To what extent shall we optimize time complexity? - time-complexity

Theory vs practice here.
Regarding time complexity, and I have a conceptual question that we didn't get to go deeper into in class.
Here it is:
There's a barbaric BROOT force algorithm, O(n^3)... and we got it down o O(n) and it was considered good enough. If we dive in deeper, it is actually O(n)+O(n), two separate iterations of the input. I came up with another way which was actually O(n/2). But those two algorithms are considered to be the same since both are O(n) and as n reaches infinity, it makes no difference, so not necessary at all once we reach O(n).
My question is:
In reality, in practice, we always have a finite number of inputs (admittedly occasionally in the trillions). So following the time complexity logic, O(n/2) is four times as fast as O(2n). So if we can make it faster, why not?

Time complexity is not everything. As you already noticed, the Big-Oh can hide a lot and also assumes that all operations cost the same.
In Practice you should always try to find a fast/the fastest solution for your problem. Sometimes this means that you use a algorithm with a bad complexity but good constants if you know that your problem is always small. Depending on your use case, you also want to implement optimizations that utilize hardware properties like cache optimizations.

Related

What is the relationship between time complexity and the number of steps in an algorithm?

For large values of n, an algorithm that takes 20000n^2 steps has better time complexity (takes less time) than one that takes 0.001n^5 steps
I believe this statement is true. But, why?
If there are more steps wouldn't that take more time?
Computational complexity is considered in the asymptotic sense because the important question is usually of scaling. Even with your clear case, the ^5 algorithm begins to take longer around 275 items - which isn't very many. See this figure from wolfram alpha:
Quoting from the wikipedia article linked above:
Usually asymptotic estimates are used because different implementations of the same algorithm may differ in efficiency. However the efficiencies of any two "reasonable" implementations of a given algorithm are related by a constant multiplicative factor called a hidden constant.
All that said, if you have two comparable algorithms and the one with less complexity has a significant constant coefficient and you're only going to process 10 items, then it very well may be a good idea to choose the less efficient one. Some common libraries even switch algorithms depending upon the size of the data being processed; this is called a hybrid algorithm and Python's sorted implementation, Timsort uses it to switch between insertion sort and merge sort.

Is lg*(n) time complexity better than lg(n)?

I am trying to understand the time complexity of the lg*(n) [log*(n) base 2] in comparison to lg(n) and I wonder which of them is faster...can someone explain it, please? Thanks in advance.
According to Wikipedia, the iterative logarithm (log*) is one of the slowest growing time complexities. In fact, of all the commonly uses complexities, it is the second slowest, beaten only by the inverse Ackerman function. This means it grows significantly slower, and as a result completes much faster, than the log function.
Source: https://en.wikipedia.org/wiki/Iterated_logarithm#Analysis_of_algorithms
I've never seen lg*(n) notation before, but I assume you're referring to log base 2 vs log base 10. It turns out that log2(N) == log10(N) * 3.32192809489..., which is a constant factor difference, and we drop constant factors when analyzing algorithmic complexity. As a result, all logarithms are considered equal, and we do not need to bother specifying the base in algorithmic complexity.
When studying actual runtimes, then log10(N) is faster than log2(N), but very rarely do developers actually analyze runtimes in this manner, they usually do it with a profiler.

OpenMP with matrices and vectors

What is the best way to utilize OpenMP with a matrix-vector product? Would the for directive suffice (if so, where should I place it? I assume outer loop would be more efficient) or would I need schedule, etc..?
Also, how would I take advantage different algorithms to attempt this m-v product most efficiently?
Thanks
The first step you should take is the obvious one, wrap the outermost loop in a parallel for directive. As you assume. It's always worth experimenting a bit to get some evidence to support your (and my) assumptions, but if you were only allowed to make 1 change that would be the one to make.
I don't know much about cache-oblivious algorithms but I understand that they, generally, work by recursive division of a problem into sub-problems. This doesn't seem to fit with the application of parallel for directives. I suspect you could implement such an algorithm with OpenMP's tasks, but I suspect that the overhead of doing this would outweigh any execution improvements on any m-v product of reasonable dimensions.
(If you demonstrate the falsity of this argument on m-v products of size N I will retort 'N's not a reasonable dimension'. As ever with these performance questions, evidence trumps argument every time.)
Finally, depending on your compiler and the availability of libraries, you may not need to use OpenMP for m-v calculations, you might find auto-parallelisation works efficiently, or already have a library implementation which multi-threads this sort of computation.

Is there any time complexity difference between recursive and iterative approach?

I am aware that we do have space complexity difference between a recursive and iterative algorithm. But , do we also have time complexity differences between them?
For example: If I have a program that counts the number of nodes in a list recursively and then I implement the same program as iterative, will I have any difference in its Time complexity i.e. O(n)? Thank you
Short answer: no.
Unless you optimize the algorithm using dynamic programming or such, there is no change to time complexity. There is also no change to space complexity, don't know where you got that idea..
However, in many programming languages there is an inherent overhead to using recursion, since they must store the stack as well, which uses more memory. This can be slower, especially if it is not tail recursion.

When is optimisation premature?

As Knuth said,
We should forget about small efficiencies, say about 97% of the time: premature optimization is the root of all evil.
This is something which often comes up in Stack Overflow answers to questions like "which is the most efficient loop mechanism", "SQL optimisation techniques?" (and so on). The standard answer to these optimisation-tips questions is to profile your code and see if it's a problem first, and if it's not, then therefore your new technique is unneeded.
My question is, if a particular technique is different but not particularly obscure or obfuscated, can that really be considered a premature optimisation?
Here's a related article by Randall Hyde called The Fallacy of Premature Optimization.
Don Knuth started the literate programming movement because he believed that the most important function of computer code is to communicate the programmer's intent to a human reader. Any coding practice that makes your code harder to understand in the name of performance is a premature optimization.
Certain idioms that were introduced in the name of optimization have become so popular that everyone understands them and they have become expected, not premature. Examples include
Using pointer arithmetic instead of array notation in C, including the use of such idioms as
for (p = q; p < lim; p++)
Rebinding global variables to local variables in Lua, as in
local table, io, string, math
= table, io, string, math
Beyond such idioms, take shortcuts at your peril.
All optimization is premature unless
A program is too slow (many people forget this part).
You have a measurement (profile or similar) showing that the optimization could improve things.
(It's also permissible to optimize for memory.)
Direct answer to question:
If your "different" technique makes the program harder to understand, then it's a premature optimization.
EDIT: In response to comments, using quicksort instead of a simpler algorithm like insertion sort is another example of an idiom that everyone understands and expects. (Although if you write your own sort routine instead of using the library sort routine, one hopes you have a very good reason.)
IMHO, 90% of your optimization should occur at design stage, based on percieved current, and more importantly, future requirements. If you have to take out a profiler because your application doesn't scale to the required load you have left it too late, and IMO will waste a lot of time and effort while failing to correct the problem.
Typically the only optimizations that are worthwhile are those that gain you an order of magnitude performance improvement in terms of speed, or a multiplier in terms of storage or bandwidth. These types of optimizations typically relate to algorithm selection and storage strategy, and are extremely difficult to reverse into existing code. They may go as deep as influencing the decision on the language in which you implement your system.
So my advice, optimize early, based on your requirements, not your code, and look to the possible extended life of your app.
If you haven't profiled, it's premature.
My question is, if a particular
technique is different but not
particularly obscure or obfuscated,
can that really be considered a
premature optimisation?
Um... So you have two techniques ready at hand, identical in cost (same effort to use, read, modify) and one is more efficient. No, using the more efficient one would not, in that case, be premature.
Interrupting your code-writing to look for alternatives to common programming constructs / library routines on the off-chance that there's a more efficient version hanging around somewhere even though for all you know the relative speed of what you're writing will never actually matter... That's premature.
Here's the problem I see with the whole concept of avoiding premature optimization.
There's a disconnect between saying it and doing it.
I've done lots of performance tuning, squeezing large factors out of otherwise well-designed code, seemingly done without premature optimization.
Here's an example.
In almost every case, the reason for the suboptimal performance is what I call galloping generality, which is the use of abstract multi-layer classes and thorough object-oriented design, where simple concepts would be less elegant but entirely sufficient.
And in the teaching material where these abstract design concepts are taught, such as notification-driven architecture, and information-hiding where simply setting a boolean property of an object can have an unbounded ripple effect of activities, what is the reason given? Efficiency.
So, was that premature optimization or not?
First, get the code working. Second, verify that the code is correct. Third, make it fast.
Any code change that is done before stage #3 is definitely premature. I am not entirely sure how to classify design choices made before that (like using well-suited data structures), but I prefer to veer towards using abstractions taht are easy to program with rather than those who are well-performing, until I am at a stage where I can start using profiling and having a correct (though frequently slow) reference implementation to compare results with.
From a database perspective, not to consider optimal design at the design stage is foolhardy at best. Databases do not refactor easily. Once they are poorly designed (this is what a design that doesn't consider optimization is no matter how you might try to hide behind the nonsense of premature optimization), is almost never able to recover from that becasue the database is too basic to the operation of the whole system. It is far less costly to design correctly considering the optimal code for the situation you expect than to wait until the there are a million users and people are screaming becasue you used cursors throughout the application. Other optimizations such as using sargeable code, selecting what look to be the best possible indexes, etc. only make sense to do at design time. There is a reason why quick and dirty is called that. Because it can't work well ever, so don't use quickness as a substitute for good code. Also frankly when you understand performance tuning in databases, you can write code that is more likely to perform well in the same time or less than it takes to write code which doesn't perform well. Not taking the time to learn what is good performing database design is developer laziness, not best practice.
What you seem to be talking about is optimization like using a hash-based lookup container vs an indexed one like an array when a lot of key lookups will be done. This is not premature optimization, but something you should decide in the design phase.
The kind of optimization the Knuth rule is about is minimizing the length the most common codepaths, optimizing the code that is run most by for example rewriting in assembly or simplifying the code, making it less general. But doing this has no use until you are certain which parts of code need this kind of optimization and optimizing will (could?) make the code harder to understand or maintain, hence "premature optimization is the root of all evil".
Knuth also says it is always better to, instead of optimizing, change the algorithms your program uses, the approach it takes to a problem. For example whereas a little tweaking might give you a 10% increase of speed with optimization, changing fundamentally the way your program works might make it 10x faster.
In reaction to a lot of the other comments posted on this question: algorithm selection != optimization
The point of the maxim is that, typically, optimization is convoluted and complex. And typically, you the architect/designer/programmer/maintainer need clear and concise code in order to understand what is going on.
If a particular optimization is clear and concise, feel free to experiment with it (but do go back and check whether that optimization is effective). The point is to keep the code clear and concise throughout the development process, until the benefits of performance outweigh the induced costs of writing and maintaining the optimizations.
Optimization can happen at different levels of granularity, from very high-level to very low-level:
Start with a good architecture, loose coupling, modularity, etc.
Choose the right data structures and algorithms for the problem.
Optimize for memory, trying to fit more code/data in the cache. The memory subsystem is 10 to 100 times slower than the CPU, and if your data gets paged to disk, it's 1000 to 10,000 times slower. Being cautious about memory consumption is more likely to provide major gains than optimizing individual instructions.
Within each function, make appropriate use of flow-control statements. (Move immutable expressions outside of the loop body. Put the most common value first in a switch/case, etc.)
Within each statement, use the most efficient expressions yielding the correct result. (Multiply vs. shift, etc)
Nit-picking about whether to use a divide expression or a shift expression isn't necessarily premature optimization. It's only premature if you do so without first optimizing the architecture, data structures, algorithms, memory footprint, and flow-control.
And of course, any optimization is premature if you don't define a goal performance threshold.
In most cases, either:
A) You can reach the goal performance threshold by performing high-level optimizations, so it's not necessary to fiddle with the expressions.
or
B) Even after performing all possible optimizations, you won't meet your goal performance threshold, and the low-level optimizations don't make enough difference in performance to justify the loss of readability.
In my experience, most optimization problems can be solved at either the architecture/design or data-structure/algorithm level. Optimizing for memory footprint is often (though not always) called for. But it's rarely necessary to optimize the flow control & expression logic. And in those cases where it actually is necessary, it's rarely sufficient.
I try to only optimise when a performance issue is confirmed.
My definition of premature optimisation is 'effort wasted on code that is not known to be a performance problem.' There is most definitely a time and place for optimisation. However, the trick is to spend the extra cost only where it counts to the performance of the application and where the additional cost outweighs the performance hit.
When writing code (or a DB query) I strive to write 'efficient' code (i.e. code that performs its intended function, quickly and completely with simplest logic reasonable.) Note that 'efficient' code is not necessarily the same as 'optimised' code. Optimisations often introduce additional complexity into code which increases both the development and maintenance cost of that code.
My advice: Try to only pay the cost of optimisation when you can quantify the benefit.
When programming, a number of parameters are vital. Among these are:
Readability
Maintainability
Complexity
Robustness
Correctness
Performance
Development time
Optimisation (going for performance) often comes at the expense of other parameters, and must be balanced against the "loss" in these areas.
When you have the option of choosing well-known algorithms that perform well, the cost of "optimising" up-front is often acceptable.
Norman's answer is excellent. Somehow, you routinely do some "premature optimization" which are, actually, best practices, because doing otherwise is known to be totally inefficient.
For example, to add to Norman's list:
Using StringBuilder concatenation in Java (or C#, etc.) instead of String + String (in a loop);
Avoiding to loop in C like: for (i = 0; i < strlen(str); i++) (because strlen here is a function call walking the string each time, called on each loop);
It seems in most JavaScript implementations, it is faster to do too for (i = 0 l = str.length; i < l; i++) and it is still readable, so OK.
And so on. But such micro-optimizations should never come at the cost of readability of code.
The need to use a profiler should be left for extreme cases. The engineers of the project should be aware of where performance bottlenecks are.
I think "premature optimisation" is incredibly subjective.
If I am writing some code and I know that I should be using a Hashtable then I will do that. I won't implement it in some flawed way and then wait for the bug report to arrive a month or a year later when somebody is having a problem with it.
Redesign is more costly than optimising a design in obvious ways from the start.
Obviously some small things will be missed the first time around but these are rarely key design decisions.
Therefore: NOT optimising a design is IMO a code smell in and of itself.
It's worth noting that Knuth's original quote came from a paper he wrote promoting the use of goto in carefully selected and measured areas as a way to eliminate hotspots. His quote was a caveat he added to justify his rationale for using goto in order to speed up those critical loops.
[...] again, this is a noticeable saving in the overall running speed,
if, say, the average value of n is about 20, and if the search routine
is performed about a million or so times in the program. Such loop
optimizations [using gotos] are not difficult to learn and, as I have
said, they are appropriate in just a small part of a program, yet they
often yield substantial savings. [...]
And continues:
The conventional wisdom shared by many of today's software engineers
calls for ignoring efficiency in the small; but I believe this is
simply an overreaction to the abuses they see being practiced by
pennywise-and-pound-foolish programmers, who can't debug or maintain
their "optimized" programs. In established engineering disciplines a
12% improvement, easily obtained, is never considered marginal; and I
believe the same viewpoint should prevail in software engineering. Of
course I wouldn't bother making such optimizations on a oneshot job,
but when it's a question of preparing quality programs, I don't want
to restrict myself to tools that deny me such efficiencies [i.e., goto
statements in this context].
Keep in mind how he used "optimized" in quotes (the software probably isn't actually efficient). Also note how he isn't just criticizing these "pennywise-and-pound-foolish" programmers, but also the people who react by suggesting you should always ignore small inefficiencies. Finally, to the frequently-quoted part:
There is no doubt that the grail of efficiency leads to abuse.
Programmers waste enormous amounts of time thinking about, or worrying
about, the speed of noncritical parts of their programs, and these
attempts at efficiency actually have a strong negative impact when
debugging and maintenance are considered. We should forgot about small
efficiencies, say 97% of the time; premature optimization is the root
of all evil.
... and then some more about the importance of profiling tools:
It is often a mistake to make a priori judgments about what parts of a
program are really critical, since the universal experience of
programmers who have been using measurement tools has been that their
intuitive guesses fail. After working with such tools for seven years,
I've become convinced that all compilers written from now on should be
designed to provide all programmers with feedback indicating what
parts of their programs are costing the most; indeed, this feedback
should be supplied automatically unless it has been specifically
turned off.
People have misused his quote all over the place, often suggesting that micro-optimizations are premature when his entire paper was advocating micro-optimizations! One of the groups of people he was criticizing who echo this "conventional wisdom" as he put of always ignoring efficiencies in the small are often misusing his quote which was originally directed, in part, against such types who discourage all forms of micro-optimization.
Yet it was a quote in favor of appropriately applied micro-optimizations when used by an experienced hand holding a profiler. Today's analogical equivalent might be like, "People shouldn't be taking blind stabs at optimizing their software, but custom memory allocators can make a huge difference when applied in key areas to improve locality of reference," or, "Handwritten SIMD code using an SoA rep is really hard to maintain and you shouldn't be using it all over the place, but it can consume memory much faster when applied appropriately by an experienced and guided hand."
Any time you're trying to promote carefully-applied micro-optimizations as Knuth promoted above, it's good to throw in a disclaimer to discourage novices from getting too excited and blindly taking stabs at optimization, like rewriting their entire software to use goto. That's in part what he was doing. His quote was effectively a part of a big disclaimer, just like someone doing a motorcycle jump over a flaming fire pit might add a disclaimer that amateurs shouldn't try this at home while simultaneously criticizing those who try without proper knowledge and equipment and get hurt.
What he deemed "premature optimizations" were optimizations applied by people who effectively didn't know what they were doing: didn't know if the optimization was really needed, didn't measure with proper tools, maybe didn't understand the nature of their compiler or computer architecture, and most of all, were "pennywise-and-pound-foolish", meaning they overlooked the big opportunities to optimize (save millions of dollars) by trying to pinch pennies, and all while creating code they can no longer effectively debug and maintain.
If you don't fit in the "pennywise-and-pound-foolish" category, then you aren't prematurely optimizing by Knuth's standards, even if you're using a goto in order to speed up a critical loop (something which is unlikely to help much against today's optimizers, but if it did, and in a genuinely critical area, then you wouldn't be prematurely optimizing). If you're actually applying whatever you're doing in areas that are genuinely needed and they genuinely benefit from it, then you're doing just great in the eyes of Knuth.
Premature optimization to me means trying to improve the efficiency of your code before you have a working system, and before you have actually profiled it and know where the bottleneck is. Even after that, readability and maintainability should come before optimization in many cases.
I don't think that recognized best practices are premature optimizations. It's more about burning time on the what ifs that are potential performance problems depending on the usage scenarios. A good example: If you burn a week trying to optimize reflecting over an object before you have proof that it is a bottleneck you are prematurely optimizing.
Unless you find that you need more performance out of your application, due to either a user or business need, there's little reason to worry about optimizing. Even then, don't do anything until you've profiled your code. Then attack the parts which take the most time.
The way I see it is, if you optimize something without knowing how much performance you can gain in different scenario IS a premature optimization. The goal of code should really making it easiest for human to read.
As I posted on a similar question, the rules of optimisation are:
1) Don't optimise
2) (for experts only) Optimise later
When is optimisation premature? Usually.
The exception is perhaps in your design, or in well encapsulated code that is heavily used. In the past I've worked on some time critical code (an RSA implementation) where looking at the assembler that the compiler produced and removing a single unnecessary instruction in an inner loop gave a 30% speedup. But, the speedup from using more sophisticated algorithms was orders of magnitude more than that.
Another question to ask yourself when optimising is "am I doing the equivalent of optimising for a 300 baud modem here?". In other words, will Moore's law make your optimisation irrelevant before too long. Many problems of scaling can be solved just by throwing more hardware at the problem.
Last but not least it's premature to optimise before the program is going too slowly. If it's web application you're talking about, you can run it under load to see where the bottlenecks are - but the likelihood is that you will have the same scaling problems as most other sites, and the same solutions will apply.
edit: Incidentally, regarding the linked article, I would question many of the assumptions made. Firstly it's not true that Moore's law stopped working in the 90s. Secondly, it's not obvious that user's time is more valuable than programmer's time. Most users are (to say the least) not frantically using every CPU cycle available anyhow, they are probably waiting for the network to do something. Plus there is an opportunity cost when programmer's time is diverted from implementing something else, to shaving a few milliseconds off something that the program does while the user is on the phone. Anything longer than that isn't usually optimisation, it's bug fixing.