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The SPY-TEC (spherical pyramid-technique) was proposed as a new indexing method for high-dimensional data spaces using a special partitioning strategy that divides a d-dimensional data space into 2d spherical pyramids. In the SPY-TEC, an efficient algorithm for processing hyperspherical range queries was introduced with a special partitioning strategy. However, the technique for processing k-nearest-neighbor queries, which are frequently used in similarity search, was not proposed. In this paper, we propose an efficient algorithm for processing nearest-neighbor queries on the SPY-TEC by extending the incremental nearest-neighbor algorithm. We also introduce a metric that can be used to guide an ordered best-first traversal when finding nearest neighbors on the SPY-TEC. Finally, we show that our technique significantly outperforms the related techniques in processing k-nearest-neighbor queries by comparing it to the R*-tree, the X-tree, and the sequential scan through extensive experiments.