Oct 18, 2001Talk @ MIT1 Survey and Recent Results: Robust Geometric Computation Chee Yap Department...

Oct 18, 2001 Talk @ MIT 1

Survey and Recent Results: Robust Geometric Computation

Chee Yap

Department of Computer Science

Courant Institute

New York University


OVERVIEW Part I: NonRobustness Survey Part II: Exact Geometric Computation

Core Library Constructive Root Bounds

Part III: New Directions Moore’s Law and NonRobustness Certification Paradigm

Conclusion


Numerical Nonrobustness Phenomenon


Part I: OVERVIEW

The Phenomenon What is Geometric? Taxonomy of Approaches EGC and relatives


Numerical Non-robustness Non-robustness phenomenon

crash, inconsistent state, intermittent Round-off errors

benign vs. catastrophic quantitative vs. qualitative


Examples Intersection of 2 lines

check if intersection point is on line Mesh Generation

point classification error (dirty meshes) Trimmed algebraic patches in CAD

bounding curves are approximated leading to topological inconsistencies

Front Tracking Physics Simulation front surface becomes self-intersecting


Responses to Non-robustness “It is a rare event” “Use more stable algorithms” “Avoid ill-conditioned inputs” “Epsilon-tweaking” “There is no solution” “Our competitors couldn’t do it, so

we don’t have to bother”


Impact of Non-robustness Acknowledged, seldom demanded Economic/productivity Impact

barrier to full automation scientist/programmer productivity mission critical computation fail

Patriot missile, Ariadne Rocket

E.g. Mesh generation a preliminary step for simulations 1 failure/5 million cells [Aftosmis] tweak data if failure


What is Special about Geometry?

Oct 18, 2001 Talk @ MIT 10

Geometry is Harder

Geometry = Combinatorics+Numerics

E.g. Voronoi Diagram

Oct 18, 2001 Talk @ MIT 11

Example: Convex Hulls

2121

33

4 4

5 56 6

7 78 8

9 9

Input Convex Hull Output

21

3

4

56

7

Oct 18, 2001 Talk @ MIT 12

Consistency

Geometric Object

Consistency Relation (P) E.g. D is convex hull or Voronoi

diagram Qualitative error inconsistency

D = (G, L,P)

G=graph, L=labeling of G

Oct 18, 2001 Talk @ MIT 13

Examples/Nonexamples

Consistency is critical matrix multiplication shortest paths in graphs (e.g.

Djikstra’s algorithm) sorting and geometric sorting Euclidean shortest paths

Oct 18, 2001 Talk @ MIT 14

Taxonomy of Approaches

Oct 18, 2001 Talk @ MIT 15

Gold Standard Must understand the dominant mode

of numerical computing “F.P. Mode” :

machine floating point fixed precision (single/double/quad) IEEE 754 Standard

What does the IEEE standard do for nonrobustness? Reduces but not eliminate it. Main contribution is cross-platform predictability.

Historical Note

Oct 18, 2001 Talk @ MIT 16

Type I Approaches

Basic Philosophy: to make the fast but imprecise (IEEE) arithmetic robust

Taxonomy arithmetic (FMA, scalar vector, sli, etc) finite resolution predicates (-tweaking, -

predicates [Guibas-Salesin-Stolfi’89]) finite resolution geometry (e.g., grids) topology oriented approach [Sugihara-Iri’88]

Oct 18, 2001 Talk @ MIT 17

Example Grid Geometry [Greene-Yao’86]

Finite Resolution Geometries

Oct 18, 2001 Talk @ MIT 18

Example What is a Finite Resolution Line?

A suitable set of pixels [graphics] A fat line [generalized intervals] A polyline [Yao-Greene, Milenkovic, etc] A rounded line [Sugihara]

fat line:polyline:

aX+bY+c=0; (a<2L, b<2L, c<22L)

rounded line:

Oct 18, 2001 Talk @ MIT 19

Example Topology Oriented Approach of

Sugihara-Iri: Voronoi diagram of 1 million points

Priority of topological part over numerical part

Identify relevant and maintainable properties: e.g. planarity

Issue: which properties to choose?

Oct 18, 2001 Talk @ MIT 20

Exact Approach

Idea: avoid all errors Big number packages (big integers, big

rationals, big floats, etc)

Only rational problems are exact Even this is a problem [Yu’92, Karasick-

Lieber-Nackman’89]

Oct 18, 2001 Talk @ MIT 21

Algebraic Computation Algebraic number:

P(x) = x2 – 2 = 0

Representation: P(x), 1, 2) Exact manipulation:

comparison arithmetic operations, roots, etc.

Most problems in Computational Geometry textbooks requires only

+, –, , ,

Oct 18, 2001 Talk @ MIT 22

Type II Approach

Basic Philosophy: to make slow but error-free computation more efficient

Taxonomy Exact Arithmetic (naïve approach) Expression packages Compiler techniques Consistency approach Exact Geometric Computation (EGC)

Oct 18, 2001 Talk @ MIT 23

Consistency Approach

Goal: ensure that no decisions are contradictory

Parsimonious Algorithms [Fortune’89] : only make tests that are independent of previous results

NP-hard but in PSPACE

Oct 18, 2001 Talk @ MIT 24

Consistency is Hard [Hoffmann, Hopcroft, Karasick’88]

Geometric Object D = (G, L) G is realizable if there exists L such that

(G, L) is consistent Algorithm AL: I G(I), L(I))

AL is geometrically exact if G(I) is the exact structure for input I

AL is consistent if G(I) is realizable Intersecting 3 convex polygons is hard

(geometry theorem proving)

Oct 18, 2001 Talk @ MIT 25

Exact Geometric Computation

Oct 18, 2001 Talk @ MIT 26

Part II: OVERVIEW Exact Geometric Computing (EGC) The Core Library Constructive Root Bounds

Oct 18, 2001 Talk @ MIT 27

How to Compute Exactly in the Geometric Sense

Algorithm = sequence of steps construction steps conditional or branching steps

Branching based on sign of predicate evaluation

Output combinatorial structure G in D=(G,L) is determined by path

Ensure all branches are correct this guarantees that G is exact!

Oct 18, 2001 Talk @ MIT 28

Exact Geometric Computation (EGC)

Exactness in the Geometry, NOT the arithmetic (cf.geometric exactness)

Simple but profound implications We can now use approximate arithmetic

[Dube-Yap’94] EGC tells us exactly how much precision is

needed No unusual geometries

No need to invent new algorithms -- “standard” algorithms apply

no unusual geometries General solution (algebraic case)

Not algorithm-specific solutions!

Oct 18, 2001 Talk @ MIT 29

Constant Expressions = set of real algebraic operators. = set of expressions over . E.g., if , x1 x2 then is

the integer polynomials over x1 x2. Assume are constant operators (no

variables like x1 x2). An expression E is a DAG

E = with and shared Value function, Val: RR where

Val(E) may be undefined

Oct 18, 2001 Talk @ MIT 30

Fundamental Problem of EGC Constant Zero Problem

CZP: given E , is Val(E)=0?

Constant Sign Problem CSP: given E , compute

the sign of Val(E).

CSP is reducible to CZP Potential exponential gap:

sum of square roots CZP is polynomial time [Bloemer-Yap]

Oct 18, 2001 Talk @ MIT 31

Complexity of CSP Hierarchy:

0 , 1 0 2 1 3 2 Root(P,i) : P(x)Z[x] 4 0 Sin(x),

Theorem: CSP(3) is decidable. Theorem: CSP(1 ) is alternating

polynomial time. Is CSP(4) is decidable?

Oct 18, 2001 Talk @ MIT 32

Root Bound A root bound for an

expression E is a value such that

E.g., the Cauchy’s bound says that because is a root of the polynomial x4 0 x2 1.

Root bit-bound is defined as log(b)

b > 0

b

=

Oct 18, 2001 Talk @ MIT 33

How to use root bounds

Let b be a root bound for Compute a numerical

approximation for . If then

sign sign( ) Else sign (E) = 0. N.B. root bound is not reached

unless sign is really zero!

E

E.

E b/2 E

( ) = EE

Oct 18, 2001 Talk @ MIT 34

Nominally Exact Inputs

EGC Inputs are exact and consistent

Why care about exactness if the input is inexact? Because EGC is the easiest method to ensure consistency.

Oct 18, 2001 Talk @ MIT 35

Core Library

Oct 18, 2001 Talk @ MIT 36

EGC Libraries GOAL: any programmer may routinely

construct robust programs* Current Libraries:

Real/Expr [Yap-Dube’94]

LEDA real [Burnikel et al’99]

Core Library [Karamcheti et al’99]

Oct 18, 2001 Talk @ MIT 37

Core Library An EGC Library

C++, compact (200 KB) Focus on “Numerical Core” of EGC precision sensitive mechanism automatically incorporates state of art

techniques

Key Design Goal: ease of use “Regular C++ Program” with preamble:

#include “CORE.h” easy to convert existing programs

Oct 18, 2001 Talk @ MIT 38

Core Accuracy API

Four Levels(I) Machine Accuracy (IEEE standard)

(II) Arbitrary Accuracy (e.g. Maple)

(III) Guaranteed Accuracy (e.g.

Real/Expr)

(IV) Mixed Accuracy (for fine control)

Oct 18, 2001 Talk @ MIT 39

Delivery System No change of programmer behavior

At the flip of a switch! Benefits: code logic verification, fast debugging

#define Level N // N=1,2,3,4#include “CORE.h”

/* ***************************** * any C/C++ Program Here * ***************************** */

Oct 18, 2001 Talk @ MIT 40

What is in CORE levels?

Numerical types: int, long, float, double BigInt, BigFloat, Expr

Promotions (+Demotions): Level II: long BigInt, double BigFloat

Level III: long, double Expr

Oct 18, 2001 Talk @ MIT 41

What is in Level III? Fundamental gap between Levels

II and III Need for iteration: consider

a = b + c; Precision sensitive evaluation

Oct 18, 2001 Talk @ MIT 42

Relative and Absolute Precision Let real X be an approximation of X. Composite precision bound [r, a]

If r , then get absolute precision a. If a , then get relative precision r. Interesting case: [r, a] = [1, ] means we

obtain the correct sign of X.

Oct 18, 2001 Talk @ MIT 43

Precision-Driven Eval of Expressions Expr’s are DAGs Each node stores: an approximate

BigFloat value; a precision; a root bound Down-Up Algorithm:

precision p is propagated down error e propagated upwards At each node, check e p Check passes automatically at leaves Iterate if check fails; use root bounds to

terminate

Oct 18, 2001 Talk @ MIT 44

Example Line intersection (2-D):

generate 2500 pairs of lines compute their intersections check if intersection lies on one line 40% failure rate at Level I

In 3-D: classify pairs of lines as skew, parallel,

intersecting, or identical. At Level I, some pairs are parallel and

intersecting, etc.

Oct 18, 2001 Talk @ MIT 45

Example: Theorem Proving Application

Kahan’s example (4/26/00) “To show that you need theorem proving, or

why significance arithmetic is doomed” F(z): if (z=0) return 1; else (exp(z)-1)/z; Q(y): |y-sqrt(y**2 +1)| - 1/(y-sqrt(y**2+1)); G(x): F(Q(x)**2). Compute G(x) for x=15 to 9999.

Theorem proving with Core Library [Tulone-Yap-Li’00] Generalized Schwartz Lemma for radical

expressions

Oct 18, 2001 Talk @ MIT 46

Constructive Root Bounds

Oct 18, 2001 Talk @ MIT 47

Problem of Constructive Root Bounds Classical root bounds (e.g.

Cauchy’s) are not constructive Wanted: recursive rules for a

family of expressions to maintain parameters p1, p2, etc, for any expression E so that B(p1, p2, ...) is a root bound for E.

We will survey various bounds

Oct 18, 2001 Talk @ MIT 48

Illustration is root of A(X) = i=0 ai Xi of degree n

Height of A(X) is A Degree-height [Yap-Dube 95]

Cauchy’s bound: maintain bound on heights but this requires the maintenance of bounds

on degrees Degree-length [Yap-Dube 95]

Landau’s bound:

A1

1||

2

1||

A

Oct 18, 2001 Talk @ MIT 49

Degree-Measure Bounds

[Mignotte 82, Burnikel et al 00]

where m( is the measure of .

It turns out that we also need to maintain the degree bound

Oct 18, 2001 Talk @ MIT 50

BFMS Bound [Burnikel-Fleischer-Mehlhorn-Schirra’99]

For radical expressions Tight for division-free expressions For those with divisions:

where E is transformed to U(E)/L(E)

Improvement [‘01]

11)( ))((||0 EDEuEE

11)( ))()((||02 ElEuEE ED

Oct 18, 2001 Talk @ MIT 51

New Constructive Bound

Applies to general - expressions. The bounding function is

: an upper bound on the absolute value of conjugates of

: an upper bound on the leading coefficient of

: an upper bound on the degree of

)(EE

)(Elc)(EIrr

)(ED)(EIrr

3

Oct 18, 2001 Talk @ MIT 52

Inductive Rules

Oct 18, 2001 Talk @ MIT 53

Comparative Study Major open question: is there a root bit-

bound that depends linearly on degree? No single constructive root bound is

always better than the others. BFMS, degree-measure and our bound Compare their behavior on interesting

classes of expressions: sum of square roots, continued fraction, etc.

In Core Library, we compute all three bounds and choose the best one.

Oct 18, 2001 Talk @ MIT 54

Experimental Results A critical predicate in Fortune’s sweepline algorithm

for Voronoi diagrams is

. Input coordinates are L-bit long, a’s, b’s and d’s are

3L-, 6L- and 2L-bit integers, respectively. The BFMS bound: (79 L + 30) bits The degree-measure bound: (64 L + 12) bits Our new bound: (19 L + 9) bits Best possible [Sellen-Yap]: (15 L + O(1)) bits

Oct 18, 2001 Talk @ MIT 55

Timing on Synthetic Input

Timings for Fortune’s predicate on degenerate inputs (in seconds)

Tested on a Sun UltraSparc (440MHz, 512 MB)

84.5410.991.620.220.03D-M

79.4311.691.630.240.03BFMS

3.900.690.120.030.01NEW

200100502010L

Oct 18, 2001 Talk @ MIT 56

Timing for Degenerate inputTimings on Fortune’s algorithm on degenerate

inputs on a uniform (32 x 32) grid with L-bit coordinates

(in seconds)

> 2 hrs1874.41681.6418.5D-M

5892.21218.11014.186.1BFMS

112.347.541.735.2NEW

50302010L

Oct 18, 2001 Talk @ MIT 57

Moore’s Law and Non-robustness Trends

Oct 18, 2001 Talk @ MIT 58

Part III: OVERVIEW New Directions Robustness as Resource

(or, how to exploit Moore’s Law) Certification Paradigm

(or, how to use incorrect algorithms)

Oct 18, 2001 Talk @ MIT 59

Moore’s Law and Robustness

Computers are becoming faster (Moore’s Law, 1965)

Software are becoming less robust crashes more often must solves larger/harder problems (e.g.

Meshes, CAD) expectation is increasing

Inverse correlation is this inevitable?

Oct 18, 2001 Talk @ MIT 60

Reversing the Trend Robustness: all-or-nothing ? Computational Paradigm

robustness as computational resource exchange some speed for robustness

Goal: a positive correlation between Moore’s Law and robustness

Oct 18, 2001 Talk @ MIT 61

Robustness Trade-off Curves

Each program P, for given input, defines a family of speed-robustness trade-off curves, one curve for each CPU speed

Trade-Off Curves

0

20

40

60

80

100

0 20 40 60 80 100

robustness

spee

d Pentium III 800MHz

Pentium III 500MHz

AMD-K6 600MHz

•What is 100% robustness?

Oct 18, 2001 Talk @ MIT 62

Computing on the Curve

P only needs to be recompiled to operate at any chosen point on a curve.

With each new hardware upgrade, we can automatically recompile program to achieve the “sweetspot” on the new curve

Oct 18, 2001 Talk @ MIT 63

Architecture

Automatically generated Makefiles a registry of programs that wants

to join sample inputs (various sizes) for

each program robustness targets/parameters optimization function (e.g., min

acceptable speed)

Oct 18, 2001 Talk @ MIT 64

Hardware Change/Upgrade For given hardware configuration, for

each program: compile different robust versions of the

program run each on the test inputs to determine trade-

off curve optimize the parameters

Layer above the usual compiler Application: libraries, CPU upgrades Can be automatic and transparent (like

the effects of Moore’s law)

Oct 18, 2001 Talk @ MIT 65

Certification Paradigm

Oct 18, 2001 Talk @ MIT 66

Another Paradigm Shift

Motivation: programs based on machine arithmetic are fast, but not always correct.

Floating point filter phenomenon Need new framework for working

with ``useful but incorrect algorithms’’

Oct 18, 2001 Talk @ MIT 67

Verification vs. Checking Program verification movement

(1970s) Another reason why it failed:

Algorithm: “always correct”

H-algorithm: “always fast, often correct”

Checking paradigm [Blum,Kannan] use H-algorithms only check specific input/output pairs, not

programs.

Oct 18, 2001 Talk @ MIT 68

Checking vs. Certifying Checker: given an input-output pair,

always accept or reject correctly Certifier (or Filter): given an input-

output pair, either accept or “unsure” Why certify?

Certifiers are easier to design Nontrivial checkers may not be

known (e.g. sign of determinant)

Oct 18, 2001 Talk @ MIT 69

Filtered Algorithms How to use Certifiers? Ingredients:

Algorithm A H-algorithm H Filter or Certifier F

Basic Architecture run H, filter with F, run A if necessary

Oct 18, 2001 Talk @ MIT 70

Some Floating Point Filters FvW Filter, BFS Filter Static, Semi-static and dynamic filters Lemma:

given expression E with k operations and floating point inputs, can compute an upper bound on E with 3k flops

Static case [FwW]: about 10 flops per arithmetic operations

Oct 18, 2001 Talk @ MIT 71

Model for Filtering Case: 2D Delaunay Triangulation

Base line Top line sigma = top line/base line = 60 phi = filtering performance = 20

efficacy = sigma/phi = 3 Synthetic vs. Realistic algorithms beta factor

= filterable fraction of work at top line estimating beta [Mehlhorn et al]

Oct 18, 2001 Talk @ MIT 72

Extensions Different levels of granularity

Checking Geometric structures [Mehlhorn, et al]

Determinant filter [Pan]

Filter compiler [Schirra]

Cascaded bank of filter [Funke et al]

skip A if necessary

Oct 18, 2001 Talk @ MIT 73

Conclusion

Oct 18, 2001 Talk @ MIT 74

REVIEW Part I: NonRobustness Survey Part II: Exact Geometric Computation

Core Library Constructive Root Bounds

Part III: New Directions NonRobustness as Resource (or, how to

exploit Moore’s Law) Certification Paradigm (or, how to use

incorrect algorithms)

Oct 18, 2001 Talk @ MIT 75

Summary Non-robustness has major

economic/productivity impact Scientifically sound and systematic

solution based on EGC is feasible Main open problem:

Is EGC for possible for non-algebraic problems? (CSP)

Program filtering goes beyond program checking

Robustness as resource paradigm can be exploited (with Moore’s Law)

Oct 18, 2001 Talk @ MIT 76

Download Software

Core Library Version 1.4 small and easy-to-use Project Homepage:

http://cs.nyu.edu/exact/

Oct 18, 2001 Talk @ MIT 77

Last Slide

Oct 18, 2001 Talk @ MIT 78

Extras

Oct 18, 2001 Talk @ MIT 79

Algebraic Degree Bound

The expression where is the radical or root node in the expression.

The degree bound , where is either the index of radical nodes or the degree in polynomial root nodes.

k

iikED

1)(

ik

)( ,,2,1 krrrQE

ir

Oct 18, 2001 Talk @ MIT 80

Leading Coefficients and Conjugates Bound Basic tool: resultant calculus Case: given two algebraic numbers and with

minimal polynomials and and a defining

polynomial for is

Represented in the Sylvester matrix form Leading coefficient: Constant term: Degree:

Oct 18, 2001 Talk @ MIT 81

(cont.)

Due to the admission of divisions, we also need to bound: Tail coefficients Lower bounds on conjugates

Measures are involved in bounding tail coefficients.

)(Etc

)(EM

)(Ev

Oct 18, 2001 Talk @ MIT 82

End of Extra Slides

Oct 18, 2001Talk @ MIT1 Survey and Recent Results: Robust Geometric Computation Chee Yap Department...

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