Etree — A Database-oriented Method for Generating Large Octree Meshes David R. O’Hallaron 1,2 Tiankai Tu 1 Julio C. López 2 1 School of Computer Science 2 Electrical and Computer Engineering Department Carnegie Mellon University

2 Motivation Goal : Make it possible to run large-scale physical simulations on PC’s with limited physical memory Mesh generation Mesh t Simulation results Physical model Visuali- zation Solver Approach : Index and store the datasets in databases and compute on the databases directly Requires research at the intersection of computer systems, scientific computing, and databases

3 Octree mesh generation abc defg h ijkl m  a compromise between structure and modeling power  balance requirement (2-to-1 constraint)  can be implemented in-core or out-of-core h1 h2h3h4 element/octant a b c d e f g h i j k l m h2 h1 h4 h3 master node slave node

4 Etree mesh generation overview A general method for manipulating large out- of-core octrees by querying databases unbalanced octree application-specific input construct etree library transform etree library balanced octree balance etree library element database node database

5 Etree library components Application (e.g.construct, balance) Etree library B-Tree Linear quadtree Auto navigation Local balancing Etree API  Etree API — a simple Application Programming Interface  Linear quadtree — an encoding scheme to assign keys to octants  Auto navigation — a mechanism for constructing an octree automatically  Local balancing — a technique that speeds up balancing operation  B-tree — a database index structure to store and access octants on disk

6 Etree API Unix file I/O style, three classes :  initialization and cleanup; example: etree_t *etree_open(const char *path, int flag, …);  octant-level operations; example: int etree_insert(etree_t *ep, location_t loc, void *value);  octree-level operations; example: int etree_balance(etree_t *ep, decom_t *baldecom);

7 Linear quadtree — how it works  Morton code: maps n-dimensional points to one- dimensional scalars  Locational code: appends an octant’s level to the Morton code of its left-lower corner abc defg h ijkl m x y a b c d e f g h i j k l m abcdefghijklm

8 Linear quadtree — how it works (cont’d) interleave the bits to obtain Morton code d’s left-lower corner (2, 2) binary form (010, 010) _11 append the level of d x y a b c d e f g h i j k l m

9 Linear quadtree — applications  an addressing scheme that clusters nearby octants  finding an octant without knowing its locational code  the order imposed by the locational code is the same as the preorder traversal of leafs in octree abc defg h ijkl m a b c d e f g h i j k l x m m

10 Navigation octree  guided by an application function  an in-memory pointer-based octree  dynamically grows in the depth- first order  leaf octants are pruned and flushed to disk in preorder (increasing locational code order)  appends the octant to the database to avoid database search Auto navigation — how it works : octants not yet processed (in memory) : non-leaf octants being decomposed (in memory) : leaf octants (flushed to database)

11 Local balancing — how it works Operational steps 1. partition the whole domain into equal-size blocks 2. conduct internal balancing to enforce 2-to-1 constraint within each block (in memory a resident blocking array) 3.perform boundary balancing to resolve interactions between adjacent blocks

12 Correctness of local balancing Claim: Interactions between adjacent blocks are always absorbed by boundary octants and will not be propagated into the blocks Proof: See the paper

13 Evaluation — questions  Is the etree method feasible?  How does the running time vary with the physical memory size?  What is the impact of auto navigation?  What is the impact of local balancing?

14 Evaluation — methodology  We implemented an etree-based mesh generator to generate a family of finite element meshes for San Fernando valley earthquake wave propagation simulations MeshElements NodesSlave nodes SF107,94012,1184,432 SF576,330105,88634,858 SF21,838,5242,213,035407,336 SF113,579,12415,097,3651,649,855

15 Evaluation — setup  All experiments are conducted on a PIII 1GHz machine running Linux  The machine’s physical memory for the experiments ranges from 128MB to 880MB  Before each experiment, two 1.5 GB files are sequential scanned to ensure that the operating system’s buffer cache is flushed

16 Evaluation — etree feasibility  Generating a mesh with 13.6 million elements and of size 4.3GB in 2.6 hours seems reasonable  The overall throughput increases with mesh size MeshElementsDB size (MB)Time(sec)Thruput(elem/s) SF107, SF576, SF21,838, ,636.71,123 SF113,579,1244,3009,448.81,439 Etree-based mesh generator running time and throughput All experiments are performed with 128MB physical memory

17 Evaluation — impact of memory size  Memory size does not have a significant impact on the running time  The etree method is not relying on the operating systems internal caching mechanism to achieve its performance

18 Evaluation — impact of auto navigation  Reducing B-tree buffer size does not increase the construction time  Auto navigation is not sensitive to B-tree buffer size

19 Evaluation — impact of local balancing  Achieves a speed-up factor ranging from 8 (SF1) to 28 (SF10)  Benefits from the one-time scan of the database and the efficient array-based neighbor finding algorithm

20 Related work  General octree algorithms: Samet 90  Octree mesh: Shepard & Geoges 91, Bern et al. 90, Young et al. 91, Wang99  Out-of-core octree method: Salmon 97  Linear quadtree: Gargantini 82, Morton 66  Space filling curve: Orenstein 84, Orenstein 86, Faloutsos & Roseman 89  Large dataset processing: Freitag & Loy 99, Seamons & Winslett 96, Ferreira et al. 99, Kurc et al. 01, Choudhary et al. 99 Parashar & Browne 97

21 Conclusion and future work  Experiment results suggest that the etree method can generate large octree meshes on memory-limited machines in a reasonable amount of time  Incorporating existing database techniques (linear quadtree and B-tree) with new algorithms (auto navigation and local balancing) in a unified design scheme (the etree) can deliver new capability  We are porting the etree method to commercial database systems such as IBM DB2