J-O-Caml (2)

(1)

J-O-Caml (2)

jean-jacques.levy@inria.fr

pauillac.inria.fr/~levy/qinghua/j-o-caml

Qinghua, November 20

dimanche 10 janvier 2010

(2)

Plan of this class

•

compiling programs

•

list processing

•

pattern-matching

•

new data types

•

labeling algorithm

(3)

Compiling programs

prog1.ml

program text ocamlc

binary a.out

• ocamlc -o prog1 produces prog1 (executable byte code)

• ocamlc -c prog1.ml produces prog1.cmo (byte code)

• ocamlopt -c prog1.ml produces prog1.cmx (binary)

•

also files prog1.cmi (compiled interfaces -- see later)

(4)

Printing

•

or if printing many times:

(5)

List processing

(6)

List processing

(7)

List processing

•

polymorphic types

• ‘a list (say ``alpha list’’) is type of list of any type, e.g. int list, bool list, ...

(8)

List processing

•

Iterators on lists (~ arrays), but no List.make

• List.map f x

• List.iter f x

• List.fold_left f a x

• List.fold_right f a x

(9)

Exercices on lists

(10)

Exercices on lists

•

^Write^{mem a x} which tests if a belongs to list x

•

^Write^{exists p x} which tests if predicate p is true for one element of x;

same forall p x which tests for all elements.

•

Arbitrary precision numbers can be implemented by lists (little endian or big endian style). Write addition and multiplication algorithms.

•

Conway sequence of lists starts with list [1]. Then next list is read from the previous one. Therefore [1; 1], [2; 1], [1; 2; 1; 1], [1; 1; 1;

2; 2; 1].... Print lists of Conway sequence. Is there any unlucky number ?

(11)

New data types

•

^products

(12)

New data types

•

^products

(13)

New data types

•

^sums

(14)

New data types

•

^sums

(15)

New data types

•

recursive types

(16)

New data types

•

recursive types

(17)

New data types

•

recursive types

•

recursive polymorphic types

(18)

New data types

•

recursive types

•

recursive polymorphic types

(19)

New data types

•

recursive polymorphic types (alternative definition for binary trees)

(20)

New data types

•

recursive polymorphic types (alternative definition for binary trees)

(21)

Caring about space

•

with an extra argument as an accumulator of the result

(22)

Combien d’objets dans une image?

Jean- Jacques L évy INRIA

(23)

Labeling

16 objects in this picture 1

1

2 3

4 5 6

7

8 9

10

11 12

13 14

(24)

Naive algorithm

1) choose an unvisited pixel

2) traverse all similar connected pixels 3) and restart until all pixel are visited

Very high complexity:

-

how to find an unvisited pixel ?

-

how organizing exploration of connected pixels (which direction?)

(25)

Algorithm

1) first pass

-

scan pixels left-to-right, top-to-bottom giving a new object id each time a new object is met

2) second pass

-

generate equivalences between ids due to new adjacent relations met during scan of pixels.

3) third pass

-

compute the number of equivalence classes

Complexity:

-

scan twice full image (linear cost)

-

try to efficiently manage equivalence classes (Union-Find by Tarjan)

(26)

Animation

with Polka system for animated algorithms

(27)

Exercise for next class

•

find an algorithm for the labeling algorithm

(28)

Equivalence classes

(29)

Equivalence classes

“Union-Find”

• ^objects

• equivalences

• find the equivalence class of x

₁

, x

₂

, ... , x

_n

x

_p

• 3 operations:

-

- -

NEW(x ) FIND(x )

UNION(x , y )

new object

find canonical representative merge 2 equivalence classes

(30)

Equivalence classes

“Union-Find”

• with tree-like structure x

₉

x

₇

x

₅

x

₁₀

x

₁

x

₄

x

₂

x

₆

x

₃

x

₈

x

₁₁

x

₁₂

x

₁₃

• given by the array of direct ancestors

7 9 9 7

9

0 2 11 1 2 8 8 13 8

ancestor

(31)

Equivalence classes

“Union-Find”

• unbalanced trees because of merges

-

UNION(x , y )

merge 2 equivalence classes

x

₉

x

₇

x

₅

x

₁₀

x

₁

x

₄

x

₃

x

₈

x

₁₁

x

₁₂

x

₁₃

x

₆

x

₂

x

₆

= x

₁₃

(32)

Equivalence classes

“Union-Find”

• try to balance

-

UNION(x , y )

merge 2 equivalence classes

x

₉

x

₇

x

₅

x

₁₀

x

₁

x

₄

x

₃

x

₈

x

₁₁

x

₁₂

x

₁₃

x

₆

x

₂

therefore keep height at each node opérations pour

O (log n)

^FIND(x⁾

(33)

Equivalence classes

“Union-Find”

• compression of path toward canonical representative

-

FIND(x )

with side-effect on tree structure

x

₉

x

₇

x

₅

x

₁₀

x

₁

x

₄

x

₆

x

₂

x

₉

x

₇

x

₅

x

₁₀

x

₁

x

₄

x

₆

x

₂

FIND(x

₄

)

(34)

Exercice for next class

•

find good primitives for bitmap graphics in Ocaml library

•

design the overall structure of the labeling program

J-O-Caml (2)

J-O-Caml (2)

Plan of this class

•

•

•

•

•

Compiling programs

•

Printing

•

List processing

List processing

List processing

•

List processing

•

Exercices on lists

Exercices on lists

•

•

•

•

New data types

•

New data types

•

New data types

•

New data types

•

New data types

•

New data types

•

New data types

•

•

New data types

•

•

New data types

•

New data types

•

Caring about space

•

Combien d’objets dans une image?

Jean- Jacques L évy INRIA

Labeling

Naive algorithm

1) choose an unvisited pixel

2) traverse all similar connected pixels 3) and restart until all pixel are visited

Very high complexity:

-

-

Algorithm

1) first pass

-

2) second pass

-

3) third pass

-

Complexity:

-

-

Animation

Exercise for next class

•

Equivalence classes

Equivalence classes

• objects

• equivalences

• find the equivalence class of x

, x

, ... , x

x

• 3 operations:

-

• ^objects