On the Length of Monotone Paths in Polyhedra

Moïse Blanchard; Jesús A. De Loera; Quentin Louveaux

doi:10.1137/20m1315646

On the Length of Monotone Paths in Polyhedra

SIAM Journal on Discrete Mathematics ◽

10.1137/20m1315646 ◽

2021 ◽

Vol 35 (3) ◽

pp. 1746-1768

Author(s):

Moïse Blanchard ◽

Jesús A. De Loera ◽

Quentin Louveaux

Keyword(s):

Monotone Paths

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Simple and optimal output-sensitive construction of contour trees using monotone paths

Computational Geometry ◽

10.1016/j.comgeo.2004.05.002 ◽

2005 ◽

Vol 30 (2) ◽

pp. 165-195 ◽

Cited By ~ 52

Author(s):

Yi-Jen Chiang ◽

Tobias Lenz ◽

Xiang Lu ◽

Günter Rote

Keyword(s):

Contour Trees ◽

Optimal Output ◽

Monotone Paths

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Monotone Descent Path Queries on Dynamic Terrains

Journal of Computing and Information Science in Engineering ◽

10.1115/1.4025780 ◽

2014 ◽

Vol 14 (1) ◽

Author(s):

Xiangzhi Wei ◽

Ajay Joneja ◽

Yaobin Tian ◽

Yan-An Yao

Keyword(s):

Data Structure ◽

Engineering Design ◽

Linear Time ◽

Monotone Path ◽

Path Queries ◽

Contour Tree ◽

Monotone Paths

Monotone paths are useful in many engineering design applications. In this paper, we address the problem of answering monotone descent path queries on terrains that are continually changing. A terrain can be represented by a unique contour tree. Such a contour tree belongs to a class of graphs called arbitrarily directed trees (ADTs). Let T be an ADT with n nodes. In this paper, we present a new linear time preprocessing algorithm for decomposing a static ADT T into a forest F, with which we can answer lowest common descendent (LCA) queries in O(1) time. This is useful in answering monotone path queries on the corresponding terrain. We show how to maintain this data structure, and thereby answer LCA queries efficiently, for dynamic ADTs. We also show how to maintain the data structure of dynamic terrains, while simultaneously maintaining the corresponding contour tree. This allows us to efficiently answer monotone path queries between any two points on dynamic terrains.

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