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Unformatted text preview: s acyclic, with v (G) vertices and v (G) − 1 edges.
Hence T is a tree (from a lemma).
(c) Assume G has n vertices. Suppose that the edges added to E (T ) by the algorithm are
e1 , e2 , . . . , en−1 so that if i < j then ei is added before ej . That is, the edges are arranged in
the order of their appearance in E (T ). For any minimum weight spanning tree (MWST) H of G,
let
τ (H ) = max{i : edges e1 , e2 , . . . , ei are in H }.
Let T ∗ be a MWST of G such that τ (T ∗ ) is the largest among all MWST of G. We say that T ∗
agrees with T the “longest time.” Assume for the purpose of contradiction that T ∗ = T . Then let
e be the ﬁrst edge, not in T ∗ , added to T by Prim’s algorithm. Consider the set S immediately
before e is added to T . Now T ∗ + e contains a cycle C and hence must contain another edge e∗
that joins a vertex in S to a vertex not in S . Now e∗ cannot be one of those edges added by the
algorithm to T before e because e∗ is one of those edges being examined by the algorithm at the
current step when e is added. By line 3 in the algorithm, we see that
wt(e) ≤ wt(e∗ ).
Now T ∗ + e − e∗ is a spanning tree of G and
wt(T ∗ + e − e∗ ) = wt(T ∗ ) + wt(e) − wt(e∗ ) ≤ wt(T ∗ ).
Since T ∗ is a MWST of G, we see that T ∗ + e − e∗ is also a MWST of G. But now T ∗ + e − e∗
agrees with T on the edges added to T before e as well as e also. This contradicts the deﬁnition of
T ∗ . Therefore, we must have T ∗ = T and hence T is a MWST of G.
NOTE: It took me a while to deﬁne all notation so that the proof is hopefully readable to you. The
deﬁnition of τ is for explaining (precisely) our choice of T ∗ . This choice of T ∗ is diﬀerent from the
one we used in proving that Kruskal’s algorithm works. If you have problems in understanding it,
please come and talk with me.
3 Question 4. (2 points) Exercise 2.3.23. Consider the following weighted graph G.
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This note was uploaded on 01/13/2014 for the course MAD 5305 taught by Professor Suen during the Spring '12 term at University of South Florida  Tampa.
 Spring '12
 Suen

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