By Eyal Ackerman, Kevin Buchin, Christian Knauer (auth.), Richard Pollack, János Pach, Jacob E. Goodman (eds.)

This commemorative publication includes the 28 significant articles that seemed within the 2008 20th Anniversary factor of the magazine Discrete & Computational Geometry, and offers a entire photo of the present nation of the sphere. shaped up to now few a long time through the merger of the classical self-discipline of combinatorial and discrete geometry with the hot box of computational geometry that sprang up within the Seventies, discrete and computational geometry now claims the allegiance of a huge variety of mathematicians and laptop scientists world wide, whose most crucial paintings has been showing considering 1986 within the pages of the journal.

The articles during this quantity, a few which resolve long-outstanding difficulties within the box, have been selected by means of the editors of DCG for the significance in their effects, for the breadth in their scope, and to teach the intimate connections that experience arisen among discrete and computational geometry and different components of either desktop technology and arithmetic. except the articles, the editors current an elevated preface, besides a suite of pictures of teams and people who have performed an important position within the heritage of the sector in past times twenty years.

Contributors include:

E. Ackerman

P.K. Agarwal

I. Aliev

I. Bárány

A. Barvinok

S. Basu

L.J. Billera

J.-D. Boissonnat

C. Borcea

E. Boros

K. Borys

B. Braun

K. Buchin

O. Cheong

D. Cohen-Steiner

M. Damian

K. Elbassioni

R. Flatland

T. Gerken

J.E. Goodman

X. Goaoc

P. Gronchi

V. Gurvich

S. Har-Peled

J. Hershberger

A. Holmsen

S.K. Hsiao

A. Hubard

J. Jerónimo

L. Khachiyan

R. Klein

C. Knauer

S. Langerman

J.-Y. Lee

M. Longinetti

E. Miller

P. Morin

U. Nagel

E. Nevo

P. Niyogi

I. Novik

J. O’Rourke

J. Pach

I. Pak

M.J. Pelsmajer

S. Petitjean

F. Pfender

R. Pinchasi

R. Pollack

J.S. Provan

K. Przeslawski

R.M. Richardson

G. Rote

M. Schaefer

Y. Schreiber

M. Sharir

J.R. Shewchuk

S. Smale

B. Solomyak

M. Soss

D. àtefankovic

G. Vegter

V.H. Vu

S. Weinberger

L. Wu

D. Yost

H. Yu

T. Zell

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**Extra resources for Twentieth Anniversary Volume:: Discrete & Computational Geometry**

**Sample text**

Narasimhan and Smid [23] study the problem of approximating the spanning ratio of an arbitrary geometric graph in Ed . They give an O(n log n)-time algorithm that computes an (1 − ε)-approximate value of the spanning ratio of a path, cycle, or tree embedded in Ed . More generally, they show that the problem of approximating the spanning ratio can be reduced to answering O(n) approximate shortest-path queries after O(n log n) preprocessing. Ebbers-Baumann et al. [10] have studied the problem of computing the detour of a planar polygonal chain G with n vertices.

Together, (ii) and (iii) imply property (i). Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D 19 Fig. 1 (Ebbers-Baumann et al. [10]) (i) Let V be the set of vertices in the polygonal chain P , and let κ ≥ 1. There is a pair (p, q) ∈ P × P so that δ(p, q) > κ if and only if there is a pair (p , q ) ∈ P × V so that δ(p , q ) > κ and p is visible from q . (ii) Assume that the detour attains a local maximum at two points, q, q that are interior points of edges e, e of P , correspondingly.

51, 169–171 (1944) 5. : Arrangements and Spreads. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, vol. 10. American Mathematical Society, Providence (1972) 6. : On the number of ordinary lines determined by n points. Can. J. Math. 1, 210–219 (1958) 7. : Die Teilung der projectiven Ebene durch Gerade oder Pseudogerade. Ber. -Phys. Kl. Sächs. Akad. Wiss. 78, 256–267 (1926) 8. : How many magic configurations are there? Am. Math. Mon. 78(9), 1000–1002 (1971) 9.