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Minimum Spanning Trees
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© 2010 Goodrich, Tamassia
Minimum Spanning Trees
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Minimum Spanning Trees
Spanning subgraph
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Subgraph of a graph G
containing all the vertices of G
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Spanning tree
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Spanning subgraph that is
itself a (free) tree
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Minimum spanning tree (MST)
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Spanning tree of a weighted
graph with minimum total
edge weight
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PIT
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STL
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DCA
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Applications
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Communications networks
Transportation networks
© 2010 Goodrich, Tamassia
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Minimum Spanning Trees
ATL
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Cycle Property
Cycle Property:
Let T be a minimum
spanning tree of a
weighted graph G
 Let e be an edge of G
that is not in T and C let
be the cycle formed by e
with T
 For every edge f of C,
weight(f)  weight(e)
Proof:
 By contradiction
 If weight(f) > weight(e) we
can get a spanning tree
of smaller weight by
replacing f with e
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© 2010 Goodrich, Tamassia
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Partition Property
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Partition Property:
Consider a partition of the vertices of
G into subsets U and V
 Let e be an edge of minimum weight
across the partition
 There is a minimum spanning tree of
G containing edge e
Proof:
 Let T be an MST of G
 If T does not contain e, consider the
cycle C formed by e with T and let f
be an edge of C across the partition
 By the cycle property,
weight(f)  weight(e)
 Thus, weight(f) = weight(e)
 We obtain another MST by replacing
f with e

© 2010 Goodrich, Tamassia
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Replacing f with e yields
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Minimum Spanning Trees
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Kruskal’s Algorithm
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Maintain a partition of the
vertices into clusters
Algorithm KruskalMST(G)
for each vertex v in G do
Create a cluster consisting of v
 Initially, single-vertex
let Q be a priority queue.
clusters
Insert all edges into Q
 Keep an MST for each
T
cluster
{T is the union of the MSTs of the clusters}
 Merge “closest” clusters
while T has fewer than n - 1 edges do
and their MSTs
e  Q.removeMin().getValue()
[u, v]  G.endVertices(e)
A priority queue stores the
A  getCluster(u)
edges outside clusters
B  getCluster(v)
 Key: weight
if A  B then
 Element: edge
Add edge e to T
mergeClusters(A, B)
At the end of the algorithm
return T
 One cluster and one MST
© 2010 Goodrich, Tamassia
Minimum Spanning Trees
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Example
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© 2010 Goodrich, Tamassia
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Campus Tour
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Example (contd.)
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Data Structures for Kruskal’s Algorithm
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The graph will be implemented using adjacency lists.
The algorithm maintains a forest of trees.
A priority queue extracts the edges by increasing
weight. The priority queue is implemented as a min
heap.
An edge is accepted it if connects distinct trees.
We need a data structure that maintains a partition,
i.e., a collection of disjoint sets, with operations:
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makeSet(u): create a set consisting of u
find(u): return the set storing u
union(A, B): replace sets A and B with their union
© 2010 Goodrich, Tamassia
Minimum Spanning Trees
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Recall of Data Structures for
Disjoint Sets
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Each set is stored as a linked list represented by its first
member.
Each list element (set member) has a reference to the set
representative.
Operation find(u) takes O(1) time, and returns the set of
which u is a member.
In operation union(A,B), we move the elements of the smaller
set to the sequence of the larger set and update their
references
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The time for operation union(A,B) is min(|A|, |B|)
Whenever an element is processed, it goes into a set of size at least
double, hence each element is processed at most log n times
© 2010 Goodrich, Tamassia
Minimum Spanning Trees
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Partition-Based Implementation
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Partition-based
version of
Kruskal’s
Algorithm
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Cluster merges
as unions
Cluster locations
as finds
Running time
O((n + m) log n)
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PQ operations
O(m log m) =O(m
log n)
UF operations
O(n log n)
© 2010 Goodrich, Tamassia
Algorithm KruskalMST(G)
Initialize a partition P
for each vertex v in G do
P.makeSet(v)
let Q be a priority queue.
Insert all edges into Q
T
{T is the union of the MSTs of the clusters}
while T has fewer than n - 1 edges do
e  Q.removeMin().getValue()
[u, v]  G.endVertices(e)
A  P.find(u)
B  P.find(v)
if A  B then
Add edge e to T
P.union(A, B)
return T
Minimum Spanning Trees
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Prim-Jarnik’s Algorithm
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Similar to Dijkstra’s algorithm
We pick an arbitrary vertex s and we grow the MST as
a cloud of vertices, starting from s
We store with each vertex v label D(v) representing
the smallest weight of an edge connecting v to a
vertex in the cloud
At each step:
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We add to the cloud the vertex u outside the cloud with the
smallest distance label
We update the labels of the vertices adjacent to u
© 2010 Goodrich, Tamassia
Minimum Spanning Trees
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Prim-Jarnik’s Algorithm (cont.)
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We will use adjacency lists for the representation of
the input graph.
We will use a priority queue to store, for each vertex
v, the pair (v,e) with key D(v) where e is the edge with
the smallest weight connecting v to the cloud and
D(v) is that weight.
The priority queue will be implemented as a min
heap.
© 2010 Goodrich, Tamassia
Minimum Spanning Trees
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Prim-Jarnik’s Algorithm (cont.)
Algorithm PrimJarnikMST(G)
Pick any vertex v of G
D[v]  0
for each vertex u  v do
D[u]  +
Initialize T  .
Initialize a priority queue Q with an entry ((u,null),D[u]) for each vertex u,
where (u,null) is the element and D[u] is the key.
while Q is not empty do
(u,e)  Q.removeMin()
Add vertex u and edge e to T.
for each vertex z adjacent to u such that z is in Q do
if weight((u,z)) < D[z] then
D[z]  weight((u,z))
Change to (z,(u,z)) the element of vertex z in Q
Change to D[z] the key of vertex z in Q
Return the tree T
© 2010 Goodrich, Tamassia
Minimum Spanning Trees
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Example
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© 2010 Goodrich, Tamassia
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Example (contd.)
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Minimum Spanning Trees
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Analysis
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Let n and m denote the number of vertices and edges
of the input graph respectively.
Since the priority queue is implemented as a heap, we
can extract the vertex u in O(log n) time.
We can update each D[z] value in O(log n) time as well
(why?). This update is done at most once for each edge
(u,z).
Hence, Prim-Jarnik’s algorithm runs in O((n + m) log n)
time.
© 2010 Goodrich, Tamassia
Minimum Spanning Trees
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Readings

M. T. Goodrich, R. Tamassia and D.
Mount. Data Structures and Algorithms
in C++. 2nd edition. John Wiley. 2011.

Chapter 13
© 2010 Goodrich, Tamassia
Minimum Spanning Trees
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