6.5.2 DavenportSchinzel Sequences
DavenportSchinzel sequences provide a powerful characterization of
the structure that arises from the lower or upper envelope of a
collection of functions. The lower envelope of five functions
is depicted in Figure 6.42. Such envelopes arise in
many problems throughout computational geometry, including many motion
planning problems. They are an important part of the design and
analysis of many modern algorithms, and the resulting algorithm's time
complexity usually involves terms that follow directly from the
sequences. Therefore, it is worthwhile to understand some of the
basics before interpreting some of the results of Section
6.5.3. Much more information on DavenportSchinzel sequences
and their applications appears in [866]. The brief
introduction presented here is based on [865].
Figure 6.42:
The lower envelope of a collection
of functions.

For positive integers and , an DavenportSchinzel
sequence is a sequence
composed from a set of
symbols such that:
 The same symbol may not appear consecutively in the sequence.
In other words,
for any such that
.
 The sequence does not contain any alternating subsequence that
uses two symbols and has length . A subsequence can be formed by
deleting any elements in the original sequence. The condition can be
expressed as: There do not exist indices
for which
and
, for some symbols and .
As an example, an sequence cannot appear as
, in which each is filled in with
any sequence of symbols. Let
denote the maximum
possible length of an DavenportSchinzel sequence.
The connection between Figure 6.42 and these sequences
can now be explained. Consider the sequence of function indices that
visit the lower envelope. In the example, this sequence is
. Suppose it is known that each pair of functions
intersects in at most places. If there are realvalued
continuous functions, then the sequence of function indices must be an
DavenportSchinzel sequence. It is amazing that such
sequences cannot be very long. For a fixed , they are close to
being linear.
The standard bounds for DavenportSchinzel sequences are
[865]^{6.9}


(6.32) 


(6.33) 


(6.34) 


(6.35) 


(6.36) 


(6.37) 


(6.38) 
In the expressions above and
are terms that
are smaller than their leading exponents. The term is the
inverse Ackerman function, which is an extremely slowgrowing function
that appears frequently in algorithms. The Ackerman function is
defined as follows. Let
and
represent
applications of . Thus, performs doubling,
performs exponentiation, and performs tower
exponentiation, which makes a stack of 's,

(6.39) 
that has height . The Ackerman function is defined as
. This function grows so fast that is already an
exponential tower of 's that has height . Thus, the
inverse Ackerman function, , grows very slowly. If
is less than or equal to an exponential tower of 's, then
. Even when it appears in exponents of the
DavenportSchinzel bounds, it does not represent a significant growth
rate.
Example 6..9 (Lower Envelope of Line Segments)
One interesting application of DavenportSchinzel applications is to
the lower envelope
of a set of line segments in
. Since
segments in general position may appear multiple times along the lower
envelope, the total number of edges is
, which is higher than one would obtain from
infinite lines. There are actually arrangements of segments in
that reach this bound; see [
866].
Steven M LaValle
20200814