This document presents research on combinatorial algorithms for nearest neighbor search, near-duplicate detection, and small-world network design. It introduces a combinatorial framework based on a comparison oracle and disorder inequality that does not require triangle inequality. This framework is used to develop new exact algorithms for nearest neighbor search and near-duplicate detection that run in polynomial time in the disorder parameter. It also shows how to construct a visibility graph for any dataset in polynomial time that allows greedy routing to find the nearest neighbor in logarithmic hops.