# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Cholesky factorization of matrices in parallel and ranking of graphs. (English) Zbl 1128.68544
Wyrzykowski, Roman (ed.) et al., Parallel processing and applied mathematics. 5th international conference, PPAM 2003, Czȩstochowa, Poland, September 7–10, 2003. Revised papers. Berlin: Springer (ISBN 3-540-21946-3/pbk). Lecture Notes in Computer Science 3019, 985-992 (2004).
Summary: The vertex ranking problem is closely related to the problem of finding the elimination tree of minimum height for a given graph. This implies that the problem has applications in the parallel Cholesky factorization of matrices. We describe the connection between this model of graph coloring and the matrix factorization. We also present a polynomial time algorithm for finding edge ranking of complete bipartite graphs. We use it to design an $O\left({m}^{2+d}\right)$ algorithm for edge ranking of graphs obtained by removing $O\left(logm\right)$ edges from a complete bipartite graph, where $d$ is a fixed number. Then we extend our results to complete $k$-partite graphs for any fixed $k>2$. In this way we give a new class of matrix factorization instances that can be optimally solved in polynomial time.
##### MSC:
 68W10 Parallel algorithms 05C85 Graph algorithms (graph theory) 65F30 Other matrix algorithms