Physics – Quantum Physics
Scientific paper
2005-06-02
Physics
Quantum Physics
7 pages
Scientific paper
Let G=(V,E) be a finite graph, and f:V->N be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f(v) is not larger than the value of f on the neighbors of v in G. In this note, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query complexity O(\log n)\cdot d+O(\sqrt{g})\cdot\sqrt{n}, so that we obtain a deterministic query complexity of d+O(\sqrt{g})\cdot\sqrt{n}, where n is the size of G, d is its maximum degree, and $g$ is its genus. We also give a quantum version of our algorithm, whose query complexity is of O(\sqrt{d})+O(\sqrt[4]{g})\cdot\sqrt[4]{n}\log\log n. Our deterministic and quantum algorithms have query complexities respectively smaller than the generic algorithms of Aldous and of Aaronson for large classes of graphs, including graphs of bounded genus and planar graphs. Independently from this work, Zhang has recently given a quantum algorithm which finds a local minimum on the planar grid over \{1,...,\sqrt{n}\}^2 using O(\sqrt[4]{n}(\log\log n)^2) queries. Our quantum algorithm can be viewed as a strongly generalized, and slightly enhanced version of this algorithm.
No associations
LandOfFree
Enhanced algorithms for Local Search does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Enhanced algorithms for Local Search, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Enhanced algorithms for Local Search will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-435383