Indexing For Function Approximation Biswanath Panda Mirek Riedewald, Stephen B. Pope, Johannes Gehrke, L. Paul Chew Cornell University VLDB 2006, Seoul
Motivation • Simulations are important in science • Large simulations computationally infeasible • Driven by complex mathematical models • Require solution to complex differential equations • Approximation techniques speed up simulations • Bounded error in the simulation • Approximate simulation steps using information from previous steps VLDB 2006, Seoul
Outline • Example scientific application • Combustion simulation • Function approximation problem • Formulation • Hardness • Algorithm • Indexing problem VLDB 2006, Seoul
Combustion Simulation High Dimensional Composition Vector Air Outflow Inflow Methane Mixing & Reaction Air + Methane VLDB 2006, Seoul
Properties Of Simulation • Composition dimensionality • 9 for simple hydrogen simulations • >50 for complex methane simulations • Cost of reaction function evaluation: 30ms • Number of function evaluations: 108 to 1010 • Total simulation time • 108 function evaluations ≈ 35 days VLDB 2006, Seoul
Function Approximation • Approximate the reaction function • Approach • Use previous function evaluations to approximate future function evaluations • ISAT (In Situ Adaptive Tabulation) [Pope’ 97] • Definition: ε-approximation of f(x) • Let f: Rm → Rn be a function, let x Rm and ε R. f*(x) is an ε-approximation of f(x) if || f*(x) –f(x)|| < ε VLDB 2006, Seoul
Example f Cost VLDB 2006, Seoul
Example f f*(x2) = f(x) + s * (x2 - x) ( x, f(x) ) ε An ε-Local Region Rf,f*(x, ε) Rm ε x1 x2 Original Cost Cost VLDB 2006, Seoul
Example f2* f f3* f1* x1 x2 x3 x4 x5 x6 Original Cost Cost VLDB 2006, Seoul
Example f2* f f3* f1* x1 x2 x3 x4 x5 x6 When should a local region be added? VLDB 2006, Seoul
Example f2* f4* f f3* Each query point can be covered by several Local Regions f1* x1 x2 x3 x4 x5 x7 x6 x8 VLDB 2006, Seoul
Challenges • Finding good f* s and corresponding Local Regions • Computing a set of Local Regions • Data management: storing Local Regions for future use • Problem: Minimize total simulation time by computing and storing a set of Local Regions VLDB 2006, Seoul
Finding The Optimal Set Of Local Regions • Simplified cost model • Both the function value and Local Region at a point can be obtained at some constant cost equal across all regions • Approximations have zero cost • Offline Problem • Given a set X={ x1, x2, … xn} of query points, find the smallest set L={ l1, l2, … lk } of Local Regions, such that for each xi X there is an lj L which contains xi • NP-Complete: Reduction from Geometric Covering By Discs • Online Problem • No online algorithm is competitive VLDB 2006, Seoul
Algorithm Illustration f2* f4* f f3* f1* x1 x2 x3 x4 x5 x7 x6 x8 VLDB 2006, Seoul
Algorithm Initialize S Retrieve Lookup x in S Simulation N Y Local Region Found? Return Approximation Evaluate function at x Add new region containing x to S Add VLDB 2006, Seoul
Possible Instantiation Of Local Regions • Local Regions can be approximated using high dimensional ellipsoids [Pope ‘97] • Based on Taylor Expansion of function • Two step approach • Initial conservative approximation • Grow x x1 VLDB 2006, Seoul
Example x ε’ < ε x1 x2 VLDB 2006, Seoul
Example x ε’ < ε x’1 x’2 VLDB 2006, Seoul
Example x ε’ < ε ε x’1 x’2 VLDB 2006, Seoul
Updating Existing Regions N Evaluate function at x Y Can existing region contain x? N Grow Update existing regions to contain x Add new region containing x to S VLDB 2006, Seoul
Outline • Example scientific application • Combustion Simulation • Function Approximation Problem • Formulation • Hardness • Algorithm • Indexing problem VLDB 2006, Seoul
Indexing Problem • Workload • Retrieve: Find ellipsoid containing query point VLDB 2006, Seoul
Indexing Problem • Workload • Retrieve: Find ellipsoid containing query point • Grow • Find ellipsoids to be grown • Update grown ellipsoids VLDB 2006, Seoul
Indexing Problem • Workload • Retrieve: Find ellipsoid containing query point • Grow • Find ellipsoids to be grown • Update grown ellipsoids • Add: Insert a new ellipsoid VLDB 2006, Seoul
New Indexing Problem • Shape of regions • Updates and queries interleaved • Additional costs: ellipsoid maintenance costs • Overall aim: Reduce total simulation time • Retrieve/grow/add are all optional • Tuning parameters at each step VLDB 2006, Seoul
Outline • Example scientific application • Combustion simulation • Function approximation problem • Formulation • Hardness • Algorithm • Indexing problem • Cost structure, tuning parameters and effects • Index structures and experiments VLDB 2006, Seoul
Grow Effects Cmiss = tf + tgrowsearch + Igrow * Cgrow + (1-Igrow)*Cadd • Tuning Parameter: Ellg • Limit on number of ellipsoids examined for growing • No pruning criteria • Affects • tgrowsearch • Chance of finding a growable ellipsoid • Tuning Parameter: Ngrown • Number of ellipsoids grown per step • Affects • Cgrow • Structure of the index (overlapping ellipsoids) VLDB 2006, Seoul
Retrieve Effects Ctot = tsearch + Iret * tla + (1-Iret) * Cmiss • Tuning Parameter: Ellr • Limit on number of ellipsoids examined during retrieve • Limits how much of the index is searched • Affects • tsearch • Chances of a current retrieve and also future retrieves VLDB 2006, Seoul
Add Effects Cmiss = tf + tgrowsearch + Igrow * Cgrow + (1-Igrow)*Cadd • Tuning parameter: Indirectly controlled by retrieves and grows • Affects • Should query point be covered by an add or grow? (-) Computing new ellipsoids is expensive (-) New ellipsoids cover smaller part of the domain (+) May lead to better ellipsoid distribution VLDB 2006, Seoul
Candidate Index Structures • Bounding Box Rtree • Point Rtree • Ellipsoid Rtree • Random Projection Rtree • Binary Tree • MRU List + Rtree VLDB 2006, Seoul
Binary Tree 1 1 B 2 A A 2 C q C B Primary Retrieve VLDB 2006, Seoul
Binary Tree 1 q 1 B 2 A A 2 C C B Secondary Retrieve VLDB 2006, Seoul
Binary Tree 1 1 B 2 A A 2 C C B VLDB 2006, Seoul
Binary Tree 1 1 B 2 A A 2 C B C 3 D 3 C D Secondary Retrieve now Primary Retrieve VLDB 2006, Seoul
Effects In Action: Binary Tree • 32 dimensional Methane simulation • 6 x 106 queries • Windows XP machine (2.4 Ghz, 2GB) VLDB 2006, Seoul
MRU List + Rtree • MRU List for retrieving • High locality • Rtree for searching growable ellipsoids Rtree MRU List VLDB 2006, Seoul
Effects In Action: MRU List + Rtree • Effects very different from Binary Tree VLDB 2006, Seoul
Total Simulation Times VLDB 2006, Seoul
Conclusion & Future Work • Formulated the function approximation problem • New class of applications for high dimensional indexing • Understand index selection for function approximation • Future work • Dynamic parameter settings • New benchmark for index structures • Evaluation of other index structures • Comparison with other function approximation techniques VLDB 2006, Seoul
Questions? VLDB 2006, Seoul
