I have a set S. It contains N subsets (which in turn contain some sub-subsets of various lengths):
1. [[a,b],[c,d],[*]]
2. [[c],[d],[e,f],[*]]
3. [[d,e],[f],[f,*]]
N. ...
I also have a list L of 'unique' elements that are contained in the set S:
a, b, c, d, e, f, *
I need to find all possible combinations between each sub-subset from each subset so, that each resulting combination has exactly one element from the list L, but any number of occurrences of the element [*] (it is a wildcard element).
So, the result of the needed function working with the above mentioned set S should be (not 100% accurate):
- [a,b],[c],[d,e],[f];
- [a,b],[c],[*],[d,e],[f];
- [a,b],[c],[d,e],[f],[*];
- [a,b],[c],[d,e],[f,*],[*];
So, basically I need an algorithm that does the following:
take a sub-subset from the subset 1,
add one more sub-subset from the subset 2 maintaining the list of 'unique' elements acquired so far (the check on the 'unique' list is skipped if the sub-subset contains the * element);
Repeat 2 until N is reached.
In other words, I need to generate all possible 'chains' (it is pairs, if N == 2, and triples if N==3), but each 'chain' should contain exactly one element from the list L except the wildcard element * that can occur many times in each generated chain.
I know how to do this with N == 2 (it is a simple pair generation), but I do not know how to enhance the algorithm to work with arbitrary values for N.
Maybe Stirling numbers of the second kind could help here, but I do not know how to apply them to get the desired result.
Note: The type of data structure to be used here is not important for me.
Note: This question has grown out from my previous similar question.
These are some pointers (not a complete code) that can take you to right direction probably:
I don't think you will need some advanced data structures here (make use of erlang list comprehensions). You must also explore erlang sets and lists module. Since you are dealing with sets and list of sub-sets, they seems like an ideal fit.
Here is how things with list comprehensions will get solved easily for you: [{X,Y} || X <- [[c],[d],[e,f]], Y <- [[a,b],[c,d]]]. Here i am simply generating a list of {X,Y} 2-tuples but for your use case you will have to put real logic here (including your star case)
Further note that with list comprehensions, you can use output of one generator as input of a later generator e.g. [{X,Y} || X1 <- [[c],[d],[e,f]], X <- X1, Y1 <- [[a,b],[c,d]], Y <- Y1].
Also for removing duplicates from a list of things L = ["a", "b", "a"]., you can anytime simply do sets:to_list(sets:from_list(L)).
With above tools you can easily generate all possible chains and also enforce your logic as these chains get generated.
Related
I have a function that does an array product:
arrayProduct(l1,l2,l3) = [[a, b, c] |
a := l1[_]
b := l2[_]
c := l3[_]
]
If I have three arrays defined as follows:
animals1 = ["hippo", "giraffe"]
animals2 = ["lion", "zebra"]
animals3 = ["deer", "bear"]
Then the output of arrayProduct(animals1, animals2, animals3) would be:
[["hippo","lion","deer"],["hippo","lion","bear"],["hippo","zebra","deer"],["hippo","zebra","bear"],["giraffe","lion","deer"],["giraffe","lion","bear"],["giraffe","zebra","deer"],["giraffe","zebra","bear"]]
If I can guarantee that the inputs will always be lists is there away I could make a function that would do the same thing except it could accept a dynamic number of lists as input instead of just 3?
I'm also exploring if it would also be possible to do this with only one argument containing all the arrays within it as opposed to accepting multiple arguments. For example:
[["hippo", "giraffe"], ["lion", "zebra"], ["deer", "bear"], ["ostrich", "flamingo"]]
Any insight into a solution with either approach would be appreciated.
There's no known way to compute an arbitrary N-way cross product in Rego without a builtin.
Why something can't be written in a language can be tricky to explain because it amounts to a proof-sketch. We need to make the argument that there is no policy in Rego that computes an N-way cross product. The formal proofs of expressiveness/complexity have not been worked out, so the best we can do is try to articulate why it might not be possible.
For the N-way cross product, it boils down to the fact that Rego guarantees termination for all policies on all inputs, and to do that it restricts how deeply nested iteration can be. In your example (using some and indentation for clarity) you have 3 nested loops with indexes i, j, k.
arrayProduct(l1,l2,l3) = [[a, b, c] |
some i
a := l1[i]
some j
b := l2[j]
some k
c := l3[k]
]
To implement an N-way cross product arrayProduct([l1, l2, ..., ln]) you would need something equivalent to N nested loops:
# NOT valid Rego
arrayProduct([l1,l2,...,ln]) = [[a, b, ..., n] |
some i1
a := l1[i1]
some i2
b := l2[i2]
...
n := ln[in]
]
where importantly the degree of nested iteration N depends on the input.
To guarantee termination, Rego restricts the degree of nested iteration in a policy. You can only nest iteration as many times as you have some (or more properly variables) appearing in your policy. This is analogous to SQL restricting the number of JOINs to those that appear in the query and view definitions.
Since the degree of nesting required for an N-way cross product is N, and N can be larger than the number of somes in the policy, there is no way to implement the N-way cross product.
As a point of contrast, the number of keys or values that are iterated over inside any one loop CAN and usually DO depend on the input. It's the number of loops that cannot depend on the input.
It's not possible to compute an n-ary product of lists/arrays (or sets or objects) in Rego without adding a built-in function.
In the scenario described above, providing a dynamic number of arrays as input to the function would be equivalent to passing an array of arrays (like you mentioned at the end):
arrayProduct([arr1, arr2, ..., arrN])
This works, except that when we try to implement arrayProduct we get stuck because Rego does not permit recursion and iteration only occurs when you inject a variable into a reference. In your original example l1[_] is a reference to the elements in the first list and _ is a unique variable referring to the array indices in that list.
OPA/Rego evaluates that expression by finding assignments to each _ that satisfy the query. The "problem" is that this requires one variable for each list in the input. If the length of the array of arrays is unknown, we would need an infinite number of variables.
If you really need an n-ary product function I would suggest you implement a custom built-in function for now.
I need a help with following:
flatten ([]) -> [];
flatten([H|T]) -> H ++ flatten(T).
Input List contain other lists with a different length
For example:
flatten([[1,2,3],[4,7],[9,9,9,9,9,9]]).
What is the time complexity of this function?
And why?
I got it to O(n) where n is a number of elements in the Input list.
For example:
flatten([[1,2,3],[4,7],[9,9,9,9,9,9]]) n=3
flatten([[1,2,3],[4,7],[9,9,9,9,9,9],[3,2,4],[1,4,6]]) n=5
Thanks for help.
First of all I'm not sure your code will work, at least not in the way standard library works. You could compare your function with lists:flatten/1 and maybe improve on your implementation. Try lists such as [a, [b, c]] and [[a], [b, [c]], [d]] as input and verify if you return what you expected.
Regarding complexity it is little tricky due to ++ operator and functional (immutable) nature of the language. All lists in Erlang are linked lists (not arrays like in C++), and you can not just add something to end of one without modifying it; before it was pointing to end of list, now you would like it to link to something else. And again, since it is not mutable language you have to make copy of whole list left of ++ operator, which increases complexity of this operator.
You could say that complexity of A ++ B is length(A), and it makes complexity of your function little bit greater. It would go something like length(FirstElement) + (lenght(FirstElement) + length(SecondElement)) + .... up to (without) last, which after some math magic could be simplified to (n -1) * 1/2 * k * k where n is number of elements, and k is average length of element. Or O(n^3).
If you are new to this it might seem little bit odd, but with some practice you can get hang of it. I would recommend going through few resources:
Good explanation of lists and how they are created
Documentation on list handling with DO and DO NOT parts
Short description of ++ operator myths and best practices
Chapter about recursion and tail-recursion with examples using ++ operator
I am using Triq (erlang quickcheck) and I am having trouble generating a set of nice rules for my program.
What I want to generate are things that looks like this:
A -> B
where I would like to provide A and the size of B, with latter not having any dupicates.
For example, if I say generate me rules with L.H.S. of [a] and R.H.S. of size 4 (ie. A = [a] and size(B) = 4) I would like to get something like this:
{rule, [a], [1,2,4,5]}
{rule, [a], [a,d,c,e]}
{rule, [a], [q,d,3,4]}
Note, I don't want any dupicates in B (this is the part I'm having trouble with). Also, it doesn't really matter what B is made up of - it can be anything as long as it is distinct and without dupicates.
My spec is far too messy to show here, so I'd rather not.
I am not familiar with Triq, but in PropEr and Quviq's Qickcheck you can use ?SUCHTHAT conditions that filter 'bad' instances.
If a generated instance does not satisfy a ?SUCHTHAT constraint it is discarded and not counted as a valid test. You could use this mechanism to generate lists of the specified size (i.e. what PropEr calls 'vectors') and then discard those that have duplicates, but I think that too many instances would then be discarded (see also the link).
It is usually more efficient to tinker with the generator so that all instances are valid, in your case by e.g. generating (3) X-times as many elements, removing duplicates and keeping as many as you need. This can still fail, and it will fail, so you need to guard against it.
Here is a generator for your case, in PropEr, together with a dummy property:
-module(dummy).
-export([rule_prop/0]).
-include_lib("proper/include/proper.hrl").
-define(X, 5).
rule_prop() ->
?FORALL(_, rule_gen(integer(), 4, integer()), true).
rule_gen(A, SizeB, TypeB) ->
?LET(
EnoughB,
?SUCHTHAT(
NoDupB,
?LET(
ManyB,
vector(?X * SizeB, TypeB),
no_dups(ManyB)
),
length(NoDupB) >= SizeB
),
begin
B = lists:sublist(EnoughB, SizeB),
{rule, A, B}
end).
no_dups([]) ->
[];
no_dups([A|B]) ->
[A | no_dups([X || X <- B, X =/= A])].
Maps, filters, folds and more : http://learnyousomeerlang.com/higher-order-functions#maps-filters-folds
The more I read ,the more i get confused.
Can any body help simplify these concepts?
I am not able to understand the significance of these concepts.In what use cases will these be needed?
I think it is majorly because of the syntax,diff to find the flow.
The concepts of mapping, filtering and folding prevalent in functional programming actually are simplifications - or stereotypes - of different operations you perform on collections of data. In imperative languages you usually do these operations with loops.
Let's take map for an example. These three loops all take a sequence of elements and return a sequence of squares of the elements:
// C - a lot of bookkeeping
int data[] = {1,2,3,4,5};
int squares_1_to_5[sizeof(data) / sizeof(data[0])];
for (int i = 0; i < sizeof(data) / sizeof(data[0]); ++i)
squares_1_to_5[i] = data[i] * data[i];
// C++11 - less bookkeeping, still not obvious
std::vec<int> data{1,2,3,4,5};
std::vec<int> squares_1_to_5;
for (auto i = begin(data); i < end(data); i++)
squares_1_to_5.push_back((*i) * (*i));
// Python - quite readable, though still not obvious
data = [1,2,3,4,5]
squares_1_to_5 = []
for x in data:
squares_1_to_5.append(x * x)
The property of a map is that it takes a collection of elements and returns the same number of somehow modified elements. No more, no less. Is it obvious at first sight in the above snippets? No, at least not until we read loop bodies. What if there were some ifs inside the loops? Let's take the last example and modify it a bit:
data = [1,2,3,4,5]
squares_1_to_5 = []
for x in data:
if x % 2 == 0:
squares_1_to_5.append(x * x)
This is no longer a map, though it's not obvious before reading the body of the loop. It's not clearly visible that the resulting collection might have less elements (maybe none?) than the input collection.
We filtered the input collection, performing the action only on some elements from the input. This loop is actually a map combined with a filter.
Tackling this in C would be even more noisy due to allocation details (how much space to allocate for the output array?) - the core idea of the operation on data would be drowned in all the bookkeeping.
A fold is the most generic one, where the result doesn't have to contain any of the input elements, but somehow depends on (possibly only some of) them.
Let's rewrite the first Python loop in Erlang:
lists:map(fun (E) -> E * E end, [1,2,3,4,5]).
It's explicit. We see a map, so we know that this call will return a list as long as the input.
And the second one:
lists:map(fun (E) -> E * E end,
lists:filter(fun (E) when E rem 2 == 0 -> true;
(_) -> false end,
[1,2,3,4,5])).
Again, filter will return a list at most as long as the input, map will modify each element in some way.
The latter of the Erlang examples also shows another useful property - the ability to compose maps, filters and folds to express more complicated data transformations. It's not possible with imperative loops.
They are used in almost every application, because they abstract different kinds of iteration over lists.
map is used to transform one list into another. Lets say, you have list of key value tuples and you want just the keys. You could write:
keys([]) -> [];
keys([{Key, _Value} | T]) ->
[Key | keys(T)].
Then you want to have values:
values([]) -> [];
values([{_Key, Value} | T}]) ->
[Value | values(T)].
Or list of only third element of tuple:
third([]) -> [];
third([{_First, _Second, Third} | T]) ->
[Third | third(T)].
Can you see the pattern? The only difference is what you take from the element, so instead of repeating the code, you can simply write what you do for one element and use map.
Third = fun({_First, _Second, Third}) -> Third end,
map(Third, List).
This is much shorter and the shorter your code is, the less bugs it has. Simple as that.
You don't have to think about corner cases (what if the list is empty?) and for experienced developer it is much easier to read.
filter searches lists. You give it function, that takes element, if it returns true, the element will be on the returned list, if it returns false, the element will not be there. For example filter logged in users from list.
foldl and foldr are used, when you have to do additional bookkeeping while iterating over the list - for example summing all the elements or counting something.
The best explanations, I've found about those functions are in books about Lisp: "Structure and Interpretation of Computer Programs" and "On Lisp" Chapter 4..
I am coding a survey that outputs a .csv file. Within this csv I have some entries that are space delimited, which represent multi-select questions (e.g. questions with more than one response). In the end I want to parse these space delimited entries into their own columns and create headers for them so i know where they came from.
For example I may start with this (note that the multiselect columns have an _M after them):
Q1, Q2_M, Q3, Q4_M
6, 1 2 88, 3, 3 5 99
6, , 3, 1 2
and I want to go to this:
Q1, Q2_M_1, Q2_M_2, Q2_M_88, Q3, Q4_M_1, Q4_M_2, Q4_M_3, Q4_M_5, Q4_M_99
6, 1, 1, 1, 3, 0, 0, 1, 1, 1
6,,,,3,1,1,0,0,0
I imagine this is a relatively common issue to deal with but I have not been able to find it in the R section. Any ideas how to do this in R after importing the .csv ? My general thoughts (which often lead to inefficient programs) are that I can:
(1) pull column numbers that have the special suffix with grep()
(2) loop through (or use an apply) each of the entries in these columns and determine the levels of responses and then create columns accordingly
(3) loop through (or use an apply) and place indicators in appropriate columns to indicate presence of selection
I appreciate any help and please let me know if this is not clear.
I agree with ran2 and aL3Xa that you probably want to change the format of your data to have a different column for each possible reponse. However, if you munging your dataset to a better format proves problematic, it is possible to do what you asked.
process_multichoice <- function(x) lapply(strsplit(x, " "), as.numeric)
q2 <- c("1 2 3 NA 4", "2 5")
processed_q2 <- process_multichoice(q2)
[[1]]
[1] 1 2 3 NA 4
[[2]]
[1] 2 5
The reason different columns for different responses are suggested is because it is still quite unpleasant trying to retrieve any statistics from the data in this form. Although you can do things like
# Number of reponses given
sapply(processed_q2, length)
#Frequency of each response
table(unlist(processed_q2), useNA = "ifany")
EDIT: One more piece of advice. Keep the code that processes your data separate from the code that analyses it. If you create any graphs, keep the code for creating them separate again. I've been down the road of mixing things together, and it isn't pretty. (Especially when you come back to the code six months later.)
I am not entirely sure what you trying to do respectively what your reasons are for coding like this. Thus my advice is more general – so just feel to clarify and I will try to give a more concrete response.
1) I say that you are coding the survey on your own, which is great because it means you have influence on your .csv file. I would NEVER use different kinds of separation in the same .csv file. Just do the naming from the very beginning, just like you suggested in the second block.
Otherwise you might geht into trouble with checkboxes for example. Let's say someone checks 3 out of 5 possible answers, the next only checks 1 (i.e. "don't know") . Now it will be much harder to create a spreadsheet (data.frame) type of results view as opposed to having an empty field (which turns out to be an NA in R) that only needs to be recoded.
2) Another important question is whether you intend to do a panel survey(i.e longitudinal study asking the same participants over and over again) . That (among many others) would be a good reason to think about saving your data to a MySQL database instead of .csv . RMySQL can connect directly to the database and access its tables and more important its VIEWS.
Views really help with survey data since you can rearrange the data in different views, conditional on many different needs.
3) Besides all the personal / opinion and experience, here's some (less biased) literature to get started:
Complex Surveys: A Guide to Analysis Using R (Wiley Series in Survey Methodology
The book is comparatively simple and leaves out panel surveys but gives a lot of R Code and examples which should be a practical start.
To prevent re-inventing the wheel you might want to check LimeSurvey, a pretty decent (not speaking of the templates :) ) tool for survey conductors. Besides I TYPO3 CMS extensions pbsurvey and ke_questionnaire (should) work well too (only tested pbsurvey).
Multiple choice items should always be coded as separate variables. That is, if you have 5 alternatives and multiple choice, you should code them as i1, i2, i3, i4, i5, i.e. each one is a binary variable (0-1). I see that you have values 3 5 99 for Q4_M variable in the first example. Does that mean that you have 99 alternatives in an item? Ouch...
First you should go on and create separate variables for each alternative in a multiple choice item. That is, do:
# note that I follow your example with Q4_M variable
dtf_ins <- as.data.frame(matrix(0, nrow = nrow(<initial dataframe>), ncol = 99))
# name vars appropriately
names(dtf_ins) <- paste("Q4_M_", 1:99, sep = "")
now you have a data.frame with 0s, so what you need to do is to get 1s in an appropriate position (this is a bit cumbersome), a function will do the job...
# first you gotta change spaces to commas and convert character variable to a numeric one
y <- paste("c(", gsub(" ", ", ", x), ")", sep = "")
z <- eval(parse(text = y))
# now you assing 1 according to indexes in z variable
dtf_ins[1, z] <- 1
And that's pretty much it... basically, you would like to reconsider creating a data.frame with _M variables, so you can write a function that does this insertion automatically. Avoid for loops!
Or, even better, create a matrix with logicals, and just do dtf[m] <- 1, where dtf is your multiple-choice data.frame, and m is matrix with logicals.
I would like to help you more on this one, but I'm recuperating after a looong night! =) Hope that I've helped a bit! =)
Thanks for all the responses. I agree with most of you that this format is kind of silly but it is what I have to work with (survey is coded and going into use next week). This is what I came up with from all the responses. I am sure this is not the most elegant or efficient way to do it but I think it should work.
colnums <- grep("_M",colnames(dat))
responses <- nrow(dat)
for (i in colnums) {
vec <- as.vector(dat[,i]) #turn into vector
b <- lapply(strsplit(vec," "),as.numeric) #split up and turn into numeric
c <- sort(unique(unlist(b))) #which values were used
newcolnames <- paste(colnames(dat[i]),"_",c,sep="") #column names
e <- matrix(nrow=responses,ncol=length(c)) #create new matrix for indicators
colnames(e) <- newcolnames
#next loop looks for responses and puts indicators in the correct places
for (i in 1:responses) {
e[i,] <- ifelse(c %in% b[[i]],1,0)
}
dat <- cbind(dat,e)
}
Suggestions for improvement are welcome.