The first upper bound for the general motion planning problem of
Formulation 4.1 came from the application of cylindrical
algebraic decomposition [852]. Let be the dimension of
. Let be the number of polynomials in , which are
used to define
. Recall from Section 4.3.3 how
quickly this grows for simple examples. Let be the maximum degree
among the polynomials in . The maximum degree of the
resulting polynomials is bounded by
, and the total
number of polynomials is bounded by
. The total
running time required to use cylindrical algebraic decomposition for
motion planning is bounded by
.^{6.10} Note that the algorithm
is doubly exponential in dimension but polynomial in and .
It can theoretically be declared to be *efficient* on a space of motion planning problems of bounded dimension
(although, it certainly is not efficient for motion planning in any
practical sense).

Since the general problem is PSPACE-complete, it appears unavoidable
that a complete, general motion planning algorithm will require a
running time that is exponential in dimension. Since cylindrical
algebraic decomposition is doubly exponential, it led many in the
1980s to wonder whether this upper bound could be lowered. This was
achieved by Canny's roadmap algorithm, for which the running time is
bounded by
. Hence, it is singly exponential,
which appears very close to optimal because it is up against the lower
bound that seems to be implied by PSPACE-hardness (and the fact that
problems exist that require a roadmap with connected
components [77]). Much of the algorithm's complexity is
due to finding a suitable deterministic perturbation to put the input
polynomials into general position. A randomized algorithm can
alternatively be used, for which the randomized expected running time
is bounded by
. For a *randomized
algorithm* [719], the *randomized expected* running time
is still a worst-case upper bound, but averaged over random ``coin
tosses'' that are introduced internally in the algorithm; it does *not* reflect any kind of average over the expected input distribution.
Thus, these two bounds represent the best-known upper bounds for the
general motion planning problem. Canny's algorithm may also be
applied to solve the kinematic closure problems of Section
4.4, but the complexity does not reflect the fact that
the dimension, , of the algebraic variety is less than , the
dimension of . A roadmap algorithm that is particularly suited
for this problem is introduced in [76], and its running
time is bounded by
. This serves as the
best-known upper bound for the problems of Section 4.4.

Steven M LaValle 2020-08-14