efkp socg slides

7/23/2019 EFKP SoCG Slides

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An output-sensitive algorithm for computing

(projections of) resultant polytopes

Vissarion FisikopoulosJoint work with I.Z.Emiris, C.Konaxis and L.Penaranda

Department of Informatics, University of Athens

SoCG, Chapel Hill, NC, USA, 18.Jun.2012


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An interesting class of polytopes: resultant polytopes

Geometry: Minkowski summands of secondary polytopes,equivalence classes of secondary vertices, generalization of Birkhoffpolytopes

Motivation: useful to express the solvability of polynomial systems

Applications: discriminant and resultant computation, implicitization

of parametric hypersurfaces

Ennepers Minimal Surface


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Existing work

Theory of resultants, secondary polytopes, Cayley trick[GKZ 94]

TOPCOM [Rambau 02]computes all vertices of secondary polytope.

[Michiels & Verschelde DCG99]define and enumerate coarse

equivalence classes of secondary polytope vertices.

[Michiels & Cools DCG00]describe a decomposition of(A) inMinkoski summands, including N(R).

Tropical geometry[Sturmfels-Yu 08]leads to algorithms for the

resultant polytope (GFan library)[Jensen-Yu 11]and thediscriminant polytope (TropLi software)[Rincn 12].


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What is a resultant polytope?

Given n+ 1 point sets A0, A1, . . . , An Zn

A0

A1

a1

a3

a2

a4


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A=

ni=0(Ai {ei}) Z2n where ei= (0, . . . , 1, . . . , 0) Zn

A

A0

A1

a1

a3

a3, 1

a1, 0

a2

a4

a4, 1

a2, 0


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A=


Given Ta triangulation ofconv(A), a cell is a-mixedif it is theMinkowski sum ofn 1-dimensional segments from Aj, j=i, andsome vertex a Ai.

A

A0

A1

a1 a2

a3 a4

a3, 1 a4, 1

a1, 0 a2, 0


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A=n

i=0(Ai {ei}) Z2n where ei= (0, . . . , 1, . . . , 0) Z

n


T(a) =

a

mixedT:a vol() N, a A

T = (0, 2, 1, 0)

A

A0

A1

a1 a2

a3 a4

a3, 1 a4, 1

a1, 0 a2, 0


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A=



T(a) =

amixed

T:avol() N, a A

Resultant polytope N(R) =conv(T:T triang. ofconv(A))

A N(R)

A0

A1


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Connection with Algebra

TheNewton polytopeoff, N(f), is the convex hull of the set ofexponents of its monomials with non-zero coefficient.

Theresultant Ris the polynomial in the coefficients of a system ofpolynomials which is zero iff the system has a common solution.

A0

A1

N(R) R(a,b,c,d,e) =ad2b+c2b2 2caeb+a2e2

f0(x) =ax2 +b

f1(x) =cx2 +dx+e


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A0

A1

N(R)

f0(x, y) =ax+by+c

f1(x, y) =dx+ey+f

f2(x, y) =gx+hy+iA2

a b c

d e f

g h i4-dimensional Birkhoff polytope

R(a ,b,c ,d ,e,f,g ,h,i) =


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A0

A1

N(R)

f0(x, y) =axy2 +x4y+c

f1(x, y) =dx+ey

f2(x, y) =gx2 +hy+iA2

NP-hardto compute the resultant

in thegeneral case


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The idea of the algorithm

Input: A Z2n defined by A0, A1, . . . , An Zn

Simplistic method:

compute the secondary polytope (A) many-to-one relation between vertices of(A) and N(R) vertices

Cannot enumerate 1 representative per class by walking on secondaryedges


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The idea of the algorithm

Input: A Z2n defined by A0, A1, . . . , An Zn

New Algorithm:

Vertex oracle: given a direction vector compute a vertex ofN(R) Output sensitive: computes only one triangulation ofA per N(R)

vertex + one per N(R) facet

ComputesprojectionsofN(R) or(A)


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The Oracle

Input: A Z2n, direction w (R|A|)

Output: vertex N(R), extremal wrt w

1. use was a lifting to construct regular subdivision SofA

w

face of (A)

S


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The Oracle




2. refine S into triangulation TofA

w

face of (A)

T

T

S


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The Oracle





3. return

TN|A|

N(R)

w

face of (A)

T

T

S

T


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The Oracle





3. return T N|A|

Oracle property: its output is a vertex of the target polytope (Lem. 5).


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Incremental AlgorithmInput: A

Output: H-rep. QH, V-rep. QVofQ= N(R)

1. initialization step

N(R)

Q

initialization:

Q N(R)

dim(Q)=dim(N(R))

I l Al i h


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1. initialization step2. all hyperplanes ofQHareillegal

Q

N(R)

2 kinds of hyperplanes ofQH:

legalif it supports facetN(R)

illegalotherwise

I l Al i h


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1. initialization step

2. all hyperplanes ofQHareillegal

3. while illegal hyperplane H QHwith outer normal wdo call oracle for wand compute v, QV QV {v}

N(R)

Qw

Extending anillegalfacet

I t l Al ith


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3. while illegal hyperplane H QHwith outer normal wdo call oracle for wand compute v, QV QV {v} if v/ QV H then QHCH(QV {v}) else H is legal

N(R)

Q

Extending anillegalfacet

I t l Al ith


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N(R)

Q

Validating alegalfacet

I t l Alg ith


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N(R)

Q

Validating alegalfacet

Incremental Algorithm


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N(R)

Q



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N(R)

Q

At any step, Q is an inner

approximation . . .



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N(R)

Q

Qo

At any step, Q is an innerapproximation . . . from which we

can compute an outer approximationQo.



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N(R)

Q



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N(R)

Q



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N(R)

Q



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N(R)

Q



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N(R)

Q



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N(R)

Q



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gInput: A




N(R)

Q



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gInput: A




N(R)

Q



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gInput: A




N(R)

Q



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gInput: A




N(R)

Q



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Input: AOutput: H-rep. QH, V-rep. QVofQ= N(R)



N(R)

Q



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N(R)

Q



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N(R)

Q



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N(R)

Q

Complexity


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TheoremWe compute the Vertex- and Halfspace-representations of N(R), as wellas a triangulation T of N(R), in

O( m5 |vtx(N(R))| |T|2),

where m= dim N(R), and|T| the number of full-dim faces of T.

Elements of proof

Computation is done in dimension m= |A| 2n+ 1.

At most vtx(N(R)) +fct(N(R)) oracle calls (Lem. 9).

Beneath-and-Beyond algorithm for converting V-rep. to H-rep[Joswig 02].

ResPol package


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C++

CGAL, triangulation [Boissonnat,Devillers,Hornus]extreme points d [Gartner](preprocessing step)

Hashing of determinantal predicates: optimizing sequences of similardeterminants

http://sourceforge.net/projects/respol

Applications of ResPol on I.Emiris talk this afternoon (CGAL, anOpen Gate to Computational Geometry!)

Output-sensitivity


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oracle calls vtx(N(R)) +fct(N(R)) output vertices bound polynomially the output triangulation size

subexponential runtime wrt to input points (L), output vertices (R)

Hashing and Gfan


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hashing determinantsspeeds 10-100x when dim(N(R)) = 3, 4 faster than Gfan [Yu-Jensen11] for dimN(R) 6, else competitive

dim(N(R)) = 4:

Ongoing and future work


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approximate resultant polytopes (dim(N(R)) 7) using approximatevolume computation

combinatorial characterization of4-dimensional resultant polytopes

computation of discriminant polytopes

More on I.Emiris talk this afternoon (CGAL, an Open Gate to

Computational Geometry!)

(figure courtesy of M.Joswig)

Facet and vertex graph of the largest 4-dimensional resultant polytope

Ongoing and future work


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(figure courtesy of M.Joswig)

Facet and vertex graph of the largest 4-dimensional resultant polytope

Thank You !

Convex hull implementations


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From V- to H-rep. ofN(R).

triangulation (on/off-line), polymake beneath-beyond,cdd, lrs

0.01

0.1

1

10

100

0 500 1000 1500 2000 2500 3000

tme

sec

Number of points

bb

cdd

lrstriang_off

triang_on

dim(N(R)) = 4

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