# Encyclopedia of Database Systems

2018 Edition
| Editors: Ling Liu, M. Tamer Özsu

# Spatial Matching Problems

Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-8265-9_80711

## Definition

A matching is a mapping from the elements of one set to the elements of another set such that each element in one set is mapped to at most one element in another set. For example, assume two sets of objects P = {p1, p2, p3} and O = {o1, o2, o3}. Then, {(p1, o1), (p2, o2), (p3, o3)} is a matching with three pairs, but {(p1, o1), (p1, o2)} is not a matching since p1 is involved in two pairs. In general, the number of possible matchings is exponential to the cardinality of P and O; e.g., if |P| = |O| = n, there are n! matchings with n pairs. Usually, among all possible matchings, the aim is to find one that optimizes/satisfies a certain criterion.

Let c(p, o) be the cost of matching pP with oO. Optimal matching  minimizes the sum of the costs of all pairs. Bottleneck matching  minimizes the maximum cost of any pair. Fair matching, also known as the stable marriage problem, returns a matching in which the following conditions cannot hold at the same time: (i) some p...

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