Say you were to create a hash table that maps every possible valid 9x9 sudoku (not yet filled in) to its solution. (as infeasible a task as this would be)
Then you were to create a simple program that takes a valid 9x9 sudoku (again, not yet filled in) as input and returns the solution that is mapped to it in the hashtable described above.
Would this not be considered a sudoku solver that works in polynomial time?
Is there something about this theoretical solution that disqualifies it from being proof that sudoku is a P class problem?
I think you're misunderstanding the problem. From Wikipedia:
The general problem of solving Sudoku puzzles on n^2×n^2 grids of n×n blocks is known to be NP-complete.
Although the game may most usually come in a 9x9 variant, the generally-stated problem characterizes the relationship between the size of the grid and the complexity of finding a solution - not any individual grid. If your hypothetical is true, it would not fundamentally change the classification of the problem.
Also, consider how you would retrieve a candidate solution from such a hash table. If you use as the keys the sequence of all initial values and their locations, then you would need to keep all possible sets of initial values (81 choose 30, 1.4e22) - for each unique solution (6.7e21). (And that's only for solutions that start with thirty values showing...)
Related
Problem
I want to find
The first root
The first local minimum/maximum
of a black-box function in a given range.
The function has following properties:
It's continuous and differentiable.
It's combination of constant and periodic functions. All periods are known.
(It's better if it can be done with weaker assumptions)
What is the fastest way to get the root and the extremum?
Do I need more assumptions or bounds of the function?
What I've tried
I know I can use root-finding algorithm. What I don't know is how to find the first root efficiently.
It needs to be fast enough so that it can run within a few miliseconds with precision of 1.0 and range of 1.0e+8, which is the problem.
Since the range could be quite large and it should be precise enough, I can't brute-force it by checking all the possible subranges.
I considered bisection method, but it's too slow to find the first root if the function has only one big root in the range, as every subrange should be checked.
It's preferable if the solution is in java, but any similar language is fine.
Background
I want to calculate when arbitrary celestial object reaches certain height.
It's a configuration-defined virtual object, so I can't assume anything about the object.
It's not easy to get either analytical solution or simple approximation because various coordinates are involved.
I decided to find a numerical solution for this.
For a general black box function, this can't really be done. Any root finding algorithm on a black box function can't guarantee that it has found all the roots or any particular root, even if the function is continuous and differentiable.
The property of being periodic gives a bit more hope, but you can still have periodic functions with infinitely many roots in a bounded domain. Given that your function relates to celestial objects, this isn't likely to happen. Assuming your periodic functions are sinusoidal, I believe you can get away with checking subranges on the order of one-quarter of the shortest period (out of all the periodic components).
Maybe try Brent's Method on the shortest quarter period subranges?
Another approach would be to apply your root finding algorithm iteratively. If your range is (a, b), then apply your algorithm to that range to find a root at say c < b. Then apply your algorithm to the range (a, c) to find a root in that range. Continue until no more roots are found. The last root you found is a good candidate for your minimum root.
Black box function for any range? You cannot even be sure it has the continuous domain over that range. What kind of solutions are you looking for? Natural numbers, integers, real numbers, complex? These are all the question that greatly impact the answer.
So 1st thing should be determining what kind of number you accept as the result.
Second is having some kind of protection against limes of function that will try to explode your calculations as it goes for plus or minus infinity.
Since we are touching the limes topics you could have your solution edge towards zero and look like a solution but never touch 0 and become a solution. This depends on your margin of error, how close something has to be to be considered ok, it's good enough.
I think for this your SIMPLEST TO IMPLEMENT bet for real number solutions (I assume those) is to take an interval and this divide and conquer algorithm:
Take lower and upper border and middle value (or approx middle value for infinity decimals border/borders)
Try to calculate solution with all 3 and have some kind of protection against infinities
remember all 3 values in an array with results from them (3 pair of values)
remember the current best value (one its closest to solution) in seperate variable (a pair of value and result for that value)
STEP FORWARD - repeat above with 1st -2nd value range and 2nd -3rd value range
have a new pair of value and result to be closest to solution.
clear the old value-result pairs, replace them with new ones gotten from this iteration while remembering the best value solution pair (total)
Repeat above for how precise you wish to get and look at that memory explode with each iteration, keep in mind you are gonna to have exponential growth of values there. It can be further improved if you lets say take one interval and go as deep as you wanna, remember best value-result pair and then delete all other memory and go for next interval and dig deep.
I am looking for a simple method to assign a number to a mathematical expression, say between 0 and 1, that conveys how simplified that expression is (being 1 as fully simplified). For example:
eval('x+1') should return 1.
eval('1+x+1+x+x-5') should returns some value less than 1, because it is far from being simple (i.e., it can be further simplified).
The parameter of eval() could be either a string or an abstract syntax tree (AST).
A simple idea that occurred to me was to count the number of operators (?)
EDIT: Let simplified be equivalent to how close a system is to the solution of a problem. E.g., given an algebra problem (i.e. limit, derivative, integral, etc), it should assign a number to tell how close it is to the solution.
The closest metaphor I can come up with it how a maths professor would look at an incomplete problem and mentally assess it in order to tell how close the student is to the solution. Like in a math exam, were the student didn't finished a problem worth 20 points, but the professor assigns 8 out of 20. Why would he come up with 8/20, and can we program such thing?
I'm going to break a stack-overflow rule and post this as an answer instead of a comment, because not only I'm pretty sure the answer is you can't (at least, not the way you imagine), but also because I believe it can be educational up to a certain degree.
Let's assume that a criteria of simplicity can be established (akin to a normal form). It seems to me that you are very close to trying to solve an analogous to entscheidungsproblem or the halting problem. I doubt that in a complex rule system required for typical algebra, you can find a method that gives a correct and definitive answer to the number of steps of a series of term reductions (ipso facto an arbitrary-length computation) without actually performing it. Such answer would imply knowing in advance if such computation could terminate, and so contradict the fact that automatic theorem proving is, for any sufficiently powerful logic capable of representing arithmetic, an undecidable problem.
In the given example, the teacher is actually either performing that computation mentally (going step by step, applying his own sequence of rules), or gives an estimation based on his experience. But, there's no generic algorithm that guarantees his sequence of steps are the simplest possible, nor that his resulting expression is the simplest one (except for trivial expressions), and hence any quantification of "distance" to a solution is meaningless.
Wouldn't all this be true, your problem would be simple: you know the number of steps, you know how many steps you've taken so far, you divide the latter by the former ;-)
Now, returning to the criteria of simplicity, I also advice you to take a look on Hilbert's 24th problem, that specifically looked for a "Criteria of simplicity, or proof of the greatest simplicity of certain proofs.", and the slightly related proof compression. If you are philosophically inclined to further understand these subjects, I would suggest reading the classic Gödel, Escher, Bach.
Further notes: To understand why, consider a well-known mathematical artefact called the Mandelbrot fractal set. Each pixel color is calculated by determining if the solution to the equation z(n+1) = z(n)^2 + c for any specific c is bounded, that is, "a complex number c is part of the Mandelbrot set if, when starting with z(0) = 0 and applying the iteration repeatedly, the absolute value of z(n) remains bounded however large n gets." Despite the equation being extremely simple (you know, square a number and sum a constant), there's absolutely no way to know if it will remain bounded or not without actually performing an infinite number of iterations or until a cycle is found (disregarding complex heuristics). In this sense, every fractal out there is a rough approximation that typically usages an escape time algorithm as an heuristic to provide an educated guess whether the solution will be bounded or not.
Basically I have a large (could get as large as 100,000-150,000 values) data set of 4-byte inputs and their corresponding 4-byte outputs. The inputs aren't guaranteed to be unique (which isn't really a problem because I figure I can generate pseudo-random numbers to add or xor the inputs with so that they do become unique), but the outputs aren't guaranteed to be unique either (so two different sets of inputs might have the same output).
I'm trying to create a function that effectively models the values in my data-set. I don't need it to interpolate efficiently, or even at all (by this I mean that I'm never going to feed it an input that isn't contained in this static data-set). However it does need to be as efficient as possible. I've looked into interpolation and found that it doesn't really fit what I'm looking for. For example, the large number of values means that spline interpolation won't do since it creates a polynomial per interval.
Also, from my understanding polynomial interpolation would be way too computationally expensive (n values means that the polynomial could include terms as high as pow(x,n-1). For x= a 4-byte number and n=100,000 it's just not feasible). I've tried looking online for a while now, but I'm not very strong with math and must not know the right terms to search with because I haven't come across anything similar so far.
I can see that this is not completely (to put it mildly) a programming question and I apologize in advance. I'm not looking for the exact solution or even a complete answer. I just need pointers on the topics that I would need to read up on so I can solve this problem on my own. Thanks!
TL;DR - I need a variant of interpolation that only needs to fit the initially given data-points, but which is computationally efficient.
Edit:
Some clarification - I do need the output to be exact and not an approximation. This is sort of an optimization of some research work I'm currently doing and I need to have this look-up implemented without the actual bytes of the outputs being present in my program. I can't really say a whole lot about it at the moment, but I will say that for the purposes of my work, encryption (or compression or any other other form of obfuscation) is not an option to hide the table. I need a mathematical function that can recreate the output so long as it has access to the input. I hope that clears things up a bit.
Here is one idea. Make your function be the sum (mod 232) of a linear function over all 4-byte integers, a piecewise linear function whose pieces depend on the value of the first bit, another piecewise linear function whose pieces depend on the value of the first two bits, and so on.
The actual output values appear nowhere, you have to add together linear terms to get them. There is also no direct record of which input values you have. (Someone could conclude something about those input values, but not their actual values.)
The various coefficients you need can be stored in a hash. Any lookups you do which are not found in the hash are assumed to be 0.
If you add a certain amount of random "noise" to your dataset before starting to encode it fairly efficiently, it would be hard to tell what your input values are, and very hard to tell what the outputs are even approximately without knowing the inputs.
Since you didn't impose any restriction on the function (continuous, smooth, etc), you could simply do a piece-wise constant interpolation:
or a linear interpolation:
I assume you can figure out how to construct such a function without too much trouble.
EDIT: In light of your additional requirement that such a function should "hide" the data points...
For a piece-wise constant interpolation, the constant intervals should be randomized so as to not reveal where the data point is. So for example in the picture, the intervals are centered about the data point it's interpolating. Instead, you might want to do something like:
[0 , 0.3) -> 0
[0.3 , 1.9) -> 0.8
[1.9 , 2.1) -> 0.9
[2.1 , 3.5) -> 0.2
etc
Of course, this only hides the x-coordinate. To hide the y-coordinate as well, you can use a linear interpolation.
Simply make it so that the "pointy" part isn't where the data point is. Pick random x-values such that every adjacent data point has one of these x-values in between. Then interpolate such that the "pointy" part is at these x-values.
I suggest a huge Lookup Table full of unused entries. It's the brute-force approach, having an ordered table of outputs, ordered by every possible value of the input (not just the data set, but also all other possible 4-byte value).
Though all of your data would be there, you could fill the non-used inputs with random, arbitrary, or stochastic (random whithin potentially complex constraints) data. If you make it convincing, no one could pick your real data out of it. If a "real" function interpolated all your data, it would also "contain" all the information of your real data, and anyone with access to it could use it to generate an LUT as described above.
LUTs are lightning-fast, but very memory hungry. Your case is on the edge of feasibility, requiring (2^32)*32= 16 Gigabytes of RAM, which requires a 64-bit machine to run. That is just for the data, not the program, the Operating System, or other data. It's better to have 24, just to be sure. If you can afford it, they are the way to go.
Suppose I have a set of tuples like this (each tuple will have 1,2 or 3 items):
Master Set:
{(A) (A,C) (B,C,E)}
and suppose I have another set of tuples like this:
Real Set: {(BOB) (TOM) (ERIC,SALLY,CHARLIE) (TOM,SALLY) (DANNY) (DANNY,TOM) (SALLY) (SALLY,TOM,ERIC) (BOB,SALLY) }
What I want to do is to extract all subsets of Tuples from the Real Set where the tuple members can be substituted to become the same as the Master Set.
In the example above, two sets would be returned:
{(BOB) (BOB,SALLY) (ERIC,SALLY,CHARLIE)}
(let BOB=A,ERIC=B,SALLY=C,CHARLIE=E)
and
{(DANNY) (DANNY,TOM) (SALLY,TOM,ERIC)}
(let DANNY=A,SALLY=B,TOM=C,ERIC=E)
Its sort of pattern matching, sort of combinatorics I guess. I really don't know how to classify this problem and what common plans of attack there are for it. What would the stackoverflow experts suggest?
Seperate your tuples into sets by size. Within each set, create a data structure that allows you to efficiently query for tuples containing a given element. The first part of this structure is your tuples as an array (so that each tuple has a cannonical index). The second set is: Map String (Set Int). This is somewhat space intensive but hopefully not prohibative.
Then, you, essentially, brute force it. For all assignments to the first master set, restrict all assignments to other master sets. For all remaining assignments to the second, restrict all assignments to the third and beyond, etc. The algorithm is basically inductive.
I should add that I don't think the problem is NP-complete so much as just flat worst-case exponential. It's not a decision problem, but an enumeration problem. And it's fairly easy to imagine scenarios of inputs that blow up exponentially.
It will be difficult to do efficiently since your problem is probably NP-complete (it includes subgraph isomorphism as a special case). That assumes the patterns and database both vary in size, though. How much data are you searching? How complicated will your patterns be? I would recommend the brute force solution first, then test if that is too slow and you need something fancier.
I'm not great with statistical mathematics, etc. I've been wondering, if I use the following:
import uuid
unique_str = str(uuid.uuid4())
double_str = ''.join([str(uuid.uuid4()), str(uuid.uuid4())])
Is double_str string squared as unique as unique_str or just some amount more unique? Also, is there any negative implication in doing something like this (like some birthday problem situation, etc)? This may sound ignorant, but I simply would not know as my math spans algebra 2 at best.
The uuid4 function returns a UUID created from 16 random bytes and it is extremely unlikely to produce a collision, to the point at which you probably shouldn't even worry about it.
If for some reason uuid4 does produce a duplicate it is far more likely to be a programming error such as a failure to correctly initialize the random number generator than genuine bad luck. In which case the approach you are using it will not make it any better - an incorrectly initialized random number generator can still produce duplicates even with your approach.
If you use the default implementation random.seed(None) you can see in the source that only 16 bytes of randomness are used to initialize the random number generator, so this is an a issue you would have to solve first. Also, if the OS doesn't provide a source of randomness the system time will be used which is not very random at all.
But ignoring these practical issues, you are basically along the right lines. To use a mathematical approach we first have to define what you mean by "uniqueness". I think a reasonable definition is the number of ids you need to generate before the probability of generating a duplicate exceeds some probability p. An approcimate formula for this is:
where d is 2**(16*8) for a single randomly generated uuid and 2**(16*2*8) with your suggested approach. The square root in the formula is indeed due to the Birthday Paradox. But if you work it out you can see that if you square the range of values d while keeping p constant then you also square n.
Since uuid4 is based off a pseudo-random number generator, calling it twice is not going to square the amount of "uniqueness" (and may not even add any uniqueness at all).
See also When should I use uuid.uuid1() vs. uuid.uuid4() in python?
It depends on the random number generator, but it's almost squared uniqueness.