Lets say I have array,and number of rows as follows and I want to decide in which order they come.
set of int: people = 1..10;
int row= 3
include "alldifferent.mzn";
constraint alldifferent(position);
array[people] of var people: position;
I have to people in the front row to come before people in behind rows; regardless of a column. Two people are in the same row if person1 mod row=person2 mod row, e.g. persons 2,5,8 are in the same row. I want to force this order in decision variable positions.
I found a naive way of doing so involving forall and exists which I believe is not efficient:
constraint forall(person in position where person>row)(exists(smallerpos in 1..person-1)(smallerpos div row = person div row));
I'm not sure I understand the problem correctly, but the model is quite faster if you add all_different(position), e.g.
include "globals.mzn"; % adding this
set of int: people = 1..10;
int row= 3
array[people] of var people: position;
constraint
forall(pos1,pos2 in people where pos1 < pos2 /\ (position[pos1] mod row)=(position[pos2] mod row))
(
position[pos1] < position[pos2]
)
/\ all_different(position) % adding this
;
Without all_different it takes 0.9s to get all 4200 solutions, with all_different it takes 0.25s.
Note: I first thought of the following solution, but since it don't give the same number of solution it's probably not correct. It only give 1201 solutions, but it's faster: 0.05s. As mentioned above, I probably don't understand your problem correctly.
constraint
forall(r in 0..row-1) (
increasing([position[p] | p in people where p mod row == r])
)
/\ all_different(position)
;
This works but is less efficient.
constraint forall(pos1,pos2 in people where pos1 < pos2 /\ (position[pos1] mod rows)=(position[pos2] mod rows))(
position[pos1] < position[pos2] );
Another proposition is, this one is not any more efficeint either
constraint forall(pos1,pos2 in people where (position[pos1] mod rows)=(position[pos2] mod rows))(
pos1 < pos2 <-> position[pos1] < position[pos2] );
Related
A little bit of background. I'm trying to make a model for clustering a Design Structure Matrix(DSM). I made a draft model and have a couple of questions. Most of them are not directly related to DSM per se.
include "globals.mzn";
int: dsmSize = 7;
int: maxClusterSize = 7;
int: maxClusters = 4;
int: powcc = 2;
enum dsmElements = {A, B, C, D, E, F,G};
array[dsmElements, dsmElements] of int: dsm =
[|1,1,0,0,1,1,0
|0,1,0,1,0,0,1
|0,1,1,1,0,0,1
|0,1,1,1,1,0,1
|0,0,0,1,1,1,0
|1,0,0,0,1,1,0
|0,1,1,1,0,0,1|];
array[1..maxClusters] of var set of dsmElements: clusters;
array[1..maxClusters] of var int: clusterCard;
constraint forall(i in 1..maxClusters)(
clusterCard[i] = pow(card(clusters[i]), powcc)
);
% #1
% constraint forall(i, j in clusters where i != j)(card(i intersect j) == 0);
% #2
constraint forall(i, j in 1..maxClusters where i != j)(
card(clusters[i] intersect clusters[j]) == 0
);
% #3
% constraint all_different([i | i in clusters]);
constraint (clusters[1] union clusters[2] union clusters[3] union clusters[4]) = dsmElements;
var int: intraCost = sum(i in 1..maxClusters, j, k in clusters[i] where k != j)(
(dsm[j,k] + dsm[k,j]) * clusterCard[i]
) ;
var int: extraCost = sum(el in dsmElements,
c in clusters where card(c intersect {el}) = 0,
k,j in c)(
(dsm[j,k] + dsm[k,j]) * pow(card(dsmElements), powcc)
);
var int: TCC = trace("\(intraCost), \(extraCost)\n", intraCost+extraCost);
solve maximize TCC;
Question 1
I was under the impression, that constraints #1 and #2 are the same. However, seems like they are not. The question here is why? What is the difference?
Question 2
How can I replace constraint #2 with all_different? Does it make sense?
Question 3
Why the trace("\(intraCost), \(extraCost)\n", intraCost+extraCost); shows nothing in the output? The output I see using gecode is:
Running dsm.mzn
intraCost, extraCost
clusters = array1d(1..4, [{A, B, C, D, E, F, G}, {}, {}, {}]);
clusterCard = array1d(1..4, [49, 0, 0, 0]);
----------
<sipped to save space>
----------
clusters = array1d(1..4, [{B, C, D, G}, {A, E, F}, {}, {}]);
clusterCard = array1d(1..4, [16, 9, 0, 0]);
----------
==========
Finished in 5s 419msec
Question 4
The expression constraint (clusters[1] union clusters[2] union clusters[3] union clusters[4]) = dsmElements;, here I wanted to say that the union of all clusters should match the set of all nodes. Unfortunately, I did not find a way to make this big union more dynamic, so for now I just manually provide all clusters. Is there a way to make this expression return union of all sets from the array of sets?
Question 5
Basically, if I understand it correctly, for example from here, the Intra-cluster cost is the sum of all interactions within a cluster multiplied by the size of the cluster in some power, basically the cardinality of the set of nodes, that represents the cluster.
The Extra-cluster cost is a sum of interactions between some random element that does not belong to a cluster and all elements of that cluster multiplied by the cardinality of the whole space of nodes to some power.
The main question here is are the intraCost and extraCost I the model correct (they seem to be but still), and is there a better way to express these sums?
Thanks!
(Perhaps you would get more answers if you separate this into multiple questions.)
Question 3:
Here's an answer on the trace question:
When running the model, the trace actually shows this:
intraCost, extraCost
which is not what you expect, of course. Trace is in effect when creating the model, but at that stage there is no value of these two decision values and MiniZinc shows only the variable names. They got some values to show after the (first) solution is reached, and can then be shown in the output section.
trace is mostly used to see what's happening in loops where one can trace the (fixed) loop variables etc.
If you trace an array of decision variables then they will be represented in a different fashion, the array x will be shown as X_INTRODUCED_0_ etc.
And you can also use trace for domain reflection, e.g. using lb and ub to get the lower/upper value of the domain of a variable ("safe approximation of the bounds" as the documentation states it: https://www.minizinc.org/doc-2.5.5/en/predicates.html?highlight=ub_array). Here's an example which shows the domain of the intraCost variable:
constraint
trace("intraCost: \(lb(intraCost))..\(ub(intraCost))\n")
;
which shows
intraCost: -infinity..infinity
You can read a little more about trace here https://www.minizinc.org/doc-2.5.5/en/efficient.html?highlight=trace .
Update Answer to question 1, 2 and 4.
The constraint #1 and #2 means the same thing, i.e. that the elements in clusters should be disjoint. The #1 constraint is a little different in that it loops over decision variables while the #2 constraint use plain indices. One can guess that #2 is faster since #1 use the where i != j which must be translated to some extra constraints. (And using i < j instead should be a little faster.)
The all_different constraint states about the same and depending on the underlying solver it might be faster if it's translated to an efficient algorithm in the solver.
In the model there is also the following constraint which states that all elements must be used:
constraint (clusters[1] union clusters[2] union clusters[3] union clusters[4]) = dsmElements;
Apart from efficiency, all these constraints above can be replaced with one single constraint: partition_set which ensure that all elements in dsmElements must be used in clusters.
constraint partition_set(clusters,dsmElements);
It might be faster to also combine with the all_different constraint, but that has to be tested.
I have a table with features = {A,B}. B is a column of integers. Scanning the table, when I have a value change in B column, I increment a variable "changes" of 1:
if data[i,B]!=data[i-1,B]
then changes=changes+1
I want to find an order that minimizes changes and at the same time keep the repetition of a value in B in [0,upper_bound].
I'm thinking to use an array as a decision variable where save the position j for the element i:
order[i]=j means i element in data is the j-th element in ordering.
How can I model with constraint? This is what I do until now:
array[1..n, Features] of int: data;
int: changes=0;
constraint
forall(i in 1..n) (
if data[i,B] != data[i-1,B] then
changes=changes+1
endif
)
;
minimize changes;
I think I'm wrong using changes as a constant variable, right? Thank you in advance.
In MiniZinc (and in constraint programming in general) you cannot increment a variable as changes=changes+1).
If changes is a variable used only for the total count of changes you can use sum instead, something like:
% ...
var 0..n: num_changes;
constraint
changes = sum([data[i,B] != data[i-1,B] | i in 2..n])
;
% ...
However, if you want to use the number of accumulated changes for each i then you have to create a changes array to collect the values for each step, e.g.
var[1..n-1] of var 0..n: changes;
% the total number of changes (to minimize)
var 0..n-1: total_changes = changes[n-1];
constraint
forall(i in 1..n-1) (
if data[i,B] != data[i-1,B] then
changes[i] = changes[i-1]+1
else
changes[i] = changes[i-1]
endif
)
;
I have a MiniZinc model for wolf-goat-cabbage in which I store the locations of each entity in its own array, e.g., array[1..max] of Loc: wolf where Loc is defined as an enum: enum Loc = {left, rght}; and max is the maximum possible number of steps needed, e.g., 20..
To find a shortest plan I define a variable var 1..max: len; and constrain the end state to occur at step len.
constraint farmer[len] == left /\ wolf[len] == left /\ goat[len] == left /\ cabbage[len] == left
Then I ask for
solve minimize len
I get all the right answers.
I'd like to display the arrays from 1..len, but I can't find a way to do it. When I try, for example, to include in the output:
[ "\(wolf[n]), " | n in 1..max where n <= len ]
I get an error message saying that I can't display an array of opt string.
Is there a way to display only an initial portion of an array, where the length of the initial portion is determined by the model?
Thanks.
Did you try to fix the len variable in the output statement like n <= fix(len)?. See also What is the use of minizinc fix function?
I am trying to use system verilog constraint solver to solve the following problem statement :
We have N balls each with unique weight and these balls need to be distributed into groups , such that weight of each group does not exceed a threshold ( MAX_WEIGHT) .
Now i want to find all such possible solutions . The code I wrote in SV is as follows :
`define NUM_BALLS 5
`define MAX_WEIGHT_BUCKET 100
class weight_distributor;
int ball_weight [`NUM_BALLS];
rand int unsigned solution_array[][];
constraint c_solve_bucket_problem
{
foreach(solution_array[i,j]) {
solution_array[i][j] inside {ball_weight};
//unique{solution_array[i][j]};
foreach(solution_array[ii,jj])
if(!((ii == i) & (j == jj))) solution_array[ii][jj] != solution_array[i][j];
}
foreach(solution_array[i,])
solution_array[i].sum() < `MAX_WEIGHT_BUCKET;
}
function new();
ball_weight = {10,20,30,40,50};
endfunction
function void post_randomize();
foreach(solution_array[i,j])
$display("solution_array[%0d][%0d] = %0d", i,j,solution_array[i][j]);
$display("solution_array size = %0d",solution_array.size);
endfunction
endclass
module top;
weight_distributor weight_distributor_o;
initial begin
weight_distributor_o = new();
void'(weight_distributor_o.randomize());
end
endmodule
The issue i am facing here is that i want the size of both the dimentions of the array to be randomly decided based on the constraint solution_array[i].sum() < `MAX_WEIGHT_BUCKET; . From my understanding of SV constraints i believe that the size of the array will be solved before value assignment to the array .
Moreover i also wanted to know if unique could be used for 2 dimentional dynamic array .
You can't use the random number generator (RNG) to enumerate all possible solutions of your problem. It's not built for this. An RNG can give you one of these solutions with each call to randomize(). It's not guaranteed, though, that it gives you a different solution with each call. Say you have 3 solutions, S0, S1, S2. The solver could give you S1, then S2, then S1 again, then S1, then S0, etc. If you know how many solutions there are, you can stop once you've seen them all. Generally, though, you don't know this beforehand.
What an RNG can do, however, is check whether a solution you provide is correct. If you loop over all possible solutions, you can filter out only the ones that are correct. In your case, you have N balls and up to N groups. You can start out by putting each ball into one group and trying if this is a correct solution. You can then put 2 balls into one group and all the other N - 2 into a groups of one. You can put two other balls into one group and all the others into groups of one. You can start putting 2 balls into one group, 2 other balls into one group and all the other N - 4 into groups of one. You can continue this until you put all N balls into the same group. I'm not really sure how you can easily enumerate all solutions. Combinatorics can help you here. At each step of the way you can check whether a certain ball arrangement satisfies the constraints:
// Array describing an arrangement of balls
// - the first dimension is the group
// - the second dimension is the index within the group
typedef unsigned int unsigned arrangement_t[][];
// Function that gives you the next arrangement to try out
function arrangement_t get_next_arrangement();
// ...
endfunction
arrangement = get_next_arrangement();
if (weight_distributor_o.randomize() with {
solution.size() == arrangement.size();
foreach (solution[i]) {
solution[i].size() == arrangement[i].size();
foreach (solution[i][j])
solution[i][j] == arrangement[i][j];
}
})
all_solutions.push_back(arrangement);
Now, let's look at weight_distributor. I'd recommend you write each requirement in an own constraint as this makes the code much more readable.
You can shorten you uniqueness constraint that you wrote as a double loop to using the unique operator:
class weight_distributor;
// ...
constraint unique_balls {
unique { solution_array };
}
endclass
You already had a constraint that each group can have at most MAX_WEIGHT in it:
class weight_distributor;
// ...
constraint max_weight_per_group {
foreach (solution_array[i])
solution_array[i].sum() < `MAX_WEIGHT_BUCKET;
}
endclass
Because of the way array sizes are solved, it's not possible to write constraints that will ensure that you can compute a valid solution using simple calls randomize(). You don't need this, though, if you want to check whether a solution is valid. This is due to the constraints on array sizes in the between arrangement and solution_array.
Try this.!
class ABC;
rand bit[3:0] md_array [][]; // Multidimansional Arrays with unknown size
constraint c_md_array {
// First assign the size of the first dimension of md_array
md_array.size() == 2;
// Then for each sub-array in the first dimension do the following:
foreach (md_array[i]) {
// Randomize size of the sub-array to a value within the range
md_array[i].size() inside {[1:5]};
// Iterate over the second dimension
foreach (md_array[i][j]) {
// Assign constraints for values to the second dimension
md_array[i][j] inside {[1:10]};
}
}
}
endclass
module tb;
initial begin
ABC abc = new;
abc.randomize();
$display ("md_array = %p", abc.md_array);
end
endmodule
https://www.chipverify.com/systemverilog/systemverilog-foreach-constraint
So I've spent hours trying to work out exactly how this code produces prime numbers.
lazy val ps: Stream[Int] = 2 #:: Stream.from(3).filter(i =>
ps.takeWhile{j => j * j <= i}.forall{ k => i % k > 0});
I've used a number of printlns etc, but nothings making it clearer.
This is what I think the code does:
/**
* [2,3]
*
* takeWhile 2*2 <= 3
* takeWhile 2*2 <= 4 found match
* (4 % [2,3] > 1) return false.
* takeWhile 2*2 <= 5 found match
* (5 % [2,3] > 1) return true
* Add 5 to the list
* takeWhile 2*2 <= 6 found match
* (6 % [2,3,5] > 1) return false
* takeWhile 2*2 <= 7
* (7 % [2,3,5] > 1) return true
* Add 7 to the list
*/
But If I change j*j in the list to be 2*2 which I assumed would work exactly the same, it causes a stackoverflow error.
I'm obviously missing something fundamental here, and could really use someone explaining this to me like I was a five year old.
Any help would be greatly appreciated.
I'm not sure that seeking a procedural/imperative explanation is the best way to gain understanding here. Streams come from functional programming and they're best understood from that perspective. The key aspects of the definition you've given are:
It's lazy. Other than the first element in the stream, nothing is computed until you ask for it. If you never ask for the 5th prime, it will never be computed.
It's recursive. The list of prime numbers is defined in terms of itself.
It's infinite. Streams have the interesting property (because they're lazy) that they can represent a sequence with an infinite number of elements. Stream.from(3) is an example of this: it represents the list [3, 4, 5, ...].
Let's see if we can understand why your definition computes the sequence of prime numbers.
The definition starts out with 2 #:: .... This just says that the first number in the sequence is 2 - simple enough so far.
The next part defines the rest of the prime numbers. We can start with all the counting numbers starting at 3 (Stream.from(3)), but we obviously need to filter a bunch of these numbers out (i.e., all the composites). So let's consider each number i. If i is not a multiple of a lesser prime number, then i is prime. That is, i is prime if, for all primes k less than i, i % k > 0. In Scala, we could express this as
nums.filter(i => ps.takeWhile(k => k < i).forall(k => i % k > 0))
However, it isn't actually necessary to check all lesser prime numbers -- we really only need to check the prime numbers whose square is less than or equal to i (this is a fact from number theory*). So we could instead write
nums.filter(i => ps.takeWhile(k => k * k <= i).forall(k => i % k > 0))
So we've derived your definition.
Now, if you happened to try the first definition (with k < i), you would have found that it didn't work. Why not? It has to do with the fact that this is a recursive definition.
Suppose we're trying to decide what comes after 2 in the sequence. The definition tells us to first determine whether 3 belongs. To do so, we consider the list of primes up to the first one greater than or equal to 3 (takeWhile(k => k < i)). The first prime is 2, which is less than 3 -- so far so good. But we don't yet know the second prime, so we need to compute it. Fine, so we need to first see whether 3 belongs ... BOOM!
* It's pretty easy to see that if a number n is composite then the square of one of its factors must be less than or equal to n. If n is composite, then by definition n == a * b, where 1 < a <= b < n (we can guarantee a <= b just by labeling the two factors appropriately). From a <= b it follows that a^2 <= a * b, so it follows that a^2 <= n.
Your explanations are mostly correct, you made only two mistakes:
takeWhile doesn't include the last checked element:
scala> List(1,2,3).takeWhile(_<2)
res1: List[Int] = List(1)
You assume that ps always contains only a two and a three but because Stream is lazy it is possible to add new elements to it. In fact each time a new prime is found it is added to ps and in the next step takeWhile will consider this new added element. Here, it is important to remember that the tail of a Stream is computed only when it is needed, thus takeWhile can't see it before forall is evaluated to true.
Keep these two things in mind and you should came up with this:
ps = [2]
i = 3
takeWhile
2*2 <= 3 -> false
forall on []
-> true
ps = [2,3]
i = 4
takeWhile
2*2 <= 4 -> true
3*3 <= 4 -> false
forall on [2]
4%2 > 0 -> false
ps = [2,3]
i = 5
takeWhile
2*2 <= 5 -> true
3*3 <= 5 -> false
forall on [2]
5%2 > 0 -> true
ps = [2,3,5]
i = 6
...
While these steps describe the behavior of the code, it is not fully correct because not only adding elements to the Stream is lazy but every operation on it. This means that when you call xs.takeWhile(f) not all values until the point when f is false are computed at once - they are computed when forall wants to see them (because it is the only function here that needs to look at all elements before it definitely can result to true, for false it can abort earlier). Here the computation order when laziness is considered everywhere (example only looking at 9):
ps = [2,3,5,7]
i = 9
takeWhile on 2
2*2 <= 9 -> true
forall on 2
9%2 > 0 -> true
takeWhile on 3
3*3 <= 9 -> true
forall on 3
9%3 > 0 -> false
ps = [2,3,5,7]
i = 10
...
Because forall is aborted when it evaluates to false, takeWhile doesn't calculate the remaining possible elements.
That code is easier (for me, at least) to read with some variables renamed suggestively, as
lazy val ps: Stream[Int] = 2 #:: Stream.from(3).filter(i =>
ps.takeWhile{p => p * p <= i}.forall{ p => i % p > 0});
This reads left-to-right quite naturally, as
primes are 2, and those numbers i from 3 up, that all of the primes p whose square does not exceed the i, do not divide i evenly (i.e. without some non-zero remainder).
In a true recursive fashion, to understand this definition as defining the ever increasing stream of primes, we assume that it is so, and from that assumption we see that no contradiction arises, i.e. the truth of the definition holds.
The only potential problem after that, is the timing of accessing the stream ps as it is being defined. As the first step, imagine we just have another stream of primes provided to us from somewhere, magically. Then, after seeing the truth of the definition, check that the timing of the access is okay, i.e. we never try to access the areas of ps before they are defined; that would make the definition stuck, unproductive.
I remember reading somewhere (don't recall where) something like the following -- a conversation between a student and a wizard,
student: which numbers are prime?
wizard: well, do you know what number is the first prime?
s: yes, it's 2.
w: okay (quickly writes down 2 on a piece of paper). And what about the next one?
s: well, next candidate is 3. we need to check whether it is divided by any prime whose square does not exceed it, but I don't yet know what the primes are!
w: don't worry, I'l give them to you. It's a magic I know; I'm a wizard after all.
s: okay, so what is the first prime number?
w: (glances over the piece of paper) 2.
s: great, so its square is already greater than 3... HEY, you've cheated! .....
Here's a pseudocode1 translation of your code, read partially right-to-left, with some variables again renamed for clarity (using p for "prime"):
ps = 2 : filter (\i-> all (\p->rem i p > 0) (takeWhile (\p->p^2 <= i) ps)) [3..]
which is also
ps = 2 : [i | i <- [3..], and [rem i p > 0 | p <- takeWhile (\p->p^2 <= i) ps]]
which is a bit more visually apparent, using list comprehensions. and checks that all entries in a list of Booleans are True (read | as "for", <- as "drawn from", , as "such that" and (\p-> ...) as "lambda of p").
So you see, ps is a lazy list of 2, and then of numbers i drawn from a stream [3,4,5,...] such that for all p drawn from ps such that p^2 <= i, it is true that i % p > 0. Which is actually an optimal trial division algorithm. :)
There's a subtlety here of course: the list ps is open-ended. We use it as it is being "fleshed-out" (that of course, because it is lazy). When ps are taken from ps, it could potentially be a case that we run past its end, in which case we'd have a non-terminating calculation on our hands (a "black hole"). It just so happens :) (and needs to ⁄ can be proved mathematically) that this is impossible with the above definition. So 2 is put into ps unconditionally, so there's something in it to begin with.
But if we try to "simplify",
bad = 2 : [i | i <- [3..], and [rem i p > 0 | p <- takeWhile (\p->p < i) bad]]
it stops working after producing just one number, 2: when considering 3 as the candidate, takeWhile (\p->p < 3) bad demands the next number in bad after 2, but there aren't yet any more numbers there. It "jumps ahead of itself".
This is "fixed" with
bad = 2 : [i | i <- [3..], and [rem i p > 0 | p <- [2..(i-1)] ]]
but that is a much much slower trial division algorithm, very far from the optimal one.
--
1 (Haskell actually, it's just easier for me that way :) )