V - CISE

Introduction to Algorithms
Minimum Spanning Trees
My T. Thai @ UF

Problem
� Find a low cost network connecting a
set of locations
� Any pair of locations are connected
� There is no cycle
� Some applications:
� Communication networks
� Circuit design
� …
My T. Thai
mythai@cise.ufl.edu
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Minimum Spanning Tree (MST) Problem
� Input: Undirected, connected graph G=(V, E),
each edge (u, v) E has weight w(u, v)
� Output: acyclic subset T E that connects all
of the vertices with minimum total weight
w(T) = (u,v) T w(u,v)
My T. Thai
mythai@cise.ufl.edu
Bold edges form a
Minimum Spanning
Tree
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Growing a minimum spanning tree
� Suppose A is a subset of some MST
� Iteratively add safe edge (u,v) s.t. A {(u,v)} is
still a subset of some MST
� Generic algorithm:
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mythai@cise.ufl.edu
Key problem:
How to find
safe edges?
Note: MST has
|V|-1 edges
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Some definitions
� An edge is a light edge crossing a cut if and
only if its weight is minimum over all edges
crossing the cut, e.g. edge (c, d)
� Observation: Any MST has at least one edge
connect S and V – S => one cross edge is safe
for A
My T. Thai
mythai@cise.ufl.edu
6

Find a safe edge
Proof: Let T be a MST that includes A
� Case 1: (u, v) T => done.
� Case 2: (u, v) not in T:
� Exist edge (x, y) T cross the cut, (x, y) A
� Removing (x, y) breaks T into two components.
� Adding (u, v) reconnects 2 components
� T´ = T - {(x, y)} {(u, v)} is a spanning tree
� w(T´) = w(T) - w(x, y) + w(u, v) w(T) => T’ is a MST => done
My T. Thai
mythai@cise.ufl.edu
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Corollary
� In GENERIC-MST
� A is a forest containing connected components.
Initially, each component is a single vertex.
� Any safe edge merges two of these components into
one. Each component is a tree.
My T. Thai
mythai@cise.ufl.edu
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Kruskal’s Algorithm
� Starts with each vertex in its own component
� Repeatedly merges two components into one by
choosing a light edge that connects them (i.e., a
light edge crossing the cut between them)
� Scans the set of edges in monotonically increasing
order by weight.
� Uses a disjoint-set data structure to determine
whether an edge connects vertices in different
components
My T. Thai
mythai@cise.ufl.edu
9

Disjoint-set data structure
� Maintain collection S = {S 1, . . . , S k} of disjoint dynamic
(changing over time) sets
� Each set is identified by a representative, which is some
member of the set
� Operations:
� MAKE-SET(x): make a new set S i = {x}, and add S i to S
� UNION(x, y): if x ∈ S x , y ∈ S y, then S ← S − S x − S y ∪ {S x ∪ S y}
� Representative of new set is any member of S x ∪ S y, often the
representative of one of S x and S y.
� Destroys S x and S y (since sets must be disjoint).
� FIND-SET(x): return representative of set containing x
� In Kruskal’s Algorithm, each set is a connected component
My T. Thai
mythai@cise.ufl.edu
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Pseudo code
� Running time: O(E lg V) ( is E is sorted)
� First for loop: |V| MAKE-SETs
� Sort E: O(E lg E) - O(E lg V)
� Second for loop: (o(E log V) (chapter 21)
My T. Thai
mythai@cise.ufl.edu
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Example
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My T. Thai
mythai@cise.ufl.edu
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Prim’s Algorithm
� Builds one tree, so A
always a tree
� Starts from an
arbitrary “root” r
� At each step, find a
light edge crossing cut
(V A, V − V A), where V A
= vertices that A is
incident on. Add this
edge to A.
My T. Thai
mythai@cise.ufl.edu
14

Prim’s Algorithm
� Uses a priority queue Q to find a light edge
quickly
� Each object in Q is a vertex in V - V A
� Key of v is minimum weight of any edge (u, v),
where u V A
� Then the vertex returned by Extract-Min is v such
that there exists u V A and (u, v) is light edge
crossing (V A, V – V A)
� Key of v is if v is not adjacent to any vertex in
V A
My T. Thai
mythai@cise.ufl.edu
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Running time: O(E lgV)
� Using binary heaps to
implement Q
� Initialization: O(V)
� Building initial queue : O(V)
� V Extract-Min’s : O(V lgV)
� E Decrease-Key’s : O(E lgV)
Note: Using Fibonacci heaps can
save time of Decrease-Key
operations to constant (chapter
19) => running time: O(E + V
lg V) My T. Thai
mythai@cise.ufl.edu
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Example
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Summary
� MST T of connected undirect graph G = (V, E):
� Is a subgraph of G
� Connected
� Has V vertices, |V| -1 edges
� There is exactly 1 path between a pair of vertices
� Deleting any edge of T disconnects T
� Kruskal’s algorithm connects disjoint sets of
connects vertices until achieve a MST
� Run nearly linear time if E is sorted:
My T. Thai
mythai@cise.ufl.edu
19

Summary
� Prim’s algorithm starts from one vertex and
iteratively add vertex one by one until achieve a
MST
� Faster than Kruskal’s algorithm if the graph is
dense O(E + V lg V) vs O(E lg V)
My T. Thai
mythai@cise.ufl.edu
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