Minimal spanning tree
What is spanning tree?
Given a connected and undirected graph, a spanning tree of that graph is a
subgraph that is a tree and connects all the vertices together. A single
graph can have many different spanning trees. A minimum spanning tree (MST)
or minimum weight spanning tree for a weighted, connected and undirected graph
is a spanning tree with weight less than or equal to the weight of every other
spanning tree. The weight of a spanning tree is the sum of weights given to
each edge of the spanning tree.
Properties of spanning tree
- A minimum spanning tree has (V – 1) edges where V is the number of vertices in
the given graph. - spanning treee is acyclic.
Kruskal algorithm
steps:
- Sort all the edges in non-decreasing order of their weight.
- pick the smallest edges.Check if if forms a cycle with the spanning tree
formed so far. If cycle is not formed, include this edge. Else, discard it.- Repeat step#2 until there are (|V|-1) edges in the spanning tree.
greedy choice and its demonstration
The algorithm is Greedy Algorithm.The greedy choice is to pick the smallest
weight edge that does not cause a cycle in the MST constructed so far.
now let’s demonstrate it.
lemma:The minimal spanning tree must include the minimal edge of the graph.
proof:
wo suppose that there is a minimal spanning tree T that does not exist the
minimal edge(u,v) of graph, and one of T’s edges is denoted by (u,x).
Now we remove the edge(u,x) but add edge(u,v), formming a new spanning tree T’.
Then W(T)-W(T’)=edge(u,x).w-edge(u,v).w>0 , contradicting to our suppose that
T is minimal spanning tree.Therefore,the lemma that the minimal spanning must
include the minimal edge of graph is corret.
optimal substructure and its demonstration
lemma: we suppose G=G’+edge(u,v),and T is the minimal spanning tree of G.We
can state that T’=T-edge(u,v) is also a minimal spanning tree of G’.
proof:
we suppose T’ is not minimal spanning tree of G’ but T” is, and T”’=edge(u,v)
+T”. So W(T”’)=W(T”)+edge(u,v) < W(T’)+edge(u,v)=W(T), which means T is not
minimal spanning tree, contradicting to our suppose.Therefore, our lemma is
proven.
prim algorithm
step
- Start by picking any vertix to be the root of the tree.
- While the tree does not contain all the vertices in the graph,
find shortest edge leaving the tree and add it to the tree.
more details
step 0: Choose any element r;set S={r} and A={}(empty set).
step 1:find a lightest edge such that one endpoint is in S and the other is
in V\S.Add this edge to A and its endpoint to S.
step 3:if V\S={}(empty),then stop and output the minimum spanning tree(S,A).
Otherwise go to step 1.
crucial questions about prim algorithm
How does the algorithm update S efficiently?
Anwser: Color the vertices. Initially all are white. Change the color to black
when the vertex is moved to S. Use color[v] to store color.How does the algorithm find the lightest edge and update A efficiently?
Anwser: Using a priority queue to find the lightest edge. We suppose that:- u is a vertex in V/S.
- key[u] is the weight of lightest edge from u to any vertex in S
- pred[u] is the endpoint of this edge in S. The array is used to build the MST
tree.
Here is the pseudocode:
the greedy choice of prim algorithm
lemma: always add the lightest edge to the tree until all verteces are included
into the tree.
proof:
we suppose T is the minmum spanning tree not including the lightest edge(u,x)
but including edge (u,y). We substitute the edge(u,v) with edge(u,x),formming a
new spanning tree T’.So,
W(T)-W(T’) = w(edge(u,y)) - w(edge(u,x)) > 0,contradicting to our suppose.
So,our lemma is corret.
the optimal substructure of prim algorithm
the same as the Kruskal’s.
Reference
- https://www.cse.ust.hk/~dekai/271/notes/L07/L07.pdf
- introduction to algorithm