Painter's Partition Problem ｜ DSA Placement Series..
Painter's Partition Problem ｜ DSA Placement Series..
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00:00Hi everyone and welcome to the complete DSA series in which we are going to do an important
00:08problem called the Painter's Partition Problem.
00:10Apart from this, if we want to learn any other concept of DSA, we will get it on this channel
00:14in this playlist.
00:15And also we have prepared a company-wise DSA sheet in which we will get the DSA questions
00:21along with the list of companies in which those questions have already been asked.
00:25You will also get the link of this sheet in the description box below.
00:28Now let's start with the problem.
00:30The problem is called Painter's Partition Problem and this question is a direct variation
00:35of book allocation problem which we covered in the last lecture.
00:40So if you have watched the last lecture, most probably we will solve this question ourselves.
00:43So the question states that we have N boards, each with a given length, which is given in
00:48the form of an array.
00:49For example, we have N is equal to 4 i.e. we have 4 boards.
00:53We can imagine these as wooden boards and the length of these 4 boards is given in the
00:58form of an array.
00:59For example, we have 4 boards and we take the length of all the 4 boards as 10 units.
01:05Along with this, we are given M painters and M painters are given in such a way that every
01:11painter paints 1 unit of board in 1 unit of time.
01:16For example, let's suppose we have 2 painters and the job of these painters is to paint
01:21our 4 boards.
01:22Now when they paint these 4 boards, then it will take them 1 unit to paint 1 unit of board.
01:28i.e. if the length of a board is 10, then they can also say that if our unit is 1 minute,
01:32then it will take them 10 minutes to paint this board.
01:35If the length is 7, then it will take them 7 minutes to paint that board and all painters
01:40paint the board with equal speed.
01:42The task is to find the minimum time to paint all the boards.
01:46We have to find out the minimum time in which we can paint all these boards using these
01:52M is equal to 2 painters, such that, there is a given condition that the painter will
01:57only paint continuous sections of the board, i.e. if the first board is painted by painter
02:021 and the second board is painted by painter 2, then painter 1 cannot paint the third board.
02:07The remaining boards will have to be painted by painter 2 only, i.e. the boards that the
02:12painters will get will be in a contiguous or continuous manner.
02:15So what will be the minimum time in which these 2 painters will paint all these boards?
02:19Obviously, we are getting to know that if we assign these 2 boards to one painter and
02:24these 2 boards to the second painter, both the painters will take 20-20 minutes to paint
02:29their boards, then our minimum value will be equal to 20.
02:33We cannot paint the boards in a value less than this, so this is going to be the answer.
02:38Now, whoever has made a book allocation problem, I would advise you to pause here and try to
02:42solve this question on your own because it is going to be very easy.
02:45Now, let's try to solve it through another example.
02:48Let's suppose we have this array of the length of our boards, n is equal to 4 value, m is
02:52equal to 2 painters.
02:54Now, binary search is not going to be the most intuitive approach on this question because
02:58neither an array, a sorted array will be given to us every time that we try to find a value
03:03in this array, so we will have to think of an alternate approach to solve this question.
03:08Basically, our problem is that we have 2 painters, p1 and p2, and our task is to allocate all
03:15of these boards among these 2 painters.
03:18Now, for the solution, we basically need to find time to paint the boards and we have to
03:24find out the minimum value of this thing.
03:26Now, how can we finish our painting work?
03:30The first method would be that we assign this board to painter 1 and assign the remaining
03:34boards to painter 2.
03:36So, how much time will painter 1 take?
03:3840 minutes.
03:39Painter 2 will have to paint all the boards, so it will take time 30 plus 10 plus 20, i.e.
03:43equal to 60 minutes.
03:46The second case can be that we assign 2 boards to painter 1 and 2 boards to painter 2, so
03:53the first painter will take 70 minutes, the second painter will take 30 minutes, the third
03:58case can be that we assign 3 boards to painter 1 and 2 boards to painter 2, so the painter
04:051 will take 80 minutes and the painter 2 will take 20 minutes.
04:09Now, there can be another case where we assign all the boards to the same painter, but we
04:14will not be going towards the minimum value of time.
04:18Why?
04:19Because we will have one painter who has not been assigned any job, painter 1 has to do
04:22all this work.
04:23So, obviously, if only one painter does all the work, then the time will increase, it
04:26can never decrease.
04:27So, these are the 3 logical cases from which we can get our minimum time.
04:31Now, if we talk about the first case, painter 1 is taking 40 minutes, painter 2 is taking
04:3460 minutes.
04:35So, how much time will it take to finish the work?
04:37In the first case, it will definitely take 60 minutes to finish the work, in the second
04:40case, painter 1 is taking 70 minutes, painter 2 is taking 30 minutes, but it will take 70
04:44minutes to finish the work.
04:45In the third case, it will take 80 minutes to finish the work.
04:49In which case has the work finished within the minimum time?
04:52Definitely the first case, in which we did the partition from here, we did the partition
04:55from here.
04:56We assigned these boards to painter 1 and we assigned the remaining boards to painter
05:002.
05:01So, this is the minimum time taken to paint all of the boards and this should be our answer,
05:06we have to return the value of 60.
05:08Now, our approach to solve the question will be that if we have to find out the minimum
05:15time taken to paint all the boards, then why not take out all the possible values of the
05:20time taken to paint the boards.
05:23That is, if we have these boards, then to paint these boards, can we take the minimum
05:29time and can we take the maximum time?
05:33If we take out the value of minimum and maximum time logically, then the answer will be within
05:38this range.
05:39So, how much time will it take us to paint the boards?
05:44The maximum time taken to paint a given board will take us to paint all the boards.
05:50We will assign only one painter.
05:51So, what will the painter do?
05:52First, he will paint this, then he will do this, then he will do this, then he will do
05:53this.
05:54In that case, we will take the maximum time.
05:55So, what is the value of the maximum time that we can take to paint all these boards?
06:01That is going to be 40 plus 30 plus 10 plus 20, which is equal to 100 units.
06:06Can we say that this maximum time is also equal to the sum of all the board lengths?
06:12If we take out the sum of the length of all the boards, then what will happen?
06:16That will be our maximum time.
06:17It cannot take more time to paint all the boards than this.
06:20And what will be the minimum time?
06:22The minimum time logically will be that we assign a different painter to paint each board.
06:26Definitely, we have a limit of painters.
06:29And if we hypothetically think that a painter will paint this board, he will do this, he
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23:58with that we will check that our mid value is it a possible answer, so for possible answer
24:06we will pass our array n, m and mid, if it is possible answer then we will save this answer,
24:14if it is possible then we will save mid as answer and with this we will try to find minimum
24:18value for which end will be mid-1 and if it is not possible answer then in that case in
24:25second half we will go to left and here we will go to right, so here it will start mid
24:32plus 1 is equal and in last we will return our possible answer, now we just have to complete
24:37this function which is a boolean function whose name is isPossible, this function will
24:44finally tell us that any mid is a possible answer or not, now here we will say this mid
24:51is a possible answer or not, if it is possible answer then we will return mid-1, if it is
24:58not possible answer then we will return mid-2, if it is possible answer then we will return
25:05mid-3, if it is possible answer then we will return mid-4, if it is possible answer then
25:12we will return mid-5, if it is possible answer then we will return mid-6, if it is possible
25:19answer then we will return mid 6, if it is able answer then we will return midt 7, if
25:27it is not possible answer then return mid 8 plus 1 and finally if it is possible answer
25:34for which we will make painters++ and reinitialize time to the current board
25:39array of i.
25:40at last we will check whether the value of painters is less than or equal to m
25:47if it is less than or equal to m then we have to return true
25:50if it is not then we have to return false
25:52so simply we can return this value from here
25:54so this is our overall code to solve the painters partition problem
25:59now if we talk about the time complexity of this code
26:02is a possible function
26:11so this function is big of n
26:13minimum time to paint
26:26can we say that this is approximately equal to
26:32log of sum
26:44for simplicity we call it log of sum
26:47because in this binary search loop
26:49we are calling this possible function
26:52whose time complexity is big of n
26:54so the overall time complexity of this loop will be log of sum into n
27:00so this will be big o of log of sum into n
27:04and this is the overall time complexity
27:06which is going to be of this function
27:08so the overall time complexity of our code is big o of n into log of sum
27:13let's save this
27:14for our values we will find out our answer
27:18and the answer is going to be equal to 60
27:20so this is how we solve the painters partition problem using binary search
27:23apart from this if we want to solve more binary search questions
27:25then we will get it in the company wise dsa sheet below
27:29link is given in the description box below
27:31if you have successfully completed the lecture
27:33you can mark your attendance in the comments and write completed
27:35and you can also share your progress with me on twitter
27:37whose link is also given in the description box below
27:40that's all for today
27:41see you in the next lecture
27:42till then keep learning and keep exploring