1 00:00:01,319 --> 00:00:07,080 Welcome back, in the last few videos, we have been studying recurrence relations. 2 00:00:07,080 --> 00:00:14,550 So recurrence relations is basically an equation that recursively defines a sequence of values. 3 00:00:14,550 --> 00:00:25,480 There are some initial terms and the nth term is defined as a function of the preceding 4 00:00:25,480 --> 00:00:26,480 terms. 5 00:00:26,480 --> 00:00:34,130 Recurrence relations have been used extensively for combinatorics, analysis of algorithms, 6 00:00:34,130 --> 00:00:41,850 in computational biology, in theoretical economics and in various other subjects. 7 00:00:41,850 --> 00:00:51,790 In the last week, sorry, couple of videos, earlier we saw how to use the recurrence relations 8 00:00:51,790 --> 00:00:58,379 for modelling some of the counting problems. 9 00:00:58,379 --> 00:01:04,449 Now once you solve, when you model some of the counting problems, you have to solve the 10 00:01:04,449 --> 00:01:07,040 recurrence relations in some way. 11 00:01:07,040 --> 00:01:17,710 So here some of the examples that appears in real life, say for example, T(1) equals 12 00:01:17,710 --> 00:01:29,710 to 1 and T(n) equals to 2 plus T(n-1) or T(1) equals to 2 and T(2) equals to 3 and T(n) 13 00:01:29,710 --> 00:01:39,920 equals to T(n-1) plus T(n-2) or this is the one that came from the Tower of Hanoi problem. 14 00:01:39,920 --> 00:01:50,140 T1=1, sorry, H(1) equals to 1, H(2) equals to 3 and H(n) equals to 2 times H (n-1) plus 15 00:01:50,140 --> 00:01:57,610 1, or this is the one that come from the Fibonacci sequence, F(1) equals to 1, F(2) equals to 16 00:01:57,610 --> 00:02:06,979 1 and F(n) equals to F(n-1) plus F(n-2), or this one that comes from the binary search 17 00:02:06,979 --> 00:02:16,730 algorithm, b(1) equals to 1 and b(n) equals to b (n over 2) + 1, or this one that comes 18 00:02:16,730 --> 00:02:24,290 from the merge sort algorithm M(1) equals to 1 and M(n) equals to 2 times M( n over 19 00:02:24,290 --> 00:02:34,510 2) + n, or this one which comes from, what is known as Catalan number C(1) equals to 20 00:02:34,510 --> 00:02:42,530 1 and C(n+1) equals to summation i= 0 to n C(i) C(n-i). 21 00:02:42,530 --> 00:02:49,540 Now these are some of the recurrence relations that appear in real life, these are the very 22 00:02:49,540 --> 00:02:52,150 small sample of them. 23 00:02:52,150 --> 00:02:55,950 Now the main question is how do you solve these recurrence relations? 24 00:02:55,950 --> 00:03:01,150 So these recurrence relations are ofcourse used to module various problems, but once 25 00:03:01,150 --> 00:03:06,530 you model them into a recurrence relations, the next step is to solve them. 26 00:03:06,530 --> 00:03:13,680 In the last video, we saw a technique of solving them. 27 00:03:13,680 --> 00:03:21,930 And we told these are the techniques that first of all guess the solution and then proves 28 00:03:21,930 --> 00:03:23,700 using induction. 29 00:03:23,700 --> 00:03:31,490 We saw that if we can guess the solution correct, then proving it by induction possibly not 30 00:03:31,490 --> 00:03:37,890 to hard a problem, it is like the typical induction problem. 31 00:03:37,890 --> 00:03:41,940 The main issue is how do you guess the solution? 32 00:03:41,940 --> 00:03:47,520 Now guessing the solution can really be a challenging problem. 33 00:03:47,520 --> 00:03:51,570 So we will be dedicating quiet a number of lectures on guessing the solution. 34 00:03:51,570 --> 00:04:03,390 Today we will be looking at the first and the simplest technique of guessing the solution. 35 00:04:03,390 --> 00:04:13,870 So here is it, so technique one, the idea is just unfolding the definitions. 36 00:04:13,870 --> 00:04:20,590 What do I mean by unfolding the definitions? 37 00:04:20,590 --> 00:04:26,120 So let us look at some of the examples and you will understand what do I mean by that? 38 00:04:26,120 --> 00:04:32,810 Note that these are not formal proofs, these are mere guessing which might work or might 39 00:04:32,810 --> 00:04:39,070 not work and whether it works or not, of course you have to go back to the induction and prove 40 00:04:39,070 --> 00:04:42,710 it and only then we get a formal proof. 41 00:04:42,710 --> 00:04:49,820 So this is whatever I am going to say it now is how to guess this step? 42 00:04:49,820 --> 00:05:01,560 Say if T(1)=1 and T(n)=2+ T(n-1), so let us write down here, T(n)= 2+ T(n-1). 43 00:05:01,560 --> 00:05:07,180 Now what is T(n-1), I can recursively now open up, right? 44 00:05:07,180 --> 00:05:18,330 so T(n-1) is 2+T(n-2), Okay, let us write down once again, 2+ T(n-3), okay, let us write 45 00:05:18,330 --> 00:05:22,590 down once again, 2 + T(n-4). 46 00:05:22,590 --> 00:05:34,260 So this is where comes up the big leap of faith, your kind of say that okay when there 47 00:05:34,260 --> 00:05:40,400 are some 4 of the 2’s here, I have a 4 here. 48 00:05:40,400 --> 00:05:46,270 When I have 3 of the 2’s here, I have 3 here. 49 00:05:46,270 --> 00:05:52,100 When I have 2 of the 2’s here, I have 2 here. 50 00:05:52,100 --> 00:05:54,650 When I have one 2 here, I have 1 here. 51 00:05:54,650 --> 00:06:09,290 So if I have keep on doing this way, I will get a 2+2+, for some k of them + T(n-k), now 52 00:06:09,290 --> 00:06:10,730 this is a leap of faith. 53 00:06:10,730 --> 00:06:16,780 Okay, again as I told it is a guessing work, right. 54 00:06:16,780 --> 00:06:22,520 Now this number is somehow to vanish. 55 00:06:22,520 --> 00:06:37,540 The idea is that, the initial things, here T(1)=1, gives us the hint, so we have to set, 56 00:06:37,540 --> 00:06:46,360 so here if k=n-1, then T(n-k) = T(1) is equals to 1. 57 00:06:46,360 --> 00:06:49,880 So I have to put this one as n-1. 58 00:06:49,880 --> 00:07:05,450 So in fact if I remove this k here and instead I write here n-1, and I remove this one n-1, 59 00:07:05,450 --> 00:07:08,200 what do I get? 60 00:07:08,200 --> 00:07:24,160 Here I should get 2 times n-1, because I have 2(n-1) + T(1), which is 2n-2+1, which is 2n-1, 61 00:07:24,160 --> 00:07:25,160 okay. 62 00:07:25,160 --> 00:07:37,760 So this by doing so we have guess that Tn=2n-1, again also it might seem very formal way of 63 00:07:37,760 --> 00:07:46,280 proving that T(n) =2(n-1), the fact is that it still not a correct proof, a complete proof, 64 00:07:46,280 --> 00:07:49,460 because we have these dot, dot, dots here. 65 00:07:49,460 --> 00:07:53,810 There was a big leap of faith from this to this. 66 00:07:53,810 --> 00:08:00,770 May be our intuitions are correct and we get the right answer and in which case we go ahead 67 00:08:00,770 --> 00:08:10,110 and prove it using induction, right; and there are times, there are examples where this leap 68 00:08:10,110 --> 00:08:15,120 of faith may not be exactly correct. 69 00:08:15,120 --> 00:08:21,990 But this is one way of kind of guessing what the number is. 70 00:08:21,990 --> 00:08:42,460 So in this case, we have T(n)= 2n-1, is the guess and you prove it by induction. 71 00:08:42,460 --> 00:08:49,110 We had seen in the last video that this is indeed right way of guessing it and we have 72 00:08:49,110 --> 00:08:51,810 or right guess by proving it by induction. 73 00:08:51,810 --> 00:09:05,089 Now let us move on to the next example, so here T(n)= n+ T (n-1), again we have to let 74 00:09:05,089 --> 00:09:12,620 us keep on unfolding a definition, so T(n)= n+ T (n-1), unfold this T(n-1), this is (n-1) 75 00:09:12,620 --> 00:09:14,120 + T(n-2). 76 00:09:14,120 --> 00:09:28,660 Now again if I unfold it more, this is (n-2) + T(n-3) and here again now let us do a leap 77 00:09:28,660 --> 00:09:42,580 of faith, as you see here, when I have 3 here, I keep on doing it till 2, when I 2 here, 78 00:09:42,580 --> 00:10:00,990 I keep on doing it till 1, so may be if I keep on doing this thing till n-k, I have 79 00:10:00,990 --> 00:10:02,950 T(n-k+1), right. 80 00:10:02,950 --> 00:10:04,910 Sorry, T(n-(k+1)). 81 00:10:04,910 --> 00:10:20,811 Now again we have to somehow disappears this term, so the idea is again that we have to 82 00:10:20,811 --> 00:10:50,550 get this n-k+1 as 1, so in other words, if I take k to be =n-2, right, what we had? 83 00:10:50,550 --> 00:11:06,029 Then n-(k+1) =1, so in that case what should we get is that, I keep on going it, +n-k, 84 00:11:06,029 --> 00:11:19,490 which is n-2 + T(1), now T(1) =1 and this keeps on going and this n-(n-2) is nothing 85 00:11:19,490 --> 00:11:30,760 but 2 and the T(1)=1, so in fact we get the sum over the first n integers, which is n(n+1)/2. 86 00:11:30,760 --> 00:11:48,550 So by doing so, we have guess that T(n)=n(n+1)/2, now again as I told you this is a leap of 87 00:11:48,550 --> 00:11:55,350 faith, because there was a leap of faith here and hence this is just a case, so formally 88 00:11:55,350 --> 00:12:08,730 proving it we have to solve it by induction and verify that our guess is indeed right. 89 00:12:08,730 --> 00:12:18,600 So again the simple idea is, keep on unfolding the definition and it will be possibly we 90 00:12:18,600 --> 00:12:26,420 are able to guess the value and in this case we did guess Tn=n(n+1)/2 and we then prove 91 00:12:26,420 --> 00:12:28,040 it by induction. 92 00:12:28,040 --> 00:12:41,070 As I told you, most of the time the guess does work correctly if we can unfold it in 93 00:12:41,070 --> 00:12:44,890 a right way. 94 00:12:44,890 --> 00:12:56,080 Let us look at one more example, is the tower of Hanoi problem where H(1) =1 and H(n)= 1+ 95 00:12:56,080 --> 00:13:02,230 2 times H(n-1), here H(n)=1+2H(n-1), so this is actually quite interesting, so this one 96 00:13:02,230 --> 00:13:17,709 is 2 times now here, 1+ 2times H(n -1), sorry (n-2), which is 1+2+2 times H(n-2), sorry 97 00:13:17,709 --> 00:13:33,580 not 2 times, this is in fact 2 times 2 is 4 times. 98 00:13:33,580 --> 00:13:52,070 Let us open it again, 1+2+ 4 times, what is H(n-2)?, is 1 + twice H(n-3), which is 1+2+4+8H(n-3). 99 00:13:52,070 --> 00:13:58,020 Now this is where you really have to take a leap of faith, so let us write this one 100 00:13:58,020 --> 00:14:01,950 here, 1+2+4+, what is 8? 101 00:14:01,950 --> 00:14:09,180 8 is 2 power 3 H(n-3). 102 00:14:09,180 --> 00:14:20,020 Note that here it was also 2 power 2 and I had 2 here, 2 power 1 and I had 1 here. 103 00:14:20,020 --> 00:14:34,360 So again by the complete leap of faith, we can write it as 1+2+4+8+2 power k+ 2 power 104 00:14:34,360 --> 00:14:39,830 k+1 and when I have k+1, I have H(n-(k+1)). 105 00:14:39,830 --> 00:14:57,220 Now here again we need to make this one disappear, so again I have H(1) equals to 1, so again 106 00:14:57,220 --> 00:14:59,600 take k to be equals to n-2. 107 00:14:59,600 --> 00:15:17,480 If I take k=n-2, then n-k+1 =1, so when this becomes T(1), this becomes T(1), which is 108 00:15:17,480 --> 00:15:28,000 1, so I get 1+2+4+8+2 power k which is n-2+2 power n-1 times T(1) and since T(1)=1, so 109 00:15:28,000 --> 00:15:33,180 I can forget this statement. 110 00:15:33,180 --> 00:15:49,750 And so I get this number, which is a GP series and the GP series as the 2 power n-1. 111 00:15:49,750 --> 00:15:59,640 So I guess that H(n)=2 power n-1, so this one clearly was slightly more complicated 112 00:15:59,640 --> 00:16:05,200 than the earlier ones, but again here there was a massive leap of faith here, when we 113 00:16:05,200 --> 00:16:17,839 guess that this 3,3 and similarly here 2, 2 and so on exist. 114 00:16:17,839 --> 00:16:24,170 And so by doing so we have managed to guess it, but we need to prove that the guess is 115 00:16:24,170 --> 00:16:28,980 right again by induction. 116 00:16:28,980 --> 00:16:35,860 It so happens that in this case, we guess it indeed right and we saw it last time that 117 00:16:35,860 --> 00:16:47,190 we can guess this one and we do get the induction, by induction we can prove the statement. 118 00:16:47,190 --> 00:16:59,350 So the basic idea that we learned from this video is that if have given a particular recurrence 119 00:16:59,350 --> 00:17:17,559 relation like this and if you can unfold it, maybe you can try to guess the number. 120 00:17:17,559 --> 00:17:31,049 The problem is that they are complicated ones like this, 121 00:17:31,049 --> 00:17:42,899 F(n)=F(n-1) + F(n-2), now we can try to unfold it by saying okay, F(n)=F(n-1) + F(n-2), where 122 00:17:42,899 --> 00:17:53,350 F(n-1)= F(n-2)+ F(n-3) + F(n-2), so I get 2 times F(n-2) + F(n-3) which will remain 123 00:17:53,350 --> 00:18:03,179 2 times F(n-3)+F(n-4)+F(n-3) which is equals to 3 times F(n-3)+ 2 times F(n-4) and of course 124 00:18:03,179 --> 00:18:16,399 it is clear if I keep on doing it and but then as you can see I can write down the next 125 00:18:16,399 --> 00:18:20,980 step directly and this will become something like 5F(n-4)+3 F(n-5). 126 00:18:20,980 --> 00:18:35,739 Since, particularly no pattern coming out in this recurrence relations. 127 00:18:35,739 --> 00:18:41,720 So in fact for this kind of recurrence relations, unfolding will not help. 128 00:18:41,720 --> 00:18:49,399 I leave you guys to check and verify and convince yourselves that here by unfolding you will 129 00:18:49,399 --> 00:18:58,179 not be able to guess the actual value, in fact guessing the actual value is quiet complicated 130 00:18:58,179 --> 00:18:59,179 here. 131 00:18:59,179 --> 00:19:08,309 Here is the actual guess and you can see by looking at this expression, it is not something 132 00:19:08,309 --> 00:19:14,690 that is easy to guess, right. 133 00:19:14,690 --> 00:19:24,789 Also we have other kind of formulas like this one, where b(1)=1 and b(n)=b(n/2)+1, where 134 00:19:24,789 --> 00:19:34,570 this is, this one denotes the integer that is bigger than or equal to n/2, so if n=9, 135 00:19:34,570 --> 00:19:40,230 then n/2, with these 2 things here is 5, right. 136 00:19:40,230 --> 00:19:48,789 So with this kind of an expression, unfortunately there is no clean guess can be made because 137 00:19:48,789 --> 00:20:02,289 of this extra bit of weird thing that are there, this what we call as ceilings right. 138 00:20:02,289 --> 00:20:08,220 So there are expressions of this form where either the guessing is too hard or we do not 139 00:20:08,220 --> 00:20:11,140 have a very clean guessing for them. 140 00:20:11,140 --> 00:20:17,279 How to attack this particular kind of recurrence relation we will be doing in the next couple 141 00:20:17,279 --> 00:20:18,279 of weeks. 142 00:20:18,279 --> 00:20:19,210 Thank you.