SYMBOLIC%20PERTURBATION

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SYMBOLIC PERTURBATION
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The problem
Algorithms in computational geometry make certain
assumptions about the input (general status). The
treatment of cases violating these assumptions
(degeneracies) is tedious, and seldom included in
the theoretical discussion. Example Algorithm
constructing convex hull in d dimensions suppose
that no d1 points lie on the same hyperplane.
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The solution

• A perturbation of the input by an infinitesimal
  factor.
• Running the algorithm on the perturb input.
• Recover the answer to the original problem from
  the output of the perturb input.

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Computational model
The input is a set of n vectors in , where
and the ith vecot is xi(xi,1,,xi,d) for .
The basic operations are assumed to be
exact. Most algorithms in computational geometry
can be modeled by extended algebraic decision
tree. The tree is a ternary tree, where each node
is labeled by a test function(branch polynomial)
and its branches labeled 1,0,1. Computation
is done by evaluating and the branch labeled by
the outcome is taken.
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Computational model cont.

• The set of arithmetic operations computing a
  branch expression together with the corresponding
  test if a primitive.
• Examples
• Ordering The comparison of coordinates. The
  branch polynomial is f xi,j xk,j for .
• Orientation The test decides which side of the
  line a third point lies. f is the
  determinant of

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Computational model cont.

• Program sequence of instructions that
  implements a specific algorithm.
• Execution path sequence of instructions
  executed on a particular input.
• Complexity measures
• Algebraic model The total cost is the number of
  instructions in the longest execution path.
• Bit model The upper bound on the bit complexity
  of two integers of size O(b) is M(b) O(b logb
  loglogb). The total bit complexity is the sum of
  the costs of every instruction, maximized over
  all paths.

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Perturbation
The set of degenerate input is represented by
and it must be with measure 0. Every input is a
point in nd dimensions. The degenerate inputs
create a union of surfaces in the nd-dimensional
space S with measure 0. For every point x in S,
every nonempty open ball around it contains a
point y which represents a nondegenerate input,
so we can perturb the input that x represents
such that all degeneracies disappear. In order
not to change any nondegenerate input an open
ball small enough is chosen so that it doesnt
intersect any of the above surfaces.
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Perturbation cont.
A Problem mapping is a mapping between
topological spaces. The input space has the
standard euclidean topology. The output space Y
is (D finite space(discrete), R
union of real spaces (euclidean)). A degeneracy
occurs when the mapping is discontinuous. For any
input , a perturbed instance of x is a curve x()
rooted at x, i.e the image of a continuous
function such that x(0)x. A perturbation
scheme Q defines a perturb instance for every
element of x.
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Perturbation cont.
Definition Given a problem mapping and a
perturbation scheme Q, the perturbed problem
mapping is a mapping from X to Y such that
assuming that every such limit
exists. The goal is that the new problem mapping
be defined on a proper superset of the original
domain, and that it will be easier than handling
the degeneracies. Proposition4 For any
perturbation scheme, if mapping ? is continuous
at , then .
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Computing with perturbations

• The goal is to obtain from a given program F that
  implements ?, another program that implements
  . The changes are
• Transforming all arithmetic operations in F in
  order to handle perturbation curves.
• Eliminating e from the output in the
  postprocessing stage.
• Every branching operation f of F is transformed
  to
• .

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Computing with perturbations cont.
Definition An input instance is degenerate with
respect to some program iff it causes some
polynomial f at a branch to vanish, where f is
not identically zero. Definition A perturbation
scheme is valid with respect to a function f iff,
for every input , the limit exists and
is nonzero. Perturbation Q is valid with the set
of functions iff it is valid with respect to
every function in this set. Q is valid with
respect to a given program iff it is valid with
respect to the set of all branch polynomials in
the program.
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Computing with perturbations cont.
Theorem7 Assume that Q is valid perturbation
scheme for a program F computing mapping ? and
that is obtained by the transformations
described above then (i) computes the
perturbed mapping and (ii) for such that ?
is continuous, yields ?(x). Statement (ii)
holds if some, or all, of the zero branches of F
are removed. Proof (i) By validity all limits
exist, hence follows the same execution path
on x(e) as F would if e were a small real
positive value. Postprocessing is possible so the
map is computed by . (ii) is proved by
proposition4, and by validity no zero branches
are taken in , so they can be pruned away.
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SOS Simulation Of Simplicity
Orientation
To decide the orientation of a sequence of d1
points in we use the matrix
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SOS-cont.
The perturbation is where
for
define
If 0ltelt1 then (3)
Lemma The set p(e) is nondegenerate if egt0 is
sufficiently small.
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SOS-cont.
Proof It is sufficient to show that for no
choice of d1 mutually distinct indices
i0,i1,,id, the determinant of the matrix
is equal to zero. We assume and sort
the terms of in order of increasing
exponents of e. Specifically, is the
first term and the last one.
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SOS-cont.
Each term is of the form bec. Because egt0 is
arbitrarily small, the absolute value of the
first term with nonzero coefficient b is bigger
than the sum of all other terms. Such a term
exists since (3) guarantees that no two terms
have the same exponent of e, and thus each term
cannot be canceled. For example, the coefficient
of the last term is not zero, so does not
vanish.
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SOS-cont.
Example d1. The goal is to calculate the sign
of Let k-fold e-product , e()1. Assume
iltj Note the last two terms have no influence
on the sign of
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SOS-cont.
If ec e((i1,j1),,(ik,jk)) then we call e(il,jl)
active for The coefficient b can be
extracted from the matrix by crossing out all
rows and columns that contain an active
e(il,jl). The sign of the coefficient is
determined by the number of transpositions needed
to sort a certain permutation (j1,j2,,jd). If it
is odd then b-1 times the determinant. The key
observations is that e((i1,j1),,(ik,jk)) is the
e-product of a relevant term iff i1ltltik and
j1ltltjk.
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SOS-cont.
The algorithm uses a vector vv1,,vdvd1,
where each vi corresponds to the ith row of .
To encode the e-product e((i1,j1),,(ik,jk)), we
set vljl. For every i such that the ith row
doesnt contain an active e(i,j) we define
with il the smallest integer in i1,,ik,d1
that is larger than i. Thus, vk implies that
e(k,vk) is active iff vkltvk1. For example,
v3,4,44 implies that the e-product of the
encoded term is e(1,3).
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SOS-cont.
In order to generate the terms in the correct
order we use the fact that vv1,,vdvd1
encodes a more significant term than iff
for j, the largest index, such that .
this implies that vd1,,d1d1 encodes the
most significant term. Function Next_v(v) returns
vector local i,k begin while vi1 do for
down to 1 do return v end
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Linear Perturbations
Linear perturbations are of the following type
xi(e)xi ebi (2) where and xi(xi,1,,xi,d) and
bi(bi,1,,bi,d) Rn. Let f(x1,,xn) be any
polynomial in n vectors, its initial form is a
homogeneous polynomial I(f) equal to the sum of
all terms in f of maximum total degree. Theorem8
Let g(x1,,xn)I(f) be the initial form of f. For
linear perturbation (2) to be valid with respect
to f, it suffices that .
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Linear Perturbations-cont.
Proof Consider f(x(e)) a univariate polynomail
in e. It is required that f(e) never vanishes on
perturbed input. The highest order coefficient in
f(e) is , which implies that f(e) has a finite
number of roots. It suffices to assume that e
takes real values smaller than the minimum
positive root of f(e). The problem is reduced to
finding a single set of input vectors on which
the initial form of all the branch polynomials do
not vanish. There is also a more generic validity
criterion with the perturbation
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Evaluation of Branch Expressions
No computation in evaluating the primitives in
perturbed programs need involve the infinitesimal
variable. In the case of tests expressed as
determinants, the complexity of the evaluation
phase dominates the overall complexity. Let
MM(t)O(t2.376) be the algebraic complexity of
multiplying two matrices and I the identity
matrix. A(e) is a matrix with polynomial entries
in a single variable e, A(e) Arer Ar-1
er-1A1e A0 where r is the maximum degree in e
of any matrix entry. Ar is always nonsingular.
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Evaluation-cont.
Ar-1A(e) IerAr-1Ar-1er-1Ar-1A1e Ar-1A0
The determinant of the right-hand side equals
the characteristic polynomial of Theorem13
Let A(e) be a matrix of order t, whose entries
are linear univariate polynomials in e then
A(e)A1eA0, where A0 and A1 are numeric
matrices. If A1 is nonsingular, determining the
sign of det(A(e)) can be reduced to computing
determinant det(A1) and the characteristic
polynomial of matrix A1-1A0.
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Evaluation-cont.
Proof By the nonsingularity of A1 we can write
A(e) as A(e) (-A1)(-Ie - A1-1A0). If we
represent sign with a value in -1,0,1, then the
sign of det(A(e)) is the product of det(A1) and
det(-Ie-A1-1A0) multiplied by (-1)t. the last
determinant is simply the characteristic
polynomial of A1-1A0.
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Modular Arithmetic
Let p0,p1,,pk-1 be a set of integers that are
pairwise relatively prime and let p be the
product of the pi. By the Chinese remainder
theorem there is one-to-one correspondence
between the integers r with and
(r0,r1,,rk-1) with where ri r mod pi,
qi p/pi, and siqi-1 mod pi. To evaluate an
expression, a set of relatively prime integers is
chosen such that the product of the primes it at
least twice the absolute value of the integral
value of the expression. Then the expression is
evaluated modulo each pi. Finally Chinese
remaindering is used to reconstruct the value of
the expression.
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Modular Arithmetic cont.
Let k denote the number of primes. The first and
third stage each have bit complexity of
O(M(k)logk). The middle stage is the actual
computation modulo each prime, and its bit
complexity is k times the algebraic complexity of
this computation. The number of primes is
determined dynamically, by choosing random primes
and constructing the answer with those primes. At
the next iteration a new prime is added and the
constructing is done again. If the answer is
stable for two or three times then with very high
probability the true answer has been calculated.
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Modular Arithmetic cont.
Corollary14 The algebraic complexity of
computing det(A1eA0) is O(MM(t)logt), where t
is the order of matrices A0 and A1. Let s be the
maximum bit size of every entry in A1 and A0.
Then the bit complexity of computing the above
determinant is , for some arbitrarily small
positive constant .
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Common Primitives

• The perturbations dealt with are
• (1)
• (5)
• Where q is the smallest prime that exceeds n.
• Two assumptions in estimating the complexity are
  made
• Sign determination of determinants is implemented
  by a determinant calculation.
• SO(logn) an upper bound on the bit size of the
  input.

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Ordering
The ordering primitive decides the order of two
quantities expressing the kth coordinate of the
i1th and i2th input points. For input perturbed
with (1) the primitive decides the sign of The
perturbation is valid because all the indices are
distinct. The perturbation (5) is valid too for
k1. Theorem The perturbation defined by (1) is
valid with respect to the ordering primitive and
doesnt change the asymptotic running time
complexity of this primitive in the algebraic as
well as the bit model. The same hold for
perturbation (5) for comparisons along the first
coordinate.
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Orientation
Given a query point and a hyperplane in Rd
spanned by xi1,,xid, orientation decides in
which one of the two half-spaces defined by this
hyperplane the query point lies. A degeneracy
occurs exactly when lies on the hyperplane.
The primitive is formulated as a test of a
determinant sign Proposition Perturbation
(1) is valid with respect to algorithms that
branch on determinants of ?d1 for , where d is
the space dimension. The perturbation increases
the asymptotic running-time complexity
(algebraic) by O(logd), and the worst-case
complexity (bit) by O(d1a).
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InSphere
InSphere decides, given d2 points, whether the
(d2)nd point lies in the interior of the
higher-dimensional sphere defined by the first
d1 points in Rd. The points are lifted by adding
a (d1)st coordinate equal to the sum of the
squares of the d-coordinates defining each point.
The lifted images define a hyperplane H in Rd1.
The query point lies within the sphere iff its
lifted image lies below H. A degeneracy occurs at
the singularities of
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InSphere-cont.
Validity follows by the nonsingularity of matrix
Descartes Rule Of Sign The number of sign
variations of a polynomials nonzero coefficients
exceeds the number of positive zeros,
multiplicities counted, by an even nonnegative
integer. Proposition Matrix Wd2 is nonsingular
for distinct positive ij, .
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InSphere-cont.
Proof If Wd2 is singular, then there is a
nonzero vector (q1,,qd2) in the kernel of the
matrix. Therefore the polynomial has at least
d2 distinct positive zeros, namely i1,,id2.
The polynomial has at most d1 sign variations
which contradicts Descartes rule.
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InSphere-cont.
For the scheme (1), the perturbed determinant
expands to Computing each of the two
polynomials can be reduced to a characteristic
polynomial computation (by theorem13). Theorem
The perturbation of (1) is valid with respect to
the InSphere primitive and increases its
algebraic complexity by an O(logd) factor. Under
the bit model, the worst-case complexity
increases by an O(d1a).
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Limitations
Primitives that decide on the relative position
of derived objects may pose a limitation. For
example the line-segment intersection problem.
The goal is to construct the planar arrangement
defined by the segments and their intersections.
The hardest primitive is to decide on which
half-plane, with respect to a query segment, the
intersection of two segments lies.
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Convex Hull
The only tests needed are ordering and
orientation, so perturbation (5) is applied to
allow arbitrary input. The CHV (convex hull
volume) problem is continuous everywhere, so it
can be obtained without any postprocessing. The
algorithm sorts all points on their first
coordinate and then proceeds incrementally by
adding each new point to the convex hull of all
previous points. Each region between the new
point and the existing hull can be partitioned
into d-simplices, each defined by the new point
and one of the visible facets. The exact volume
is computed by summing all simplex volumes.
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Convex Hull cont.
Perturbation (5) guarantees that the output
polytopes vertex set is a superset of the
vertices on the original convex hull. The
postprocessing is done by comparing the normals
of every to adjacent facets. The normal of a
facet can be computed in O(MM(d)) and there are d
tests per facet, hence the postprocessing does
not affect the asymptotic complexity of the
program.
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