Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

RolyPoly

  • 11 months ago

Consider f(n) = lg(n). What is the largest size n of the problem that can be solved in 1 second, assuming that the time required to solve the problem takes f(n) microseconds? How to start?

  • This Question is Closed
  1. whereismymind49
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So you want the value that will make lg(n) = 1 second or 1000000 microsecond

  2. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I am still not sure how to proceed..

  3. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    say n = 10^(10^6) then u wil get f(10^(10^6)) = log(10^(10^6)) us = 10^6 us = 1s

  4. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so the largest size of problem that can be solved in 1s is \(10^{1000000}\)

  5. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    10^6 us?? And why did you pick n = 10^(10^6)?

  6. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    i just guessed, u can setup an equation aswell

  7. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    we're given :- time required to solve the problem takes f(n) microseconds

  8. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so, if the problem size is n, then it takes f(n) microseconds

  9. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    you need to solve the value of n, that makes f(n) = 1s

  10. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    below eq'n :- 10^6 us = log(n)

  11. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    remember that 10^6us = 1s

  12. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    us = micro second

  13. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ah! So, if it is one minute instead of one second, then it will be 60 x 10^6 us = lg(n) ?

  14. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    thats it, all bigOh problems are just primitive accounting problems lol

  15. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Accounting is way easier than this. I just can't comprehend the problem. :(

  16. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    did u get the phrase 'size of problem' ?

  17. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    No, not at all.

  18. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    okay, suppose i give u \(10\) different numbers and ask you to sort them

  19. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    we say, the size of this problem is \(10\)

  20. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    and lets say, u take 1 minute to solve them.

  21. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    next, suppose i give u 1000000 numbers and ask u to sort them this time u wil take more time, but how much time u wil take depends on wat algorithm u use to sovle them.

  22. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    here problem size is 1000000

  23. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    'problem size' and 'data size' both refer to same thing.

  24. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    see if that seems plausible

  25. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ugh! Why is it not this: \[\frac{\lg n}{1}=\frac{1}{\lg n \times 10^{-6}}\]

  26. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    dint get u ?

  27. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ah! Not that one.. No./Size of problem Time Given 1 lg(n) x 10^(-6) Find n 1 So, we get \[\frac{1}{n} = \frac{\lg n\times 10^{-6}}{1}\]

  28. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    n = 1 is undefined i guess, cuz it wud make the time 0

  29. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    consider the cases n > 1, and use bit common sense

  30. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    if a thing is done in 1us, and we're given the relation that f(n) = log(n) give time for doing n things, finding how many things we can do in 1s should be a high-school grade problem. i dont see any point in you pondering over it :|

  31. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    " required to solve the problem takes f(n) microseconds" Isn't it lg(n) us instead of 1 us?

  32. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes ur right

  33. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Also, if 1 thing is done in lg(n) us, and n things done in 1s, then we can set the equation \[\frac{1}{\lg (n) \times 10^{-6}}=\frac{n}{1}\]Right? Wrong?

  34. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    oops sorry, looks i was sounding bit offensive earlier

  35. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    It's ok...

  36. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    10^6 us = log(n) whats the problem wid this equation ?

  37. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I don't understand how you get this equation, and I don't understand why mine is not correct.

  38. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    same is the case wid me, gimme a sec to digest ur equation

  39. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    wid ur equaiton, wats the largest size you're getting ?

  40. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    we may plug that number in reverse and see if it works or not

  41. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Getting something weird with my equation. \[\frac{1}{\lg (n) \times 10^{-6}}=\frac{n}{1}\]\[n\lg (n) \times 10^{-6}=1\]\[\lg (n^n) =10^6\]\[n^n=10^{10^6}\] Don't know how to solve this thing..

  42. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    humm ok, ima have dinner... wil get back and have a look at this again :)

  43. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    read it like below :- Also, if \(\color{red}{n}\) things are done in lg(n) us, then find how many things can be done in 1s

  44. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\frac{n}{\lg n \times 10^{-6}} = \frac{N}{1}\] N = Number of things done in one second.

  45. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    still stuck on this ha ?

  46. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes...

  47. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    okay, i tried but im not getting ur equation, wat u trying to do wid that equation can u explain ?

  48. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    we're given n things take log(n) micro seconds, and asked to find out how many things we can do in in 1 second so we simply sovle the equation log(n) = 10^6us

  49. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    that gives the 'n' value corresponding to 10^6 us

  50. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    It's like, you need 4 dollars to buy a bar of chocolate, how many bars of chocolate you can buy when you have 120 dollars? We can have \[\frac{1}{4}=\frac{n}{120}\]

  51. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    thats right, wat if it takes log(n) dollars to buy n chocolates ?

  52. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    and you're asked to find how many u can buy wid a million dollars money ?

  53. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    the price per chocolate is not fixed here, clearly it depends on how many u buy

  54. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\frac{n}{\log n}=\frac{N}{1,000,000}\]?

  55. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    its not a linear scaling, i dont think you can do it that way

  56. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    first tell me this, are u convinced 10^(10^6) is the largest problem size yet ?

  57. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    How do you know if it is the largest?

  58. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    plugin that number in the given function

  59. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    It works, but that does not guarantee there is no other (larger) solution, does it?

  60. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    add 1 to that that, and plugin, wat do u get ?

  61. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    u will get 1s + for any value larger than 10^(10^6) cuz log(n) is a strictly increasing function

  62. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    but u want to complete job within 1s, so..

  63. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ah! Ok...

  64. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    the question is begging us to solve f(n) = log(n) = 10^6 no more than that.

  65. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    if it helps seeing this, let me plot the time Vs problem-size graph quick

  66. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1389594869323:dw|

  67. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1389594972424:dw|

  68. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    time to complete the job, depends on teh size of job

  69. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so ive put time in y-axis and size in x-axis. see if that makes more/less sense

  70. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    It makes more sense :) Let say, if f(n) = n, will the graph then be |dw:1389595232578:dw|

  71. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yup, if its linear then we can use ur proportion :)

  72. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    however most algorithms we deal with are exponential/logarthmic/polynomial there will be very few that scale linearly

  73. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    How... do... you... know.... it ... is related to algorithms?

  74. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    here, f(n) = log(n) represents how time scales as size increases for doing a 'particular job' using a 'particular algorithm'

  75. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I should have re-worded the question in a better way :\ So, give a function f(n) and time t seconds, we have f(n) = t x 10^6 Can I generalize the solution to this problem in this way?

  76. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    dint get u

  77. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    given : t = log(n) the inverse wud be :n = 10^t

  78. RolyPoly
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 0

    The question can be generalized in this way: Consider f(n). What is the largest size n of the problem that can be solved in t seconds, assuming that the time required to solve the problem takes f(n) microseconds? And is the general solution/equation f(n) = t x 10^6 ?

  79. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    general solution is to take inverse of time function

  80. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    for our original probel :- given : t = log(n) the general solution (inverse )wud be :n = 10^t units are in microseconds

  81. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    *problem

  82. ganeshie8
    • 11 months ago
    Best Response
    You've already chosen the best response.
    Medals 1

    this may enhance ur understanding on complexity analysis http://pages.cs.wisc.edu/~vernon/cs367/notes/3.COMPLEXITY.html

  83. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.