ECLiPSeELearning HelmutSimonis Chapter11:LimitsofPropagation(CostasArray)

Problem Program Search Improvements

Chapter 11: Limits of Propagation (Costas Array)

Helmut Simonis

Cork Constraint Computation Centre Computer Science Department

University College Cork Ireland

ECLiPSe ELearning

Problem Program Search Improvements

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Problem Program Search Improvements

Outline

1 Problem

2 Program

3 Search

4 Improvements

Problem Program Search Improvements

What we want to introduce

Improving propagation does not always pay For some problems, simple backtracking is best CP may not always be the best method

CP should always be fastest way to model problem Consider time to target

Time required to run program Time required to write program

Problem: Costas Array (Antenna design, sonar systems)

Problem Program Search Improvements

Outline

1 Problem

2 Program

3 Search

4 Improvements

Problem Program Search Improvements

Problem Definition

Costas Array (Wikipedia)

A Costas array (named after John P. Costas) can be regarded

geometrically as a set of N points lying on the squares of a NxN

checkerboard, such that each row or column contains only one

point, and that all of the N(N - 1)/2 vectors between each pair of

dots are distinct.

Problem Program Search Improvements

Example (Size 6)

X1 X2 X3 X4 X5 X6

1

2

3 4

5

Problem Program Search Improvements

Model

A variable for each column, ranging from 1 to N A list of N variables for the columns

A difference variable between each ordered pair of variables

alldifferent constraint between variables

alldifferent constraints for all differences

Problem Program Search Improvements

Model

X ₁ X ₂ X ₃ X ₄

X ₅

D ₅₆ D ₄₅ D ₃₄ D ₂₃ D ₁₂

D ₄₆ D ₃₅ D ₂₄ D ₁₃

D ₃₆ D ₂₅ D ₁₄

D ₂₆ D ₁₅

D ₁₆

Problem Program Search Improvements

Model

X ₁ X ₂ X ₃ X ₄ X ₅ X ₆

D ₅₆ D ₄₅ D ₃₄ D ₂₃ D ₁₂

D ₄₆ D ₃₅ D ₂₄ D ₁₃

D ₃₆ D ₂₅ D ₁₄

D ₂₆ D ₁₅

D ₁₆

Problem Program Search Improvements

Model

X ₁ X ₂ X ₃ X ₄

X ₅

D ₅₆ D ₄₅ D ₃₄ D ₂₃ D ₁₂

D ₄₆ D ₃₅ D ₂₄ D ₁₃

D ₃₆ D ₂₅ D ₁₄

D ₂₆ D ₁₅

D ₁₆

Problem Program Search Improvements

Model

X ₁ X ₂ X ₃ X ₄ X ₅ X ₆

D ₅₆ D ₄₅ D ₃₄ D ₂₃ D ₁₂

D ₄₆ D ₃₅ D ₂₄ D ₁₃

D ₃₆ D ₂₅ D ₁₄

D ₂₆ D ₁₅

D ₁₆

Problem Program Search Improvements

Model

X ₁ X ₂ X ₃ X ₄

X ₅

D ₅₆ D ₄₅ D ₃₄ D ₂₃ D ₁₂

D ₄₆ D ₃₅ D ₂₄ D ₁₃

D ₃₆ D ₂₅ D ₁₄

D ₂₆ D ₁₅

D ₁₆

Problem Program Search Improvements

Model

X ₁ X ₂ X ₃ X ₄ X ₅ X ₆

D ₅₆ D ₄₅ D ₃₄ D ₂₃ D ₁₂

D ₄₆ D ₃₅ D ₂₄ D ₁₃

D ₃₆ D ₂₅ D ₁₄

D ₂₆ D ₁₅

D ₁₆

Problem Program Search Improvements

Model

X ₁ X ₂ X ₃ X ₄

X ₅

D ₅₆ D ₄₅ D ₃₄ D ₂₃ D ₁₂

D ₄₆ D ₃₅ D ₂₄ D ₁₃

D ₃₆ D ₂₅ D ₁₄

D ₂₆ D ₁₅

D ₁₆

Problem Program Search Improvements

Example

X1 X2 X3 X4 X5 X6

1

2

3 4

5

6

1 2 5 4 6 3

-3 2 -1

3 1

-1 1 2 4

-2 4 3

1 5

2

Problem Program Search Improvements

Example

X1 X2 X3 X4 X5 X6

1

2

3 4

5

6

1 2 5 4 6

-3 2 -1

3 1

-1 1 2 4

-2 4 3

1 5

2

Problem Program Search Improvements

Example

X1 X2 X3 X4 X5 X6

1

2

3 4

5

6

1 2 5 4 6 3

-3 2 -1

3 1

-1 1 2 4

-2 4 3

1 5

2

Problem Program Search Improvements

Example

X1 X2 X3 X4 X5 X6

1

2

3 4

5

6

1 2 5 4 6

-3 2 -1

3 1

-1 1 2 4

-2 4 3

1 5

2

Problem Program Search Improvements

Example

X1 X2 X3 X4 X5 X6

1

2

3 4

5

6

1 2 5 4 6 3

-3 2 -1

3 1

-1 1 2 4

-2 4 3

1 5

2

Problem Program Search Improvements

Example

X1 X2 X3 X4 X5 X6

1

2

3 4

5

6

1 2 5 4 6

-3 2 -1

3 1

-1 1 2 4

-2 4 3

1 5

2

Problem Program Search Improvements

Example

X1 X2 X3 X4 X5 X6

1

2

3 4

5

6

1 2 5 4 6 3

-3 2 -1

3 1

-1 1 2 4

-2 4 3

1 5

2

Problem Program Search Improvements

Outline

1 Problem

2 Program

3 Search

4 Improvements

Problem Program Search Improvements

Declarations

:-module(costas).

:-export(top/0).

:-lib(ic).

Problem Program Search Improvements

Main Program

top:-

(for(N,3,20) do costas(N,_) ).

costas(N,L):-

length(L,N), L :: 1..N,

alldifferent(L),

L = [_|L1],

diffs(L,L1),

Problem Program Search Improvements

Differences

diffs(_,[]).

diffs(L,[H|T]):-

diff_pairs(L,[H|T],Diffs), alldifferent(Diffs),

diffs(L,T).

diff_pairs(_,[],[]).

diff_pairs([X|X1],[Y|Y1],[D|D1]):- X #= Y+D,

diff_pairs(X1,Y1,D1).

Problem Program Search Improvements

Outline

1 Problem

2 Program

3 Search

4 Improvements

Problem Program Search Improvements

Basic Model

Problem Program Search Improvements

Basic Model

1

1

Problem Program Search Improvements

Basic Model

1 2

2

1

Problem Program Search Improvements

Basic Model

1 2 3

4

2

1

Problem Program Search Improvements

Basic Model

1 2 3 4

3

4

2

1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

6 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5 6

10 6 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5 6

10 11 6 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5 6

10 11 12 6 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5 6

10 11

8

5 12 6 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5 6

10 11

8

5 7 12 6 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2

6 7 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7

6 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7 6

11 6 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7 6

11 12 6 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7

1 2

6 7 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7

1 2

6

6

11 7 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7

1 2

6

6

11 12 7 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7

1

6

1

7 9 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7

1 2

6

1 2

7 9 10 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7

1 2

6

1 2

7 9

8

5 10 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7

1 2

6

1 2

7 9

8

5 6 10 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

7

1 2

6

1 2

7 9

8

5 6 7 10 8 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6 7 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

10

11 7 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

10

11 12 7 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

1

7 10 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

1 2

7

12

5 10 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

1 2

7

12

5 6 10 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

1 2

7

12

5 6 7 10 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

1 2

7

12

5 6

8

6 7 10 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

1 2

7

12

5 6

8

6 11 7 10 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

1 2

7

12

5 6

8

6 11 12 7 10 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

11

5 6

1 2

7

2 5

10 11 8 9 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

4

8 10 9

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

4 10

8 10 11 9

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

4 10

8 10

8

5 11 9

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

7

6 7

4 10

8 10

8

5 6 11 9

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2

6 7

4

8

6

9 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

6

5 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

6

5 6 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

6

5 6 7 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

6

5 6

8

5 7 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

6

5 6

8

5 6 7 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

6

5 6

1

7 8 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

6

5 6

1 2

7

8

5 8 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

6

5 6

1 2

7

8

5 6 8 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

6

5 6

1 2

7

8

5 6 7 8 10 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2

6 7

4

8

6

9

3

10 11 3

4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

6

5 11 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

6

5 6 11 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

6

5 6 7 11 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

6

5 6 7 8 11

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

6

5 6 7 8 9 11

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11 12 3

4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6

5 12 3

4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6 8

9 5 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6 8

9 10 5 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6 8

9 10 11 5 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6

1 3

5 6 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6

1 3

5 6 7 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6

1 3

5 6

7

10 7 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6

1 3

5 6

7

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

7

10

9

5 11 7 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

7

10

9

5 6 11 7 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11 6

1 3

5 6 7 10 9 5 6 8

11 7 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6

1 3

5 6

2 4

7 8 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11

6

1 3

5 6

2 4

7

7

6 8 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

2 4

7

7

6 7 8 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11 6

1 3

5 6

2 4

7 7 6 7 9 8 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

2 4

7

7

6 7 9 10 8 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

2 4

7 7 6 7 9 10 11

8 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

2 4

7 7 6 7 9 10

8 5 11 8 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

2 4

7 7 6 7 9 10

8 5 6 11 8 12 3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

2 4

7

2 6 8 9 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

2 4

7

2 6 8

9

5 9 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11 6

1 3

5 6

2 4

7

2 6

8 9 5 6 9 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11 6

1 3

5 6

2 4

7

2 6

8

1 2

9 10 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7

10

1 5

11 6

1 3

5 6

2 4

7

2 6

8

1 2

9 7

6 10 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4 5

2 4

6 7

4 8

8

6 15

9

3 7 10

1 5 11

6

1 3

5 6

2 4

7

2 6

8

1 2 9

7

6 7 10 12

3 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25

3 5

4

2

1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

4

2

1

Problem Program Search Improvements

Basic Model

1 2 3 4

25

3 5 6 7

4

2

1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

8 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7 5

6 8 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7 5

6 9 8 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7 5

6 9 10 8 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7 5

6 9 10 11 8 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1

8 9 10 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

8

5 10 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

8

5 6 10 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

8

5 6 8 10 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

8

5 6 8 9 10 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

6 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

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1 4

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11 8

5 8 6 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11 8

5 8 9 6 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11 8

5 8 9 10 6 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1 4

6 8 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1 4

6

9

6 8 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1 4

6

9

6 9 8 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1 4

6

9

6 9 10 8 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1 4

6

1 3

8 9 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1

6

1

8 9 10 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1 4

6

1 3

8 9

8

5 10 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1 4

6

1 3

8 9

8

5 6 10 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1 4

6

1 3

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8

5 6 8 10 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

11

1 4

6

1 3

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5 6 8 9 10 11 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

4 12

11 12 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

4 12

11

10

5 12 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

4 12 11

10 8

10 5 12 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

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1 4

8 9

1 4

10

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

4 12

11

10

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5 6 12 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

4 12

11

10

1

5 6 8 12 3 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

4 12

11

10

1 2

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

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10

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

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10

1 2

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

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1 4

8 9

1 4

10

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11

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1

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

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1 4

8 9

1 4

10

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10

1 2

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6 7

1 4

8 9

1 4

10

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10

1 2

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8

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10 11 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

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6 11 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

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3 6 11 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

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3 8 6 11 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

8

1

6 8 11 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

8

1 2

6 8 9 11 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

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11 12 5

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Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

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11

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6 12 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

2 4 11

8 7

9 6 12 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

2 4 11

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9 10 6 12 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

2 4 11

8 7

9 10 11 6 12 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

2 4

11

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1 3

6 8 12 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

2 4

11

8

1

6 8 9 12 5 7 4 2 1

Problem Program Search Improvements

Basic Model

1 2 3 4

25 58

3 5 6

6

11 29

3

5

6 8 9 10

2 4 11

8

1 3

6 8

9

3 9 12 5 7 4 2 1