Algorithm Design

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Chapter 9 PSPACE: A Class of Problems beyond NP
(0,0) (I,0)
Figure 9.3 Three distinct self-avoiding walks of length 4. Note that although walks (a)
and (b) involve the same set of points, they are considered different walks because they
pass through them in a different order.
and so it appears in theories concerning the rates of certain metabolic and
organic synthesis reactions.
Despite its importance, no simple formula is known for the value A(n). Indeed,
no algorithm is known for computing A(n) that runs in time polynomial
0,0)
(I,I)
(I,0)
(a) Show that A(n) > 2n-1 for all natural numbers n >_ 1.
(b) Give an algorithm that takes a number n as input, and outputs A(n) as a
number in binary notation, using space (i.e., memory) that is polynomial
Solved Exercises
(Thus the running time of your algorithm can be exponential, as long as its
space usage is polynomial. Note also that polynomial here means "polynomial
in n," not "polynomial in log n." Indeed, by part (a), we see that it will take
at least n - 1 bits to write the value of A(n), so clearly n - 1 is a lower bound
on the amount of space you need for producing a correct answer.)
Solution We consider part (b) first. One’s first thought is that enumerating all
self-avoiding walks sounds like a complicated prospect; it’s natural to imagine
the search as grov#ing a chain starting from a single bead, exploring possible
conformations, and backtracking when there’s no way to continue growing
and remain self-avoiding. You can picture attention-grabbing screen-savers that
do things like this, but it seems a bit messy to write down exactly what the
algorithm would be.
So we back up; polynomial space is a very generous bound, and we
can afford to take an even more brute-force approach. Suppose that instead
of trying just to enumerate all self-avoiding walks of length n, we simply
enumerate all walks of length n, and then check which ones turn out to be selfavoiding.
The advantage of this is that the space of all walks is much easier
to describe than the space of self-avoiding walks.
Pn) on the set 2~ of grid points in the plane
can be described by the sequence of directions it takes. Each step frompi to
in the walk can be viewed as moving in one of four directions: north, south,
east, or west. Thus any walk of length n can be mapped to a distinct string
of length n - 1 over the alphabet [N, S, E, W}. (The three walks in Figure 9.3
would be ENW, NES, and EEN.) Each such string corresponds to a walk of
length n, but not al! such strings correspond to walks that are self-avoiding:
for example, the walk NESW revisits the point (0, 0).
We can use this encoding of walks for part (b) of the question as follows.
Using a counter in base 4, we enumerate all strings of length n - 1 over the
alphabet {N, S, E, W}, by viewing this alphabet equivalently as {0, 1, 2, 3}. For
each such string, we construct the corresponding walk and test, in polynomial
space, whether it is self-avoiding. Finally, we increment a second counter A
(initialized to 0) if the current walk is self-avoiding. At the end of this algorithm,
A will hold the value of A(n).
Now we can bound the space used by this algorithm is follows. The first
counter, which enumerates strings, has n - 1 positions, each of which requires
two bits (since it can take four possible values). Similarly, the second counter
holding A can be incremented at most 4 n-1 times, and so it too needs at most
2n bits. Finally, we use polynomial space to check whether each generated
walk is self-avoiding, but we can reuse the same space for each walk, and so
the space needed for this is polynomial as well.
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