Dynamic smooth compressed quadtrees. / Ivor Hoog, V. D.; Khramtcova, Elena; Löffler, Maarten.
34th International Symposium on Computational Geometry, SoCG 2018. ed. / Csaba D. Toth; Bettina Speckmann. Vol. 99 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. p. 45:1-45:15.Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review
}
TY - GEN
T1 - Dynamic smooth compressed quadtrees
AU - Ivor Hoog, V. D.
AU - Khramtcova, Elena
AU - Löffler, Maarten
PY - 2018/6/1
Y1 - 2018/6/1
N2 - We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with worst-case constant time updates in constant dimensions. We distinguish two versions of the problem. First, we show that quadtrees as a space-division data structure can be made smooth and dynamic subject to split and merge operations on the quadtree cells. Second, we show that quadtrees used to store a set of points in ℝd can be made smooth and dynamic subject to insertions and deletions of points. The second version uses the first but must additionally deal with compression and alignment of quadtree components. In both cases our updates take 2O(dlogd) time, except for the point location part in the second version which has a lower bound of Ω(logn); but if a pointer (finger) to the correct quadtree cell is given, the rest of the updates take worst-case constant time. Our result implies that several classic and recent results (ranging from ray tracing to planar point location) in computational geometry which use quadtrees can deal with arbitrary point sets on a real RAM pointer machine.
AB - We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with worst-case constant time updates in constant dimensions. We distinguish two versions of the problem. First, we show that quadtrees as a space-division data structure can be made smooth and dynamic subject to split and merge operations on the quadtree cells. Second, we show that quadtrees used to store a set of points in ℝd can be made smooth and dynamic subject to insertions and deletions of points. The second version uses the first but must additionally deal with compression and alignment of quadtree components. In both cases our updates take 2O(dlogd) time, except for the point location part in the second version which has a lower bound of Ω(logn); but if a pointer (finger) to the correct quadtree cell is given, the rest of the updates take worst-case constant time. Our result implies that several classic and recent results (ranging from ray tracing to planar point location) in computational geometry which use quadtrees can deal with arbitrary point sets on a real RAM pointer machine.
KW - Alignment
KW - Compression
KW - Data structure
KW - Dynamic
KW - Quadtree
KW - Real ram
KW - Smooth
UR - http://www.scopus.com/inward/record.url?scp=85048954481&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2018.45
DO - 10.4230/LIPIcs.SoCG.2018.45
M3 - Conference contribution
AN - SCOPUS:85048954481
VL - 99
SP - 45:1-45:15
BT - 34th International Symposium on Computational Geometry, SoCG 2018
A2 - Toth, Csaba D.
A2 - Speckmann, Bettina
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 34th International Symposium on Computational Geometry, SoCG 2018
Y2 - 11 June 2018 through 14 June 2018
ER -
ID: 38614052