# Algorithms for solving Zebra and Sherlock

## What is in this article

## Preface

This section contains pseudocode of the algorithms used in the paper as well as a brief discussion of each algorithm.

## Definitions for the Algorithms

These definition are based on the descriptions in [Prosser93] and [Kondrak97].

*v*An array of values such that v[i] is the value assigned to V_{i}in the binary constraint satisfaction problem.*n*The number of variables actually in the problem. v[1] is the first variable and the last variable is v[n].*domain*An array such that domain[i] is the domain of of i (D_{i}from the bcsp).*current-domain*An array such that current-domain[i] is the set of values in D_{i}which has not yet been shown to be inconsistent in the current search process. current-domain[i] is initialized to domain[i].*C*An n x n array such that C[i,j] contains the set of constraints between V_{i}and V_{j}in the bcsp. It corresponds to C_{i,j}in the bcsp.*check(i,j)*A function that returns true if the constraint between V_{i}and V_{j}is satisfied. (If there is no constraint between i and j then constraint is considered to be satisfied). Each call to check(i,j) is considered a consistency check for the measurements elsewhere in this paper.*mcl*An n x m (where m is the maximum domain size) array which holds the deepest variable that the instantiation v[i] = k was checked against (that is, mcl[i,k] = h was assigned when check(i,h) was performed)*mbl*In BM-CBJ an array of size n which holds the shallowest past variable that changed value since v[i] was the current variable. In BM-CBJ2 mbl is an n x m array in which mbl[i,j] holds the shallowest past variable that has changed value since v[i] last held the jth value in it’s domain.*conf-set*An array of size n such that conf-set[i] holds the subset of of the past variables in conflict (failed consistency checks) with V_{i}.*row-permutes*A set containing all possible permutations of a row. The size of the set should be |D_{r}|! where D_{r}is the domain of any variable in the row (the domain should be identical for each item in a row).

### Achieving Arc Consistency: AC-3

An algorithm used extensively in preparing in this paper is AC-3, as presented by Kumar in [Kumar92]. This algorithm eliminates values from the domains of a given variable which cannot work based on the constraints and domains of the rest of the variables. The resulting constraint network is said to be arc consistent for all arcs (constraints).

The algorithm achieves arc consistency for all arcs by eliminating the
values from the domain of a given variable which cannot be made
consistent with the rest of the variables. That is, for each x €
D_{i}
, if there is no y € D_{j}
such that
C_{i,j}
is true, x is removed from D_{i}
.

Procedure REVISE makes a given arc arc-consistent with the current set of constraints. Procedure AC-3 executes REVISE for every arc in the constraint problem. Note that it is not sufficient to execute REVISE once for each arc as eliminating an arc affects the validity of other arcs. AC-3 reruns REVISE only for those arcs possibly affected by the elimination of a given arc.

#### procedure AC-3

```
begin AC-3;
G := { All (i,j) such that Ci,j is not null }
Q := { All (Vi, Vj) such that (i, j) € G, i ≠ j }
while Q not empty
select and delete any arc (Vk, Vm) from Q;
if (REVISE(Vk, Vm) then
Q = Q union { (Vi, Vk) such that (i, k) € G, i ≠ k, i ≠ m) };
end if;
end while;
end AC-3;
```

#### procedure REVISE

`REVISE(V`

_{i}
, V_{j}
)

```
begin REVISE(Vi, Vj)
DELETE = false;
for each x is an element of Di do
if there is no such vj €; Dj such that (x, vj) is consistent,
then
delete x from Di;
DELETE = true;
end if;
end for;
return DELETE;
end REVISE;
```

### Backmarking with Conflict-Directed Backjumping 2: BM-CBJ2

BM-CBJ2 is a modification of the algorithm in [Prosser93] based on information in [Kondrak97] and [Kondrak94]. It is a backtracking algorithm which keeps track of conflicts between variables and, when the current variable cannot be made consistent with a prior variable, jumps (skips multiple nodes rather than simply backtracking) back until a consistent node is reached. The algorithm also keeps track of successful and unsuccessful instantiations of a given variable with other variables and, when possible, uses this information to skip re-checking for consistency.

The original algorithm, BM-CBJ, as it appears in [Prosser93], is presented below. Note that it consists of three functions: bcssp, bm-cbj-label, and bm-cbj-unlabel. Prosser used this format for conceptual clarity. It is easier to see the backtracking actions in this format than in a recursive function.

#### BM-CBJ

```
1 PROCEDURE bcssp (n, status)
2 BEGIN
3 consistent = true;
4 status = "unknown";
5 i = 1;
6 WHILE status = "unknown"
7 DO BEGIN
8 IF consistent
9 THEN i = label(i,consistent)
10 ELSE i = unlabel(i,consistent);
11 IF i > n
12 THEN status = "solution"
13 ELSE IF i = 0
14 THEN status = "impossible"
15 END
16 END;
```

Note label and unlabel in lines 9 and 10 are replaced by bm-cbj-label and bm-cbj-unlabel (which are presented below).

```
1 FUNCTION bm-cbj-label (i,consistent): INTEGER
2 BEGIN
3 consistent = false;
4 FOR K = EACH ELEMENT OF current-domain[i] WHILE not consistent
5 DO BEGIN
6 consistent = mcl[i,k] ≥ mbl[j];
7 FOR h = mbl[i] TO i - 1 WHILE consistent
8 DO BEGIN
9 v[i] = k;
10 consistent = check(i, h);
11 mcl[i,k] = h
12 END;
13 If not consistent
14 THEN BEGIN
15 pushnew(mcl[i,k],conf-set[i]);
16 current-domain[i] = remove(v[i],current-domain[i])
17 END
18 END;
19 IF consistent THEN return(i+1) ELSE return(i)
20 END;
```

```
1 FUNCTION bm-cbj-unlabel(i,consistent): INTEGER
2 BEGIN
3 h = max-list(conf-set[i]);
4 conf-set[h] = remove(h,union(conf-set[h],conf-set[i]));
4.1 mbl[i] = h;
4.2 FOR j = h + 1 TO n DO mbl[j] = min(mbl[j],h);
5 FOR j = h + 1 TO i
6 DO BEGIN
7 conf-set[j] = {0}
8 current-domain[j] = domain[j];
9 END;
10 current-domain[h] = remove(v[h],current-domain[h]);
11 consistent = current-domain[h] ≠ nil;
12 return(h)
13 END;
```

The modified algorithm, BM-CBJ2 is presented below. It has been changed from BM-CBJ as follows: Lines 9 and 10 of bcssp are modified to used bm-cbj2-label and bm-cbj2-unlabel instead of bm-cbj-label and bm-cbj-unlabel, in bm-cbj2-label lines 6, and 7 have been modified to use mbl[i][k] instead of mbl[i], line 9 has been moved to 6.1, line 6.2 has been added, line 4.1 has been deleted, and in bm-cbj2-unlabel line 4.2 has been replaced with lines 4.2.1 - 4.2.7.

The purpose of these changes is to correct a deficiency in BM-CBJ.
BM-CBJ fails to account for the fact that backjumping means that not all
values in the domain of a variable are tested (since backjumping occurs
immediately when a conflict is found) and therefore redundant checks are
performed. To correct this the mbl array needs to maintain a separate
record of the shallowest variable whose value has changed since
x_{i}
was last instantiated with that value, for each
variable-value pair. This is achieved by making mbl a two-dimensional
array and recording instantiation points at bm-cbj2-label 6.2 instead of
in bm-cbj2-unlabel (at 4.1).

The need for these changes and a general description of the changes required can be found in [Kondrak97], however there is no example in that paper. Code for a similar change to BMJ (backmarking with backjumping) can be found in [Kondrak94] and was invaluable in making the changes described in this paper.

#### BM-CBJ2

```
1 PROCEDURE bcssp (n, status)
2 BEGIN
3 consistent = true;
4 status = "unknown";
5 i = 1;
6 WHILE status = "unknown"
7 DO BEGIN
8 IF consistent
9 THEN i = bm-cbj2-label(i,consistent)
10 ELSE i = bm-cbj2-unlabel(i,consistent);
11 IF i > n
12 THEN status = "solution"
13 ELSE IF i = 0
14 THEN status = "impossible"
15 END
16 END;
```

```
1 FUNCTION bm-cbj2-label (i,consistent): INTEGER
2 BEGIN
3 consistent = false;
4 FOR K = EACH ELEMENT OF current-domain[i] WHILE not consistent
5 DO BEGIN
6 consistent = mcl[i,k] ≥ mbl[j][k];
6.1 v[i] = k;
6.2 mbl[i][k] = i;
7 FOR h = mbl[i][k] TO i - 1 WHILE consistent
8 DO BEGIN
10 consistent = check(i, h);
11 mcl[i,k] = h
12 END;
13 IF not consistent
14 THEN DO BEGIN
14.1 pushnew(mcl[i,k],conf-set[i]);
14.2 current-domain[i] = remove(v[i],current-domain[i])
14.3 END
15 END;
16 IF consistent THEN return(i+1) ELSE return(i)
17 END;
```

```
1 FUNCTION bm-cbj2-unlabel(i,consistent): INTEGER
2 BEGIN
3 h = max-list(conf-set[i]);
4 conf-set[h] = remove(h,union(conf-set[h],conf-set[i]));
4.2.1 FOR j = h + 1 TO n
4.2.2 DO BEGIN
4.2.3 FOR 0 to m
4.2.4 DO BEGIN
4.2.5 mbl[j][m] = min(mbl[j][m],h);
4.2.6 END
4.2.7 END
5 FOR j = h + 1 TO i
6 DO BEGIN
7 conf-set[j] = {0}
8 current-domain[j] = domain[j];
9 END;
10 current-domain[h] = remove(v[h],current-domain[h]);
11 consistent = current-domain[h] ≠ nil;
12 return(h)
13 END;
```

## Genetic Algorithms

### Summary

This section describes the three genetic algorithms designed by this author to solve the Zebra and Sherlock problems for Assignment #3 for CIS*4750 at the University of Guelph. The first algorithm uses only mutation and is called, aptly enough, Mutate, the second algorithm uses crossover with mutation and is called Xover, while the third algorithm uses double crossover and is called DoubleX.

### Definitions for Genetic Algorithms

*Genetic Algorithm*A genetic algorithm is paradigm modelled after evolutionary processes. A ‘population’ of possible solutions is generated and evaluated for fitness. A partially-random selection of solutions is then taken (possibly multiple times, with the ‘most fit’ algorithms being used most often) and from that selection new solutions are generated and evaluated. This process is repeated until a solution which meets some terminal criterion is found.*Mutation*In the context of genetic algorithms mutations refers to a random change in some part of the individual (solution) such that a new individual (solution) is created.*Crossover*Crossover occurs during reproduction involving two parents. Some random point is selected up to which one parent’s ‘genetic’ information is is used and after which the other parent’s information is used.*Death*In a genetic algorithm death occurs when the individual is removed from the population (e.g. in a fixed size array with the least fit on the bottom, children replace the least fit individual which can be said to have died).*calculate-distance-from-consistent*This is the heuristic used to calculate fitness in the genetic algorithms discussed in this paper. When a solution violates a constraint, the number of columns one of the variables would have to move to be consistent is calculated (e.g. if there is a constraint that says the coffee is next to the zebra, but the coffee is in the same column as the zebra a distance of one is calculated).*fitness*An attempt measure of how close the solution is to a solution. In the following algorithms a higher number for fitness is worse and a lower value is better.*chromosome*For this paper chromosome is an array of size n whose contents correspond to the value of the variables in the constraint problem to be solved.

### Mutate

This is a simple algorithm that does asexual reproduction. A selection of the best fit, as well as a small number of the ‘least fit but not dying’, individuals are copied and then mutated to create children. The random selection of ‘least fit but not dying’ individuals is done to help prevent the algorithm from getting ‘stuck’ on a local minimum by boosting the probability a diverse population will exist. Experience gathered while watching the debug output during development reveals that this, unfortunately, has had limited success and that the mutation-only algorithm still tends to land a local minimum, and has poor chances of getting out.

The pseudocode for the algorithm is outlined below.

```
1 PROCEDURE Mutate()
2 FOR 1 TO max_population
3 DO BEGIN
4 select a solution from the set of possible solutions
5 END FOR
6 evaluate-fitness()
7 fitness-sort()
8 WHILE best-solution-is-not-the-solution()
9 DO BEGIN
10 select-fit-individuals()
11 reproduce()
12 evaluate-fitness()
13 fitness-sort()
14 END WHILE
15 output-solution()
16 END PROCEDURE
1 PROCEDURE evaluate-fitness()
2 FOR EACH variable i
3 DO BEGIN
4 IF domain[i] does not contain v[i]
5 THEN BEGIN
6 fitness += 200
7 END IF
8 END FOR
9 FOR EACH constraint
10 DO BEGIN
11 IF constraint ≠ "not-equal"
12 THEN fitness += calculate-distance-to-consistent
13 ELSE
14 DO BEGIN
15 fitness += 25
16 END IF
17 END FOR
18 END evaluate-fitness
1 PROCEDURE fitness-sort
2 place lowest fitness in population[0], next in population[1], etc.
3 END fitness-sort
1 PROCEDURE select-fit-individuals
2 calculate-relative-probability()
3 FOR 0 to (num-children - num-poor-fit)
4 DO BEGIN
5 add population[get-parent()] to parents
6 END FOR
7 FOR 0 to (num-poor-fit)
8 DO BEGIN
9 add population[get-poor-parent()] to parents
10 END FOR
11 END select-fit-individuals
1 PROCEDURE calculate-relative-probability
2 total-times-better = 0
3 FOR i = 0 TO population-size - num-to-die
4 DO BEGIN
5 times-better[i] = (max-fitness + 1 - population[i].fitness)
6 total-times-better += times-better[i]
7 END FOR
8 x = 1 ÷ totalTimesBetter
9 FOR i = 0 TO times-better.size
10 DO BEGIN
11 cancel-reproducing = times-better[i] × x
12 END FOR
13 END calculate-relative-probability
1 PROCEDURE get-parent()
2 parent = first accumulated-chance-reproducing > random number between 0 and 1
3 END get-parent
1 PROCEDURE get-poor-parent()
2 parent = first (1 - accumulated-chance-reproducing) > random number between 0 and 1
3 END get-parent
1 PROCEDURE reproduce()
2 FOR i = 1 to num-parents
3 DO BEGIN
4 child = parents[i]
5 FOR j = 1 TO n
6 DO BEGIN
7 IF (random number between 0 and 1 < mutation-probability)
8 THEN BEGIN
9 child.chromosome[j] = random number between 1 and max-domain
10 END IF
11 END FOR
12 population[max-population - num-children + i] = child
13 END FOR
14 END reproduce
```

### Xover

This genetic algorithm replaces the asexual reproduction of Mutate with a two parent reproduction scheme using a combination of crossover and a chance of mutation of the result of the parent’s combined contributions.

Xover() is identical to Mutate(), except the name while select() and reproduce() have been modified to use two parents and crossover.

Xover proves to be much more robust than Mutate, solving problems much more quickly and usually being able to solve the problem in a somewhat reasonable time-frame, rather than getting stuck on a local minimum and only getting out after an unlikely mutation.

```
1 PROCEDURE select-fit-individuals
2 calculate-relative-probability()
3 FOR 0 to (num-children ÷ 2)
4 DO BEGIN
5 parent-num = get-parent()
6 add population[parent-num] to parents
7 add population[population-size - num-children - parent-num] to parents
8 END FOR
9 IF (parents.size < num-children + 1)
10 THEN BEGIN
11 add population[get-parent()]
12 END IF
13 END select-fit-individuals
1 PROCEDURE reproduce()
2 FOR i = 1 to (num-parents - 1) STEP 2
3 DO BEGIN
4 k = random number between 1 and n
5 FOR j = 1 to k
6 DO BEGIN
7 child1.chromosome[j] = parents[i].chromosome[j]
8 child2.chromosome[j] = parents[i + 1].chromosome[j]
9 END FOR
10 FOR j = k to n
11 DO BEGIN
12 child1.chromosome[j] = parents[i + 1].chromosome[j]
13 child2.chromosome[j] = parents[i].chromosome[j]
14 END FOR
15 FOR j = 1 TO n
16 DO BEGIN
17 IF (random number between 0 and 1 < mutation-probability)
18 THEN BEGIN
19 child1.chromosome[j] = random number between 1 and max-domain
20 child2.chromosome[j] = random number between 1 and max-domain
21 END IF
22 END FOR
23 population[max-population - num-children + i] = child1;
24 population[max-population - num-children + i + 1] = child2;
25 END FOR
26 END reproduce
```

### DoubleX

This is a variation on Xover which uses two crossover points instead of only one. The only change to the algorithm from Xover is in the reproduce() function in which the second crossover point is added (lines 4.1, and 9.1-9.5 are added while lines 9 and 10 are modified). DoubleX() is exactly same as Xover() except the name.

DoubleX does not seem to be much different than Xover, and in fact may experience decreased performance. Unfortunately, due to run times required to test this question it has not been answered for this paper.

```
1 PROCEDURE reproduce()
2 FOR i = 1 to (num-parents - 1) STEP 2
3 DO BEGIN
4 k = random number between 1 and n
4.1 l = random number between k + 1 and n
5 FOR j = 1 to k
6 DO BEGIN
7 child1 = parents[i].chromosome[j]
8 child2 = parents[i + 1].chromosome[j]
9 END FOR
9.1 FOR j = k to l
9.2 DO BEGIN
9.3 child1 = parents[i + 1].chromosome[j]
9.4 child2 = parents[i].chromosome[j]
9.5 END FOR
10 FOR j = l to n
11 DO BEGIN
12 child1 = parents[i + 1].chromosome[j]
13 child2 = parents[i].chromosome[j]
14 END FOR
15 FOR j = 1 TO n
16 DO BEGIN
17 IF (random number between 0 and 1 < mutation-probability)
18 THEN BEGIN
19 child1.chromosome[j] = random number between 1 and max-domain
20 child2.chromosome[j] = random number between 1 and max-domain
21 END IF
22 END FOR
23 population[max-population - num-children + i] = child1;
24 population[max-population - num-children + i + 1] = child2;
25 END FOR
26 END reproduce
```