Combinatorial Optimization for Text Layout

1
Combinatorial Optimization for Text Layout

• Richard Anderson
• University of Washington

Microsoft Research, Beijing, September 6,
2000 http//www.cs.washington.edu/homes/anderson/
msrcn.ppt
2
Biography

• Background
• Education
• PhD Stanford (1985), Post Doc MSRI, Berkeley
• Experience
• University of Washington, since 1986. Associate
  Chair for outreach. Visiting prof. IISc,
  Bangalore, 1993-1994
• Professional Interests
• Algorithms
• Parallel algorithms, N-Body Simulation, Model
  Checking for Software, Text Layout
• Distance Learning
• Tutored Video Instruction, Professional Masters
  Program

3
Optimization for Text Layout

• Express text placement as a geometric
  optimization problem.
• Why???
• Generate best layouts
• Body of algorithmic research to build on, as well
  as high performance hardware
• Problem specification and formalization
• Flexibility via parameterization

4
TeX Knuth

• Typography as optimization
• Optimal paragraphing via dynamic programming
  algorithm
• Flexibility
• Tradeoff between uneven lines and hyphenation
  frequency
• Penalty weighted sum of whitespace and
  hyphenation penalties

5
Outline

• Survey of problems studied
• 1) Generating all paragraphs of text
• 2) Picture layout with anchors to text
• 3) Optimal table layout
• 4) Customized content compression

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Paragraphing problem

• Given geometric constraints, find line breaks
• Fixed width, find minimum height
• Greedy Algorithm
• Fixed height, find minimum width
• Only need to consider n2 widths O(n3) algorithm.
• Most practical approach binary search on width.
  O(nlog W) algorithm
• Theoretical O(n) algorithm

7
All minimal paragraph sizes

• Find minimum width paragraph for a given height.
• Solve for each height best known O(n3/2)

Malfoy couldnt believe his eyes when he saw that
Harry and Ron were still at Hogwarts the next
day, looking tired but perfectly cheerful.
Malfoy couldnt believe his eyes when he saw that
Harry and Ron were still at Hogwarts the next
day, looking tired but perfectly cheerful.
Malfoy couldnt believe his eyes when he saw that
Harry and Ron were still at Hogwarts the next
day, looking tired but perfectly cheerful.
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All minimal paragraph sizes

• Motivation
• Placement of floating text
• Formatting tables with text entries
• Basic approach
• Break into segments of roughly n1/2 words each
• Compute possibilities for these, and then combine
• Much work still to do on this problem

9
Placement of text and pictures

• Given text with embedded pictures and tables
• Place pictures close to their references
  (anchors)
• This is a major headache when using LaTeX!
• Futher complications
• Multi-column layouts
• Partial column width pictures
• Typographic considerations for text and headings
• Other graphical layout considerations

10
Placement of text and pictures

• Given text and pictures, where each picture has a
  location in the text, find a layout which
  minimizes the sum of the text-anchor distances
• Single page and multi page problems
• Horizontal placement of pictures fixed wrt column
  boundaries
• May require that picture order is consistent with
  text order

11

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Results

• 2-d bin packing problem do the pictures fit on
  the page.
• May not be the problem of interest simper cases
  pictures fit in columns, align with text rows,
  fixed horizontal position in columns.
• Easy for one column.
• NP-complete for three or more columns.
• NP-complete even if picture area is very small.

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Fixed horizontal bin packing

• Two-d bin packing, except that rectangles have
  fixed horizontal positions
• Motivated by picture placement
• Best known result 3-approximation algorithm
• Problem arises in memory allocation

14
Practical results

• The number of pictures and columns is small.
  (columns lt 5, pictures lt 10).
• Enumeration works well for pictures lt 3.
• Branch and bound works well for pictures lt6.
• Heuristics BB work well for given range.
• Prototypes developed, including typography and
  aesthetic considerations.
• Very interesting layouts generated

15
Tables

• General Problem
• Given a set of configurations for each cell, find
  the maximum value table that satisfies size
  constraints
• Special Cases
• Layout Problem
• No values, minimize table height for fixed width
• Compression Problem
• Configurations for a cell satisfy nesting
  property
• Value decreases with size

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Layout Problem (with S. Sobti)
Divination. Sybill Trelawney Defense against
dark arts. R. J. Lupin
Potions. Severus Snape Care of magical
creatures. Rubeus Hagrid
Divination. Sybill Trelawney Defense against
dark arts. R. J. Lupin
Potions. Severus Snape Care of magical
creatures. Rubeus Hagrid

• NP complete
• Restricted instances (1,2), (2,1), (1,1)

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Layout Problem results

• Fixed W, minimize H, NP complete
• Minimize aWbH solvable with mincut algorithm
• Compute convex hull of feasible table
  configurations
• Heuristic algorithm

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Table compression problem

• Display a table in less than the required area,
  with a penalty for shrinking cells

Divination. Sybill Trelawney Defense against
dark arts. R. J. Lupin
Potions. Severus Snape Care of magical
creatures. Rubeus Hagrid
Divin. Sybill T. Defense against dark arts.
Lupin
Potions. Severus Snape Care of magical
creatures. Hagrid
Divin. Sybill T. Def. dark arts. Lupin
Potions. Severus Snape Care of magical
critters. Hagrid
Divin. Sybill T. Def. dark arts. Lupin
Potions. S. Snape Care of creatures. Hagrid
Divin. Sybill T. Dark arts. Lupin
Potions. S. Snape Critr care. Hagrid
Div D. arts. Lupin
Pot Critters.Hagrid
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Compression Problem

• NP complete for simple case
• Choice cells 1 x 1 (value 1), 0 x 0 (value 0)
• Dummy cells 0 x 0 (value 0)
• Maximize number of full size choice cells in when
  table n x n table compressed to n/2 x n/2.
• Reduction from clique problem
• Incidence matrix reduction

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Attacking the 0-1 problem
Equivalent problem maximum density
(n/2,n/2)-subgraph of a (n,n)-bipartite graph
1
1
2
2
3
3
4
4
Choose n/2 vertices from each side to maximize
the number of edges between chosen vertices
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Greedy Algorithm

• Find MDS of G(X,Y,E)
• Choose X, the set of n/2 vertices of highest
  degree w.r.t. Y
• Choose Y, the set of n/2 vertices of highest
  degree w.r.t. X
• Claim
• (X,Y) is a 1/2 approximation of the MDS
• Proof
• (X,Y) has at least as many edges as the MDS.
  (X,Y) has at least half as many edges as (X,Y)

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Greedy Algorithms

• Non-bipartite graphs
• Add vertices of maximum degree starting with
  empty graph
• Remove vertices of minimum degree, starting with
  full graph
• 4/9 approximation algorithm (Asahiro et al.)
• Open problem generalize and analyze greedy
  algorithms for tables

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Semidefinite programming

• Maxcut problem divide vertices of a graph into
  two sets to maximize number of edges between the
  sets.
• Goemans-Williamson SDP result
• Improved approximation bound from 0.5 to 0.878
• Introduced new technique to the field
• Idea - solve the problem on an n-dimensional
  sphere, use a random projection to divide
  vertices.
• MDS problem can also be attacked with SDP.
• Technical problems with bipartiteness and equal
  division
• lead to a weak result.

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Research directions

• Can semidefinite programming beat the greedy
  algorithm on the 0-1 problem?
• Develop greedy algorithms for the general case.
• Linear programming fractional solution to table
  problems has a natural interpretation.
• Results on rounding?
• Combinatorial algorithms for the fractional
  problem.
• Develop/analyze fast heuristic algorithms

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Content Choice

• If information does not fit, allow substitutions

The Dark Forces A Guide to Self-Protection,
Quenton Trimble, Hogwarts Academic Press,
Hogsmeade, 1999, 2nd Edition, 238 pages, Albus
Dumbledore editor.
The Dark Forces A Guide to Self-Protection,
Quenton Trimble, Hogwarts Ac. Press, Hogsmeade,
1999, 2nd Ed., 238 pp, Albus Dumbledore ed.
The Dark Forces A Guide to Self-Protection,
Quenton Trimble, Hogwarts Ac. Press, Hogsmeade,
1999, 2nd Edition, 238 pages
The Dark Forces A Guide to Self-Protection,
Quenton Trimble, Hogwarts, Hogsmeade, 1999, 2nd
Ed., 238 pp.
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The Dark Forces A Guide to Self-Protection, Q.
Trimble, HAP, Hogs., 99, 2nd, 238 pp.
Dark Forces, Q. Trimble, HAP, 99, 2nd.
The Dark Forces Self-Protection, Q. Trimble,
HAP, 1999, 2nd, 238 pp.
Dark Forces, Q. Trimble, HAP, 1999.
The Dark Forces, Q. Trimble, HAP, Hogs., 1999,
2nd, 238 pp.
Dk. Forces, Q. Trimble, HAP, 1999.
The Dark Forces Q. Trimble, HAP, 99, 2nd, 238 pp.
Dark Forces, Trimble.
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Source representation
lttextgt ltchoicegt ltfragment val90gt The Dark
Forces A Guide to Self-Protection
lt/fragmentgt ltfragment val50gt The Dark
Forces Self-Protection lt/fragmentgt ltfragment
val30gt The Dark Forceslt/fragmentgt ltfragment
val20gt Dark Forceslt/fragmentgt ltfragment
val10gt Dk. Forceslt/fragmentgt lt/choicegt
ltchoicegt ltfragment val30gt Hogwarts Academic
Press lt/fragmentgt ltfragment val20gt Hogwarts
Ac. Press lt/fragmentgt ltfragment val15gt
Hogwarts lt/fragmentgt ltfragment val10gt HAP
lt/fragmentgt ltfragment val0gt lt/fragmentgt
lt/choicegt . . . lt/textgt
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Typography with content choice

• Problem 1
• Given a fixed area for the text, find the optimal
  choice of content
• Problem 2
• Find the set of all maximal configurations
• Problem 3
• Find a good approximation to the set of all
  maximal configurations

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Content Choice

• Algorithmic choice rectangles with values.
  Place one rectangle from each set to maximize
  value.

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40
15
25
20
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Warm up problem Lists

• Optimally display the list for a fixed height
• Set of configurations for each list item.
  (height, value)
• Solvable with knapsack dynamic programming
  algorithm

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List compression
Harry Potter and the Prisoner of Azkaban J. K.
Rowling / Hardcover / Published 1999 Our Price
9.98 Harry Potter and the Sorcerer's Stone J.
K. Rowling / Hardcover / Published 1998 Our
Price 8.98 Harry Potter and the Chamber of
Secrets J. K. Rowling / Hardcover / Published
1999 Our Price 8.98
Harry Potter and the Prisoner of Azkaban
Usually ships in 24 hours J. K. Rowling /
Hardcover / Published 1999 Our Price 9.98
You Save 9.97 (50) Harry Potter and the
Sorcerer's Stone Usually ships in 24 hours J.
K. Rowling / Hardcover / Published 1998 Our
Price 8.98 You Save 8.97 (50) Harry
Potter and the Chamber of Secrets J. K. Rowling /
Hardcover / Published 1999 Our Price 8.98
You Save 8.97 (50)
Harry Potter and the Prisoner of Azkaban J. K.
Rowling / HC / Publ 1999 Our Price 9.98 Harry
Potter and the Sorcerer's Stone J. K. Rowling /
HC / 1998 8.98 Harry Potter and the Chamber of
Secrets J. K. Rowling / HC / 1999 8.98
Harry Potter and the Prisoner of Azkaban J. K.
Rowling 9.98 Harry Potter and the Sorcerer's
Stone Rowling HP Chamber of Secrets
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Implementation goal

• Real time resizing of lists
• Maintain optimal display as window size changes.
• Recompute at refresh rate
• Knapsack/dynamic programming algorithm
• http//www.cs.washington.edu/homes/anderson/demo2/
  Page1.htm

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Customization

• Choice-content generation
• Generate choices for fields
• Automatic abbreviations
• Dictionary lookup
• Assign weights
• Based on compression and component
• Based on user profile

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Browsing applications

• Browsing book lists
• User sets degree of compression
• Issues query
• Source gives default weights
• Value of field
• Strength of match
• Value of item
• Weights modified based on user profile
• Optimal list display done for given compression
  factor

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Display of 2-d time tables

• Show most likely routes and times at highest
  precision
• Based on user profile and travel data
• Memory of user interactions (expanding items)

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Summary

• Graphical layout as geometric optimization
• Theoretical background
• Basic algorithms for rectangle placement
• Algorithm implementation
• Performance requirements are significant
• Application
• Do these techniques work for universal,
  customized display?