Efficient Algorithms for

Large-Scale GIS Applications

Laura Toma

Duke University

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

GIS and the EnvironmentMonitoring: 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

Precipitation in Tropical South America

Lots of rain

Dry

H. Mitasova

Nitrogen in Chesapeake Bay

High nitrogen concentrationsH. Mitasova

Jockey’s Ridge evolution

N

H. Mitasova

Combining IR-DOQQ, LIDAR and RTK GPS to assess the change: decreasing elevation, extending towards homes and a road

A

B

C

Bald Head Island Renourishment

1998: LIDAR shoreline 1998

2000: LIDAR shoreline 2000

2001, Dec.: RTK GPS shoreline

surface is 1998 LIDAR

H. Mitasova

Sediment flow

H. Mitasova

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

DEM Representations

3 2 47 5 87 1 9

3 2 47 5 87 1 9

3 2 47 5 87 1 9

3 2 47 5 87 1 9

TIN

GridContour lines

Sample points

Panama DEM

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)

Panama DEM - Flow Accumulation

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

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

Example: LIDAR Terrain Data

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

Example: Jockey’s ridge (NC coast)

It’s 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)

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 • Doesn’t work for datasets bigger than 2GB

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

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

Flow accumulation with MFD

Flow accumulation with SFD

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

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

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)

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)

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).

Scalability Problem

We can compute FD and FA using simple O(N)-time algorithms

..but.. for large sets..??

Dataset Size (log)

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!

track

magnetic surface

read/write armread/write head

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 one’s desk or by taking an airplane to the other side of the world and using a sharpener on someone else’s desk.” (D. Comer)

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 71 5 2 6 3 8 9 4 7 10

Algorithm 2: Loads 5 blocksAlgorithm 1: Loads 10 blocks

N blocks >> N/B blocks Block size is large (32KB, 64KB) N >> 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

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

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

Sorting

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

them)

)log(BN

BN

BMO

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

I/O-Efficient Flow Accumulation

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 neighbor’s 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

[ATV00]

TerraFlow TerraFlow is our suite of programs for flow routing and

flow accumulation on massive grids [ATV`00,AC&al`02]

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!! (>2GB data)

Flexible • Allows multiple methods flow modeling

http://www.cs.duke.edu/geo*/terraflow

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!)Haw

aii

56M

Cumber

lands

80M Lower

NE

256M

East-C

oast

491M M

idwes

t

561M

Was

hingto

n

2G

0

10

20

30

40

50

60

70

80

90

Run

ning

Tim

e (H

ours

)

TerraFlow 512TerraFlow 128ArcInfo 512ArcInfo 128

500 MHz Alpha, FreeBSD 4.0

I/O-Model Parameters

N = # elements in problem instance

B = # elements that fit in disk blockM = # elements that fit in main memory

Fundamental bounds:• Sorting: sort(N) =

D

P

M

Block I/O

)log(BN

BN

BMO

NBN

BN

BN

BM <<< log

In practice block and main memory sizes are big

I/O-Efficient Graph AlgorithmsGraph 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 Ω (sort(V)), best known Ω (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

SSSP on Grid Graphs [ATV’00]

Lemma: The portion of δ(s,t) between

intersection points with boundaries of subgrids is the shortest path within the subgrid

Grid graphO(N) vertices, O(N) edges

Dijskstra’s algorithm: O(N) I/Os

Goal: compute shortest path δ(s,t) in O(sort(N)) I/Os

SSSP on Grid Graphs [ATV’00]

Divide grid into subgrids of size BxB (assume M > 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

Idea: Compute shortest paths locally in each subgrid then compute the shortest way to combine them together

SSSP on Grid Graphs [ATV’00]Algorithm

1. Compute SSSP on GR from s to all boundary vertices

2. Find SSSP from s to all interior vertices: for any subgrid σ, for any t in σ

δ(s,t) = min v in Bnd(σ) {δ(s,v) + δ σ(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

Results on Planar graphsPlanar graph G with N vertices Separators can be computed in O(sort(N)) I/Os I/O-efficient reductions [ABT’00, AMTZ’01]

BFS, DFS, SSSP in O(sort(N)) I/Os

O(sort(N)) I/Os [AMTZ’01]

O(sort(N)) I/Os [ABT’00]O(sort(N)) I/Os [ABT’00]

DFS

BFS SSSPε-separators

SSSP on Planar Graphs Similar with grid graphs. Assume M > B2, bounded degree Assume graph is separated

• O(N/B2) subgraphs, O(B2) vertices each, S=O(N/B) separators

• each subgraph adjacent to O(B) separators

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• Dijkstra’s 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

On I/O-Efficient DFS

DFS upper bounds• Internal memory algorithm: O(V+E) time, O(V+E) 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 [AMTZ’01]

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 <= 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

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

Other Graphs Results

Grid graphs [ATV’00]• MST, SSSP in O(sort(N)) I/Os

• CC in O(scan(N)) I/Os

Planar graphs [ABT’00, AMTZ’01]• Planar reductions

• DFS

General graphs [ABT’00]• 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

..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!

Collaboration

Rewarding, good response • Duke Nicholas School of the Environment• NCSU Dept. of Marine, Earth and Atmospheric Sciences• GRASS, ESRI

TerraFlow • Incorporated in GRASS [AMT’02]• 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