Planar Point Location Using Persistent Search Trees

Neil Sarnak, Robert E. Tarjan

Research output: Contribution to journalArticlepeer-review

343 Scopus citations


A classical problem in computational geometry is the planar point location problem. This problem calls for preprocessing a polygonal subdivision of the plane defined by n line segments so that, given a sequence of points, the polygon containing each point can be determined quickly on-line. Several ways of solving this problem in O(log n) query time and O(n) space are known, but they are all rather complicated. We propose a simple O(log n)-query-time, O(n)-space solution, using persistent search trees. A persistent search tree differs from an ordinary search tree in that after an insertion or deletion, the old version of the tree can still be accessed. We develop a persistent form of binary search tree that supports insertions and deletions in the present and queries in the past. The time per query or update is O(log m), where m is the total number of updates, and the space needed is O(1) per update. Our planar point location algorithm is an immediate application of this data structure. The structure also provides an alternative to Chazelle's “hive graph” structure, which has a variety of applications in geometric retrieval.

Original languageEnglish (US)
Pages (from-to)669-679
Number of pages11
JournalCommunications of the ACM
Issue number7
StatePublished - Jul 1 1986

All Science Journal Classification (ASJC) codes

  • General Computer Science


  • persistent data structure
  • planar point location
  • post-office problem
  • search tree


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