I have a SMT application (built on Haskell SBV library), which solves some complex equation against single s variable in Real logic using Z3. Finding solution takes about 30 seconds in my case.
Trying to speed things up, I added additional constraint s < 40000, as I have some estimation of solution. I was thinking that such constraint would shrink the search space and make solver return the result faster. However, this only made it slower (it didn't even finished in 10 minutes, actually).
The question is: can it be assumed that additional constraints always slows down/speeds up solution process, or there are no general rules and it always depends on circumstances?
I'm worried that even my 30-seconds algorithm may contain some extra constraint that isn't necessarily needed, but just slows the process.
I don't think you can make any general assumptions about this. It may or may not impact solving time, assuming sat/unsat status doesn't change.
Equalities usually help (as they propagate freely), but for anything else, it's anybody's guess. Also, different solvers can exhibit differing behavior for the same addition.
One way to think about this is that the underlying DPLL(T) algorithm is essentially a very smart glorified search algorithm. It keeps producing "learned lemmas" with the hope that it will find a contradiction with a previously known fact. The new "constraint" you add might cause it to generate a ton of correct but irrelevant lemmas that makes it go down the deep-end with no useful result. (Quantified formulae are usually like this: You can instantiate them in a million different ways; but unless you find the "correct" instantiation, all they do is end up polluting your search space.)
At least that's been my experience!
Related
Is there anything mathematically interesting about the CAP theorem? Looking at the proof, there seem to be four different cases for two different statements across two formalisms. The CAP theorem holds in the three trivial cases and not in the fourth one. All of which use extremely roundabout proof techniques to say something extraordinarily simple.
3.1 Thm 1. If two machines have no communication whatsoever, they cannot contain consistent data.
3.1 Corollary 1.1 If two machines are not allowed to wait to receive messages from each other and the communication line between them is arbitrarily slow, you get an inconsistent result if you write to one and then immediately query the other.
4.2 Thm 2. If two machines that are allowed to wait-with-timeout have no connection whatsoever, they still cannot contain consistent data.
... but if the communication line between them has guarantees about worst-case transmission time, then you can just wait for the timeout each time you perform a write and CAP theorem doesn't apply.
Am I missing something here? The proof techniques used in the paper seem to be more like the kind of thing you find in the generals-on-a-hill problem (which IS nontrivial) where the generals can set a time to coordinate their attack and agree they're going to do it, but they can't agree that they agree. But I just can't see how that applies here.
recently my friend attended intv, he faced this question(intviewer made this up from my fren's answer to another question)
Say, we have option to use either
1) recursion --> uses system stack, i think OS takes care of everything
2) use our own stack for only data part and get things done.
to fix something. Which one do you prefer? and why?
assume stack size wouldn't grow beyond 100.
I would use the system stack. Why re-invent the wheel?
Function calls, while not really slow per se, do take non-zero time. Therefore an iterative solution can be slightly faster.
More often thatn not, simplicity is better than a slight performance gain.
Dont overkill a solution, and loose maitainability/readability for 1ms if you are not going to use that 1ms.
Just remember that whatever clever little hack you put together has to be maintained (and proven to work first for that matter) where as many standard/system solutions are available, that has been proven. (see Reinventing the wheel).
If it is really system crytical that you reduce memory allocation and enhance performance, you have your work cut out for you, and be prepared to spend some time proving that your solution is better/faster and stable.
Interesting to see the general preference for recursion on here, and a few who assume that the recursive implementation will necessarily be clearer or more maintainable... maybe, maybe not :-).
recursion typically avoids an explicit loop
recursion can sometimes simply use local variables inside the function to avoid a container storing results as they're calculated
recursion can make it trivial to reverse the order in which sub-results are gathered
recursion means there's a limit to the depth of information being processed, where-as often a loop implementation easily avoids this, or at least has memory requirements that more accurately reflect the data-processing needs
the more widely applicable you want your software to be, the more important it is to remove arbitrary limits (e.g. UNIX software like modern vim, less, GNU grep etc. make minimal assumptions about file/line/expression length and dynamically attempt whatever they're asked / many here will remember old editors and vendor-specific utilities e.g. one "celestial" company's grep that would never match results at the end of a too-long line, editors that SIGSEGVed, shutdown, corrupted or slowed down into uselessness on long lines or files)
naive recursion can result in spectacularly inefficiently combined sub-results
some people find recursion easier to understand, some find it harder - definitely it suits how we think about some problems better than others
Depends on the algorithm. Small stack usage, system stack. Lot of stack needed, go on the heap. Stack size is limited by OS beyond which OS throws stackoverflow ;-) If algo uses more stack space then I would go with stack data structure and push the data on the heap
Hm, I think it deppends the problem...
The stack size, if I got your point, is not only what limits you from using one or another.
But wanting to use recursion... well, no bads, really, for the length of the stack, but I'd rather make my own solution.
Avoid recursion when you can. :)
Recursion may be the simplest way to solve a particular problem. An iterative solution can required more code and more opportunities for errors. The testing and maintenance cost may be greater than the performance benefit.
I would go with the first, use the system stack. That being said the language FORTH there are two system stacks. One is the return stack and the other is the parameters stack. This offers some nice flexibility.
<tl;dr>
In source version control diff patch generation, would it be worth it to use the optimizations listed at the very bottom of this writing (see <optimizations>) in my Ruby implementation of diff for making diff patches?
</tl;dr>
<introduction>
I am programming something I have never done before and there might already be tools out there to do the exact thing I am programming but at this point I am having too much fun to care so I am still going to do it from scratch, even if there is a tool for this.
So anyways, I am working on a Ruby on Rails app and need a certain feature. Basically I want each entry in a table of mine, let's say for example a table of video games, to have a stored chunk of text that represents a review or something of the sort for that table entry. However, I want this text to be both editable by any registered user and also keep track of different submissions in a version control system. The simplest solution I could think of is just implement a solution that keeps track of the text body and the diff patch history of different versions of the text body as objects in Ruby and then serialize it, preferably in human readable form (so I'll most likely use YAML for this) for editing if needed due to corruption by a software bug or a mistake is made by an admin doing some version editing.
So at first I just tried to dive in head first into this feature to find that the problem of generating a diff patch is more difficult that I thought to do efficiently. So I did some research and came across some ideas. Some I have implemented already and some I have not. However, it all pretty much revolves around the longest common subsequence problem, as you would already know if you have already done anything with diff or diff-like features, and optimization the function that solves it.
Currently I have it so it truncates the compared versions of the text body from the beginning and end until non-matching lines are found. Then it solves the problem using a comparison matrix, but instead of incrementing the value stored in a cell when it finds a matching line like in most longest common subsequence algorithms I have seen examples of, I increment when I have a non-matching line so as to calculate edit distance instead of longest common subsequence. Although as far as I can tell between the two approaches, they are essentially two sides of the same coin so either could be used to derive an answer. It then back-traces through the comparison matrix and notes when there was an incrementation and in which adjacent cell (West, Northwest, or North) to determine that line's diff entry and assumes all other lines to be unchanged.
Normally I would leave it at that, but since this is going into a Rails environment and not just some stand-alone Ruby script, I started getting worried about needing to optimize at least enough so if a spammer that somehow knew how I implemented the version control system and knew my worst case scenario entry still wouldn't be able to hit the server that bad. After some searching and reading of research papers and articles through the internet, I've come across several that seem decent but all seem to have pros and cons and I am having a hard time deciding how well in this situation that the pros and cons balance out. So are the ones listed here worth it? I have listed them with known pros and cons.
</introduction>
<optimizations>
Chop the compared sequences into multiple subsequences by splitting where lines are unchanged, and then truncating each section of unchanged lines at the beginning and end of each section. Then solve the edit distance of each subsequence.
Pro: Changes the time increase as the changed area gets bigger from a quadratic
increase to something more similar to a linear increase.
Con: Figuring out where to split already seems like you have to solve edit distance
except now you don't care how it is changed. Would be fine if this was solvable by
a process closer to solving hamming distance but a single insertion would throw this
off.
Use a cryptographic hash function to both convert all sequence elements into integers and ensure uniqueness. Then solve the edit distance comparing the hash integers instead of the sequence elements themselves.
Pro: The operation of comparing two integers is faster than the operation of comparing
two strings, so a slight performance gain is received after every comparison, which
can be a lot overall.
Con: Using a cryptographic hash function takes time to convert all the sequence
elements and may end up costing more time to do the conversion that you gain back from
the integer comparisons. You could use the built in hash function for a string but
that will not guarantee uniqueness.
Use lazy evaluation to only calculate the three center-most diagonals of the comparison matrix and then only calculate additional diagonals as needed. And then also use this approach to possibly remove the need on some comparisons to compare all three adjacent cells as desribed here.
Pro: Can turn an algorithm that always takes O(n * m) time and make it so only worst
case scenario is that time, best case becomes practically linear, and average case is
somewhere between the two.
Con: It is an algorithm I've only seen implemented in functional programming languages
and I am having a difficult time comprehending how to convert this into Ruby based on
how it is described at the site linked to above.
Make a C module and do the hard work at the native level in C and just make a Ruby wrapper for it so Ruby can make all the calls to it that it needs.
Pro: I have to imagine that evaluating something like this in could be a LOT faster.
Con: I have no idea how Rails handles apps with ruby code that has C extensions and it
hurts the portability of the app.
This is an optimization for after the solving of edit distance, but idea is to store additional combined diffs with the ones produced by each version to make a delta-tree data structure with the most recently made diff as the root node of the tree so getting to any version takes worst case time of O(log n) instead of O(n).
Pro: Would make going back to an old version a lot faster.
Con: It would mean every new commit, the delta-tree would get a new root node that
will cost time to reorganize the delta-tree for an operation that will be carried out
a lot more often than going back a version, not to mention the unlikelihood it will be
an old version.
</optimizations>
So are these things worth the effort?
With regard to item 4 in your list, this seems to be ( from what I can tell ) how most gems work if there is any heavy lifting to be done by the code. Rails plays nice with the gem system, so you should find that if you need to incorporate this - probably alongside other optimisations you have suggested here - it should be fine, although you may need to recompile for different platforms.
At one point while traveling the web, I came across a great page which contrasted the clarity and terseness of different methods of doing a sequence of operations without having to make a bunch of throwaway variables, e.g., Var1, Var2, Var3. It tried list comprehensions, folds, maps, etc. For some reason, now matter what I google, I can't find it again. Anyone have any idea what I'm talking about? Or want to explore the topic anyway?
Your question doesn't make much sense.
List comprehensions, fold, and map aren't for avoiding variables (nor are they interchangeable), they're the right ways to process data depending on what you're trying to do.
This is the article you were looking for:
http://erlanganswers.com/web/mcedemo/VersionedVariables.html
It is probably more of an art than a science. In a nutshell my advice is to lean away from using throw-aways as a general habit, but equally, do not be afraid of using them intelligently and sparingly where you feel appropriate or necessary.
When you are starting to learn then by all means use throw-away variables if it helps you break things down into understandable chunks. But try to break away from that sooner rather than later, as using throw-aways may at times make your code harder to maintain and modify. On the other hand, even when you are experienced you may sometimes find that it is worth using throwaways for the same reason : keep things readable and manageable for less experienced programmers. Purists may say that you should never use them, but I believe that when you consider the lifetime costs of software maintenance it is important to remember that readability is very important. Maybe this argument doesn't apply if you are lucky enough to work in an environment that only hires the best of the best, but for the rest of us that's simply not a reflection of the real world.
The bottom line : what is "right" depends on your skill level, the skill level of your peers, what you are doing, and the likely volatility, complexity, and lifetime of the code. Use your best judgement.
In response to the answer saying the question doesn't make sense, you would certainly think it made sense if you saw the article to which I'm referring. The point is to elegantly process a series of statements without redundant intermediate variables. Zed is right on target. I really wish I could find the original link because it was super detailed and went through 5 or 6 methods, some of which were referenced from the erlang mailing list, and weighed the pros and cons of each.
I've started reading Algorithms and I keep wondering, when dealing with primitives of the same type, which is the more expensive operation, assignment or comparison? Does this vary a great deal between languages?
What do you think?
At the lowest level one does two reads, the other does a read and a write.
But why should you really care? You shouldn't care about performance at this level. Optimize for Big-O
Micro-optimization is almost always the wrong thing to do. Don't even start on it unless the program runs too slowly, and you use a profiler to determine exactly where the slow parts are.
Once you've done that, my advice is to see about improving code and data locality, because cache misses are almost certainly worse than suboptimal instructions.
That being done, in the rather odd case that you can use either an assignment-based or comparison-based approach, try both and time them. Micro-optimization is a numbers game. If the numbers aren't good enough, find out why, then verify that what you're doing actually works.
So, what do you mean by a comparison? Conditional jumps cause problems to any vaguely modern processor, but different processors do different things, and there's no guarantee that any given one will slow things down. Also, if either causes a cache miss, that's probably the slower one no matter what.
Finally, languages are normally compiled to machine code, and simple things like comparisons and assignments will normally be compiled the same. The big difference will be the type of CPU.