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Greedy AlgorithmsPart Two

Announcements

Problem Set Three graded, will be

returned at end of lecture.

Problem Set Four due on Monday, or on

Wednesday if you're using a late period.

Outline for Today

Minimum Spanning Trees

What's the cheapest way to connect a graph?

Prim's Algorithm

A simple and efficient algorithm for finding

minimum spanning trees.

Exchange Arguments

Another approach to proving greedy

algorithms work correctly. Trees

A tree is an undirected,

acyclic, connected graph.

An undirected graph is called minimally

connected iff it is connected and removing any edge disconnects it.

Theorem: An undirected graph is a tree iff

it is minimally connected.

An undirected graph is called maximally

acyclic iff adding any missing edge introduces a cycle.

Theorem: An undirected graph is a tree iff

it is maximally acyclic.

Theorem: An undirected graph is a tree iff

it is connected and |E| = |V| - 1. Trees

A tree is an undirected graph G = (V, E)

that is connected and acyclic.

All the following are equivalent:

G is a tree.

G is connected and acyclic.

G is minimally connected (removing any

edge from G disconnects it.)

G is maximally acyclic (adding any edge

creates a cycle)

G is connected and |E| = |V| - 1.

Theorem: Let T be a tree and (u, v) ∉ T. The graph T ∪ {(u, v)} contains a cycle. For any edge (x, y) on the cycle, the graph T' = T ∪ {(u, v)} - {(x, y)} is a tree. Proof: Since (u, v) ∉ T and (x, y) ∈ T ∪ {(u, v)}, we know |T'| = |T| + 1 - 1 = |T| = |V| - 1. Therefore, we will show that T' is connected to conclude T' is a tree. Consider any s, t ∈ V. Since T is connected, there is some path from s to t in T. If that path does not cross (x, y), or if (x, y) = (u, v), then this path is also a path from s to t in T', so s and t are connected in T'. Otherwise, suppose the path from s to t crosses (x, y). Assume without loss of generality that the path starts at s, goes to x, crosses (x, y), then goes from y to t. Since (u, v) and (x, y) are part of the same cycle, we can modify the original path from s to t so that instead of crossing (x, y), it goes around the cycle from x to y. This new path is then a path from s to t in T', so s and t are connected in T'. Thus any arbitrary pair of nodes are connected in T', so T' is connected. ■

Minimum Spanning Trees

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4 8 76 7

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Spanning Trees

Let G = (V, E). A spanning tree (or ST) of G

is a graph (V, T) such that (V, T) is a tree. For notational simplicity: we'll identify a spanning tree with just the set of edges T. Suppose that each edge (u, v) ∈ E is assigned a cost c(u, v).

The cost of a tree T, denoted c(T), is the sum

of the costs of the edges in T:

A minimum spanning tree (or MST) of G is

a spanning tree T* of G with minimum cost.c(T)=∑(u,v)∈T c(u,v)

Minimum Spanning Trees

There are many greedy algorithms for finding

MSTs:

Borůvka's algorithm (1926)

Kruskal's algorithm (1956)

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