Genetic Algorithms for Optimized Book Embedding

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Genetic Algorithms for Optimized Book Embedding
by Katie Dahmen and Ann Kilzer

• Gonzaga University Center for Evolutionary
  Algorithms

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The Traveling Salesman Problem (TSP)

• The salesman visits each city once, and returns
  to the starting point (Hamiltonian Circuit)
• We want to find the shortest route.

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Traveling Salesman Problem

• NP-Complete
• O(n!) - for a problem of size n (n cities), the
  solution requires c n! operations for some
  constant c and n gt N0.
• For a computer running 1 billion operations per
  sec, it would take 9.83 billion years to solve
  the 24 city problem with a brute-force algorithm
  (check every possible tour)

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Genetic Algorithms

• Also called Evolutionary Algorithms
• A form of artificial intelligence based on the
  theory of evolution
• Best solutions are combined to form better
  solutions
• An attempt to find optimal or near optimal
  solutions to NP-Complete problems more quickly,
  without checking every possible solution

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Genetic Algorithm Terminology

• Gene Contains a single element of a solution
• Ex One city in the tour, such as Butte (B).
• Chromosome Contains a solution to the problem,
  often in the form of a sequence
• Ex A travel route of Butte, Coeur DAlene,
  Edmonton, Davenport, and Anaconda can be
  represented as B-C-E-D-A.
• It is assumed that we will return to the starting
  city.

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Genetic Algorithm Terminology

• Population A set of possible solutions
  (chromosomes)
• Cost The efficiency of a solution
• Ex In a tour of cities, the total distance
  traveled.

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Genetic Algorithms applied to TSP

• Randomly generate a population of solutions
  (chromosomes)

SOLUTION ADBCE BACED EABDC EBACD BDCEA DECAB CEAD
B DABCE
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Genetic Algorithms applied to TSP

• Evaluate cost Find distance traveled

SOLUTION COST ADBCE 28 BACED 25 EABDC
25 EBACD 21 BDCEA 23 DECAB 25 CAEDB 31 DABCE
23
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Genetic Algorithms applied to TSP

• Sort population
• Throw out worst solutions
• Pair parents

SOLUTION COST EBACD 21
BDCEA 23 DABCE 23 BACED 25 EABDC 25 DECAB 25
ADBCE 28 CAEDB 31
SOLUTION COST EBACD 21
BDCEA 23 DABCE 23 BACED 25
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Genetic Algorithms applied to TSP

• Mate
• Two parents create two children
• Combine solutions, avoiding duplicates
• Place children in bottom half of population

BDACE EBCDA DABEC BACDE
SOLUTION COST EBACD 21
BDCEA 23 DABCE 23 BACED 25
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Mate Function How it Works

• Partially Matched Crossover A pivot point is
  randomly chosen between two genes
• The genes on one side of the pivot are swapped
  between the parents
• Duplicates are eliminated
• Now we have created two children from two
  parents.

BDACE EBCDA

• EBACD
• BDCEA

BDACD EBCEA
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Genetic Algorithms applied to TSP

• Mutate
• Avoid duplication by swapping genes
• We can experiment with the percent of the
  population to mutate, though its usually around
  5

SOLUTION COST EBACD 21
BDCEA 23 DABCE 23 BACED 25 BDACE EBCDA DAB
EC BACDE
DCBAE 22
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Genetic Algorithms applied to TSP

• 8. Redo Steps 2 - 7
• Until population converges
• or a certain number of generations

SOLUTION COST DABEC 18 EBCDA 20 BACDE
21 EBACD 21 DCBAE 22 BDCEA 23 BDACE 23 BACED
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SOLUTION COST EBACD 21
BDCEA 23 DCBAE 22 BACED 25 BDACE 23 EBCDA 20
DABEC 18 BACDE 21
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Graph Theory

• A Graph is made up of edges and vertices
• Applications include social networks, travel
  routes, and circuits

K5 graph (Complete graph with five vertices)
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Complete Graphs
K4 graph
K5 graph
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Complete Bipartite Graphs

• A Complete Bipartite Graph Kn,m can be drawn with
  n vertices on one side and m vertices on the
  other side.
• Each vertex on the left is connected to each
  vertex on the right

• K2,3 Bipartite Graph

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Square Grids
3 x 3
4 x 4
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X - Trees
X Trees are formed by creating a binary tree,
and then connecting each level.
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Squares, Cubes, Hypercubes
Q2 Graph
Q3 Graph
Q4 Graph
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Book Embedding

• Part of Graph Theory
• NP-Complete Problem
• Our work involves applying Genetic Algorithms to
  the Book Embedding Problem
• This works very similar to the TSP application

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Embedding a Graph
2
1
3
4
5

• We can make a list of edges in the graph
• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Book Embedding

• The spine is a line containing all the vertices.
  A page is a half-plane extruding from the spine.

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Book Embedding
Edge 2
Edge 1
1
4
3
5
2
Edge 3

• We draw edges of a graph on a page so that they
  do not cross. To avoid a collision, we begin
  embedding on a new page.

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Book Embedding
Edge on Page 3
1
4
3
5
2

• Here, the first page is drawn above the spine,
  and the second page below it. Additional pages
  can be denoted by using patterned lines for the
  edges.

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Embedding a Graph
2
1
3
4
5

• We can make a list of edges in the graph
• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4
• A 2 Page Embedding

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Embedding a Graph Reverse Order of Edge
Placement
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph Reverse Order of Edge
Placement
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph Reverse Order of Edge
Placement
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph Reverse Order of Edge
Placement
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph Reverse Order of Edge
Placement
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph Reverse Order of Edge
Placement
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph Reverse Order of Edge
Placement
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4

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Embedding a Graph Reverse Order of Edge
Placement
2
1
3
4
5

• 5 4, 5 3, 4 3, 1 2, 1 4, 2 5, 2 3,
  2 4
• A 3 Page Embedding of the same graph

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Book Embedding and Genetic Algorithms

• The ordering of the edge placement affects the
  number of pages in the book
• Our genetic algorithm tries to find the minimal
  page embedding by alternating the order of the
  edges placed in the book
• It is also possible to adjust the page number by
  manipulating the vertex ordering on the spine

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Results Complete Graphs
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Results X-Trees
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Results Square Grids
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Results Hypercube
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Results Complete Bipartite
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Future Plans

• Try to make the genetic algorithm adjust the
  spine (vertex) ordering and the edge placement
  ordering
• Experiment with different types of graphs

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Questions?

• Katie Dahmen
• kdahmen1_at_gonzaga.edu
• Ann Kilzer
• akilzer_at_gonzaga.edu

Thank you to the McDonald Family for sponsoring
our research project.