I've to make an optimisation for the manufacturing of wood plank.
I have base planks with two possible dimensions: L1 x l1 and L2 x l2.
I was asked to produce planks of different known dimensions (in 1 or more copies for some) from these basic planks, the goal being to optimize the process and to use as few base planks as possible. Note that each plank must be made entirely from a same base board and thus cannot be made in 2 or more parts.
I first made a simple excel calculation on a single variable dimension but it did not take into account that a plank could be made under another one on a same base plank.
I then got interested in optimization. If I'm not mistaken I'm looking to optimize the difference between :
the number of base boards x total area of a base board - sum of planks to make
However, I have great difficulties to write mathematically the condition "each board is entirely made from one and the same base board L1 x l1 or L2 x l2".
Does someone would know how to do that? I've looked on internet but I found nothing.Also do you think it's possible to do that on basic Excel (I've never used VBA)?
Feel free to ask any questions (my english isn't the best, sorry if my explaination isn't clear enough :) )
Thanks and have a good day :)
The best way for me to understand J is emulating the interpreter. Since the language is compact and has little rules, it's been easy... with the exception of how ranks affect function evaluation.
I want to be able to see an expression and know what's J doing to get the result, step by step.
Is there a doc, or someone could give me an algorithm so I can calculate myself how a f " n m b is evaluated?
Thanks in advance.
For learning about Rank the most accessible text is probably chapter 6 of J for C Programmers. The section of Eric Iverson's Primer that begins with Atom and goes through Checkpoint E covers the topic more concisely. Chapter 7 of Learning J is another place Rank is covered. All are valuable.
The most in-depth examination of Rank is Roger Hui's essay Rank and Uniformity. Hui's paper will make better reading after you've studied the other texts on this topic. Should it come down to wanting the nitty-gritty of implementation, you could dive into the interpreter source code. Personally, I'd not do that last one. Were I wanting to look at implementation algorithms I'd build a little model, and check it against the results of a J interpreter to make sure that my understanding of Rank matches.
Rank, in my view, is the most important concept in J. It is quite abstract in that it applies across all the shapes that nouns can take. The associated concepts are important to learn. These include shape, frame, cell, and agreement. These are explained individually in the Primer, but they're explained in some manner every time the topic is dealt with in depth.
The better your understanding of the Rank conjunction, and the broader world of noun Rank and verb Rank in which it applies, the more useful you'll find the three sections of the Vocabulary that deal with this conjunction. (Those sections are m"n , u"n , and m"v u"v .)
If you do come to write any algorithms that help you examine things in a step-by-step fashion, other J programmers will enjoy seeing them, I'm sure. I don't know of anything along those lines other than the actual interpreter source code.
This is a problem appeared in today's Pacific NW Region Programming Contest during which no one solved it. It is problem B and the complete problem set is here: http://www.acmicpc-pacnw.org/icpc-statements-2011.zip. There is a well-known O(n^2) algorithm for LCS of two strings using Dynamic Programming. But when these strings are extended to rings I have no idea...
P.S. note that it is subsequence rather than substring, so the elements do not need to be adjacent to each other
P.S. It might not be O(n^2) but O(n^2lgn) or something that can give the result in 5 seconds on a common computer.
Searching the web, this appears to be covered by section 4.3 of the paper "Incremental String Comparison", by Landau, Myers, and Schmidt at cost O(ne) < O(n^2), where I think e is the edit distance. This paper also references a previous paper by Maes giving cost O(mn log m) with more general edit costs - "On a cyclic string to string correcting problem". Expecting a contestant to reproduce either of these papers seems pretty demanding to me - but as far as I can see the question does ask for the longest common subsequence on cyclic strings.
You can double the first and second string and then use the ordinary method, and later wrap the positions around.
It is a good idea to "double" the strings and apply the standard dynamic programing algorithm. The problem with it is that to get the optimal cyclic LCS one then has to "start the algorithm from multiple initial conditions". Just one initial condition (e.g. setting all Lij variables to 0 at the boundaries) will not do in general. In practice it turns out that the number of initial states that are needed are O(N) in number (they span a diagonal), so one gets back to an O(N^3) algorithm.
However, the approach does has some virtue as it can be used to design efficient O(N^2) heuristics (not exact but near exact) for CLCS.
I do not know if a true O(N^2) exist, and would be very interested if someone knows one.
The CLCS problem has quite interesting properties of "periodicity": the length of a CLCS of
p-times reapeated strings is p times the CLCS of the strings. This can be proved by adopting a geometric view off the problem.
Also, there are some additional benefits of the problem: it can be shown that if Lc(N) denotes the averaged value of the CLCS length of two random strings of length N, then
|Lc(N)-CN| is O(\sqrt{N}) where C is Chvatal-Sankoff's constant. For the averaged length L(N) of the standard LCS, the only rate result of which I know says that |L(N)-CN| is O(sqrt(Nlog N)). There could be a nice way to compare Lc(N) with L(N) but I don't know it.
Another question: it is clear that the CLCS length is not superadditive contrary to the LCS length. By this I mean it is not true that CLCS(X1X2,Y1Y2) is always greater than CLCS(X1,Y1)+CLCS(X2,Y2) (it is very easy to find counter examples with a computer).
But it seems possible that the averaged length Lc(N) is superadditive (Lc(N1+N2) greater than Lc(N1)+Lc(N2)) - though if there is a proof I don't know it.
One modest interest in this question is that the values Lc(N)/N for the first few values of N would then provide good bounds to the Chvatal-Sankoff constant (much better than L(N)/N).
As a followup to mcdowella's answer, I'd like to point out that the O(n^2 lg n) solution presented in Maes' paper is the intended solution to the contest problem (check http://www.acmicpc-pacnw.org/ProblemSet/2011/solutions.zip). The O(ne) solution in Landau et al's paper does NOT apply to this problem, as that paper is targeted at edit distance, not LCS. In particular, the solution to cyclic edit distance only applies if the edit operations (add, delete, replace) all have unit (1, 1, 1) cost. LCS, on the other hand, is equivalent to edit distances with (add, delete, replace) costs (1, 1, 2). These are not equivalent to each other; for example, consider the input strings "ABC" and "CXY" (for the acyclic case; you can construct cyclic counterexamples similarly). The LCS of the two strings is "C", but the minimum unit-cost edit is to replace each character in turn.
At 110 lines but no complex data structures, Maes' solution falls towards the upper end of what is reasonable to implement in a contest setting. Even if Landau et al's solution could be adapted to handle cyclic LCS, the complexity of the data structure makes it infeasible in a contest setting.
Last but not least, I'd like to point out that an O(n^2) solution DOES exist for CLCS, described here: http://arxiv.org/abs/1208.0396 At 60 lines, no complex data structures, and only 2 arrays, this solution is quite reasonable to implement in a contest setting. Arriving at the solution might be a different matter, though.
I am a big fan of Stephen Wolfram, but he is definitely one not shy of tooting his own horn. In many references, he extols Mathematica as a different symbolic programming paradigm. I am not a Mathematica user.
My questions are: what is this symbolic programming? And how does it compare to functional languages (such as Haskell)?
When I hear the phrase "symbolic programming", LISP, Prolog and (yes) Mathematica immediately leap to mind. I would characterize a symbolic programming environment as one in which the expressions used to represent program text also happen to be the primary data structure. As a result, it becomes very easy to build abstractions upon abstractions since data can easily be transformed into code and vice versa.
Mathematica exploits this capability heavily. Even more heavily than LISP and Prolog (IMHO).
As an example of symbolic programming, consider the following sequence of events. I have a CSV file that looks like this:
r,1,2
g,3,4
I read that file in:
Import["somefile.csv"]
--> {{r,1,2},{g,3,4}}
Is the result data or code? It is both. It is the data that results from reading the file, but it also happens to be the expression that will construct that data. As code goes, however, this expression is inert since the result of evaluating it is simply itself.
So now I apply a transformation to the result:
% /. {c_, x_, y_} :> {c, Disk[{x, y}]}
--> {{r,Disk[{1,2}]},{g,Disk[{3,4}]}}
Without dwelling on the details, all that has happened is that Disk[{...}] has been wrapped around the last two numbers from each input line. The result is still data/code, but still inert. Another transformation:
% /. {"r" -> Red, "g" -> Green}
--> {{Red,Disk[{1,2}]},{Green,Disk[{3,4}]}}
Yes, still inert. However, by a remarkable coincidence this last result just happens to be a list of valid directives in Mathematica's built-in domain-specific language for graphics. One last transformation, and things start to happen:
% /. x_ :> Graphics[x]
--> Graphics[{{Red,Disk[{1,2}]},{Green,Disk[{3,4}]}}]
Actually, you would not see that last result. In an epic display of syntactic sugar, Mathematica would show this picture of red and green circles:
But the fun doesn't stop there. Underneath all that syntactic sugar we still have a symbolic expression. I can apply another transformation rule:
% /. Red -> Black
Presto! The red circle became black.
It is this kind of "symbol pushing" that characterizes symbolic programming. A great majority of Mathematica programming is of this nature.
Functional vs. Symbolic
I won't address the differences between symbolic and functional programming in detail, but I will contribute a few remarks.
One could view symbolic programming as an answer to the question: "What would happen if I tried to model everything using only expression transformations?" Functional programming, by contrast, can been seen as an answer to: "What would happen if I tried to model everything using only functions?" Just like symbolic programming, functional programming makes it easy to quickly build up layers of abstractions. The example I gave here could be easily be reproduced in, say, Haskell using a functional reactive animation approach. Functional programming is all about function composition, higher level functions, combinators -- all the nifty things that you can do with functions.
Mathematica is clearly optimized for symbolic programming. It is possible to write code in functional style, but the functional features in Mathematica are really just a thin veneer over transformations (and a leaky abstraction at that, see the footnote below).
Haskell is clearly optimized for functional programming. It is possible to write code in symbolic style, but I would quibble that the syntactic representation of programs and data are quite distinct, making the experience suboptimal.
Concluding Remarks
In conclusion, I advocate that there is a distinction between functional programming (as epitomized by Haskell) and symbolic programming (as epitomized by Mathematica). I think that if one studies both, then one will learn substantially more than studying just one -- the ultimate test of distinctness.
Leaky Functional Abstraction in Mathematica?
Yup, leaky. Try this, for example:
f[x_] := g[Function[a, x]];
g[fn_] := Module[{h}, h[a_] := fn[a]; h[0]];
f[999]
Duly reported to, and acknowledged by, WRI. The response: avoid the use of Function[var, body] (Function[body] is okay).
You can think of Mathematica's symbolic programming as a search-and-replace system where you program by specifying search-and-replace rules.
For instance you could specify the following rule
area := Pi*radius^2;
Next time you use area, it'll be replaced with Pi*radius^2. Now, suppose you define new rule
radius:=5
Now, whenever you use radius, it'll get rewritten into 5. If you evaluate area it'll get rewritten into Pi*radius^2 which triggers rewriting rule for radius and you'll get Pi*5^2 as an intermediate result. This new form will trigger a built-in rewriting rule for ^ operation so the expression will get further rewritten into Pi*25. At this point rewriting stops because there are no applicable rules.
You can emulate functional programming by using your replacement rules as function. For instance, if you want to define a function that adds, you could do
add[a_,b_]:=a+b
Now add[x,y] gets rewritten into x+y. If you want add to only apply for numeric a,b, you could instead do
add[a_?NumericQ, b_?NumericQ] := a + b
Now, add[2,3] gets rewritten into 2+3 using your rule and then into 5 using built-in rule for +, whereas add[test1,test2] remains unchanged.
Here's an example of an interactive replacement rule
a := ChoiceDialog["Pick one", {1, 2, 3, 4}]
a+1
Here, a gets replaced with ChoiceDialog, which then gets replaced with the number the user chose on the dialog that popped up, which makes both quantities numeric and triggers replacement rule for +. Here, ChoiceDialog as a built-in replacement rule along the lines of "replace ChoiceDialog[some stuff] with the value of button the user clicked".
Rules can be defined using conditions which themselves need to go through rule-rewriting in order to produce True or False. For instance suppose you invented a new equation solving method, but you think it only works when the final result of your method is positive. You could do the following rule
solve[x + 5 == b_] := (result = b - 5; result /; result > 0)
Here, solve[x+5==20] gets replaced with 15, but solve[x + 5 == -20] is unchanged because there's no rule that applies. The condition that prevents this rule from applying is /;result>0. Evaluator essentially looks the potential output of rule application to decide whether to go ahead with it.
Mathematica's evaluator greedily rewrites every pattern with one of the rules that apply for that symbol. Sometimes you want to have finer control, and in such case you could define your own rules and apply them manually like this
myrules={area->Pi radius^2,radius->5}
area//.myrules
This will apply rules defined in myrules until result stops changing. This is pretty similar to the default evaluator, but now you could have several sets of rules and apply them selectively. A more advanced example shows how to make a Prolog-like evaluator that searches over sequences of rule applications.
One drawback of current Mathematica version comes up when you need to use Mathematica's default evaluator (to make use of Integrate, Solve, etc) and want to change default sequence of evaluation. That is possible but complicated, and I like to think that some future implementation of symbolic programming will have a more elegant way of controlling evaluation sequence
As others here already mentioned, Mathematica does a lot of term rewriting. Maybe Haskell isn't the best comparison though, but Pure is a nice functional term-rewriting language (that should feel familiar to people with a Haskell background). Maybe reading their Wiki page on term rewriting will clear up a few things for you:
http://code.google.com/p/pure-lang/wiki/Rewriting
Mathematica is using term rewriting heavily. The language provides special syntax for various forms of rewriting, special support for rules and strategies. The paradigm is not that "new" and of course it's not unique, but they're definitely on a bleeding edge of this "symbolic programming" thing, alongside with the other strong players such as Axiom.
As for comparison to Haskell, well, you could do rewriting there, with a bit of help from scrap your boilerplate library, but it's not nearly as easy as in a dynamically typed Mathematica.
Symbolic shouldn't be contrasted with functional, it should be contrasted with numerical programming. Consider as an example MatLab vs Mathematica. Suppose I want the characteristic polynomial of a matrix. If I wanted to do that in Mathematica, I could do get an identity matrix (I) and the matrix (A) itself into Mathematica, then do this:
Det[A-lambda*I]
And I would get the characteristic polynomial (never mind that there's probably a characteristic polynomial function), on the other hand, if I was in MatLab I couldn't do it with base MatLab because base MatLab (never mind that there's probably a characteristic polynomial function) is only good at calculating finite-precision numbers, not things where there are random lambdas (our symbol) in there. What you'd have to do is buy the add-on Symbolab, and then define lambda as its own line of code and then write this out (wherein it would convert your A matrix to a matrix of rational numbers rather than finite precision decimals), and while the performance difference would probably be unnoticeable for a small case like this, it would probably do it much slower than Mathematica in terms of relative speed.
So that's the difference, symbolic languages are interested in doing calculations with perfect accuracy (often using rational numbers as opposed to numerical) and numerical programming languages on the other hand are very good at the vast majority of calculations you would need to do and they tend to be faster at the numerical operations they're meant for (MatLab is nearly unmatched in this regard for higher level languages - excluding C++, etc) and a piss poor at symbolic operations.
In reading Haskell-related stuff I sometimes come across the expression “tying the knot”, I think I understand what it does, but not how.
So, are there any good, basic, and simple to understand explanations of this concept?
Tying the knot is a solution to the problem of circular data structures. In imperative languages you construct a circular structure by first creating a non-circular structure, and then going back and fixing up the pointers to add the circularity.
Say you wanted a two-element circular list with the elements "0" and "1". It would seem impossible to construct because if you create the "1" node and then create the "0" node to point at it, you cannot then go back and fix up the "1" node to point back at the "0" node. So you have a chicken-and-egg situation where both nodes need to exist before either can be created.
Here is how you do it in Haskell. Consider the following value:
alternates = x where
x = 0 : y
y = 1 : x
In a non-lazy language this will be an infinite loop because of the unterminated recursion. But in Haskell lazy evaluation does the Right Thing: it generates a two-element circular list.
To see how it works in practice, think about what happens at run-time. The usual "thunk" implementation of lazy evaluation represents an unevaluated expression as a data structure containing a function pointer plus the arguments to be passed to the function. When this is evaluated the thunk is replaced by the actual value so that future references don't have to call the function again.
When you take the first element of the list 'x' is evaluated down to a value (0, &y), where the "&y" bit is a pointer to the value of 'y'. Since 'y' has not been evaluated this is currently a thunk. When you take the second element of the list the computer follows the link from x to this thunk and evaluates it. It evaluates to (1, &x), or in other words a pointer back to the original 'x' value. So you now have a circular list sitting in memory. The programmer doesn't need to fix up the back-pointers because the lazy evaluation mechanism does it for you.
It's not quite what you asked for, and it's not directly related to Haskell, but Bruce McAdam's paper That About Wraps It Up goes into this topic in substantial breadth and depth. Bruce's basic idea is to use an explicit knot-tying operator called WRAP instead of the implicit knot-tying that is done automatically in Haskell, OCaml, and some other languages. The paper has lots of entertaining examples, and if you are interested in knot-tying I think you will come away with a much better feel for the process.