# Reference Point Hyperplane Trees

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## Abstract

Our context of interest is tree-structured exact search in metric spaces. We make the simple observation that, the deeper a data item is within the tree, the higher the probability of that item being excluded from a search. Assuming a fixed and independent probability *p* of any subtree being excluded at query time, the probability of an individual data item being accessed is \( (1-p)^d\) for a node at depth *d*. In a balanced binary tree half of the data will be at the maximum depth of the tree so this effect should be significant and observable. We test this hypothesis with two experiments on partition trees. First, we force a balance by adjusting the partition/exclusion criteria, and compare this with unbalanced trees where the mean data depth is greater. Second, we compare a generic hyperplane tree with a monotone hyperplane tree, where also the mean depth is greater. In both cases the tree with the greater mean data depth performs better in high-dimensional spaces. We then experiment with increasing the mean depth of nodes by using a small, fixed set of reference points to make exclusion decisions over the whole tree, so that almost all of the data resides at the maximum depth. Again this can be seen to reduce the overall cost of indexing. Furthermore, we observe that having already calculated reference point distances for all data, a final filtering can be applied if the distance table is retained. This reduces further the number of distance calculations required, whilst retaining scalability. The final structure can in fact be viewed as a hybrid between a generic hyperplane tree and a LAESA search structure.

## Keywords

Generic Hyperplane Tree Reference Point Pair Partition Tree Individual Data Items Permutation Tree## Notes

### Acknowledgements

Richard Connor would like to acknowledge support by the National Research Council of Italy (CNR) for a Short-term Mobility Fellowship (STM) in June 2015, which funded a stay at ISTI-CNR in Pisa where some of this work was done. The work has also benefitted considerably from conversations with Franco Alberto Cardillo, Lucia Vadicamo and Fausto Rabitti, as well as feedback from the anonymous referees. Thanks also to Jakub Lokoč for pointing out his earlier invention of parameterised hyperplane partitioning!

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