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
Dec 16, 2015
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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
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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 Master’s Program
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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
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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
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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
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All minimal paragraph sizes Find minimum width paragraph for a given height. Solve for each height: best known: O(n3/2)
Malfoy couldn’t believe his eyes when he saw that Harry and Ron were still at Hogwarts the next day, looking tired but perfectly cheerful.
Malfoy couldn’t believe his eyes when he saw that Harry and Ron were still at Hogwarts the next day, looking tired but perfectly cheerful.
Malfoy couldn’t 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
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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
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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
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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
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Practical results The number of pictures and columns is small.
(columns <= 5, pictures <= 10). Enumeration works well for pictures <= 3. Branch and bound works well for pictures
<=6. Heuristics + B&B work well for given range. Prototypes developed, including typography
and aesthetic considerations. Very interesting layouts generated
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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)
NP complete Restricted instances: {(1,2), (2,1)}, {(1,1)}
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
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Layout Problem: results
Fixed W, minimize H, NP complete
Minimize W+H 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 cellsDivination. 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
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2
1
3 3
2
4 4
Choose n/2 vertices from each side to maximize the number of edges between chosen vertices
Equivalent problem: maximum density (n/2,n/2)-subgraph of a (n,n)-bipartite graph
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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 substitutionsThe 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, Hogsmeade, 1999, 2nd Ed., 238 pp.
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 Ac. Press, Hogsmeade, 1999, 2nd Ed., 238 pp, Albus Dumbledore ed.
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The Dark Forces: A Guide to Self-Protection, Q. Trimble, HAP, Hogs., `99, 2nd, 238 pp.
The Dark Forces, Q. Trimble, HAP, Hogs., 1999, 2nd, 238 pp.
The Dark Forces: Self-Protection, Q. Trimble, HAP, 1999, 2nd, 238 pp.
The Dark Forces Q. Trimble, HAP, `99, 2nd, 238 pp.
Dark Forces, Q. Trimble, HAP, `99, 2nd.
Dark Forces, Q. Trimble, HAP, 1999.
Dk. Forces, Q. Trimble, HAP, 1999.
Dark Forces, Trimble.
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Source representation
<text> <choice> <fragment val=90> The Dark Forces: A Guide to Self-Protection </fragment> <fragment val=50> The Dark Forces: Self-Protection </fragment> <fragment val=30> The Dark Forces</fragment> <fragment val=20> Dark Forces</fragment> <fragment val=10> Dk. Forces</fragment> </choice> <choice> <fragment val=30> Hogwarts Academic Press </fragment> <fragment val=20> Hogwarts Ac. Press </fragment> <fragment val=15> Hogwarts </fragment> <fragment val=10> HAP </fragment> <fragment val=0> </fragment> </choice> . . . </text>
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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.
4040
25 20 15
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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?