Cylindrical algebraic decomposition II: an adjacency algorithm for the plane

Dennis S. Arnon*, George E. Collins, Scott McCallum

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

77 Citations (Scopus)


Given a set of r-variate integral polynomials, a cylindrical algebraic decomposition (cad) of euclidean r-space Epartitions Er into connected subsets compatible with the zeros of the polynomials. Each subset is a cell. Informally, two cells of a cad are adjacent if they touch each other; formally, they are adjacent if their union is connected. In applications of cads one often wishes to know the adjacent pairs of cells. Previous algorithms for cad construction (such as that given in Part I of this paper) have not actually determined them. The authors give here in Part II an algorithm which determines the pairs of adjacent cells as it constructs a cad of E2.

Original languageEnglish
Pages (from-to)878-889
Number of pages12
JournalSIAM Journal on Computing
Issue number4
Publication statusPublished - 1 Jan 1984
Externally publishedYes


  • polynomial zeros
  • computer algebra
  • computational geometry
  • semi-algebraic geometry
  • real closed fields
  • decision procedures
  • real algebraic geometry


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