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Let f be a drawing in the Euclidean plane of a graph G, which is understood to be a 1-dimensional simplicial complex. We assume that every edge of G is drawn by f as a curve of constant algebraic complexity, and the ratio of the length of the longest simple path to the the length of the shortest edge is poly(n). In the drawing f, a path P of G, or its image in the drawing π = f(P), is β-stretch if π is a simple (non-self-intersecting) curve, and for every pair of distinct points p ∈ P and q ∈ P , the length of the sub-curve of π connecting f(p) with f(q) is at most β∥f(p) − f(q)∥, where ∥.∥ denotes the Euclidean distance. We introduce and study the β-stretch Path Problem (βSP for short), in which we are given a pair of vertices s and t of G, and we are to decide whether in the given drawing of G there exists a β-stretch path P connecting s and t. We also output P if it exists. The βSP quantifies a notion of “near straightness” for paths in a graph G, motivated by gerrymandering regions in a map, where edges of G represent natural geographical/political boundaries that may be chosen to bound election districts. The notion of a β-stretch path naturally extends to cycles, and the extension gives a measure of how gerrymandered a district is. Furthermore, we show that the extension is closely related to several studied measures of local fatness of geometric shapes. We prove that βSP is strongly NP-complete. We complement this result by giving a quasi-polynomial time algorithm, that for a given ε > 0, β ∈ O(poly(log |V (G)|)), and s, t ∈ V (G), outputs a β-stretch path between s and t, if a (1 − ε)β-stretch path between s and t exists in the drawing.
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Bulk, shear and scattering attenuation beneath Hawaiian Volcanos and in the oceanic crust extending to the Aloha Cabled Observatory
SUMMARY Seismic attenuation is measured from a swarm of 50 earthquakes in Kīlauea volcano in 2018, associated with caldera collapse. The traverse extends at nearly constant azimuth to the saddle between Mauna Loa and Mauna Kea, continuing to Maui beneath the distal flanks of three dormant volcanos. From Maui the traverse then extends seaward to the Aloha Cabled Observatory (ACO) on the seafloor north of O‘ahu. The effective attenuation is measured with respect to an $${\omega ^{ - 2}}$$ earthquake source model. Frequency dependent $${Q_P}$$ and $${Q_S}$$ are derived. The initial path is shallow and uphill, the path to Maui propagates at mid-crustal depths, and the path to ACO extends through oceanic crust. The observations of $${Q_P} \le {Q_S}$$ over each traverse are modelled as bulk attenuation $${Q_K}$$. Several attenuation processes are observed, including $${Q_\mu }$$, $${Q_K}$$, $$Q\sim f$$, constant Q and scattering. The observation of bulk attenuation is ascribed to contrasting physical properties between basalt and water saturated vesicles. The ratio of Q values between shallow and mid-crustal propagation is used to derive an activation energy E* for the undetermined shear attenuation mechanism. A Debye relaxation peak is fit to the $${Q_S}( f )$$ and $${Q_K}( f )$$ observed for the mid-crustal pathway. A prior high-frequency attenuation study near Wake Island compares well with this Hawaiian Q data set, which in general shows lower values of Q than observed for Wake.
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- PAR ID:
- 10203559
- Date Published:
- Journal Name:
- Geophysical Journal International
- Volume:
- 223
- Issue:
- 1
- ISSN:
- 0956-540X
- Page Range / eLocation ID:
- 543 to 560
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/gji/ggaa309
