Download Approximation Algorithms

Introduction to Algorithms
Minimum Spanning Trees
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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
…
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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)
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:
Key problem:
How to find
safe edges?
Note: MST has
|V|-1 edges
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Some definitions
 A cut (S, V - S) is a partition of vertices into
disjoint sets S and V - S
 An edge crosses the cut (S, V - S) if it has one
end point in S, one end point in V - S
 A cut respects a set A of edges if and only if no edge in A
crosses the cut, e.g. A is the set of bold 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
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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
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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.
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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
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Disjoint-set data structure
 Maintain collection S = {S1, . . . , Sk} 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 Si = {x}, and add Si to S
 UNION(x, y): if x ∈ Sx , y ∈ Sy, then S ← S − Sx − Sy ∪ {Sx ∪ Sy}

Representative of new set is any member of Sx ∪ Sy, often the
representative of one of Sx and Sy.

Destroys Sx and Sy (since sets must be disjoint).
 FIND-SET(x): return representative of set containing x
 In Kruskal’s Algorithm, each set is a connected component
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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)
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Example
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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
(VA, V − VA), where VA
= vertices that A is
incident on. Add this
edge to A.
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Prim’s Algorithm
 Uses a priority queue Q to find a light edge
quickly
 Each object in Q is a vertex in V - VA
 Key of v is minimum weight of any edge (u, v),
where u  VA
 Then the vertex returned by Extract-Min is v such
that there exists u  VA and (u, v) is light edge
crossing (VA, V – VA)
 Key of v is  if v is not adjacent to any vertex in
VA
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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)
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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:
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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)
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