Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 513 — #521
12.10. References 513
0 0.04 0.08
1
1.04 1.08 1.12
u
0.01 0.05 0.09
w
1.01
1.05
1.09
1.13
0.02 0.06 0.10
1.02
1.06
1.10
0.03 v 0.07 0.11
1.14
Figure 12.35
The 4x4 array graph G
For any step in Prim’s procedure, describe a black-white coloring of the graph
components so that the edge Prim chooses is the minimum weight “gray edge”
according to Lemma 12.9.11.
(c) The 6.042 “parallel” MST algorithm can grow an MST for G by starting with
the upper left corner vertex along with the vertices labelled v and w. Regard each
of the three vertices as one-vertex trees. Successively add, for each tree in parallel,
the minimum weight edge among the edges with exactly one endpoint in the tree.
Stop working on a tree when it is within distance two of another tree. Continue
until there are no more eligible trees—that is, each tree is within distance two of
some other tree—then go back to applying the general gray-edge method until the
parallel trees merge to form a spanning tree of G.
(d) Verify that you got the same MST each time. Problem 12.63 explains why
there is a unique MST for any finite, connected, weighted graph where no two
edges have the same weight.