Minimum Spanning Trees
Subgraph
• A graph G is a subgraph of graph H if
– The vertices of G are a subset of the vertices
of H, and
– The edges of G are a subset of the edges of
H.
Spanning Tree
• A spanning tree of a graph G is a subgraph
of G that is a tree and includes all the
vertices of G.
• Every connected graph has a spanning tree
– Start with G. If G is a tree then it is a ST of G. If
not, G has a cycle. Remove any edge of the
cycle. The result is a connected subgraph of G.
Repeat. Process must terminate since the
number of edges is finite.
Spanning Trees
Spanning Trees
a spanning tree
Spanning Trees
another spanning tree
(can have many)
Weighted Graphs
• A weighted graph is a graph with a
mapping of edges to R+ assigning a
weight to each edge
• A minimum spanning tree of a weighted
graph is a spanning tree of minimum total
weight (sum of the weights of the edges in
the tree)
Minimum Spanning Trees
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1. Choose any vertex
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2. Choose minimum weight edge
connected to that vertex
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3. Choose minimum weight edge
connected to chosen vertices
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4. Choose minimum weight edge
connected to chosen vertices
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5. Choose minimum weight edge
connected to chosen vertices
without creating a cycle
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6. Choose minimum weight edge
connected to chosen vertices
without creating a cycle
Prim’s algorithm
• To find a MST of a connected weighted
graph G, we build a tree one edge at a time
– Start with T being any node of G
– Pick a minimum weight edge that does not
create a cycle and that connects a node in T to
a node not in T. Add that edge to T.
– Repeat until all nodes of G have been added to
T.
• A “Greedy algorithm”
Proof that Prim’s algorithm works
• Invariant: At every stage, T is a subgraph
of a MST
• True initially since T is just a single node
• The algorithm runs until all nodes have
been added to T, so in the end the
invariant says that T is a MST.
• So we just need to show that each stage
of the algorithm preserves the invariant.
Proof that the algorithm maintains
the invariant
• Proof by contradiction. Suppose there is a
stage at which the invariant is true before
and false after.
• Let e be the edge added when the invariant
is first violated. Suppose e joins node v in T
to node w not in T.
• So T is a subgraph of a MST but T+e is not.
• Let T’ be a MST of which T is a subgraph.
• Edge e is not in T’.
Proof that the algorithm maintains
the invariant
• Since T’ is a spanning tree, there is a path p from v to w
in T’. Path p does not go through e since e is not part of
T’. So p+e is a cycle consisting of two different paths
from u to v.
• Path p must include an edge f which has one end in T
and the other not in T.
• Edge f must have weight ≥ the weight of e, otherwise the
algorithm would have chosen to add edge f to T instead
of e.
• Removing f from T’ and adding e results in a spanning
tree of weight ≤ the weight of T’ and including T as a
subgraph, contradicting the assumption that T+e was not
a subgraph of a MST. ✔
Finis