Discuss how to report on alternate algorithms

[mike-barber]

I think we need to have a discussion to figure out how to deal with reporting for different algorithms, as this adds an extra dimension of fun.

Currently, all of the implementations (except for the C ones) are very similar to the "reference" C++ version, where we consider all odd numbers as potential primes.

The C implementations from @mckoss and @danielspaangberg are really interesting additions, and make reference to this wikipedia section. The same applies to the C# implementation by @Kinematics. Thanks for these! The important thing to note here is that these are algorithmically quite different from all other implementations as they avoid needing to check higher known primes factors besides just 2: variants include 3,5,7, and upwards.

However, this does raise some issues:

• we should not just exclude these from reporting - I think it's great work, and it would be a disservice to not include them. We definitely should continue showcasing these.
• we also can't compare these results directly to every other implementation, as it would be a disservice to all the work done in other languages
• one of the C solutions, 1of2, is the equivalent of the "reference" C++ implementation, so this one is comparable
• considering that 1of2 is part of a more general set of algorithms, I don't think it would make sense to move it to a separate project.
• there's a potential arms race as other languages implement this, resulting in a lot of code :)

In essence, I think we just need to work out what do to on the reporting side. My initial thoughts are along the lines of this:

• have a category for "normal" solutions, which is where most stuff lives, including C 1of2
• have a special "different algorithm" category where we can put the more interesting stuff like 8of30 and up

I'm not sure if this is the right approach, or how to deal with tagging specific lines in the output as "different algorithm". It'll be great to get everyone's input! I know you're working on reporting currently @marghidanu and it's looking pretty cool so far.

[marghidanu]

Hello! We do include a label in the standard output which should account for the solution "category". Right now this label is not taken into account but as we progress with multiple solution types we will include that as a dimension in the report.

[Kinematics]

I was experimenting with breaking up the sieves into components, and run things as an arbitrary combination of those components. The components in question were:

• Storage: BitArray (bit manipulation class provided by the base library), bool array (simple array of booleans (1 byte), no bit manipulation), raw allocation with manual bit manipulation, and rented array (common memory space that you can borrow from without needing to explicitly allocate your own memory block) used for bit manipulation. Also possible variations between the base storage unit sizes, such as 1 byte (8 bit) vs uint (32 bit) vs ulong (64 bit).
• Algorithm: Flat (check every number), 1 of 2 (skip all even numbers; Dave's algorithm), 2 of 6 (skipping all multiples of 3, which leads to checking only 1/3 of all numbers), 8 of 30 (skips all multiples of 5), etc.
• Parallelism: Linear (no parallel work), by factor (parallel threads each handle an individual factor), by region (parallel threads each handle a region of sieve space). Also, bitmasking is a pseudo-parallelization method.

The parallel stuff was set aside as it was getting overly complicated, but even the storage/algorithm pairings had issues of adding a lot of overhead when trying to componentize things.

I still have the partially completed work on a separate branch, but I wasn't sure it was worth trying to build them for the purpose of this repo. At least, not as a part of my existing solution. I'd need to build an entirely new solution to make it work, and I don't know yet how it would perform. It would still feel nice to be able to run it as something like primes.exe --2of6 --arraypool instead of some bastardized parameter name that I had to make up for each combo.

[danielspaangberg]

I also experimented a bit with running things more in parallel, but it is very difficult to get it to run faster if we're only doing the primes up to 1'000'000. I found this nice site that got me a bit inspired (although my code is very different from the java on that page): http://compoasso.free.fr/primelistweb/page/prime/eratosthene_en.php. So I have a working segmented prime implementation using the 5760of30030 map, which works fine for larger number of primes than the ones below 1'000'000. Doesn't use much memory, but since it needs more operations, it is still slower, since the bitmap for the sieve below 1'000'000 fits in L2 cache.

[rbergen]

I think there is indeed a case to be made to separate the basic comparison of languages (which is what @davepl originally created the repo for) from the comparison between algorithms/alternate solutions (in part across languages, by now). I do not believe reusing the free-form "username_optionaltag" field to make the distinction is the way to go, though.

At the moment, my proposal would be to add a category field to the "drag-race" output format, with the following defined values:

• base_faithful. This would be the category label for implementations that use the algorithm that @davepl used in his original solutions, and are in line with the definition of "faithful" that is included in CONTRIBUTING.md.
• base_unfaithful. This covers implementations that use @davepl's algorithm and are in line with the definition of "unfaithful" that is included in CONTRIBUTING.md.
• alternate for all other implementations. All implementations that use a category label other than base_faithful or base_unfaithful, or do not include a category in their output would also be considered alternate.

Of course, we would need to do some work to update existing base_faithful and base_unfaithful solutions so that they add the right label to their output.

In all this, the definition of the "base" algorithm is of course relevant. I would propose to define it as follows:

• When seeking factors (the first inner loop in @davepl's original implementations), the algorithm sequentially checks all odd numbers, starting with 3.
• When clearing non-primes (the second inner loop in @davepl's original implementations), the algorithm clears all non-primes individually, increasing the number with 2 * factor on each cycle.
• As the previous points indicate, no bit masks are used for seeking or clearing.
• Not seeking factors beyond the square root of the sieve size is permissible.
• Changing the order of the two mentioned loops is permissible.
• Inverted prime marking logic (marking primes as zero/false and non-primes as one/true) is permissible.
• Starting the clearing loop at factor factor (instead of the original 3 factor) is permissible.

[mike-barber]

Thanks @rbergen!

As a pragmatic suggestion, we could append a new (optional) column to the output for the category label. The rationale here is that the vast majority of existing implementations are just base_faithful translations of Dave's C++ code into new languages, and this could be inferred if the column is missing.

Solutions that have interesting work done on algorithms could simply implement the final column with the label, with some runs like 1of2 marked as base_faithful and other more interesting ones labelled appropriately. This would allow us to progress without altering a whole bunch of existing stuff :)

I'm not sure what base_unfaithful means in practice, but I do think that alternate would be useful in this case. It's also easier to decide between just two categories and should result in a lot less confusion around subtle implementation details.

I agree that the definition of the base algorithm hasn't been buttoned down, but I haven't personally noticed anything too strange implemented yet luckily. I need some clarification on some of the points, though, as I haven't quite understood them. Maybe they'll make sense after some more coffee :)

As the previous points indicate, no bit masks are used for seeking or clearing.

To be honest, I think it will need some clarification. In my understanding, all solutions that are using bits for storage will need to use bit masks as bits aren't addressable on most hardware. But you might be trying to make a different point that I haven't picked up on :)

Changing the order of the two mentioned loops is permissible.

I'm not sure what this would permit people to do. I suspect that any solution that exchanges inner/outer loops in some way would be different enough to the original algorithm that it wouldn't be comparable with it, and would need to be labelled appropriately. But I also might have misunderstood what you mean by it.

[rbergen]

As a pragmatic suggestion, we could append a new (optional) column to the output for the category label. The rationale here is that the vast majority of existing implementations are just base_faithful translations of Dave's C++ code into new languages, and this could be inferred if the column is missing.

That's basically what I was thinking too, although my suggestion would be to make alternate the default; this makes the "base-faithful" claim an explicit one, and I do think that's appropriate at a conceptual level. And yes, that does mean that a large number of existing implementations would need to be changed, but that's basically (just) work for me. As I've gone over the majority of the solutions once already to make them compliant with the guidelines, I've become comfortable with the exercise.

I'm not sure what base_unfaithful means in practice, but I do think that alternate would be useful in this case. It's also easier to decide between just two categories and should result in a lot less confusion around subtle implementation details.

There are examples of languages where the nature of the language is such that its "natural form" is not faithful, and there are also languages that just don't include the requirements for a faithful implementation. This is true because for a faithful implementation, some form of "true/false" array, some form of class/object/structure, and means to explicitly choose to create a new instance of the latter are required. If the algorithm is the base algorithm but the implementation does not include things like recreation and initialization of the sieve state then it is not "fair" to compare them with other, faithful, implementations. At the same time, I think it is also not fair to put them in the same category as completely different algorithms. If people are confused or unsure, they can always choose alternate. Also, I and others are here to help make the distinction, if/where needed.

As the previous points indicate, no bit masks are used for seeking or clearing.

To be honest, I think it will need some clarification. In my understanding, all solutions that are using bits for storage will need to use bit masks as bits aren't addressable on most hardware. But you might be trying to make a different point that I haven't picked up on :)

I think I agree with you. In fact, I think this whole point can be dropped, as what it's trying to express is already covered by other points.

Changing the order of the two mentioned loops is permissible.

I'm not sure what this would permit people to do.

It would allow people to run the clearing loop first with 3 as a factor, and then seek for the first (next) factor. I know by experience that for some languages, this makes a real difference for how "language-like" the implementation is, while it actually doesn't change the algorithm, as such. What is missing in my proposed phrasing is the word "inner", as in "Changing the order of the two mentioned inner loops is permissible."

[danielspaangberg]

As a pragmatic suggestion, we could append a new (optional) column to the output for the category label. The rationale here is that the vast majority of existing implementations are just base_faithful translations of Dave's C++ code into new languages, and this could be inferred if the column is missing.

That's basically what I was thinking too, although my suggestion would be to make alternate the default; this makes the "base-faithful" claim an explicit one, and I do think that's appropriate at a conceptual level. And yes, that does mean that a large number of existing implementations would need to be changed, but that's basically (just) work for me. As I've gone over the majority of the solutions once already to make them compliant with the guidelines, I've become comfortable with the exercise.

I'm not sure what base_unfaithful means in practice, but I do think that alternate would be useful in this case. It's also easier to decide between just two categories and should result in a lot less confusion around subtle implementation details.

There are examples of languages where the nature of the language is such that its "natural form" is not faithful, and there are also languages that just don't include the requirements for a faithful implementation. This is true because for a faithful implementation, some form of "true/false" array, some form of class/object/structure, and means to explicitly choose to create a new instance of the latter are required. If the algorithm is the base algorithm but the implementation does not include things like recreation and initialization of the sieve state then it is not "fair" to compare them with other, faithful, implementations. At the same time, I think it is also not fair to put them in the same category as completely different algorithms. If people are confused or unsure, they can always choose alternate. Also, I and others are here to help make the distinction, if/where needed.

As the previous points indicate, no bit masks are used for seeking or clearing.

To be honest, I think it will need some clarification. In my understanding, all solutions that are using bits for storage will need to use bit masks as bits aren't addressable on most hardware. But you might be trying to make a different point that I haven't picked up on :)

I think I agree with you. In fact, I think this whole point can be dropped, as what it's trying to express is already covered by other points.

Changing the order of the two mentioned loops is permissible.

I'm not sure what this would permit people to do.

It would allow people to run the clearing loop first with 3 as a factor, and then seek for the first (next) factor. I know by experience that for some languages, this makes a real difference for how "language-like" the implementation is, while it actually doesn't change the algorithm, as such. What is missing in my proposed phrasing is the word "inner", as in "Changing the order of the two mentioned inner loops is permissible."

I agree with what both of you are saying. When I submitted the various algorithms in C I was surprised that faithful just stated "uses sieve of Eratosthenes" and "does not use external dependencies", so they all were included into faithful. As long as you drop the point about the bit masks, which it seems you are already saying, it all makes sense to me. If bit masks are not allowed it is not possible to express smaller than byte storage, unless one concocts code with divisions and modulus just to avoid the "bit masks not allowed" rule.

[marghidanu]

I see this as two separate problems, one refers to the output format for the current solutions and the other is coming up with a proper ruleset for classifying the solution types.

Problem 1

I propose adding another column that can store multiple labels with values. Something like label1=value1;label2=value2... should give us enough flexibility to support any additional tags that might appear in the future

Problem 2

Using the proposed format above should remove the single classification problem since we can now support any classification based on tags.

[rbergen]

That's an interesting idea, indeed. A few things cross my mind thinking about it:

• This can be considered a technical detail, but it is one that can bite us in the backside, later: if we use a semicolon between key/value pairs, then in the context of the overall output format, each of those pairs effectively becomes a column of its own. If we want to treat the combination of key/value pairs for an implementation as one column, using a different separator (like a comma) may be better.
• With the previous point addressed (or dismissed, which is also a decision we can make), we would have a situation that I think resembles the JSON blob field in a relational database table. That's a concept that can be very powerful, but also demands a lot of discipline to work in a context like this one.
• As you indicate yourself, adopting this format does itself not address the problem of defining recognized labels (keys) and values. I'm wondering what your thoughts are on how we would project the "multiple key/value" concept onto the categorization topic?
• Kind of in follow-up to/extension of the previous two points, I feel that every new label/key we recognize, basically adds a dimension to the reporting we need to implement on the benchmark runs. As in, people will want to slice and dice the reported results on every label, and probably on every possible combination of labels as well. Do you think that this is something we will be able to support?

[marghidanu]

Maybe we should just switch to JSON :)

[mike-barber]

Wow there's been a lot of great discussion!

@marghidanu, I'd avoid probably json unless it's really necessary; while we're used to it in the higher-level language world, it's not part of the standard library of many other languages and will require external dependencies. Or you could just hard-code the string I guess, although it'll get pretty ugly potentially.

I think the bigger question is whether we're going to benefit from having the flexibility of arbitrary key=value pairs. Definitely it's more flexible. But I'm concerned it will just add confusion unless we're very opinionated about what the keys are that we're going to accept. So we'd need to do some thinking about the things we could support:

• As an example, this discussion revolves around which category a given run belongs to, so that we can compare language implementations, e.g. base_faithful, alternate, etc.
• I do think the category tag would be mandatory; we can't compare language implementations without it. Would we then specify that the final column must contain category=base_faithful or category=alternate etc.?
• What other keys would make sense? language=x, storage=bits/bytes etc.

I don't really have a clear idea of what kind of reporting we want to do, or what attributes we want to extract from it. But if you've got a pretty good use case in mind, it'll help guide us on what sort of keys/etc you'd be happy to support. I think this echoes what @rbergen has said above too.

Also consider the option of just having comma-separated tags instead of key=value pairs; these work much like the way you'd tag a github issue or something, and just represent membership of a set. It's maybe a bit simpler, and I like simple things when coordinating between a large number of different developers :)

Talking to some of the above comments too.

@danielspaangberg

I agree with what both of you are saying. When I submitted the various algorithms in C I was surprised that faithful just stated "uses sieve of Eratosthenes" and "does not use external dependencies", so they all were included into faithful. As long as you drop the point about the bit masks, which it seems you are already saying, it all makes sense to me. If bit masks are not allowed it is not possible to express smaller than byte storage, unless one concocts code with divisions and modulus just to avoid the "bit masks not allowed" rule.

This makes sense :) We'll get the fine-grained reporting we need by having labelling each "run" appropriately rather than marking the whole solution as faithful/otherwise. It's great having different implementations in the same solution as it's a lot easier for readers to understand and compare.

I think we're all in agreement on the bit masking as it's essential for bit-level storage. Even C++ does this internally behind vector<bool> even if it isn't immediately obvious :) The only potential loophole here is that one could mask off multiple bits in a single word and apply the mask once. This definitely would skew results. But for single bit mask/apply, it's ident to the C++ reference implementation.

@rbergen

There are examples of languages where the nature of the language is such that its "natural form" is not faithful, and there are also languages that just don't include the requirements for a faithful implementation. This is true because for a faithful implementation, some form of "true/false" array, some form of class/object/structure, and means to explicitly choose to create a new instance of the latter are required. If the algorithm is the base algorithm but the implementation does not include things like recreation and initialization of the sieve state then it is not "fair" to compare them with other, faithful, implementations. At the same time, I think it is also not fair to put them in the same category as completely different algorithms. If people are confused or unsure, they can always choose alternate. Also, I and others are here to help make the distinction, if/where needed.

This is an interesting one, but I'm not sure I can think of an easy example. I can imagine fun stuff with functional/recursive code, but that's not my area of expertise. In my head, you'd have to allocate space for the sieve and initialise it to apply Erastothene's sieve algorithm to it. Do you have something in mind you can point me at?

Changing the order of the two mentioned loops is permissible. It would allow people to run the clearing loop first with 3 as a factor, and then seek for the first (next) factor. I know by experience that for some languages, this makes a real difference for how "language-like" the implementation is, while it actually doesn't change the algorithm, as such. What is missing in my proposed phrasing is the word "inner", as in "Changing the order of the two mentioned inner loops is permissible."

Ah, thanks! I see what you mean, and I definitely agree. It won't have much impact on performance. We'll need to be careful with the phrasing as it's quite hard to convey clearly.

Maybe something along the lines of: there are two operations within the outer loop: 1) searching for the next prime from in the sieve, and 2) clearing this prime's multiples in the sieve. In order to allow for idiomatic code, it is permissible to perform these steps in either order, provided the results are correct.

[marghidanu]

@mike-barber That’s why I suggested storing the tags in an arbitrary format, it’s impossible to think of all dimensions right now. We need something that is extensible. As for the output of the implementations, I can tell you that all output will eventually end up in JSON since I’m writing the report generator :) Because the parser actually will convert any output to JSON for generating the HTML files.

[Kinematics]

I think we're all in agreement on the bit masking as it's essential for bit-level storage. Even C++ does this internally behind vector<bool> even if it isn't immediately obvious :) The only potential loophole here is that one could mask off multiple bits in a single word and apply the mask once. This definitely would skew results. But for single bit mask/apply, it's ident to the C++ reference implementation.

To me, 'bitmasking' is different from 'bit manipulation'. Bit manipulation is setting a single bit at a time in a way that is specific to a single operation, whereas bitmasking is setting multiple values at a time in a way that can be repeatedly applied (a rubberstamping effect). So bitmasking is a parallelization tool, whereas bit manipulation is merely a way of saving memory.

Bit manipulation may use a mask/bitmask (noun) to apply its effect, because the languages and CPU's don't allow direct addressing of bits, but it's different from bitmasking (verb) as an operation concept.

It's basically what you describe, but with separate terms describing the two methods, rather than overloading a single term to mean both types of operations (and confusing people).

[rbergen]

@mike-barber

This is an interesting one, but I'm not sure I can think of an easy example. I can imagine fun stuff with functional/recursive code, but that's not my area of expertise. In my head, you'd have to allocate space for the sieve and initialise it to apply Erastothene's sieve algorithm to it. Do you have something in mind you can point me at?

Assembly is a prime example, in my view. I've now coded 4 implementations in various incarnations of that language, 3 of which are now in the drag-race branch (the arm64 version is on its way). In my opinion/experience, the "natural flow" of assembly is to just thunder through the algorithm, and not spend coding time/bytes/processing cycles on things like subroutine calls and heap allocations unless absolutely necessary. In fact, for a proper "faithful" comparison, one basically needs to step out to libc function calls. In C and other languages these are very much "native" to the language, but from the perspective of assembly, one could argue they are "external".

[rbergen]

@marghidanu As I now understand what we're discussing, we would end up with a definition of "official" labels, similar to this:

Label	Values
algorithm	base for the algorithm used in the original C#/C++ implementations <insert reference/code snippets>
faithful	yes or no <insert reference to definitions>
...	...

These would then be combined with the existing standardized output format, to create something like this:

rbergen;2567;5.012;1;algorithm=base,faithful=no

Would this be in line with your thinking?

[mike-barber]

Assembly is a prime example, in my view. I've now coded 4 implementations in various incarnations of that language, 3 of which are now in the drag-race branch (the arm64 version is on its way). In my opinion/experience, the "natural flow" of assembly is to just thunder through the algorithm, and not spend coding time/bytes/processing cycles on things like subroutine calls and heap allocations unless absolutely necessary. In fact, for a proper "faithful" comparison, one basically needs to step out to libc function calls. In C and other languages these are very much "native" to the language, but from the perspective of assembly, one could argue they are "external".

Thanks, that makes a lot of sense! I wasn't thinking about assembly :) And I see what you mean -- you've basically got ~100 extra lines to deal with the housekeeping to handle the setup and memory allocation. Where the "slightly unfaithful" one just has reserved space in the data section.

Nice work, by the way!

My understanding is that believe heap allocation is always external in some form or another. At some point, you've still got to do a syscall to get some pages from the OS. I wouldn't worry about that being external too much :)

[rbergen]

Nice work, by the way!

Thank you! After tripping over myself a few times at the beginning, I did enjoy coding both the x64 and the arm64 implementations. I did find out that documentation on arm64 assembly is quite a bit behind that on x64. Particularly finding out how to execute a square root with conversions from and to "integer" registers was quite a bigger puzzle than I expected.

My understanding is that believe heap allocation is always external in some form or another. At some point, you've still got to do a syscall to get some pages from the OS. I wouldn't worry about that being external too much :)

Indeed, which is why I had the audacity to still label the implementations using those calls as faithful. :)

[mike-barber]

I did find out that documentation on arm64 assembly is quite a bit behind that on x64. Particularly finding out how to execute a square root with conversions from and to "integer" registers was quite a bigger puzzle than I expected.

I know what you mean! I had the same experience doing some simd stuff on arm64 before. Risc is fun though :)

Indeed, which is why I had the audacity to still label the implementations using those calls as faithful. :)

I think it is, on the basis that it's doing the same work and allocating memory and so on.

More relevant to the general thread, though, I think we're in a good place on the ideas for labels and so on. Thanks for fixing the format in all the solutions, including mine! Happy to help with any other changes needed.

The next thorny issue to work out is how to do the benchmark runs outside of the default Azure runners that GitHub provides -- results are all over the place between runs. But I think that'll be a subject for a different thread :)

[rbergen]

I know what you mean! I had the same experience doing some simd stuff on arm64 before. Risc is fun though :)

RISC, or the arm64 implementation of it, is no-nonsense, clean and consistent. I really do like that. The only instructions I found oddly absent at a conceptual level were push and pop. (I know, the 16-byte SP alignment requirement makes it practically impossible to implement them, but still...)

More relevant to the general thread, though, I think we're in a good place on the ideas for labels and so on. Thanks for fixing the format in all the solutions, including mine! Happy to help with any other changes needed.

Well, now that we have a proposal for a structured distinction between implementations, we will at some point have to implement it. I say that in part because @davepl has been very supportive of doing so in recent e-mail communication I've had with him. So, it may not be too long before we indeed need to go over the whole set of solutions, again.

The next thorny issue to work out is how to do the benchmark runs outside of the default Azure runners that GitHub provides -- results are all over the place between runs. But I think that'll be a subject for a different thread :)

We may be closer to an actual solution for that issue than one might think. It's still work in progress, but in my opinion some good progress on externalizing the benchmark runs is being made. If you have any systems laying around that could act as benchmark runners, I think it's not too soon to start dusting them off. :)

[mike-barber]

We may be closer to an actual solution for that issue than one might think. It's still work in progress, but in my opinion some good progress on externalizing the benchmark runs is being made. If you have any systems laying around that could act as benchmark runners, I think it's not too soon to start dusting them off. :)

Awesome! At work, we use self-hosted runners for a bunch of things, so my initial instinct was to do that. I can't reconcile how to make it secure enough on a public repo though: It's a bit scary running code you don't control. Even if you put the runner itself in a Docker container, it still needs network access to pull dependencies etc. I haven't thought of a good way to solve this yet, but definitely happy to help if I can.

[marghidanu]

Runners on a public repo are not really an option for us. We are now testing the bit of code that allows any of us to run the benchmark on any machine externally. Additionally, we will provide a way to publish the local results to a web service and gather the data into a web app so anybody could see it. More to come in the following few days.

[mckoss]

The problem with the original definition of "faithful" is that it did not exclude other sieve-algorithms that employed further optimizations. So, those algorithms, like mine, were no longer apples-to-apples comparisons of languages and language features, but rather of algorithmic optimizations.

I think both are interesting.

To my mind, we should have a category of "direct ports" of the original algorithm to other languages, used for language comparisons - perhaps clarifying "faithful" as using Dave's original algorithm in all respects.

And then, more interesting to me are other algorithmic optimizations - still using a sieve, but applying different techniques to optimize cpu or memory. Actually, any algorithm that could produce the list of primes to 1M by any method would be interesting even if not using a sieve (though I don't know if there are more efficient algorithms for producing bulk primes within a range).

A tag like "algorithmic" could be applied to these entries.

We don't gather statistics like memory use - though I think that could be interesting.

[rbergen]

In line with what was discussed in earlier comments, I think one-dimensional tagging is going to be too limited. I therefore still think that working with key-value pairs is the way to go. For one, from a perspective of comparing algorithms, I think it would be very useful for the benchmark reports to not only indicate that solutions are intended for an algorithmic comparison, but also what algorithm each solution implements. To phrase it this way: you're going to want to make comparisons between (the best performers in) the range of algorithms that are in the set of solutions, as well as compare the various implementations of "8 of 30" with each other, to name one example.

In that context, I think that once we start identifying implementations that stick to @davepl's original algorithm as "base", then the exact meaning of faithful vs unfaithful will become somewhat less relevant (but not meaningless). It will then make a statement on the (technical) implementation style, instead of defining the implementation as a whole.