• Yet more rigorous about computational complexity..

    From Wisdom90@21:1/5 to All on Sat Jan 11 15:02:04 2020
    Hello...

    Read this:

    Yet more rigorous about computational complexity..

    I said previously(read below) that for example a time complexity such
    as n*(log(n)) and n are fuzzy, because we can say that n*(log(n) is
    an average resistance(read below to understand the analogy with material resistance) or we can say that n*log(n) is faster than a quadratic
    complexity or than an exponential complexity, but we can not say giving
    a time complexity of n or n*log(n) how fast it is giving the input of
    the n of the time complexity, so since it is not exact prediction, so it
    is fuzzy, but this level of fuzziness, like in the example below of the
    obese person, permits us to predict important things in the reality, and
    this level of fuzziness of computational complexity is also science,
    because it is like probability calculations that permits us to predict.

    Read my previous thoughts to understand:

    What is science? and is computational complexity science ?

    You just have seen me talking about computational complexity,
    but we need to answer the questions of: What is science ?
    and is computational complexity science ?

    I think that we have to be more smart because there is like
    higher level abstractions in science, and we can be in those
    abstractions exact precisions of science, but we can be more fuzzy
    precisions that are useful and that are also science, to understand me
    more, let me give you an example:

    If i say that a person is obese, so he has a high risk to get a disease
    because he is obese.

    Now you are understanding more that with this abstraction we are not
    exact precision, but we are more fuzzy , but this fuzziness
    is useful and its level of precision is also useful, but is it
    science ? i think that this probabilistic calculations are
    also science that permits us to predict that the obese person
    has a high risk to get a disease. And this probabilistic calculations
    are like a higher level abstractions that lack exact precision but
    they are still useful precisions. This is how look like computational complexity and its higher level abstractions, so you are immediately understanding that a time complexity of O(n*log(n)) or a O(n)
    is like a average level of resistance(read below to know why i am
    calling it resistance by analogy with material resistance) when n grows
    large, and we can immediately notice that an exponential time complexity
    is a low level resistance when n grows large, and we can immediately
    notice that a log(n) time complexity is a high level of resistance
    when n grows large, so those time complexities are like a higher level abstractions that are fuzzy but there fuzziness, like in the example
    above of the obese person, permits us to predict important things in the reality, and this level of fuzziness of computational complexity is also science, because it is like probability calculations that permits us
    to predict.

    Read the rest of my previous thoughts to understand better:

    The why of computational complexity..


    Here is my previous answer about computational complexity and the rest
    of my current answer is below:


    =====================================================================
    Horand gassmann wrote:

    "Where your argument becomes impractical is in the statement "n becomes
    large". This is simply not precise enough for practical use. There is a break-even point, call it n_0, but it cannot be computed from the Big-O
    alone. And even if you can compute n_0, what if it turns out that the
    breakeven point is larger than a googolplex? That would be interesting theoretically, but practically --- not so much."


    I don't agree, because take a look below at how i computed the binary
    search time complexity, it is a divide and conquer algorithm, and it is
    log(n), but we can notice that a log(n) is good when n becomes large, so
    this information is practical because a log(n) time complexity is
    excellent in practice when n becomes large, and when you look at an
    insertion sort you will notice that it is a quadratic time complexity of
    n^2, here again, you can feel that it is practical because an quadratic
    time complexity is not so good when n becomes large, so you can say that
    n^2 is not so good in practice when n becomes large, so
    as you are noticing having time complexities of log(n) and n^2
    are useful in practice, and for the rest of the the time complexities
    you can also benchmark the algorithm in the real world to have an idea
    at how it is performing. =================================================================



    I think i am understanding better Lemire and Horand gassmann,
    they say that if it is not exact needed practical precision, so it is
    not science or engineering, but i don't agree with this, because
    science and engineering can be like working with more higher level
    abstractions that are not exact needed practical precision calculations,
    but they can still be useful precision in practice, it is like being a
    fuzzy precision that is useful, this is why i think that probabilistic calculations are also scientific , because probabilistic calculations
    are useful in practice because they can give us important informations
    on the reality that can also be practical, this is why computational
    complexity is also useful in practice because it is like a higher level abstractions that are not all the needed practical precision, but it is precision that is still useful in practice, this is why like
    probabilistic calculations i think computational complexity is also science.


    Read the rest of my previous thoughts to understand better:


    More on computational complexity..

    Notice how Horand gassmann has answered in sci.math newsgroup:

    Horand gassmann wrote the following:

    "You are right, of course, on one level. An O(log n)
    algorithm is better than an O(n) algorithm *for
    large enough inputs*. Lemire understands that, and he
    addresses it in his blog. The important consideration
    is that _theoretical_ performance is a long way from
    _practical_ performance."


    And notice how what Lemire wrote about computational complexity:

    "But it gets worse: these are not scientific models. A scientific model
    would predict the running time of the algorithm given some
    implementation, within some error margin. However, these models do
    nothing of the sort. They are purely mathematical. They are not
    falsifiable. As long as they are mathematically correct, then they are
    always true. To be fair, some researchers like Knuth came up with models
    that closely mimic reasonable computers, but that’s not what people
    pursuing computational complexity bounds do, typically."


    So as you are noticing that both of them want to say that computational complexity is far from practical, but i don't agree with them,
    because time complexity is like material resistance, and it
    informs us on important practical things such as an algorithm of log(n)
    time complexity is like much more resistant than O(n) when n becomes
    large, and i think this kind of information of time complexity is
    practical, this is why i don't agree with Lemire and Horand gassmann,
    because as you notice that time complexity is scientific and it is
    also engineering. Read the rest of my post to understand more what i
    want to say:


    More precision about computational complexity, read again:

    I have just read the following webpage of a PhD Computer Scientist and researcher from Montreal, Canada where i am living now from year 1989,
    here is the webpage and read it carefully:

    Better computational complexity does not imply better speed

    https://lemire.me/blog/2019/11/26/better-computational-complexity-does-not-imply-better-speed/

    And here is his resume:

    https://lemire.me/pdf/resume/resumelemire.pdf

    So as you are noticing on the webpage above he is saying the following
    about computational complexity:

    "But it gets worse: these are not scientific models. A scientific model
    would predict the running time of the algorithm given some
    implementation, within some error margin. However, these models do
    nothing of the sort. They are purely mathematical. They are not
    falsifiable. As long as they are mathematically correct, then they are
    always true. To be fair, some researchers like Knuth came up with models
    that closely mimic reasonable computers, but that’s not what people
    pursuing computational complexity bounds do, typically."

    But i don't agree with him because i think he is not understanding
    the goal of computational complexity, because when we say that
    an algorithm has a time complexity of n*log(n), you have to
    understand that it is by logical analogy like saying in physics what is
    the material resistance, because the n*log(n) means how well the
    algorithm is "amortizing"(it means reducing) the "time" that it takes
    taking as a reference of measure the average time complexity of an
    algorithm, i said that the time complexity is like the material
    resistance in physics, because if the time complexity n grows large, it
    is in physics like a big force that you apply to the material that is by logical analogy the algorithm, so if the time complexity is log(n), so
    it will amortize the time that it takes very well, so it is in physics
    like the good material resistance that amortizes very well the big force
    that is applied to it, and we can easily notice that an algorithm
    becomes faster, in front of the data that is giving to him, by going
    from an exponential time complexity towards a logarithmic time
    complexity, so we are noticing that the time complexity is "universal"
    and it measures how well the algorithm amortizes(that means it reduces)
    the time that it takes taking as a reference of measure the average time complexity of an algorithm, so this is is why computational complexity
    is scientific and also it is engineering and it gives us information on
    the physical world.

    So to give an interesting example of science in computing, we can ask of
    what is the time complexity of a binary search algorithm, and here
    is my mathematical calculations of its time complexity:

    Recurrence relation of a binary search algorithm is: T(n)=T(n/2)+1

    Because the "1" is like a comparison that we do in each step of
    the divide and conquer method of the binary search algorithm.

    So the calculation of the recurrence equation gives:

    1st step=> T(n)=T(n/2) + 1

    2nd step=> T(n/2)=T(n/4) + 1 ……[ T(n/4)= T(n/2^2) ]

    3rd step=> T(n/4)=T(n/8) + 1 ……[ T(n/8)= T(n/2^3) ]

    .

    .

    kth step=> T(n/2^k-1)=T(n/2^k) + 1*(k times)

    Adding all the equations we get, T(n) = T(n/2^k) + k times 1

    This is the final equation.

    So how many times we need to divide by 2 until we have only one element
    left?

    So it must be:

    n/2^k= 1

    This gives: n=2^k

    this give: log n=k [taken log(base 2) on both sides ]

    Put k= log n in the final equation above and it gives:

    T(n) = T(1) + log n

    T(n) = 1 + log n [we know that T(1) = 1 , because it’s a base condition
    as we are left with only one element in the array and that is the
    element to be searched so we return 1]

    So it gives:

    T(n) = O(log n) [taking dominant polynomial, which is n here)

    This is how we got “log n” time complexity for binary search.



    Thank you,
    Amine Moulay Ramdane.

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