Permutation graphsandapplications

Permutation Graphs and Applications

Joe Krall

West Virginia University

In this Presentation:

Overview Permutation Graphs

Applications

Sorting Permutations

4/26/2011Permutation Graphs and Applications2

⇨ Let us consider the sequence of natural numbers:

N = [1, 2, 3, …, n]

⇨ Then a permutation is a sequence as follows:

π = [π1, π2, π3, …, πn], Where, πi = Nj

⇨ Ex: N = [1, 2, 3, 4], π = [3, 4, 2, 1].

Note there are many permutations for N.

IntroductionPermutationsInversion GraphsPermutation GraphsProperties


⇨ Given a permutation π, we can construct a graph:

⇨ Example:

N = [1, 2, 3, 4, 5, 6]

π = [4, 3, 6, 1, 5, 2]

Given a permutation π, then G[π] is called the Inversion Graph of π. There is an edge between πi and πj IFF their order is inverted.

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⇨ Consider an ordinary graph, G.

A permutation always exists.

But, the inversion graph may not be isomorphic to G.

An undirected graph G is called a Permutation Graph if there exists a permutation π, such that G ≅ G[π].



⇨ Interesting Properties

Reversing the order of π gives us the complement

The complement is also a permutation graph

Transitively Orientable

Orient each edge towards its “larger” endpoint

Permutation Graphs are Perfect

Also:

Theorem 7.1: A graph is a permutation graph IFF G

and G’ are comparability graphs.


Applications


⇨ Matching Diagram

Suppose a parallel listing of numbers in both N and π.

Matching Diagrams characterize Permutation Graphs

Because adjacencies only happen due to inversions

1 2 43 65

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Matching Diagram of π = [4, 3, 6, 1, 5, 2] Permutation Graph for π = [4, 3, 6, 1, 5, 2]

A Matching Diagram is drawn by adjoining pairs of matching elements from the sets N

and π. Intersections depict adjacency.

Applications


⇨ Flight Altitudes Problem

Suppose two collections of cities - airports

⇨ Assign flight altitudes to connecting cities

Prevent intersecting flights colliding mid-air

Omaha Des Moines Chicago Fort Wayne Cleveland New York

Kansas City St. Louis Louisville Charleston Richmond

Applications


⇨ Draw a matching diagram

Each edge of the flight diagram is an “element”

Gives us a permutation and a graph

1 2 43 65 7 8 9 10 11

1 24 3 65 78 910 11

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Applications


⇨ This is now a coloring problem

Distinct colors need distinct altitudes

⇨ An algorithm is given later

Topic of sorting a permutation

⇨ Digression:

Flying Cars?

Problem of mid-air collisions

Solvable with “dynamic” coloring

Car knows to change altitude

Using a little radar from surroundings

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Sorting a Permutation

⇨ Consider the problem of sorting a permutation

⇨ We use n-many queues in parallel to demonstrate:

⇨ Each element goes into one of the n-queues But cannot go “behind” a larger element

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4/26/201110 Permutation Graphs and Applications


⇨ The “unpacking” stage is simple

Pull out elements in proper order

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4/26/201111 Permutation Graphs and Applications



⇨ How does this relate to Coloring Problem?

⇨ Proof (Prop 7.3):

We only allow non-inverted pairs in same queue

⇨ Decreasing Subsequence:

Proposition 7.3: There is a one-to-one

correspondence between the proper k-colorings of p

and the sorting strategies.

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Corollary 7.4: The following are equivalent:I. The chromatic number

II. Minimum number of queues needed to sort

III. Length of longest decreasing subsequence


⇨ Proof (Corollary 7.4):

Equivalence of (i) and(ii): (Prop 7.3)

Equivalence of (i) and (iii): (Perfect Graph Theorem)

Clique Number = Longest Decreasing Subsequence

Clique Number = Chromatic Number

Perfect Graph Theorem


⇨ Algorithm:

Input: A permutation

Output: A coloring of the vertices

Algorithm 7.1:begin𝑘 ← 0;For 𝑗 ← 1 to 𝑛 do

begin𝑖 ← FIRST allowable queue;COLOR(π𝑗) ← 𝑖;LAST(i) ← π𝑗;𝑘 ← max{𝑘, 𝑖};

endχ ← 𝑘;

end

procedure FIRST allowable queuebegin𝑖 ← 1; 𝑡 ← 𝑘 +1;until 𝑖 = 𝑡 do

begin𝑟 ← ⌊(𝑖 + 𝑡)/2⌋;if π𝑗 ≥ LAST(𝑟)

then 𝑡 ← 𝑟;else 𝑖 ← 𝑟 + 1;

endreturn 𝑖;

end


⇨ Pass 1

⇨ Pass 2

𝑘

j

𝑖

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LAST

3 6 1 5 24π𝑗 4 Queue #1

𝑘

j

𝑖

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1 2 3 4 5 6

4 3 0 0 0 0

LAST

6 1 5 24π𝑗 3 Queue #1

3Queue #2

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Permutation graphsandapplications