DocEng2010 Bilauca Healy - A New Model for Automated Table Layout

DocEng2010, September 21– 24, 2010, Manchester, United Kingdom

A New Model for
Automated Table Layout
Mihai Bilauca

Patrick Healy

Department of Computer Science and Information Systems
University of Limerick, Ireland

Supported by Science Foundation Ireland under the research programme 01/P1.2/C009,
Mathematical Foundations, Practical Notations, and Tools for Reliable Flexible Software.

A New Model for Automated Table Layout
Overview
• An exact combinatorial optimization modelling method
for tables that do not contain spanning cells and
provide examples using OPL:
– Mixed Integer Programming model - MIP
– Constraint Programming Model - CP

• Report experimental results for tables with a size of up
to 40x40 (1,600 cells and 9,000 text configurations);
• Conclusions

Slide 2 of 35

Why a new method?
• For applications where finding the layout with the minimum height for a
given width is important.

• Because it is exact (not based on heuristics)

• Priority should be given to user constraints imposed by space limitations or
other aesthetic criteria;

• To the best of our knowledge this is the first attempt to report on run time for
large table sizes and specific (rather than heuristic) cell configurations

Slide 3 of 35

The Table Layout Problem
Find a layout λ of a table ℑ with minimum
such that

height (λ )

width(λ ) < W
W – a given page width


Slide 4 of 35

Definitions (Anderson and Sobti)
ℑ , m x n table, m rows, n columns, λ

is a layout of ℑ

k
k
Ci , j : {( wij , hij ) | 1 ≤ k ≤ K ij

is the set of configurations for cell i,j with

1 ≤ ki , j ≤ K ij

the index of the configuration selected from Ci,j
m

height (λ ) = ∑ hi
i =1

n

width(λ ) = ∑ w j
j =1

where hi = max(hi,j) for each row i, wj = max(wi,j) for each column j
ki , j
ij

hij = h

kij
wij = wij
Slide 5 of 35

Cell configurations
Example:
k
k
Ci , j : {( hij , wij ) | 1 ≤ k ≤ K ij
Cell configurations for cell i,j
this blue sky


this blue
grey

this
blue
sky

Slide 6 of 35

Integer Programming definition
m

n

minimize∑ max hi , j ,k ⋅ xi , j ,k
i =1

j =1

subject to
n

1)

m

∑ max w
i =1

j =1

2)
where

∑x

i , j ,k

i , j ,k

⋅ xi , j ,k ≤ W

= 1,

x ∈ {0,1}
1 ≤ ki , j ≤ K i , j
Slide 7 of 35

Table Layout problem is NP-complete
Demonstrated by:
• 1996 Wang - demonstration using large
integers;
• 1999 Anderson and Sobti - using
reductions of the clique problem to the
table layout, on simple tables.


Slide 8 of 35

OPL
OPL – Optimization Programming Language
designed for solving combinatorial optimization
problems.
• support for:
– MIP and constraint programming including search specification;
– logical and higher order constraints;
– support for scheduling and resource allocation applications;

• shares structure and syntax features with mathematical
programming languages such as AMPL or GAMS
• problems can be formulated in a language similar to their
algebraic notation


Slide 9 of 35

OPL Keywords
dvar – decision variable. The purpose of an OPL

model is to find values for the decision variables
such that all constraints are satisfied
dexpr - to express decision variables in a more

compact way
{dataType} - set of type dataType
<x,y> - represents a tuple value;

Slide 10 of 35

Data types
tuple Conf {int w; int h;}
tuple CellConf {int i; int j; Conf c;}
int pageW;
// page width
{CellConf} configs; // set of cell configurations
sample data: configs = {<0,0, <127,10>>, <0,0, <92,20>>,
<0,1, <75,20>>,<0,2, <65,10>>,…,}

{Cell} cells = {<i,j> | <i,j,k> in configs}
{int} rows = {i | <i,j> in cells}
{int} cols = {j | <i,j> in cells}

Slide 11 of 35

Basic MIP model - BMIP
dvar int cellSel[configs] in 0..1
minimize tableH
constraints
{
ct1:
tableW <= pageW

}

ct2: // select only one cell configuration
forall(i in rows,j in cols)
sum(<i,j,k> in configs) cellSel[<i,j,k>]== 1


Slide 12 of 35

BMIP – Table width/height
// cell width
dexpr int cellW[<i,j> in cells] =
sum(<i,j,k> in configs) cellSel[<i,j,k>] * k.w
// column width
dexpr int colW[j in cols] =
max(i in rows) cellW[<i,j>]
// table width
dexpr int tableW = sum(j in cols) colW[j]
• Table height is computed in a similar manner

Slide 13 of 35

MIP model
MIP model is based on two observations
a) The minimum column width minW[j] is the
maximum of minimum cell widths for a column j;
a) for each column, its width is one of the values
selected from the union of its cell configuration
widths colWset<j,k.w>;


Slide 14 of 35

MIP model
// compute minW[j]

int minW[j in columns] = max(i in rows)
min(<i,j,k> in configs) k.w
int minH[i in rows] = … //and similarly minH[i]
// for each column j, the set of possible widths

{ColW} colWset = {<j, k.w> | <i,j,k> in configs:
k.w >= minW[j] &&
k.h >= minH[i]}


Slide 15 of 35

MIP model
Two decision variables:
// column width selector
dvar int colSel[colWset] in 0..1
// cell configuration selector
dvar int cellSel[configs] in 0..1
// objective function
minimize tableH


Slide 16 of 35

MIP model
constraints{
ct1:… // table width < page width
ct2: // only one configuration per column
forall(j in columns)
sum(<j,k> in colWset) colSel[<j,k>] == 1;
ct3: // cell width <= column width
forall(j in columns, i in rows)
cellW[<i,j>] <= colW[j];
ct4:…} //select only one cell configuration

Slide 17 of 35

Constraint Programming model - CP
Advantage is given by support for two main activities of
combinatorial optimization algorithms:
– constraint reasoning for domain reduction
– search strategy specification for improved performance

Binary decision variables collSel and cellSel in
the MIP model become integer decision variables
colW[j] and rowH[i]


Slide 18 of 35

Constraint Programming model - CP
Strategy: find colW[j] and rowH[i] from the set of
possible widths and heights colWset and
rowHset such that
• the height of the table layout is minimized;
• table width is less than page width;
• for each cell i,j there is at least one configuration k
with a width less than or equal to colW[j] and
the height less than or equal to rowH[i]

wi , j ,k ≤ colW [ j ]

hi , j ,k ≤ rowH [ j ]
Slide 19 of 35

CP model - definitions
tuple Dim{int index; int value;}
Reduce the set of possible widths an heights:
{Dim} colWset = {<j, k.w> | <i,j,k> in
configs : k.w >= minW[j] && k.h >= minH[i]}
{Dim} rowHset = {<i, k.h> | <i,j,k> in
configs : k.w >= minW[j] && k.h >= minH[i]}


Slide 20 of 35

CP model - definitions
// two decision variables
dvar int colW[j in columns] in min(<j,w> in
colWset) w .. max(<j,w> in colWset) w;
dvar int rowH[i in rows] in min(<i,h> in
rowHset) h .. max(<i,h> in rowHset) h;
// define table width/height
dexpr int tableW = sum(j in columns) colW[j];
dexpr int tableH = sum(i in rows) rowH[i];
//objective function
minimize tableH;

Slide 21 of 35

CP model - constraints
constraints{
ct1:… // table width < page width
ct2:
forall(j in columns)
sum(<j,w> in colWset) (colW[j]==w) == 1
ct3:
forall(i in rows)
sum(<i,h> in rowHset) (rowH[i]==h) == 1
ct4:
forall(i in rows, j in columns)
sum(<i,j,k> in fConfigs)
(k.w <= colW[j] && k.h <= rowH[i]) >= 1}

Slide 22 of 35

CP model – search strategy
In OPL 5.5 there is no search language. The default
search can be tuned, by specifying search phases.
Maximal regret heuristic: the regret of a column is the
difference between its first and second choice of
possible width value


Slide 23 of 35

CP model – setting search strategy
var f = cp.factory
var phase0 = f.searchPhase(colW,
f.selectLargest(f.regretOnMax()),
f.selectLargest(f.value()));
var phase1 = f.searchPhase(rowH,
f.selectLargest(f.regretOnMin()),
f.selectSmallest(f.value()));
cp.setSearchPhases(phase0, phase1);


Slide 24 of 35

Adding user imposed constraints
Adding the user constraint that all columns must be of
equal width is easy:
constraints{
…
ct6:
forall(ordered j1, j2 in columns )
colW[j1] == colW[j2] ;
…}
the forall expression is equivalent to:
forall(j1,j2 in 1..n : j1 < j2).

Slide 25 of 35

Experimental results
What are the factors that impact on computational
time?
We tested the proposed models on two types of tables:
• tables with a small number of cells but a large
number of cell configurations
• large tables with cells that contain text with up to 6
words (size up to 40x40).


Slide 26 of 35

Tables with large amounts of text

Running time for a 3x3 table
and up to 200 words per
cell.


MIP model running time for
a 3x3 table; word-count per
cell varies from 200 to 600.

Slide 27 of 35


Slide 28 of 35

Tables with increasing size and configurations

Running time for a 10x10
table and up to 6 words per
cell.


MIP model running time for
10x10 tables; word count
per cell varies from 6 to 100.

Slide 29 of 35

Tables with fixed number of cell configurations
Table size
2x3x200
4x3x100
10x10x12
20x20x3


wSet
1,160
1,113
844
356

Slide 30 of 35

Conclusions
1. OPL provides viable solutions. The MIP model
which uses the CPLEX optimization engine
generally provides faster results than CP
2. The CP model can find a feasible solution faster
than the MIP model, but it continues to explore the
solution space until it finds whether the solution
found is the final objective;
3. Performance depends highly on the input data.
When the page width is closer to the minimum
table width the problem is harder to solve


Slide 31 of 35

Conclusions
1. It is more difficult to solve the table layout problem
when there is a large number of rows, columns and
column configurations than when the table has a
large number of cell configurations;
1. Hardware is essential:The CP engine reported that
1.76 Gb of memory was used to solve a table with
10.000 cells;


Slide 32 of 35

Conclusions
1. The constraint programming model may be slower
than the MIP model due to the version of the CP
engine used in testing. ILOG stopped supporting
the CP engine in 2005 but reintroduced it in 2007.
Not all search features are available
1. Using a modelling language allows user imposed
constraints to be easily added to the model
1. Paragraphing – is an essential step as it impacts on
the quality and computational time

Slide 33 of 35

Future work
To develop a model for tables with
• spanning cells
• inner tables
• other user constraints, i.e. constraints on group of
rows/columns
Note: We already developed a model for tables with
spanning cells ☺


Slide 34 of 35

Acknowledgements

We would like to express our gratitude to
Prof. David Parnas
for motivating much of this work


Slide 35 of 35

Acknowledgements

Questions ?

www.tabularlayout.org


DocEng2010 Bilauca Healy - A New Model for Automated Table Layout

More Related Content

What's hot (17)

Recently uploaded (20)

DocEng2010 Bilauca Healy - A New Model for Automated Table Layout