The.Algorithm.Design.Manual.Springer-Verlag.1998

Bandwidth Minimization
immediately eliminate half of our search space. The reverse copies are easily removed by placing the
leftmost and rightmost elements of the permutation as the first two elements of the vector and then
pruning if . Because we are dealing with an exponential search, removing a single factor of two
can only be of limited usefulness. Such symmetry elimination might add one to the size of the problem
we can do within one CPU minute.
For more serious speedups, we need to prune partial solutions. Say we have found a permutation p that
yields a longest edge of . By definition, . Suppose that among the elements in a partial layout
, where k < n, there is an edge that is at least in length. Can this partial solution expand to
provide a better bandwidth solution? Of course not! By pruning the search the instant we have created a
long edge, typical instances of 15 to 20 vertices can be solved in one CPU minute, thus providing a
substantial improvement.
Efforts to further improve the search algorithm must strive for even greater pruning. By using a heuristic
method to find the best solution we can before starting to search, we save time by realizing early length
cutoffs. In fact, most of the effort in a combinatorial search is typically spent after the optimal solution is
found, in the course of proving that no better answer exists. By observing that the optimal bandwidth
solution must always be at least half the degree of any vertex (think about the incident edges), we have a
lower bound on the size of the optimal solution. We can terminate search soon as we find a solution
matching the lower bound.
One limitation of this pruning strategy is that only partial solutions of length >b can be pruned, where b
is the bandwidth of the best solution to date, since we must place b+1 vertices before we can generate
any edges of length at least b. To achieve earlier cutoffs, we can alternately fill in the leftmost and
rightmost slots of the configuration, instead of always proceeding from the left. This way, whenever
there is an edge between a vertex on the left side and a vertex on the right side, this edge is likely long
enough to achieve a cutoff. Pruning can easily occur while positioning the second vertex in the solution
vector.
Using these enhancements, top-notch programs are capable of solving typical problems on up to 30
vertices consistently within one CPU minute, operating literally millions of times faster than unpruned,
untuned efforts. The speed difference between the final and initial versions of the program dwarf the
difference between a supercomputer and a microcomputer. Clever search algorithms can easily have a
bigger impact on performance than expensive hardware.
Next: War Story: Covering Chessboards Up: Combinatorial Search and Heuristic Previous: Search
Pruning
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