Nearest Neighbor Queries, Nick Roussopoulos

Nearest Neighbor Queries, Nick Roussopoulos

Summary of the paper:

Introduce the NN problem
There is a wide variety of spatial access methods, however, very few have been used for NN. Although, NN algorithms have been proposed for quad-tree and k-d-tree, a NN algorithm for R-tree is required.

Branch-and-bound
The main idea in this NN algorithm is branch-and-bound, which decreasing unnecessary queries by calculating the proximity of the nearest distance. This pruning strategy is widely used in AI area. 

Core of the branch-and bound method: Metrics Choosing
How to estimate the nearest distance is the key to successful pruning. In this paper, the author introduced two metrics for heuristic search
MINDIST
The minimum distance from the point P to all the faces of a MBR.
MINDIST is a lower bound for nearest distance.

MINMAXDIST
The minimun of the maximun distance from the point P to any faces of a MBR.
Choose "min" to guarantee at least one object exist, larger will make the bound not tight enough, smaller might not guarantee one.
MINMAXDIST is an upper bound for nearest distance.

MINDIST<= || (P,o) || <= MINMAXDIST

Searching Order
MINDIST is optimistic distance while MINMAXDIST is a pessimistic one.
In most of the cases, the MINDIST ordering behaves well ( which is verified in later experiments), but in cases where the MBR is sparse, then MINMAXDIST may over-perform MINDIST ordering. 

Nearest Neighbor Algorithm for R-treesInitialize the nearest distance as infinite distance

Traverse the tree depth-first starting from the root. At each Index node, sort all MBRs using an ordering metric and put them in an Active Branch List (ABL).

Apply pruning rules 1 and 2 to ABL

Visit the MBRs from the ABL following the order until it is empty

If Leaf node, compute actual distances, compare with the best NN so far, update if necessary.

At the return from the recursion, use pruning rule 3

When the ABL is empty, the NN search returns.

Generalization: Finding the k Nearest Neighbors
Keep a sorted buff of at most k current nearest neighbors

Questions:
The third step of the algorithm is a little big vague. It only says to apply pruning rules 1 and 2, but rule 2 can only applied to leaf. 

Is there any possibility to combine the two metrics orderings to give a overall better solution?

Is there any other metrics could be used?

Apart from spatial database, k-NN could be also used for classify and clustering of different patterns or objects.