Loop-free algorithms for traversing binary trees

Abstract. Depressions – inwardly draining regions – are common to many landscapes. When there is sufficient moisture, depressions take the form of lakes and wetlands; otherwise, they may be dry. Hydrological flow models used in geomorphology, hydrology, planetary science, soil and water conservation, and other fields often eliminate depressions through filling or breaching; however, this can produce unrealistic results. Models that retain depressions, on the other hand, are often undesirably expensive to run. In previous work we began to address this by developing a depression hierarchy data structure to capture the full topographic complexity of depressions in a region. Here, we extend this work by presenting the Fill–Spill–Merge algorithm that utilizes our depression hierarchy data structure to rapidly process and distribute runoff. Runoff fills depressions, which then overflow and spill into their neighbors. If both a depression and its neighbor fill, they merge. We provide a detailed explanation of the algorithm and results from two sample study areas. In these case studies, the algorithm runs 90–2600 times faster (with a reduction in compute time of 2000–63 000 times) than the commonly used Jacobi iteration and produces a more accurate output. Complete, well-commented, open-source code with 97 % test coverage is available on GitHub and Zenodo.

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“…2a. Geolinks are similar to the connections used in a threaded binary tree (Fenner and Loizou, 1984).…”

Section: Terminologymentioning

confidence: 99%

Computing water flow through complex landscapes – Part 3: Fill–Spill–Merge: flow routing in depression hierarchies

2021

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“…These are the dashed lines shown in Figure 2. Geolinks are similar to the connections used in a threaded binary tree (Fenner and Loizou, 1984).…”

Section: Each Node In the Dh Is Associated With Several Propertiesmentioning

confidence: 99%

Computing water flow through complex landscapes, Part 3: Fill-Spill-Merge: Flow routing in depression hierarchies

Barnes¹,

Callaghan²,

Wickert³

2020

Preprint

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Abstract. Depressions – inwardly-draining regions – are common to many landscapes. When there is sufficient moisture, depressions take the form of lakes and wetlands; otherwise, they may be dry. Hydrological flow models used in geomorphology, hydrology, planetary science, soil and water conservation, and other fields often eliminate depressions through filling or breaching; however, this can produce unrealistic results. Models that retain depressions, on the other hand, are often undesirably expensive to run. In previous work we began to address this by developing a depression hierarchy data structure to capture the full topographic complexity of depressions in a region. Here, we extend this work by presenting a Fill-Spill-Merge algorithm that utilizes our depression hierarchy to rapidly process and distribute runoff. Runoff fills depressions, which then overflow and spill into their neighbors. If both a depression and its neighbor fill, they merge. We provide a detailed explanation of the algorithm as well as results from two sample study areas. In these case studies, the algorithm runs 90–2600× faster (with a 2000–63 000× reduction in compute time) than the commonly-used Jacobi iteration and produces a more accurate output. Complete, well-commented, open-source code is available on Github and Zenodo.

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“…define a set of geolinks that tie an overflowing depression with the depression into which it overflows. As in a threaded binary tree (Fenner and Loizou, 1984), these links can be used to accelerate traversals by eliminating recursion.…”

Section: The Depression Hierarchymentioning

confidence: 99%

Computing water flow through complex landscapes, Part 2: Finding hierarchies in depressions and morphological segmentations

Barnes¹,

Callaghan²,

Wickert³

2019

Preprint

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Abstract. Depressions – inwardly-draining regions of digital elevation models – present difficulties for terrain analysis and hydrological modeling. Analogous depressions also arise in image processing and morphological segmentation where they may represent noise, features of interest, or both. Here we provide a new data structure – the depression hierarchy – that captures the full topologic and topographic complexity of depressions in a region. We treat depressions as networks, in a way that is analogous to surface-water flow paths, in which individual sub-depressions merge together to form meta-depressions in a process that continues until they begin to drain externally. The hierarchy can be used to selectively fill or breach depressions, or to accelerate dynamic models of hydrological flow. Complete, well-commented, open-source code and correctness tests are available on Github and Zenodo.

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Loop-free algorithms for traversing binary trees

Cited by 7 publications

References 15 publications

Computing water flow through complex landscapes – Part 3: Fill–Spill–Merge: flow routing in depression hierarchies

Computing water flow through complex landscapes – Part 3: Fill–Spill–Merge: flow routing in depression hierarchies

Computing water flow through complex landscapes, Part 3: Fill-Spill-Merge: Flow routing in depression hierarchies

Computing water flow through complex landscapes, Part 2: Finding hierarchies in depressions and morphological segmentations

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