Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

c.
Now suppose that
is weighted and assume the following:
1:
for 1 ≤ i ≤ n,M(i,i)=0.
2:
for 1 ≤ i,j ≤ n, M(i, j) = weight(i, j) if (i, j) is in E.
3:
for 1 ≤ i, j ≤ n, M(i, j) = ∞ if (i, j) is not in E.
Also, let M 2 be defined, for 1 ≤ i,j ≤ n, as follows:
M 2 (i, j) = min{M(i, 1) +M(1,j),... ,M(i,n) +M(n,j)}.
If M 2 (i, j) = k, what may we conclude about the relationship between vertices i
and j?
C-13.27
A graph G is bipartite if its vertices can be partitioned into two sets X and Y
such that every edge in G has one end vertex in X and the other in Y. Design
and analyze an efficient algorithm for determining if an undirected graph G is
bipartite (without knowing the sets X and Y in advance).
C-13.28
An old MST method, called Baruvka's algorithm, works as follows on a graph
G having n vertices and m edges with distinct weights:
Let Tbe a subgraph of G initially containing just the vertices in V.
while T has fewer than n − 1 edges do
for each connected component Q of T do
Find the lowest-weight edge (v, u) in E with v in Ci and u not in Ci.
Add (v, u) to T (unless it is already in T).
return T
Argue why this algorithm is correct and why it runs in O(mlogn) time.
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