Nearest Neighbor Interchange

Another general optimization method is that of hill-climbing. The basic idea is to define a neighborhood relation for the search-space, and given such a relation, use one of several well-known heuristic algorithms - such as the greedy algorithm, simulated annealing, etc. - to find a local optimum. There are many ways to define neighborhood among trees. For instance - we can say that two unrooted^9.3 trees T₁ and T₂ are neighbors, if T₁ can be presented as in figure 9.5, and T₂ can be presented as one of the trees in figure 9.6.

  

Figure 9.5: An unrooted tree with 4 subtrees.

  

Figure 9.6: Neighbors of the tree in figure 9.5. The subtrees A, B, C, and D are common to all three trees.

A similar relation can be defined over 5-branch trees, resulting in a more complex neighborhood graph, also known as Peterson's graph. Another possibility is to say that the neighbors of a tree are those trees created by detaching any subtree and attaching it in some other place than its original location.