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Data Structures and Algorithms
with Object-Oriented Design Patterns in C# |
The depth-first and breadth-first backtracking algorithms described in the preceding sections both naıvely traverse the entire solution space. However, sometimes we can determine that a given node in the solution space does not lead to the optimal solution--either because the given solution and all its successors are infeasible or because we have already found a solution that is guaranteed to be better than any successor of the given solution. In such cases, the given node and its successors need not be considered. In effect, we can prune the solution tree, thereby reducing the number of solutions to be considered.
For example, consider the scales balancing problem
described in Section 
.
Consider a partial solution 
in which
we have placed k weights onto the pans ( 
) and,
therefore, n-k weights remain to be placed.
The difference between the weights of the left and right pans is given by
and the sum of the weights still to be placed is
Suppose that 
.
That is, the total weight remaining is less than the difference between
the weights in the two pans.
Then, the best possible solution that we can obtain
without changing the positions of the weights that have already been placed is
The quantity 
is a lower bound
on the value of the objective function for all the solutions
in the solution tree below the given partial solution .
In general, during the traversal of the solution space we may
have already found a complete, feasible solution for which
the objective function is less than .
In that case,
there is no point in considering any of the solutions below .
That is, we can prune the subtree rooted at node
from the solution tree.
A backtracking algorithm that prunes the search space in this manner
is called a branch-and-bound algorithm.
Copyright © 2001 by Bruno R. Preiss, P.Eng. All rights reserved.