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  • pg 1
									     Finding the max & min
             of a set

A naive algorithm would

    find the max in n-1
     comparisons, then
    find the min in n-1

for a total of 2n-2
  One might suspect that an
improvement is possible if
the max & min are found
simultaneously -- since the
information found by the
naïve algorithm in the first
step isn't used in its second

  A divide & conquer
approach will give an
improvement, although not
an asymptotic one.
  For n a power of 2,
consider the strategy of
 dividing the set in half
 finding the max & min of
  each half
 comparing the two max
 comparing the two min
  Let T(n) be the number of
comparisons needed to find
the max & min of a set of
size n. Then
      T(n) = 2T(n/2) + 2
           T(2) = 1

  An easy induction shows
that T(n) >= n for n>= 4, so
the algorithm gives no
asymptotic improvement
over the naive version.
  A related algorithm
isolates one pair, compares
its elements, and proceeds
recursively. Here two
additional comparisons are

     The resulting recurrence
   T(n) = 1 + T(n-2) + 2
          T(2) = 1
 Again we can see that
T(n) >= n for n>=4.
  In fact, no asymptotic
improvement should be
expected in either case,
since any algorithm would
need to look at all n items.
  But the argument
suggests that T(n) = pn + q
for some p and q.

  If this is true, then the
first recurrence becomes
   pn + q = 2(pn/2 + q) + 2
from which one may
conclude that q = -2.
Then T(2) = 1 implies that
2p + q = 1, so that p = 3/2.

  The same conclusion
follows from the second

  That T(n) = 3n/2 - 2 can
be checked by induction for
even n.
  If n is odd, the result
remains true if the right-
hand side is rounded up.
  Note that T(n) = 3n/2 - 2
does give an improvement
over naive version,
although not an asymptotic

  It's possible to show that
that no algorithm for this
problem can use fewer
comparisons, by using an
adversary argument.

  It will help to talk in
terms of elements
"winning" and
"losing" comparisons.

 Any comparison-
based algorithm for the
problem implicitly
manipulates 4 sets:

  N: the elements that
have neither won nor
lost a comparison
 W: the elements that
have won but not lost
 L: the elements that
have lost but not won
 B: the elements that
have both won and lost

  Initially, all elements
are in N.

  The algorithm can't
terminate until 1
element is in W, 1 is in
L, and n-2 are in B.
  Elements must move
from N to W or L to B,
one step at a time.

 In all, 2(n-2) + 1 + 1 =
2n-2 steps must be

  The adversary can
arrange that at most
one step is made,
except for comparisons
between elements of N.
 In this case, two steps
must be allowed.

  At most n/2 such
comparisons are
  So there are at most
n/2 more steps than

  And so the number of
comparisons is at least
2n-2 - n/2 = 3n/2 - 2.
   Closest pairs of points

  One problem for which a
naïve approach is improved
by a divide-and-conquer
algorithm is that of finding
the closest pair of points in
a set P = {(xi,yi)} of points.
  A naïve approach would
find the (n ) distances

between the n points.
  But a divide-and-conquer
approach needs only time
(n log n).

 In this approach, we may
assume that the points are
sorted both by x-coordinate
and by y-coordinate, and
stored in two lists Px and Py.
  We may also assume that
no two points agree in
either coordinate (perhaps
by performing a rotation).

 With these assumptions,
we may, in linear time,

 1. split Px into a "left"
half Q and a "right" half R

  2. create lists Qx and Qy
containing the elements of
Q sorted by x-coordinate
and y-coordinate
  3. create lists Rx and Ry
containing the elements of
R sorted by x-coordinate
and y-coordinate

 Now we may process Q
and R recursively.

  Suppose that we can
combine the processed
solutions into an overall
solution in linear time.
  Then an argument like
that for mergesort will give
us an overall (n log n)
time complexity.

  Note that the linear
processing will somehow
have to compare elements
of Q with those of R.

  In doing so, we may
assume that we have found
the smallest distance 
between any two points in
Q or any two points in R.

 The question becomes
whether there are two
points q  Q and r  R,
whose distance is < .

  If so, both points must be
closer than  to the vertical
line dividing Q from R.

  So we only need to search
the set S of points in P
whose x coordinate is
within of this line.

  Claim: any point s in S
has O(1) neighbors in S
within a distance of .
  These points are within 15
elements of s in Sy.

  From the claim it follows
that searching S for
sufficiently close pairs of
points takes time O(n).
   So the algorithm's
"conquer" phase of the
takes time (n), and overall
it takes time (n log n).

  The proof of the claim is
easiest to understand
geometrically, as in Figure
5.7 of Kleinberg & Tardos.

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