Lecture Notes Discrete Optimization - Applied Mathematics

2.4 Prim’s Algorithm
Prim’s algorithm grows a single blue tree, starting at an arbitrary node s ∈ V. In every
step, it chooses among all edges that are incident to the current blue tree T containing s an
uncolored edge e i of minimum cost and colors it blue. The algorithm stops if T contains
all nodes. We implicitly assume that all edges that are not part of the final tree are colored
red in a post-processing step. The algorithm is summarized in Algorithm 2.
Input: undirected graph G=(V,E) with edge costs c : E →R
Output: minimum spanning tree T
1 Initialize: all edges are uncolored
(Remark: we implicitly maintain a forest of blue trees below)
2 Choose an arbitrary node s
3 for i←1to n−1 do
4 Let T be the current blue tree containing s
5 Select a minimum cost edge e i ∈ δ(V(T)) incident to T and color it blue
6 end
7 Implicitly: color all remaining edges red
8 Output the resulting tree T of blue edges
Algorithm 2: Prim’s MST algorithm.
Note that the pre-conditions are met whenever the algorithm applies one of the two coloring
rules: If the blue rule applies, then the node set V(T) of the current blue tree T
containing s induces a cut (X, ¯X) with X = V(T). No blue edge crosses (X, ¯X) by construction.
Moreover, e i is among all uncolored edges crossing the cut one of minimum
cost and can thus be colored blue. If the red rule applies to edge e=(u,v), both endpoints
u and v are contained in the final tree T . The path P uv in T together with e induce a cycle
C. All edges in C∩ P uv are blue and we can thus color e red.
The time complexity of the algorithm depends on how efficiently we are able to identify
a minimum cost edge e i that is incident to T . To this aim, good implementations use a
priority queue data structure. The idea is to keep track of the minimum cost connections
between nodes that are outside of T to nodes in T . Suppose we maintain two data entries
for every node v /∈ V(T): π(v) = (u,v) refers to the edge that minimizes c(u,v) among
all u∈ V(T) and d(v)=c(π(v)) refers to the cost of this edge; we define π(v)=nil and
d(v)=∞ if no such edge exists. Initially, we have for every node v∈ V \{s}:
π(v)=
nil
{
(s,v) if (s,v)∈E
otherwise.
and
d(v)=
∞
{
c(s,v) if(s,v)∈E
otherwise.
The algorithm now repeatedly chooses a node v /∈ V(T) with d(v) minimum, adds it to
the tree and colors its connecting edge π(v) blue. Because v is part of the new tree,
we need to update the above data. This can be accomplished by iterating over all edges
(v,w) ∈ E incident to v and verifying for every adjacent node w with w /∈ V(T) whether
the connection cost from w to T via edge (v,w) is less than the one stored in d(w) (via
π(w)). If so, we update the respective data entries accordingly. Note that if the value of
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