1
00:00:01,140 --> 00:00:06,240
So we saw in the last video
why if you represent scheduling

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00:00:06,240 --> 00:00:08,580
constraints among
courses by a digraph

3
00:00:08,580 --> 00:00:11,900
that it's critical that that
digraph in fact be a DAG.

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00:00:11,900 --> 00:00:15,810
And let's now look at this
scheduling issue represented

5
00:00:15,810 --> 00:00:17,700
by DAGs in more detail.

6
00:00:17,700 --> 00:00:23,670
So here's a chart of a selection
of Course 6 prerequisites--

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00:00:23,670 --> 00:00:26,610
some of them obsolete, but
they serve the purposes

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00:00:26,610 --> 00:00:28,790
of being an
illustrative example--

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00:00:28,790 --> 00:00:30,630
and the little arrows
here are indicating

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00:00:30,630 --> 00:00:31,780
arrows in the digraph.

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00:00:31,780 --> 00:00:34,490
So what this tells
me is that 18.01

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00:00:34,490 --> 00:00:37,150
is listed as an
immediate prerequisite

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00:00:37,150 --> 00:00:40,180
in the catalog for 6.042.

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00:00:40,180 --> 00:00:45,940
18.01 is also an immediate
prerequisite of 18.02.

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00:00:45,940 --> 00:00:52,890
6.001 and 6.004 are both
prerequisites of 6.033,

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00:00:52,890 --> 00:00:58,200
and 6.042 of 6.046,
and 6.046 of 6.840.

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00:00:58,200 --> 00:01:01,560
So we're seeing here this
indirect prerequisite issue

18
00:01:01,560 --> 00:01:05,050
that I mentioned before, which
is that even though the only

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00:01:05,050 --> 00:01:08,790
thing listed as a prerequisite
for 6.840 in the catalog is

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00:01:08,790 --> 00:01:12,850
6.046, as a matter of fact in
order to take 6.046 you have

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00:01:12,850 --> 00:01:14,720
to have taken 6.042.

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00:01:14,720 --> 00:01:20,020
So 6.042 is an indirect
prerequisite of 6.840.

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00:01:20,020 --> 00:01:24,120
So in terms of graph
language and path language,

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00:01:24,120 --> 00:01:27,470
a subject u is an
indirect prerequisite of v

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00:01:27,470 --> 00:01:28,880
when there's is
a positive length

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00:01:28,880 --> 00:01:31,930
path from u to v
in the digraph that

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00:01:31,930 --> 00:01:35,950
describes the prerequisite
structure among the classes.

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00:01:35,950 --> 00:01:39,270
It simply means that-- using
our notation for R plus

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00:01:39,270 --> 00:01:41,460
is the positive
length path relation

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00:01:41,460 --> 00:01:44,150
of a digraph or a
binary relation R--

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00:01:44,150 --> 00:01:47,470
it simply means
u R plus v, which

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00:01:47,470 --> 00:01:52,880
is read as there is a positive
length path from u to v.

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00:01:52,880 --> 00:01:56,020
Now, a key idea that we're
going to be examining

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00:01:56,020 --> 00:01:58,490
in learning how to
do scheduling is

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00:01:58,490 --> 00:02:00,500
the idea of a minimal subject.

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00:02:00,500 --> 00:02:02,730
So the definition
of a minimal subject

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00:02:02,730 --> 00:02:05,060
is a subject that
has no prerequisites,

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00:02:05,060 --> 00:02:08,229
no arrows in, a
freshman subject.

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00:02:08,229 --> 00:02:09,400
So nothing comes in.

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00:02:09,400 --> 00:02:12,150
There are three
examples of subjects

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00:02:12,150 --> 00:02:15,380
with no prerequisites
in the preceding chart,

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00:02:15,380 --> 00:02:18,790
namely 18.01, 8.02, and 6.001.

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00:02:18,790 --> 00:02:21,380
Let me say a word about where
this funny terminology minimal

44
00:02:21,380 --> 00:02:22,900
comes from.

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00:02:22,900 --> 00:02:26,150
It's because another
way to talk about DAGs

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00:02:26,150 --> 00:02:29,690
is in terms of things that are
like order relations called

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00:02:29,690 --> 00:02:33,520
partial orders, which we'll
be looking at shortly.

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00:02:33,520 --> 00:02:38,870
And so you think of
the later subjects

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00:02:38,870 --> 00:02:42,340
as being bigger than
the earlier subjects.

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00:02:42,340 --> 00:02:44,800
So a minimal
subject is one where

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00:02:44,800 --> 00:02:47,620
there is nothing less than it.

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00:02:47,620 --> 00:02:49,727
Now, there might be
several minimal subjects,

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00:02:49,727 --> 00:02:51,560
because it might be
that neither one of them

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00:02:51,560 --> 00:02:53,351
is less than the other,
but there's nothing

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00:02:53,351 --> 00:02:54,900
less than 18.01.

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00:02:54,900 --> 00:02:59,010
There's no other subject that
you have to take before 18.01.

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00:02:59,010 --> 00:03:02,250
So that's the
definition of minimal.

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00:03:02,250 --> 00:03:03,600
Nothing smaller.

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00:03:03,600 --> 00:03:05,350
Now, you could ask
what's a minimum, which

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00:03:05,350 --> 00:03:07,250
you may be more familiar with.

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00:03:07,250 --> 00:03:11,020
A minimum means that not only
is there nothing before it,

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00:03:11,020 --> 00:03:13,220
but it comes before
everything else.

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00:03:13,220 --> 00:03:16,790
It would be the earliest
of all possible subjects

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00:03:16,790 --> 00:03:18,710
in the indirect
prerequisite chain.

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00:03:18,710 --> 00:03:20,890
There isn't any in this
example, but there actually

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00:03:20,890 --> 00:03:22,970
used to be one at MIT.

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00:03:22,970 --> 00:03:25,980
For a while, we experimented
with giving an orientation week

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00:03:25,980 --> 00:03:29,190
summer assignment, that is,
an assignment over the summer

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00:03:29,190 --> 00:03:31,730
for newly admitted
students in order for them

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00:03:31,730 --> 00:03:34,340
to take a subject
during orientation

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00:03:34,340 --> 00:03:37,250
week in which they discussed
some book that they had all

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00:03:37,250 --> 00:03:39,240
been assigned to
read beforehand.

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00:03:39,240 --> 00:03:41,880
Seemed like a great idea to kind
of pull the freshman community

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00:03:41,880 --> 00:03:44,710
together, but it turned
out to be unsustainable

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00:03:44,710 --> 00:03:47,180
because they couldn't
find enough faculty

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00:03:47,180 --> 00:03:50,380
and others willing to
conduct these seminars.

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00:03:50,380 --> 00:03:53,703
So MIT stopped having
a minimum subject.

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00:03:56,094 --> 00:03:57,510
So let's look at
the prerequisites

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00:03:57,510 --> 00:04:00,292
again, and discuss how
to do a scheduling.

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00:04:00,292 --> 00:04:02,500
And the first thing we're
going to do in the schedule

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00:04:02,500 --> 00:04:04,820
is, as I say, identify
the minimal elements.

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00:04:04,820 --> 00:04:07,650
There are the three of
them that we mentioned.

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00:04:07,650 --> 00:04:11,840
And we're going to start
by deciding that we'll take

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00:04:11,840 --> 00:04:14,070
those three in the first term.

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00:04:14,070 --> 00:04:16,570
So we're going to be operating
with basically what's

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00:04:16,570 --> 00:04:17,784
called a greedy strategy.

87
00:04:17,784 --> 00:04:19,200
We're going to
take as many things

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00:04:19,200 --> 00:04:23,414
as we possibly can take at any
term given the constraints.

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00:04:23,414 --> 00:04:25,830
So we can take all the freshman
subjects in our first term

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00:04:25,830 --> 00:04:27,740
because they have
no prerequisites.

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00:04:27,740 --> 00:04:30,040
Well, the next step, then,
is just get rid of them

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00:04:30,040 --> 00:04:31,570
because they're
scheduled already.

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00:04:31,570 --> 00:04:35,380
So we can get rid of all those
occurrences of 18.01, 8.02,

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00:04:35,380 --> 00:04:40,420
and 6.001, not only-- there are
other occurrences as well here

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00:04:40,420 --> 00:04:43,280
where 18.01 is a
prerequisite for things.

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00:04:43,280 --> 00:04:46,080
So they're all gone, and
we get a simplified diagram

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00:04:46,080 --> 00:04:48,760
where we've removed
the minimal elements.

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00:04:48,760 --> 00:04:51,710
Now in the new
diagram, there are now

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00:04:51,710 --> 00:04:53,560
things that didn't
used to be minimal

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00:04:53,560 --> 00:04:55,200
before but are minimal now.

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00:04:55,200 --> 00:04:56,760
These are the new
minimal elements,

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00:04:56,760 --> 00:04:58,490
and we can identify those.

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00:04:58,490 --> 00:05:01,480
Here are five subjects--
four here and one there--

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00:05:01,480 --> 00:05:03,790
that now have no
more prerequisites.

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00:05:03,790 --> 00:05:06,770
These are kind of the second
level minimal elements,

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00:05:06,770 --> 00:05:09,670
and we're going to
schedule them next.

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00:05:09,670 --> 00:05:12,210
So those are all the
subjects that we can possibly

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00:05:12,210 --> 00:05:16,660
take after we've taken the
first set of minimal subjects.

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00:05:16,660 --> 00:05:18,160
They're the second
level minimals.

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00:05:18,160 --> 00:05:19,930
And we'll schedule
them in the next term.

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00:05:19,930 --> 00:05:25,140
This is our five subject
second term schedule.

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00:05:25,140 --> 00:05:27,760
Likewise, you delete
these guys, and then you

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00:05:27,760 --> 00:05:31,660
discover that 6.046 and 6.004
are the resulting minimal ones,

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00:05:31,660 --> 00:05:34,620
which it's now possible to take
because all their prerequisites

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00:05:34,620 --> 00:05:35,740
have been satisfied.

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00:05:35,740 --> 00:05:39,450
So we schedule them in the
third term, 6.840 and 6.033,

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00:05:39,450 --> 00:05:41,650
by the same reasoning,
in the fourth term,

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00:05:41,650 --> 00:05:44,180
and 6.857 in the fifth term.

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00:05:44,180 --> 00:05:48,050
There is our complete
term schedule obtained

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00:05:48,050 --> 00:05:49,460
in this particular way.

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00:05:49,460 --> 00:05:51,584
There's, of course, many
other ways to schedule it,

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00:05:51,584 --> 00:05:54,700
but this is a particular orderly
way where the strategy, again,

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00:05:54,700 --> 00:05:55,260
is greedy.

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00:05:55,260 --> 00:05:56,980
You take as many
things as you possibly

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00:05:56,980 --> 00:05:59,616
can take in a given term.

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00:05:59,616 --> 00:06:00,990
Now, there are
some concepts that

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00:06:00,990 --> 00:06:02,781
come up when you're
talking about schedules

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00:06:02,781 --> 00:06:04,880
that are worth introducing.

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00:06:04,880 --> 00:06:06,940
So one of them is an antichain.

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00:06:06,940 --> 00:06:10,000
An antichain is-- in
this particular example

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00:06:10,000 --> 00:06:12,060
means a set of
subjects where there

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00:06:12,060 --> 00:06:15,080
are no indirect
prerequisites among them.

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00:06:15,080 --> 00:06:19,320
They can be taken in any order,
because it doesn't matter

134
00:06:19,320 --> 00:06:22,820
whether you've taken one
or not when you're thinking

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00:06:22,820 --> 00:06:24,660
about taking the others.

136
00:06:24,660 --> 00:06:28,230
In technical language,
again motivated

137
00:06:28,230 --> 00:06:31,860
by the idea of thinking
of there being a path

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00:06:31,860 --> 00:06:34,270
as though it was less than
or equal to something,

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00:06:34,270 --> 00:06:36,170
these are elements
that are incomparable.

140
00:06:36,170 --> 00:06:38,680
Neither one is less than
or equal to another.

141
00:06:38,680 --> 00:06:41,730
So in terms of
the path relation,

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00:06:41,730 --> 00:06:44,670
u is incomparable
to v if and only

143
00:06:44,670 --> 00:06:47,740
if there is no path from u to v
of positive length and there's

144
00:06:47,740 --> 00:06:51,030
no positive length
path from v to u.

145
00:06:51,030 --> 00:06:53,180
So let's look at some
antichains-- and the part

146
00:06:53,180 --> 00:06:54,750
of the point of
defining it is we

147
00:06:54,750 --> 00:06:59,500
have chosen antichains as
our schedule for each term.

148
00:06:59,500 --> 00:07:02,060
So the freshman subjects
with no prerequisites,

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00:07:02,060 --> 00:07:04,420
clearly there's no
path among them,

150
00:07:04,420 --> 00:07:06,090
because there is no
path to them at all.

151
00:07:06,090 --> 00:07:08,160
So they are an antichain.

152
00:07:08,160 --> 00:07:11,400
The next level we chose
were the second level

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00:07:11,400 --> 00:07:15,870
minimal elements, which
only had as prerequisites

154
00:07:15,870 --> 00:07:19,500
the original minimal elements,
and so certainly none of them

155
00:07:19,500 --> 00:07:22,390
was a prerequisite
of the others.

156
00:07:22,390 --> 00:07:26,777
So that's another
example of an antichain.

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00:07:26,777 --> 00:07:28,860
And of course the third
level and the fourth level

158
00:07:28,860 --> 00:07:30,318
and the fifth level
are antichains.

159
00:07:30,318 --> 00:07:34,090
But not all antichains
are there in our schedule.

160
00:07:34,090 --> 00:07:38,060
So for example here, is a
diagonal lying antichain.

161
00:07:38,060 --> 00:07:42,720
6.840, 6.004, and 6.034
have no paths between them.

162
00:07:42,720 --> 00:07:46,290
So in fact it's possible to
take them simultaneously,

163
00:07:46,290 --> 00:07:48,550
because you could have taken
all their prerequisites

164
00:07:48,550 --> 00:07:51,110
in the upper left here and
then take the three of them.

165
00:07:51,110 --> 00:07:54,200
So that's what an
antichain means here.

166
00:07:54,200 --> 00:07:57,700
So the technical definition
is no path between any two

167
00:07:57,700 --> 00:08:01,160
of them, but in terms of
the scheduling of courses,

168
00:08:01,160 --> 00:08:04,930
it means it's possible to
take them in the same term

169
00:08:04,930 --> 00:08:06,960
if you've satisfied all
their prerequisites,

170
00:08:06,960 --> 00:08:10,220
which it is possible to do.

171
00:08:10,220 --> 00:08:15,860
So let's ask about the
various patterns of scheduling

172
00:08:15,860 --> 00:08:16,890
that are possible.

173
00:08:16,890 --> 00:08:20,160
We've discovered this
particular greedy one,

174
00:08:20,160 --> 00:08:22,630
where we take as many
things as we can each term.

175
00:08:22,630 --> 00:08:24,950
But suppose that I was
constrained to only take

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00:08:24,950 --> 00:08:26,050
one subject per term.

177
00:08:26,050 --> 00:08:28,390
I was going to-- I
have an outside job,

178
00:08:28,390 --> 00:08:30,430
I'm too busy to take more
than one class a term,

179
00:08:30,430 --> 00:08:32,990
and if MIT will let
me dawdle so long,

180
00:08:32,990 --> 00:08:34,450
that's what I'd like to do.

181
00:08:34,450 --> 00:08:35,409
So can I do this?

182
00:08:35,409 --> 00:08:36,150
Yeah, well sure.

183
00:08:36,150 --> 00:08:39,480
Just schedule all the
minimal elements first

184
00:08:39,480 --> 00:08:40,950
in any order, one, two, three.

185
00:08:40,950 --> 00:08:43,770
And then schedule the five
second level minimal elements

186
00:08:43,770 --> 00:08:46,150
next, and the third
level, and so on.

187
00:08:46,150 --> 00:08:47,860
And it's perfectly
possible, then,

188
00:08:47,860 --> 00:08:50,830
to modify the
schedule that we found

189
00:08:50,830 --> 00:08:54,610
into a schedule in which you
only take one subject per term,

190
00:08:54,610 --> 00:08:56,990
and of course you
only take a subject

191
00:08:56,990 --> 00:09:00,400
after you've taken
all of its indirect

192
00:09:00,400 --> 00:09:02,270
and direct prerequisites.

193
00:09:02,270 --> 00:09:04,830
This is called a
topological sort.

194
00:09:04,830 --> 00:09:08,020
Again, the sorting word
comes from the motivation

195
00:09:08,020 --> 00:09:11,170
of thinking of there being
a path as like a less

196
00:09:11,170 --> 00:09:12,640
than or equal to relation.

197
00:09:12,640 --> 00:09:16,630
So we're sorting things in
order of increasing size.

198
00:09:16,630 --> 00:09:20,200
18.01 would be, in this
case, a smallest element

199
00:09:20,200 --> 00:09:25,640
and 6.857 a biggest in
this list of elements.

200
00:09:25,640 --> 00:09:28,750
A chain is kind of
technically, literally,

201
00:09:28,750 --> 00:09:31,420
a thing called the
dual of an antichain.

202
00:09:31,420 --> 00:09:33,480
A chain is a
sequence of subjects

203
00:09:33,480 --> 00:09:36,350
that must be taken in order.

204
00:09:36,350 --> 00:09:38,960
That is, these are subjects
where for any two of them,

205
00:09:38,960 --> 00:09:41,030
you know which one
has to come first.

206
00:09:41,030 --> 00:09:43,110
That is, between any
two of them there is

207
00:09:43,110 --> 00:09:45,070
a path in one way or the other.

208
00:09:45,070 --> 00:09:46,810
Now of course, it's
a DAG, so there can't

209
00:09:46,810 --> 00:09:48,480
be paths in both directions.

210
00:09:48,480 --> 00:09:52,690
So a chain is simply a set
of comparable elements, which

211
00:09:52,690 --> 00:09:55,800
implies that there's an order
in which they have to be taken.

212
00:09:55,800 --> 00:09:58,550
So here are some chains.

213
00:09:58,550 --> 00:10:01,160
This one was shown pictorially
as a vertical chain

214
00:10:01,160 --> 00:10:03,860
with five courses in it.

215
00:10:03,860 --> 00:10:05,370
Here's a vertical chain of four.

216
00:10:05,370 --> 00:10:07,260
And not all of
them are vertical.

217
00:10:07,260 --> 00:10:10,800
Here's a chain where you have
to take 18.01 before you take

218
00:10:10,800 --> 00:10:13,810
18.03 before you take 6.004.

219
00:10:13,810 --> 00:10:16,830
So they form a chain.

220
00:10:16,830 --> 00:10:18,530
It's important to
realize that this

221
00:10:18,530 --> 00:10:20,190
is a chain with
five subjects in it,

222
00:10:20,190 --> 00:10:24,780
but a chain doesn't have to
have every possible element that

223
00:10:24,780 --> 00:10:25,530
could be in it.

224
00:10:25,530 --> 00:10:28,620
It's still a chain even if it's
only got these three subjects,

225
00:10:28,620 --> 00:10:31,580
because there's a path
from 8.02 to 6.004

226
00:10:31,580 --> 00:10:36,060
and a path from 6.004 to 6.857.

227
00:10:36,060 --> 00:10:38,410
But maximum length
chains, chains

228
00:10:38,410 --> 00:10:42,700
that are as full as possible,
are important theoretically.

229
00:10:42,700 --> 00:10:46,710
And so this in particular
is a maximal length chain.

230
00:10:46,710 --> 00:10:50,320
The longest chain
here is of length 5.

231
00:10:50,320 --> 00:10:51,490
Now, it's not the only one.

232
00:10:51,490 --> 00:10:54,350
There's another chain of length
5 here if you look for it.

233
00:10:54,350 --> 00:10:57,620
But no chain is of
length longer than 5

234
00:10:57,620 --> 00:11:00,700
and there is one of length
5, and that leads us

235
00:11:00,700 --> 00:11:04,710
to the question of how many
terms is it necessarily

236
00:11:04,710 --> 00:11:06,280
going to take to graduate.

237
00:11:06,280 --> 00:11:11,970
Well, we saw that you
can graduate in five.

238
00:11:11,970 --> 00:11:16,720
But given that there's a
maximum chain of length 5,

239
00:11:16,720 --> 00:11:18,940
it means that you
can't do it in fewer,

240
00:11:18,940 --> 00:11:24,860
because those five courses
have to be taken consecutively.

241
00:11:24,860 --> 00:11:26,650
The third has to
be taken in a term

242
00:11:26,650 --> 00:11:28,440
after the first two
have been taken.

243
00:11:28,440 --> 00:11:30,470
The second has to be
taken after the first.

244
00:11:30,470 --> 00:11:33,180
If you have a chain of any
size, actually, the number

245
00:11:33,180 --> 00:11:36,430
of terms to graduate has
to be at least as big

246
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as that chain,
which means it has

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to be at least as many terms
as a maximum size chain.

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So five terms are
necessary, and we

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saw using our minimal
strategy of being

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greedy that you can always do
it in maximum chain length.

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So five are also sufficient.

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This is providing that you
can take an unlimited number

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of subjects per term.

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Remember our strategy to
graduate in five terms

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was to take as many subjects
as we possibly could each term.

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So there's the sufficient
way to take subjects

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to graduate in five terms.

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And of course,
one consequence is

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that in my second
term freshman year,

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I was taking five subjects
because it was possible.

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But that leaves me with a
kind of heavily loaded term

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compared to-- here's a
term with two subjects,

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and there's a term with only
one subject at the very end.

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So it's possible, in fact, to
somewhat adjust the term load.

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Let's just shift taking
18.02 to the third term.

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It's perfectly
feasible to do that,

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because I will have satisfied
all the prerequisites of 18.02

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after the first term,
but I don't have

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to take it in the second term.

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00:12:43,560 --> 00:12:44,820
Let's shift it off.

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So now I've lightened the
load in the second term

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to four subjects,
somewhat increasing

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00:12:49,340 --> 00:12:52,070
the load-- I had to do it
somewhere-- in the third term

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to three subjects.

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So now I have to take no more
than four subjects a term.

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And as a matter of
fact, if you fiddle,

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you can actually find
a graduating schedule

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in which you can only take
three subjects per term.

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And we will examine what's
the minimum number of subjects

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per term in the next segment.