Data Structures for dealing with permutations

1
Data Structures for dealing with permutations

• Based on a paper by Fredman, Johnson, McGeoch and
  Ostheimer

2
Talk Outline

• We Define the TSP problem and consider the 2-opt
  algorithm
• We present the modified splay tree an
  interesting DS for implementing 2-opt.
• We show the connection to another problem on
  permutations

3
TSP is cool
Taken from the TSP Homepage
4
TSP Of Ireland
Taken from the TSP Homepage
5
TSP

• Given a complete graph with weights on the edges,
  find a minimum-weight hamiltonian tour
• A tour can be viewed as a
• permutation of the vertices

6
Local Improvement Heuristics

• A nice approach to get a heuristic is to use
  local improvements
• Start with a random tour
• Repeatedly find a local improvement, and perform
  it
• Return the tour when there are no more local
  improvements

7
A 2-move

• (a, v1,v2,,vi,c,d,u1,,uj,b)
• (a, v1,v2,,vi,c,b,uj,uj-1,,u1,d)

8
2-opt

• Repeatedly performing 2-moves is called the 2-opt
  heuristic.
• What is its running time?
• O(nlogn) for producing the first tour
• O(1) to draw a random improvement and check if it
  is good
• O(n) for generating the new tour

9
Requirements from the DS

• Operations we want to perform
• Reverse part of the permutation
• Return the successor of an element in the
  permutation

10
Data Structure for holding the tour

• Tour in linked list performing a 2-move is
  easy. Accessing the vertices is hard.
• Tour in array Access is easy, performing a
  2-move is hard.
• Solution Hold the tour in the nodes of a
  balanced search tree!

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The Modified Splay Tree

• Each node of the tree can be marked as flipped
• The tour is an in-order traversal of the tree,
  altered by the flipped markers

IFJCG A DHBE
12
Requirements from the DS

• Operations we want to perform
• Reverse part of the permutation
• Return the successor of an element in the
  permutation
• op 2 is trivially done in O(logn) in a balanced
  search tree (even with flipped markers)
• op 1 can be done in O(logn) time if we use AVL
  trees or splay trees

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(No Transcript)
14
Splay Trees

• Invented by Sleator Tarjan
• Support
• Search
• Insert, Delete
• Splitting and Concatenation
• all in logarithmic amortized time

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Splay Trees

• To perform an operation on an element we first
  splay it.
• Splaying a node Performing (composite) rotations
  until it gets to the root
• Rotations

x
y
x
y
A
C
B
C
A
B
16
Splay Trees With flipped markers

• Observe that we can push the
• flipped markers down

A
B
B
A
So, before performing any operation (even
rotation) just push the markers down, and were
O.K.
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Reversing a Segment
x

• If we want to reverse between x and y (non
  inclusive)
• Splay y
• Splay x
• Then we can do the flip using O(1) operations

y
A
B
C
x
y
A
B
C
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Signed Sorting by reversals
Given a signed permutation, find a shortest
sequence of reversals that transforms it to (1
2 n)
(
-
-
-
(3 4 -2 -5 1)
3
4
2
5
1)
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Example

• (3 4 -2 -5 1)
• (-4 -3 -2 -5 1)
• (-4 -3 -2 -1 5)
• ( 1 2 3 4 5)

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In SBR

• We want to maintain a (signed) permutation under
  reversals this is the Modified Splay Tree (with
  signs)
• More involved version We want to actually solve
  the problem. We maintain some reversals in a
  similar, but more involved, way.

21
Conclusions

• We showed a Data Structure for holding
  permutations under reversals that is usefull both
  for TSP and for SBR
• The idea of the flipped markers has further
  uses.