to t in the MST, but this s-t path is created with considerations of all the rest nodes, therefore, this path can be longer than the s-t path found by the Dijstra's algorithm. Below is a quick example with 3 nodes:
2 2
(s) o ----- o ----- o (t)
| |
-----------------
3
Let's say each of the top edges has the cost of 2, and the bottom edge has cost of 3, then Dijktra will tell us to the take the bottom path, since we don't care about the middle node. On the other hand, Prim will return us a MST with the top 2 edges, discarding the bottom edge.
Such difference is also reflected from the subtle difference in the implementations: in Dijkstra's algorithm, one needs to have a book keeping step (for every node) to update the shortest path from s, after absorbing a new node, whereas in Prim's algorithm, there is no such need.
The simplest explanation is in Prims you don't specify the Starting Node, but in dijsktra you (Need to have a starting node) have to find shortest path from the given node to all other nodes.
The key difference between the basic algorithms lies in their different edge-selection criteria. Generally, they both use a priority queue for selecting next nodes, but have different criteria to select the adjacent nodes of current processing nodes: Prim's Algorithm requires the next adjacent nodes must be also kept in the queue, while Dijkstra's Algorithm does not:
def dijkstra(g, s):
q <- make_priority_queue(VERTEX.distance)
for each vertex v in g.vertex:
v.distance <- infinite
v.predecessor ~> nil
q.add(v)
s.distance <- 0
while not q.is_empty:
u <- q.extract_min()
for each adjacent vertex v of u:
...
def prim(g, s):
q <- make_priority_queue(VERTEX.distance)
for each vertex v in g.vertex:
v.distance <- infinite
v.predecessor ~> nil
q.add(v)
s.distance <- 0
while not q.is_empty:
u <- q.extract_min()
for each adjacent vertex v of u:
if v in q and weight(u, v) < v.distance:// <-------selection--------
...
The calculations of vertex.distance are the second different point.
Dijkstras algorithm is used only to find shortest path.
In Minimum Spanning tree(Prim's or Kruskal's algorithm) you get minimum egdes with minimum edge value.
For example:- Consider a situation where you wan't to create a huge network for which u will be requiring a large number of wires so these counting of wire can be done using Minimum Spanning Tree(Prim's or Kruskal's algorithm) (i.e it will give you minimum number of wires to create huge wired network connection with minimum cost).
Whereas "Dijkstras algorithm" will be used to get the shortest path between two nodes while connecting any nodes with each other.
Dijkstra's algorithm is a single source shortest path problem between node i and j, but Prim's algorithm a minimal spanning tree problem. These algorithm use programming concept named 'greedy algorithm'
If you check these notion, please visit
Greedy algorithm lecture note : http://jeffe.cs.illinois.edu/teaching/algorithms/notes/07-greedy.pdf
Minimum spanning tree : http://jeffe.cs.illinois.edu/teaching/algorithms/notes/20-mst.pdf
Single source shortest path : http://jeffe.cs.illinois.edu/teaching/algorithms/notes/21-sssp.pdf
#templatetypedef has covered difference between MST and shortest path. I've covered the algorithm difference in another So answer by demonstrating that both can be implemented using same generic algorithm that takes one more parameter as input: function f(u,v). The difference between Prim and Dijkstra's algorithm is simply which f(u,v) you use.
At the code level, the other difference is the API.
You initialize Prim with a source vertex, s, i.e., Prim.new(s); s can be any vertex, and regardless of s, the end result, which are the edges of the minimum spanning tree (MST) are the same. To get the MST edges, we call the method edges().
You initialize Dijkstra with a source vertex, s, i.e., Dijkstra.new(s) that you want to get shortest path/distance to all other vertices. The end results, which are the shortest path/distance from s to all other vertices; are different depending on the s. To get the shortest paths/distances from s to any vertex, v, we call the methods distanceTo(v) and pathTo(v) respectively.
They both create trees with the greedy method.
With Prim's algorithm we find minimum cost spanning tree. The goal is to find minimum cost to cover all nodes.
with Dijkstra we find Single Source Shortest Path. The goal is find the shortest path from the source to every other node
Prim’s algorithm works exactly as Dijkstra’s, except
It does not keep track of the distance from the source.
Storing the edge that connected the front of the visited vertices to the next closest vertex.
The vertex used as “source” for Prim’s algorithm is
going to be the root of the MST.
One can use Prim's algorithm or Kruskal's algorithm to find the minimum spanning tree/graph of a collection of vertices/nodes and edges/links. What I want though, is an algorithm that finds the minimum spanning graph of this collection, but the resulting graph needs to include only arbitrarily chosen nodes, instead of all nodes. It's okay if the resulting graph includes more nodes than just those needed.
Does such an algorithm exist? Perhaps one could just use Prim's (or Kruskal's) algorithm after modifying the graph to include only the needed nodes? But, I'm not sure how to modify the graph to do so while maintaining its connectedness.
For example, say we have a diamond shaped starting graph (with costs of links in brackets):
A
(2)/ \(1)
B C
(2)\ /(5)
D
Now, we arbitrarily decide that only nodes A and D are needed. If we started at A, we'd still want it to take the left path, because ((2 + 2) < (1 + 5)).
Say we modify the graph slightly:
A
(2)/ \(1) (2)
B C ------E
(2)\ /(5)
D
If we decide that only nodes A, D, and E are needed, we realize that the path with the minimum cost is not necessarily the one with the fewest links. Taking A--B--D and A--C--E costs 7, but A--C--D and C--E costs 8.
What you want to find is a discrete Steiner tree. When not all vertices in the graph are mandatory but the tree is allowed to split at the optional vertices, the problem is NP-hard.
Wikipedia says (linked above) of this problem: it is believed that arbitrarily good approximation ratios cannot in general be achieved in polynomial time. There is a polynomial-time algorithm that finds a factor 1.39 approximation of a minimum Steiner tree.
I have an unweighted, connected graph. I want to find a connected subgraph that definitely includes a certain set of nodes, and as few extras as possible. How could this be accomplished?
Just in case, I'll restate the question using more precise language. Let G(V,E) be an unweighted, undirected, connected graph. Let N be some subset of V. What's the best way to find the smallest connected subgraph G'(V',E') of G(V,E) such that N is a subset of V'?
Approximations are fine.
This is exactly the well-known NP-hard Steiner Tree problem. Without more details on what your instances look like, it's hard to give advice on an appropriate algorithm.
I can't think of an efficient algorithm to find the optimal solution, but assuming that your input graph is dense, the following might work well enough:
Convert your input graph G(V, E) to a weighted graph G'(N, D), where N is the subset of vertices you want to cover and D is distances (path lengths) between corresponding vertices in the original graph. This will "collapse" all vertices you don't need into edges.
Compute the minimum spanning tree for G'.
"Expand" the minimum spanning tree by the following procedure: for every edge d in the minimum spanning tree, take the corresponding path in graph G and add all vertices (including endpoints) on the path to the result set V' and all edges in the path to the result set E'.
This algorithm is easy to trip up to give suboptimal solutions. Example case: equilateral triangle where there are vertices at the corners, in midpoints of sides and in the middle of the triangle, and edges along the sides and from the corners to the middle of the triangle. To cover the corners it's enough to pick the single middle point of the triangle, but this algorithm might choose the sides. Nonetheless, if the graph is dense, it should work OK.
The easiest solutions will be the following:
a) based on mst:
- initially, all nodes of V are in V'
- build a minimum spanning tree of the graph G(V,E) - call it T.
- loop: for every leaf v in T that is not in N, delete v from V'.
- repeat loop until all leaves in T are in N.
b) another solution is the following - based on shortest paths tree.
- pick any node in N, call it v, let v be a root of a tree T = {v}.
- remove v from N.
loop:
1) select the shortest path from any node in T and any node in N. the shortest path p: {v, ... , u} where v is in T and u is in N.
2) every node in p is added to V'.
3) every node in p and in N is deleted from N.
--- repeat loop until N is empty.
At the beginning of the algorithm: compute all shortest paths in G using any known efficient algorithm.
Personally, I used this algorithm in one of my papers, but it is more suitable for distributed enviroments.
Let N be the set of nodes that we need to interconnect. We want to build a minimum connected dominating set of the graph G, and we want to give priority for nodes in N.
We give each node u a unique identifier id(u). We let w(u) = 0 if u is in N, otherwise w(1).
We create pair (w(u), id(u)) for each node u.
each node u builds a multiset relay node. That is, a set M(u) of 1-hop neigbhors such that each 2-hop neighbor is a neighbor to at least one node in M(u). [the minimum M(u), the better is the solution].
u is in V' if and only if:
u has the smallest pair (w(u), id(u)) among all its neighbors.
or u is selected in the M(v), where v is a 1-hop neighbor of u with the smallest (w(u),id(u)).
-- the trick when you execute this algorithm in a centralized manner is to be efficient in computing 2-hop neighbors. The best I could get from O(n^3) is to O(n^2.37) by matrix multiplication.
-- I really wish to know what is the approximation ration of this last solution.
I like this reference for heuristics of steiner tree:
The Steiner tree problem, Hwang Frank ; Richards Dana 1955- Winter Pawel 1952
You could try to do the following:
Creating a minimal vertex-cover for the desired nodes N.
Collapse these, possibly unconnected, sub-graphs into "large" nodes. That is, for each sub-graph, remove it from the graph, and replace it with a new node. Call this set of nodes N'.
Do a minimal vertex-cover of the nodes in N'.
"Unpack" the nodes in N'.
Not sure whether or not it gives you an approximation within some specific bound or so. You could perhaps even trick the algorithm to make some really stupid decisions.
As already pointed out, this is the Steiner tree problem in graphs. However, an important detail is that all edges should have weight 1. Because |V'| = |E'| + 1 for any Steiner tree (V',E'), this achieves exactly what you want.
For solving it, I would suggest the following Steiner tree solver (to be transparent: I am one of the developers):
https://scipjack.zib.de/
For graphs with a few thousand edges, you will usually get an optimal solution in less than 0.1 seconds.