Write a python program and HTML table for this problem. 1-1...

Write a python program and HTML table for this problem. 1-1 Comparison of running times For each function f(n) and time t in the following table, determine the largest size n of a problem that can be solved in time t, assuming that the algorithm to solve the problem takes $f(n)$ microseconds. \lg n $\sqrt{n}$ $\lg^2 n$ $n$\lg n$ n$^3$ $2^n$ n$^n$ 1 1 1 1 1 1 second minute hour day month year century Assume 30 days/month and 365 days a week. If you write a Python program, please consider using 'numpy' or 'math' module. Your output should be in an HTML text format which includes a table with your answers as shown in the textbook. You may use any HTML module or use print() to write your own html file. One way or the other you will learn how to write an html file. However, when the HTML file is loaded on a browser, it should display the table in a reasonably nice table format. The objectives of this homework: 1. Demonstrating basic understanding of algorithms efficiency 2. Learning and exercising Python \quad \text{Math and/or numpy modules} \quad \text{Basic HTML syntax of } \langle \text{table} \rangle Deliverable: 1. Program source code 2. Output in HTML file

Transcript

00:01 In this problem we are going to determine the largest size n of a problem that can be solved in time t for various time values and various algorithms.

00:16 For instance, one algorithm may lead to a function in form of, let's say, logarithm of n.

00:24 One may lead to squared of n, etc.

00:28 So we will have lots of functions, f of n, we are going to construe various times, like one second, one minute, one hour, up to one century.

00:39 So let me just show you one specific row and i will just complete the table for the other functions because it is just a numerical, it is all about numerical solution of an equation in the following form.

00:58 So i will just focus on one specific case in great detail.

01:04 So here it goes.

01:17 So what is going on here? we have this function and the size of the problem is n.

01:26 It can be like 100 or 1 ,000 etc.

01:31 So when you insert that value of n into this function, it will give you the time.

01:43 It will take for this problem to be solved in units of microseconds.

01:51 So f of n has the units of microseconds.

02:00 So what is the largest size for this problem, for a problem that goes like logarithm of n to be completed within a second? for instance, how do you find this number? we do it like this.

02:18 We take our number, this unknown number, and whatever it is, we insert it to this function, and we will have some value here and it should go like microseconds.

02:33 So whatever the time value we see here, we should convert it to microseconds.

02:40 For instance, for one second we have 10 to the power 6.

02:48 So then we ignore the unit because now we know what we are dealing with while we know the output of this function.

02:56 And we saw the rest of the problem numerically like this.

03:03 So for a problem that goes like logarithm of n, the maximum, the largest size n for this problem is 10 to the power, 10 to the power.

03:17 Six for instance let's do it also for a minute it will become i think clearer in that case so we have this size and we insert it into the logarithm function it is usually based on i mean in numerical in such problems we don't consider the ln function we always focus on the usual logarithm function so for a minute first we convert one minute into microseconds.

03:54 So we have 1 minute is equal to 60 seconds.

04:05 And this is equal to 60 times 10 to the power 6 microseconds.

04:12 So what is the equation that we need to solve? it is low rate of n equal to 60 times 10 to power 6.

04:21 And the solution is, or the largest size is, 10 to the power 60 times 10 to the power 6 so this shows indeed the power of the logarithmic behavior let's say so if your problem goes like log of n then you can solve lots of you can have a real large system that can be sold within a minute so this is the moral of the story so in this fashion we will go from one second to one century and now i will fill the table for you.

05:01 So this is just the first one i will do the other functions separately.

05:06 So okay let's do it like this because i don't have much space here so i will just quote the results in this fashion.

05:14 So okay we have one second so we have seen that it is 10 to the power.

05:31 Okay.

05:31 Notation to avoid you know clutter so it is 10 to the power 10 to power 6 for one minute it is 10 to the power 60 times 10 to power 6 for one hour 10 to the power 3 .6 times 10 to the power 9 one day 10 to the power sixty, uh 86 point four times 10 to the power nine one month we have 10 to the power 2 .992 times 10 to the power 12 and continue over here for a year still the same function we have 10 to the power 31 point one of one of 4 times 10 to the power.

07:02 Okay, you know, let's erase these lines.

07:06 1 .15.

07:09 And for one century, we have 10 to the power 3 .11 .04 times 10 to the power 15.

07:26 Wait a second.

07:28 Okay, this should be 12.

07:32 Okay, like this.

07:34 So these are the values for log n.

07:43 Just paint this result so that we see it i think more clearly at the end of today okay now do let us do the function f of an equal to skirt of n same idea we have a skirt of an equal to one second but we need all the time we use in use of microseconds so 1 second is equal to 10 to power 6 microseconds.

08:20 After writing this time, after converting time value into microseconds, we ignore this unit and sold the rest of the equation.

08:29 That's it.

08:31 For a second again, we have 10 to power 6.

08:34 For a minute, we have 60 times 10 to power 6.

08:38 For an hour, we have 60 times 60 times 10 to power 6.

08:43 For a day 24 times 60 times 60 times 10 to power 6 for a month we include a factor of 30 here for a year we include a factor of 12 and for a century we include a factor of 100 so this number gets bigger and bigger in this fashion let me do it over here so skirt of n one seconds we have 10 to the power 12 one minute 3 .6 times 10 to the power 15 one hour 1 .2 .96 times 10 to power 19 one day 7 .446 6 times 10 to power 21.

09:59 1 month.

10:04 6 .7184 6 times 244...

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