Efficient%20Algorithms%20for%20Large-Scale%20GIS%20Applications

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Efficient Algorithms for Large-Scale GIS
Applications

• Laura Toma
• Duke University

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Why GIS?

• How it all started..
• Duke Environmental researchers
• computing flow accumulation for Appalachian
  Mountains took 14 days (with 512MB memory)
• 800km x 800km at 100m resolution ? 64 million
  points
• GIS (Geographic Information Systems)
• System that handles spatial data
• Visualization, processing, queries, analysis
• Indispensable tool
• Modeling, analysis, prediction, decision making
• Rich area of problems for Computer Science
• Graphics, graph theory, computational geometry etc

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GIS and the Environment

• Monitoring keep an eye on the state of earth
  systems using satellites and monitoring stations
  (water, ecosystems, urban development)
• Modeling, simulation predict consequences of
  human actions and natural processes
• Analysis and risk assessment find the problem
  areas and analyse the possible causes (soil
  erosion, groundwater pollution, traffic jams)
• Planning and decision support provide
  information and tools for better management of
  natural and socio-economic resources

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Precipitation in Tropical South America
Lots of rain Dry
H. Mitasova
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Nitrogen in Chesapeake Bay
High nitrogen concentrations
H. Mitasova
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Jockeys Ridge evolution
Combining IR-DOQQ, LIDAR and RTK GPS to assess
the change decreasing elevation, extending
towards homes and a road
C
A
B
N
H. Mitasova
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Bald Head Island Renourishment
1998 LIDAR shoreline 1998 2000
LIDAR shoreline 2000 2001, Dec. RTK GPS
shoreline
surface is 1998 LIDAR
H. Mitasova
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Sediment flow
H. Mitasova
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Computations on Terrains

• Reality
• Height of terrain is a continuous function of
  two variables f(x,y)
• Estimate, predict, simulate
• Flooding, pollution
• Erosion, deposition
• Vegetation structure
• .

• GIS
• DEM (Digital Elevation Model) is a set of sample
  points and theirheights ? x, y, hxy?

Compute indices
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DEM Representations
TIN
Grid
Contour lines
Sample points
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Panama DEM
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Modeling Flow on Terrains

• What happens when it rains?
• Predict areas susceptible to floods.
• Predict location of streams.
• Compute watersheds.
• Flow is modeled using two basic attributes
• Flow Direction (FD)
• The direction water flows at a point
• Flow Accumulation (FA)
• Total amount of water that flows through a point
  (if water is distributed according to the flow
  directions)

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Panama DEM - Flow Accumulation
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Uses

• Flow direction and flow accumulationare used
  for
• Computing other hydrological attributes
• river network
• moisture indices
• watersheds and watershed divides
• Analysis and prediction of sediment and
  pollutant movement in landscapes.
• Decision support in land management, flood and
  pollution prevention and disaster management

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Massive Terrain Data

• Remote sensing technology
• Massive amounts of terrain data
• Higher resolutions (1km, 100m, 30m, 10m, 1m,)

• NASA-SRTM
• Mission launched in 2001
• Acquired data for 80 of earth at 30m resolution
• 5TB
• USGS
• Most of US at 10m resolution
• LIDAR
• 1m res

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Example LIDAR Terrain Data

• Massive (irregular) point sets (1-10m resolution)
• Relatively cheap and easy to collect

Example Jockeys ridge (NC coast)
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Its Growing!

• Appalachian Mountains
• Area if approx. 800 km x 800 km
• Sampled at
• 100m resolution ? 64 million points (128MB)
• 30m resolution ? 640
  (1.2GB)
• 10m resolution ? 6400 6.4 billion (12GB)
• 1m resolution ? 600.4 billion (1.2TB)

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Computing on Massive Data

• GRASS (open source GIS)
• Killed after running for 17 days on a 6700 x 4300
  grid (approx 50 MB dataset)
• TARDEM (research, U. Utah)
• Killed after running for 20 days on a 12000 x
  10000 grid (appox 240 MB dataset)
• CPU utilization 5, 3GB swap file
• ArcInfo (ESRI, commercial GIS)
• Can handle the 240MB dataset
• Doesnt work for datasets bigger than 2GB

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Outline

• Introduction
• Flow direction and flow accumulation
• Definitions, assumptions, algorithm outline.
• Scalability to large terrains
• Why not?
• I/O-efficient algorithms
• I/O-efficient flow accumulation
• TerraFlow
• Theoretical results
• Conclusion

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Flow Direction (FD) on Grids

• Water flows downhill
• follows the gradient
• On grids Approximated using 3x3 neighborhood
• SFD (Single-Flow Direction)
• FD points to the steepest downslope neighbor
• MFD (Multiple-Flow direction)
• FD points to all downslope neighbors

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Flow accumulation with MFD
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Flow accumulation with SFD
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Computing FD

• Goal compute FD for every cell in the grid (FD
  grid)
• Algorithm
• For each cell compute SFD/MFD by inspecting 8
  neighbor cells
• Analysis O(N) time for a grid of N cells
• Is this all?
• NO! flat areas Plateas and sinks

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FD on Flat Areas

• no obvious flow direction
• Plateaus
• Assign flow directions such that each cell flows
  towards the nearest spill point of the plateau
• Sinks
• Either catch the water inside the sink
• Or route the water outside the sink using uphill
  flow directions
• model steady state of water and remove (fill)
• sinks by simulating flooding uniformly
• pouring water on terrain until steady state
• is reached
• Assign uphill flow directions on the original
  terrain by assigning downhill flow directions on
  the flooded terrain

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Flow Accumulation (FA) on Grids

• FA models water flow through each cell with
  uniform rain
• Initially one unit of water in each cell
• Water distributed from each cell to neighbors
  pointed to by its FD
• Flow conservation If several FD, distribute
  proportionally to height difference
• Flow accumulation of cell is total flow through
  it
• Goal compute FA for every cell in the grid (FA
  grid)

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Computing FA

• FD graph
• node for each cell
• (directed) edge from cell a to b if FD of a
  points to b
• FD graph must be acyclic
• ok on slopes, be careful on plateaus
• FD graph depends on the FD method used
• SFD graph a tree (or a set of trees)
• MFD graph a DAG (or a set of DAGs)

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Computing FA Plane Sweeping

• Input flow direction grid FD
• Output flow accumulation grid FA (initialized to
  1)
• Process cells in topological order. For each
  cell
• Read its flow from FA grid and its direction from
  FD grid
• Update flow for downslope neighbors (all
  neighbors pointed to by cell flow direction)
• Correctness
• One sweep enough
• Analysis
• O(sort) O(N) time for a grid of N cells
• Note Topological order means decreasing height
  order (since water flows downhill).

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Scalability Problem

• We can compute FD and FA using simple O(N)-time
  algorithms
• ..but.. for large sets..??

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Scalability Problem Why?

• Most (GIS) programs assume data fits in memory
• minimize only CPU computation
• But.. Massive data does not fit in main memory!
• OS places data on disk and moves data between
  memory and disk as needed
• Disk systems try to amortize large access time by
  transferring large contiguous blocks of data
• When processing massive data disk I/O is the
  bottleneck, rather than CPU time!

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Disks are Slow

• The difference in speed between modern CPU
  and disk technologies is analogous to the
  difference in speed in sharpening a pencil using
  a sharpener on ones desk or by taking an
  airplane to the other side of the world and using
  a sharpener on someone elses desk. (D. Comer)

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Scalability to Large Data

• Example reading an array from disk
• Array size N 10 elements
• Disk block size 2 elements
• Memory size 4 elements (2 blocks)

1 2 10 9 5 6 3 4 8 7
1 5 2 6 3 8 9 4 7 10
Algorithm 2 Loads 5 blocks
Algorithm 1 Loads 10 blocks
N blocks gtgt N/B blocks

• Block size is large (32KB, 64KB) ? N gtgt N/B
• N 256 x 106, B 8000 , 1ms disk access time
• ? N I/Os take 256 x 103 sec 4266 min
  71 hr
• ? N/B I/Os take 256/8 sec 32 sec

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• I/O model
• I/O-operation
• Read/write one block of data from/to disk
• I/O-complexity
• number of I/O-operations (I/Os) performed by the
  algorithm
• External memory or I/O-efficient algorithms
• Minimize I/O-complexity

• RAM model
• CPU-operation
• CPU-complexity
• Number of CPU-operations performed by the
  algorithm
• Internal memory algorithms
• Minimize CPU-complexity

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I/O-Efficient Algorithms

• O(N) I/Os is bad!!
• Improve to O(N/B) I/Os (if possible)
• Minimize the number of blocks transferred between
  main memory and disk
• Compute on whole block while it is in memory
• Avoid loading a block each time
• Use techniques from PRAM algorithms

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Sorting

• Mergesort illustrates often used features
• Main memory sized chunks (for N/M runs)
• Multi-way merge (repeatedly merge M/B of them)

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Computing FAI/O-Analysis

• Algorithm O(N) time
• Process (sweep) cells in topological order. For
  each cell
• Read flow from FA grid and direction from FD grid
• Update flow in FA grid for downslope neighbors
• Problem Cells of same height distributed over
  the terrain
• ? scattered access to FA grid and FD grid ?O(N)
  blocks

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I/O-Efficient Flow Accumulation
ATV00

• Eliminating scattered accesses to FD grid
• Store FD grid in topological order
• Eliminating scattered accesses to FA grid
• Obs flow to neighbor cell is only needed when
  its time comes to be processed
• Topological rank time when cell is
  processed priority
• Push flow by inserting flow increment in
  priority queue with
• priority equal to neighbors priority
• Flow of cell obtained using DeleteMin operations
• Note Augment each cell with priority of 8
  neighbors
• Obs Space (9N) traded for I/O
• Turns O(N) grid accesses into O(N) priority queue
  operations
• Use I/O-efficient priority queue A95,BK97
• Buffered B-tree with with lazy updates

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TerraFlow

• TerraFlow is our suite of programs for flow
  routing and flow accumulation on massive grids
  ATV00,ACal02
• Flow routing and flow accumulation modeled as
  graph problems and solved in optimal I/O bounds
• Efficient
• 2-1000 times faster on very large grids than
  existing software
• Scalable
• 1 billion elements!! (gt2GB data)
• Flexible
• Allows multiple methods flow modeling

http//www.cs.duke.edu/geo/terraflow
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TerraFlow

• Significant speedup over ArcInfo for large
  datasets
• East-Coast
• TerraFlow 8.7 Hours
• ArcInfo 78 Hours
• Washington state
• TerraFlow 63 Hours
• ArcInfo
• GRASS cannot handle
• Hawaii dataset (killed
• after (17 days!)

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I/O-Model
D

• Parameters
• N elements in problem instance
• B elements that fit in disk block
• M elements that fit in main memory
• Fundamental bounds
• Sorting sort(N)

Block I/O
M
P
In practice block and main memory sizes are big
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I/O-Efficient Graph Algorithms

• Graph G(V,E)
• Basic graph (searching) problems
• BFS, DFS, SSSP, topological sorting
• ..are big open problems in the I/O-model!
• Standard internal memory algorithms O(E) I/Os
• No I/O-efficient algorithms are known for any of
  these problems on general graphs!
• Lower bound O (sort(V)), best known O (V/sqrt(B))
• O(sort(E)) algorithms for special classes of
  graphs
• Trees, grid graphs, bounded-treewidth graphs,
  outerplanar graphs, planar graphs
• Exploit existence of small separators or
  geometric structure

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SSSP on Grid Graphs ATV00
Grid graph O(N) vertices, O(N) edges
Dijskstras algorithm O(N) I/Os Goal compute
shortest path d(s,t) in O(sort(N)) I/Os

• Lemma
• The portion of d(s,t) between intersection
  points with boundaries of subgrids is the
  shortest path within the subgrid

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SSSP on Grid Graphs ATV00
Idea Compute shortest paths locally in each
subgrid then compute the shortest way to
combine them together

• Divide grid into subgrids of size BxB (assume M gt
  B2)
• Replace each BxB subgrid with complete graph on
  boundary nodes
• Edge weight shortest path between the two
  boundary vertices within the subgrid
• Reduced graph GR
• O(N/B) vertices, O(N) edges

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SSSP on Grid Graphs ATV00

• Algorithm
• Compute SSSP on GR from s to all boundary
  vertices
• Find SSSP from s to all interior vertices for
  any subgrid s, for any t in s
• d(s,t) min v in Bnd(s) d(s,v) d s(v,t)
• Correctness
• easy to show using Lemma
• Analysis O(sort(N)) I/Os
• Dijkstra algorithm using I/O efficient priority
  queue and graph blocking

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Results on Planar graphs

• Planar graph G with N vertices
• Separators can be computed in O(sort(N)) I/Os
• I/O-efficient reductions ABT00, AMTZ01
• ? BFS, DFS, SSSP in O(sort(N)) I/Os

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SSSP on Planar Graphs

• Similar with grid graphs. Assume M gt B2, bounded
  degree
• Assume graph is separated
• O(N/B2) subgraphs, O(B2) vertices each, SO(N/B)
  separators
• each subgraph adjacent to O(B) separators

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SSSP on Planar Graphs

• Reduced graph GR
• S O(N/B) vertices
• O(N/B2) x O(B2) O(N) edges
• Compute SSSP on GR
• Dijkstras algorithm and I/O-efficient priority
  queue
• Each vertex is accessed once by its O(B)
• adjacent vertices ?O(N) I/Os
• Use boundary sets
• O(N/B2) boundary sets, each
• accessed once by its O(B) adjacent
• vertices ? O(N/B) I/Os

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On I/O-Efficient DFS

• DFS upper bounds
• Internal memory algorithm O(VE) time, O(VE)
  I/Os
• Best upper bound
• O(V E/B log V) I/Os on general graphs
• DFS on general graphs is a big open problem
• Note PRAM DFS is P-complete
• DFS on planar graphs uses O(sort(N)) I/Os
• DFS to BFS reduction AMTZ01

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DFS to BFS Reduction on Planar Graphs

• Idea Partition the faces of G into levels around
  a source face containing s and grow DFS
  level-by-level
• Levels can be obtained from BFS in dual graph
• Denote
• Gi union of the boundaries of faces at level lt
  i
• Ti DFS tree of Gi
• Hi Gi \ G i-1
• Algorithm Compute a spanning forest of Hi and
  attach it onto T i-1
• Structure of levels is simple
• The bicomps of the Hi are the boundary cycles of
  Gi
• Glueing onto T i-1 is simple
• A spanning tree is a DFS tree if and only if it
  has no cross edges

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DFS to BFS Reduction on Planar Graphs

• Idea Partition the faces of G into levels around
  a source face containing s and grow DFS
  level-by-level

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Other Graphs Results

• Grid graphs ATV00
• MST, SSSP in O(sort(N)) I/Os
• CC in O(scan(N)) I/Os
• Planar graphs ABT00, AMTZ01
• Planar reductions
• DFS
• General graphs ABT00
• MST in O(sort(N) log log N) I/Os
• Planar directed graphs submitted
• Topological sorting and ear decomposition in
  O(sort(N)) I/Os

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..In Conclusion

• I have tried to convince you of a few of things
• Massive data is available and in order to process
  it scalable algorithms are necessary
• I/O-efficient algorithms have applications
  outside computer science and have big potential
  for (interdisciplinary) collaboration
• I/O-efficient algorithms are theory and practice
  put together and support educational efforts
• Challenging, rewarding, fun!

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Collaboration

• Rewarding, good response
• Duke Nicholas School of the Environment
• NCSU Dept. of Marine, Earth and Atmospheric
  Sciences
• GRASS, ESRI
• TerraFlow
• Incorporated in GRASS AMT02
• Current work with U. Muenster GE
• 2 MS students port TerraFlow to VisualC under
  Windows and make it ArcInfo extension
• Extends projects and brings up new problems
• LIDAR data