TY - GEN
T1 - Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs
AU - Bauer, Ulrich
AU - Munch, Elizabeth
AU - Wang, Yusu
PY - 2015/6/1
Y1 - 2015/6/1
N2 - The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has been commonly used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several metrics on the set of Reeb graphs have been proposed. In this paper, we focus on two: the functional distortion distance and the interleaving distance. The former is based on the Gromov Hausdorff distance, while the latter utilizes the equivalence between Reeb graphs and a particular class of cosheaves. However, both are defined by constructing a near-isomorphism between the two graphs of study. In this paper, we show that the two metrics are strongly equivalent on the space of Reeb graphs. Our result also implies the bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.
AB - The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has been commonly used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several metrics on the set of Reeb graphs have been proposed. In this paper, we focus on two: the functional distortion distance and the interleaving distance. The former is based on the Gromov Hausdorff distance, while the latter utilizes the equivalence between Reeb graphs and a particular class of cosheaves. However, both are defined by constructing a near-isomorphism between the two graphs of study. In this paper, we show that the two metrics are strongly equivalent on the space of Reeb graphs. Our result also implies the bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.
KW - Functional distortion distance
KW - Interleaving distance
KW - Reeb graph
UR - http://www.scopus.com/inward/record.url?scp=84958046345&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SOCG.2015.461
DO - 10.4230/LIPIcs.SOCG.2015.461
M3 - Conference contribution
AN - SCOPUS:84958046345
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 461
EP - 475
BT - 31st International Symposium on Computational Geometry, SoCG 2015
A2 - Pach, Janos
A2 - Pach, Janos
A2 - Arge, Lars
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 31st International Symposium on Computational Geometry, SoCG 2015
Y2 - 22 June 2015 through 25 June 2015
ER -