Foundations of Data Science

simple.
We supply some intuition before going to the proof. Imagine maintaining a queue of
1 and 2-clauses. A 3-clause enters the queue when one of its literals is set to false and
it becomes a 2-clause. SC always picks a 1 or 2-clause if there is one and sets one of its
literals to true. At any step when the total number of 1 and 2-clauses is positive, one of
the clauses is removed from the queue. Consider the arrival rate, that is, the expected
number of arrivals into the queue at a given time t. For a particular clause to arrive into
the queue at time t to become a 2-clause, it must contain the negation of the literal being
set to true at time t. It can contain any two other literals not yet set. The number of
such clauses is ( )
n−t 2 2 . So, the probability that a particular clause arrives in the queue at
time t is at most
2
( n−t
(
2
n
3
)
2
2
)
2
3
≤
3
2(n − 2) .
Since there are cn clauses in total, the arrival rate is 3c , which for c < 2/3 is a constant
2
strictly less than one. The arrivals into the queue of different clauses occur independently
(Lemma 4.21), the queue has arrival rate strictly less than one, and the queue loses one
or more clauses whenever it is nonempty. This implies that the queue never has too many
clauses in it. A slightly more complicated argument will show that no clause remains as
a 1 or 2-clause for ω(ln n) steps (Lemma 4.22). This implies that the probability of two
contradictory 1-length clauses, which is a precursor to a 0-length clause, is very small.
Lemma 4.21 Let T i be the first time that clause i turns into a 2-clause. T i is ∞ if clause
i gets satisfied before turning into a 2-clause. The T i are mutually independent over the
randomness in constructing the formula and the randomness in SC, and for any t,
Prob(T i = t) ≤
3
2(n − 2) .
Proof: For the proof, generate the clauses in a different way. The important thing is
that the new method of generation, called the method of “deferred decisions”, results in
the same distribution of input formulae as the original. The method of deferred decisions
is tied in with the SC algorithm and works as follows. At any time, the length of each
clause (number of literals) is all that we know; we have not yet picked which literals are
in each clause. At the start, every clause has length three and SC picks one of the clauses
uniformly at random. Now, SC wants to pick one of the three literals in that clause to
set to true, but we do not know which literals are in the clause. At this point, we pick
uniformly at random one of the 2n possible literals. Say for illustration, we picked ¯x 102 .
The literal ¯x 102 is placed in the clause and set to true. The literal x 102 is set to false. We
must also deal with occurrences of the literal or its negation in all other clauses, but again,
we do not know which clauses have such an occurrence. We decide that now. For each
clause, independently, with probability 3/n include either the literal ¯x 102 or its negation
x 102 , each with probability 1/2. In the case that we included ¯x 102 (the literal we had set
to true), the clause is now deleted, and if we included x 102 (the literal we had set to false),
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