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The Volume of a Lattice Polyhedron to Enumerate Processors and Parallelism Philippe CLAUSS

ICPS, Universite Louis Pasteur, Strasbourg, P^ole API, Bd Sebastien Brant, 67400 Illkirch France [email protected] October 11, 1996

Abstract. In the area of automatic parallelization of programs, restructuration of nested loops or parallelization of systems of ane recurrence equations use classically a geometrical model based on convex polyhedra of lattices. Since these algorithms are speci ed for parametric sized calculation domains, parametric lattice polyhedra have to be considered. Computing on such domains is a grand challenge in today's discrete mathematics research. In particular, the number of lattice points in a polyhedron has several useful meanings in automatic parallelization. At the end of the 50's, E. Ehrhart introduced the concept of homothetic polyhedra suprisingly tting so well to our context. We recall in this paper Ehrhart's powerful results allowing to compute the number of lattice points in integral or rational polyhedra of any dimension which may be nonconvex, and apply these results to the computation of the number of processors resulting from linear allocations of the calculation domain, and of the potential parallelism occurring in parallel algorithms. 1. Introduction The type of programs to be considered in our context of automatic parallelization, is a set of nested loops, where all loop bounds are ane functions of loop counters of surrounding loops and of one size parameter, and all loop increments are equal to 1. The only dependencies are supposed to be ow-dependencies and the nested loops may not contain other control structures such as if statements. So, such programs can also be expressed as a system of ane recurrence equations and are modeled by { a convex lattice polyhedron in Zd, where Zd is the standard integer lattice, called the iteration space, with points corresponding to computed values, and d denotes the depth of the loop nest,

2

Ph. Clauss

{ vectors denoting dependencies, i.e., the parents of a point are the values used to compute this point.

Given an iteration space, a multiprocessor mapping assigns point z for processing during step t(z) on a virtual processor alloc(z). This mapping is subject to two constraints : a point can be computed only when its parents have been computed at previous time steps (causality). A processor cannot compute two dierent points during the same time step, i.e., t(z) = t(z 0 ) ) alloc(z) 6= alloc(z 0 ). We only consider here schedules de ned by an ane function t(z) = 
z +  assuming that the computation time of any point is one. We choose  = ?minz2D f
z ? 1g in order to start the computation at time step 1. The most classical method to allocate the points onto a virtual mesh connected array considers a linear allocation function of the form alloc(z) = 
z where  is a (d ? 1)  d integer matrix of linearly independent row-vectors. Such an allocation can be seen as a projection of the iteration space along a projection direction de ned by a unimodular vector p. This method respects implicitely the constraints of locality and regularity imposed by the target architecture. The convenience of such a resulting mapping may be de ned through many criteria : e.g. the potential parallelism the schedule allows, or the number of processors the allocation requires. These criteria can be de ned by using nice results on polyhedra. At the end of the 50's, E. Ehrhart introduced the concept of homothetic polyhedra [11], i.e. polyhedra Pn de ned by a parameterized system of linear equalities and inequalities of one parameter n, such that Pn is the polyhedron resulting from the n-homothetic transformation of P of center the origin O of Rd . This model seems very opportune in our context where loop bounds are de ned as ane functions of some size parameters. We recall in this paper Ehrhart's powerful results allowing to compute the number of lattice points in integral or rational polyhedra of any dimension (which may be non-convex), and apply these results to the computation of the number of processors resulting from linear allocations and of the potential parallelism occurring in parallel algorithms. It is shown that Ehrhart's method allows the determination of symbolic expressions of the number of integral points depending on the size parameter. Many of Ehrhart's valuable results are unknown by the mathematician and computer science community at this time, in particular those concerning rational polyhedra, since many of these results were published in local reports and in french language [14]. Hence, the aim of this paper is also to nally make them known by everybody. We have already presented in previous works [7, 9, 8] a computation method of the number of integral points in integral polytopes. Since this method is closely related to the activity model of the processors recalled in section 4, the parameterized expressions of the number of integral points are de ned at time intervals, as parameterized sums of parameterized polynomials. Moreover, this method needs the parametric coordinates of all the vertices which must be integral. Hence it yields more complex computations than the powerful mathematical results used in this paper, but applies with any number of parameters. 1

The Volume of a Lattice Polyhedron

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2. De nitions and assumptions We rst recall some basic notions dealing with geometry of numbers [18, 24, 6] and enumerative combinatorics [26, 27]. Then some more speci c concepts, closely dedicated to the scope of the paper, are introduced. 2.1. Some basic notions

A convex polyhedron P is a subset of Rd that is the intersection of a nite number of closed halfspaces. A bounded convex polyhedron is called a convex polytope. The ane span of P is the ane subspace of Rd entirely containing P. The dimension, dimP, of a polyhedron P is the dimension of its ane span, and a kdimensional polyhedron is called a k-polyhedron for brevity. The iteration spaces considered by our method are lattice polyhedra in the sense of the following de nitions. De nition 1. (lattice) Let g ; :::; gd be linearly independent vectors of the real

numerical space Rd. The set of points generated by these vectors is called a lattice, 1

Ld = f1

g + ::: + dgd j  ; :::; d 2 Zg 1

1

and vectors g ; :::; gd are called a basis for the lattice. The set of all the integral points is called standard lattice denoted by Zd. The lattice Zdn = n Zd = fz 2 Qd j nz 2 Zdg is called associated lattice. Let G be 1

1

the matrix whose columns are the vectors g ; :::; gd . Hence in the general case, any lattice Ld consists of the points G
z, z 2 Zd, i.e. Ld = G
Zd. So we get the most general lattice Ld by subjecting the standard lattice Zd to an arbitrary non-singular transformation G. Some mathematical results studied in this work seem to be reduced to the standard and associated lattices. But the fact that any lattice can be deduced from the standard lattice [18] allows to generalize these results. De nition 2. (mesh of a lattice) The mesh of a lattice Ld is any closed paral1

lelotope of same dimension whose unique lattice points are vertices. De nition 3. (enumerator) The enumerator of any set S is the number of points of S \ Ld. De nition 4. (lattice polytope and lattice simplex) The intersection between a lattice Ld and the convex hull of a nite number of points z , : : : , zm in Rd is called a lattice polytope P of Ld . It is then homeomorphic to a ball B k . We write k = dimP and call P a k-polytope. If z , : : : , zm are anely independent points of Rd, which means that z ? z , z ? z , : : : , zm ? z are linearly independent, then the convex hull of z , : : : , zm is called a m-dimensional lattice simplex S of Ld with vertices z , : : : , zm . A lattice polytope or a lattice simplex is then de ned by a set of linear inequalities of the form fz 2 Ld j A
z  bg where A is a m  d integral matrix and b a m-dimensional integral vector. 1

1

2

1

+1

3

1

1

1

+1

+1

1

+1

+1

Note that the maximum number of linearly independent vectors in Rd is d and, hence, there could be no lattice simplex in Ld with more than d+1 vertices.

4

Ph. Clauss

De nition 5. (supporting hyperplane) If c is a nonzero vector and is de ned by = MAX fc
z j A
z  bg, the ane hyperplane fz j c
z = g is called a supporting hyperplane of P. The faces of P are , P, and the intersections of P with its supporting hyperplanes. Each face of P is itself a polyhedron. If P is a k-polyhedron, faces of P of dimension 0, 1 and k ? 1 are called vertices, edges and facets, respectively. De nition 6. (implicit equality) An inequality Ai
z  bi from A
z  b is called an implicit equality (in A
z  b) if Ai
z = bi for all z satisfying A
z  b. We use the following notation : { A
z  b is the system of implicit equalities in A
z  b { A
z  b is the system of all other inequalities in A
z  b. =

=

+

+

Proposition 1. (facet) There exists a one-to-one correspondence between the facets of P and the inequalities in A
z  b , given by Fi = fz 2 P j Ai
z = big for any facet Fi of P and any inequality Ai
z  bi from A
z  b . The proof can be found in [24]. It comes that any row Ai from A
z  b de nes the normal vector Ai to a facet Fi of P, and that any equation Ai
z = bi from A
z  b de nes the supporting hyperplane spanning the facet Fi of P. De nition 7. (poset) The partially ordered set of the faces of P, ordered by +

+

+

+

+

+

+

+

inclusion, is called the poset of P. 2.2. Some more speci c notions

In this work, we consider lattice polyhedra which may be non-convex according to the following de nition. This de nition corresponds to the one given by Ehrhart in [12, 14] and [15, p. 47] when de ning normal polyhedra. De nition 8. (lattice polyhedron) A lattice polyhedron P is a closed polyhedron

which admits a nite rectilinear simplicial subdivision. It is then the union of a nite number of simplices S ; S ; :::; Sn of same dimension k such that, { The intersection between two simplices either is empty or is a common face, { Each facet of one of these simplices belongs to at most one other simplex, faces belonging to only one simplex being the border of the polyhedron, { These simplices can be ordered such that for any h < n, if Ph = S [ S [ ::: [ Sh , the intersection Ph \ Sh either is empty, or is the union of common facets of Sh and the border of Ph . It is then homeomorphic to a ball B k . We write k = dimP and call P a kpolyhedron. 1

2

1

2

+1

+1

The reader can refer to [1, 26] for investigations on combinatorial and algebraic topology.

The Volume of a Lattice Polyhedron

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De nition 9. (Euler-Poincare characteristic) The Euler-Poincare characteristic of any k-polyhedron P is de ned by

(P) =

k X i

(?1)i fi

=0

where fi denotes the number of i-dimensional faces of P .

It is well-known that the Euler-Poincare characteristic (P) is equal to 1 if P is convex. Some results given in the following are reduced to the class of lattice polyhedra whose vertices all belong to the considered lattice. De nition 10. (Ld-polyhedron, integral polyhedron, rational polyhedron) A lattice polyhedron P of Ld is a Ld-polyhedron if all its vertices are in Ld. If Ld = Zd, P is an integral polyhedron. If some vertex of P do not belong to Ld and have rational coordinates, P is a rational polyhedron. Since by de nition, any lattice Ld can be deduced from Zd by a non-singular transformation G, i.e. Ld = G
Zd, any Ld-polyhedron is anely equivalent to an integral polyhedron. Hence, results given in the following and concerning integral polyhedra, can be directly extended to Ld-polyhedra. From now on and without loss of generality, the generic term of integral polyhedra will be used. De nition P 11. (homothetic system) if its relations are P A system is homothetic P of the form ai zi = bi n + ci , aizi < bin + ci , ai zi  bi n + ci , where the ai 's, the bi 's and the ci 's are given integers and n a positive integral parameter. Such a system Hn is signi cant if and only if H de nes a polyhedral domain P (obviously convex) . In this case, Hn de nes the domain Pn, which corresponds to P by the n-homothetic transformation of center O. P is called the primitive domain of Hn. In the following, we consider any k-polyhedron Pn for which we know the poset and/or de ned by a union of homothetic systems of the form fz 2 Ld j P A
z  Bn +PC g, since any equality Paizi = bi n + ci is equivalent to the two inequalities ?ai zi  ?bi n ? ci and ai zi  bi n + ci. Notation. We denote by jn the enumerator of Pn, denote by in the enumerator of Pn ? @Pn where @Pn denotes the boundary sub-polyhedron of Pn, i.e. its facets, and denote by pn the enumerator of @Pn . In particular, @Pn is homeomorphic to the (k ? 1)-sphere Sd? . De nition 12. (reticulate ane span) An ane span is reticulate if it supports a k-dimensional lattice simplex. Then it supports a lattice Lk. De nition 13. (reticulate volume) The reticulate volume V of a k-polyhedron Pn in Ld is the volume of P where the unit of measure is the mesh of Ld. We denote by S the peripheral reticulate volume, i.e. the sum of all the reticulate volumes of its facets. If Ld = Zd, i.e. Pn is an integral k-polyhedron in Zd, then V is just the usual volume of P . 1

1

1

1

1

1

1

1

1

The term signi cant means that all the vertices of the domain

P1

must be separated.

6

Ph. Clauss

A classic relation in combinatorics links the reticulate volume of P to its enumerator jn . The proof can be found in [27]. Theorem 1. Let Pn be a homothetic k-polyhedron, k X V = k!1 ((?1)k + Ckq (?1)k?q jq ) q 1

=1

where Ckq denotes the number of q-combinations of

k objects, i.e. Ckq = q kk?q . Ehrhart in [12] and [15, Corollary 31] has proved the following theorem for any integral 3-polyhedron. Theorem 2. For any integral 3-polyhedron Pn, S = p ? 2. The enumerators will be determined by polynomials and pseudo-polynomials according to the following de nitions. De nition 14. (periodic number) A periodic number un = [u ; u ; : : :; up] is equal to the item whose rank is equal to n modulo p, p is called the period of un. 8 u if n = 1 modulo p > < un = > :u: : if n = 2 modulo p : up if n = p modulo p Example 1. We give some examples of periodic numbers. { The nth decimal number of the periodic decimal number 4; 285 285 : : : is the integral periodic number [2; 8; 5]. { The general term of the serie 1+ ? ? + + ? : : : = p is ; ;n??;? . { (?1)n = [?1; 1]. { cos n  = ? ;? ; . ut { sin n ? cos n = [1; 1; ?1; ?1]. Ehrhart in [14, 15] gives the basic properties of periodic numbers. De nition 15. (pseudo-polynomial) A polynomial f(n) is a pseudo-polynomial !

!(

1

)!

2

1

2

2

3

2

[

1

1

1

1

1

1

3

5

7

9

11

[1 1

2

2

2

1

1]

1

1 2]

2

2

if some of its coecients are periodic numbers instead of being constants. The lowest common multiple of the periods of its periodic coecients is the pseudoperiod of f(n). So a pseudo-polynomial has two characteristics, its degree and its pseudo-period. Their general determination concerns the theory of rational generating functions. De nition 16. (rational generating function) A rational generating function is of the form,

F(x) =

X

n

0

P(x) f(n)xn = Q(x)

It is assumed that Q(0) 6= 0, so that Q(x)? exists. 1

The reader can refer to [27] for further investigations on rational generating functions.

The Volume of a Lattice Polyhedron

7

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3. The volume of a lattice polyhedron The results given in this section are due to E. Ehrhart who made a study of the enumeration of lattice points in polyhedra, by means of polynomials in the way of Euler. All the proofs can be found in [11, 12, 13, 15]. Some aspects of Ehrhart's work were corrected, streamlined, and expanded by I. G. Macdonald [19, 20] and R. P. Stanley [26, 27]. De nition 17. (Eulerian recurrence relation) Let us consider the product (u) = (1 ? ua1 )(1 ? ua2 ) : : :(1 ? ua ) (1) where any ai 2 Z . When this product is simpli ed, if each power ur is replaced by un?r , it is obtained a linear and homogeneous recurrence relation, denoted by f(u)g = 0 and called eulerian recurrence relation. The general solution of such s

+

a relation is a pseudo-polynomial. It is determined by some boundary conditions, P by a rational generating function P tt = 1 f(n)tn , and it can be reduced to a real polynomial in n of degree k, if (t) = (1 ? t)k . ( )

( )

0

+1

P

Euler has shown that the enumerator of any system ai Xi = n, Xi  0, i.e. the number of integral solutions, is determined by the characteristic product (1 ? ta ). De nition 18. (denominator) Theddenominator of a rational point is the lowest index a such that it is integral in Za. The denominator of a rational polyhedron i

is the lowest common multiple of the denominators of its vertices. De nition 19. (characteristic product) The characteristic product of any rational polyhedron is (1) where the a's are the denominators of its vertices.

Ehrhart's results are based on his following fundamental theorems. Theorem 3. (fundamental theorem) Any enumerator dn of any k-polyhedron Pn, whose characteristic product is (t), veri es the eulerian recurrence relation f(d)g = 0. Hence it is a polynomial in n of degree k if Pn is integral, and it is a pseudo-polynomial in n of degree k whose pseudo-period is the denominator of Pn if Pn is rational. Theorem 4. (reciprocity law) The enumerator jn of the k-polyhedron Pn in Ld is a polynomial or pseudo-polynomial j(n) of degree k. The enumerator in of Pn is linked to jn by the following reciprocity law, in = (?1)d j(?n) jn = (?1)d i(?n) Ehrhart has determined those polynomials for the integral lattice polyhedra of dimensions 2, 3 and 4. Theorem 5. (Ehrhart polynomial) For any integral polyhedron Pn of dimension 2, 3 or 4, we have respectively,

jn = V n + p n + (P) 2

2

jn = V n + S n + (i + p ? V )n + (P) 2

3

2

2

jn = V n + S n + (i + p ? V ? 1)n + p?S n + (P) 3

4

2

2

2

2

8

Ph. Clauss

More generally, the following results can be used, allowing to determine the polynomials of any enumerators from k of its initial values. Theorem 6. (Ehrhart polynomial) For any integral k-polyhedron Pn, the enumerators verify the following identities,

f(j ? j )k g = 0 f(i ? i )k g = 0 f(p ? p )k g = 0 P jn = kq Cnq (j ? j )q + j j r  jr ; ir  ir ; pr  pr 0

where j r

+1

+1

0

0

=1

+1

0

0

 jr means that in the expansion of any symbolic power, any power j r

must be replaced by jr .

in = n n? k::: n?k (i ? i )k

jn = n n? k::: n?k (j ? j )k (

1)

(

pn = n n? k:::? n?k (

1) (

(

(

)

0

!

+1)

(

)

0

!

(p ? p )k? j r  nj?r ; ir  ni?r ; pr  np?r 0

1)!

1)

r

1

r

r

where j r  nj?r means that in the expansion of any symbolic power, any power j r must be replaced by nj?r r

r

In the general case of rational lattice polyhedron, the determination of the pseudo-polynomial of any enumerator is rather delicate and subtle and needs the use of a rational generating function.

Theorem 7. The rational generating function associated with the enumerator of any rational k-polyhedron P tt is of degree less than 0. ( )

( )

The proof can be found in [27]. It comes that the numerator P(t) of the rational generating function associated with any enumerator dn can be deduced by inspection of some initial values of this enumerator. Let (t) being of degree , the method consists P in rst counting  initial values of dn allowingPto express the  rst terms of n dn tn in order to determine P(t) = (t) n dntn , and then nally expanding by Laurent expansion the rational generating function Pt th  t . The pseudo-polynomial of the enumerator will then be the n general term in this expansion, i.e the constant term. We illustrate this method with a simple example. Example 2. Consider the rational 2-polytope Pn de ned by, 0 ?1 0 1 0 0 1 Pn = fz 2 Z j B @ ?11 00 CA
z  B@ 0 CA  ng 1 0 0

0

( )

( )

1

2

2 1 2

P is the closed rectangle whose vertices are (0; 0), ( ; 0), (0; ), ( ; ). Hence their denominators are 1; 2; 2; 2 and the characteristic product of Pn is, (t) = (1 ? t)(1 ? t ) = 1 ? t ? 3t + 3t + 3t ? 3t ? t + t The recurrence f(d) = 0g is deduced, dn ? dn? ? 3dn? + 3dn? + 3dn? ? 3dn? ? dn? + dn? = 0 1

2 3

1

2

2

3

4

3

1

1

1

1

2

2

2

2

4

5

5

6

6

7

7

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From P to P , we count jn = 1; 1; 4; 4; 9; 9;16. It follows that, P(t) = X j tn = 1 + t + 4t + 4t + 9t + 9t + 16t + : : : (t) n n 0

6

2

3

4

5

6

0

Hence we deduce the numerator of the rational generating function, X P(t) = (t) jn tn = 1 ? t 4

n

0

The rational generating function can then be reduced to 1+t P(t) = (t) (1 ? t ) (1 ? t) By expansion and simpli cation of this rational generating fraction, there is obtained a pseudo-polynomial of degree 2 and of period 2, jn = 41 n + [ 21 ; 1]n + [ 14 ; 1] The reduced expression of the rational generating function shows that (t) is not its least denominator. That is the reason why Ehrhart introduced in [12] the concept of reduced characteristic product, denoted by !(t), which is for our example, !(t) = (1 ? t ) (1 ? t) = 1 ? t ? 2t + 2t + t ? t yielding the following recurrence f!(d) = 0g, dn ? dn? ? 2dn? + 2dn? + dn? ? dn? = 0 From this reduced version, only 5 instead of 7 initial values of jn have to be counted. ut The reduced characteristic product can be determined by use of a Ehrhart's conjecture [15, p. 53] later solved independently by R. Stanley [25, Thm 2.8] and P. McMullen [22]. But we rather give some other Ehrhart's results allowing a more direct and simplier resolution in our context. Theorem 8. Let Pn be a rational k-polyhedron and let the Ehrhart pseudo2

2

2

2

2 2

2

1

2

3

3

4

4

5

5

polynomial be,

jn = ck (n)nk + ck? (n)nk? + : : : + c (n) where c ; : : :; ck are periodic numbers of n. If for some q 2 [0; k], the ane span of every q-dimensional face of P contains a lattice point, then for any p  q, cp (n) is constant, i.e. of period one. De nition 20. (grade) The value g = q ? 1 where q is the lowest value verifying 1

1

0

0

1

the last theorem is called the grade of the pseudo-polynomial. Then a pseudopolynomial is said to be of characteristics (k; g; p) where k is its degree and p its pseudo-period.

The following theorem on pseudo-polynomials given by Ehrhart in [14] and [15, Thm 10] allows an easy determination of the reduced characteristic product.

10

Ph. Clauss

Theorem 9. Any pseudo-polynomial fn of characteristics (k; g; p) veri es the recurrence relation, f(1 ? f)k?g (1 ? f p )g g = 0 +1

Example 3. By looking at P , we observe that there is always an ane span of a q-dimensional face, with q  1, containing no integer point. The only ane span containing integer points is the 2-dimensional one of the rectangle itself, containing for example point (0; 0). The pseudo-period is the denominator of Pn. Hence, jn is a pseudo-polynomial of characteristics (2; 1; 2) and veri es the recurrence relation deduced from the last theorem. This recurrence relation is associated with the reduced characteristic product !(t). ut The fact that any face of P contains a lattice point occurs mainly in the considered k-polyhedra, when they represent iteration spaces of regular nested loops with integral loop counters. The case of rational k-polyhedra whose vertices do not belong to the considered lattice, and having ane spans with no integral point, has to be frequently considered when considering cutting hyperplanes of Pn. But the use of rational generating function yields, by Laurent expansion, to formulas with harmonic functions (sinus, cosinus, : : :) which are rather not tting to the domain of rational numbers. Ehrhart's proofs allow the use of a more adapted method : Due to these last theorems, the coecients which are periodic are known, and since the period is known equal to the denominator of the k-polyhedron Pn and the degree is known equal to k, the determination of the Ehrhart pseudopolynomial can be done by simply resolving a system of rational linear equations. Since the number of equations ne of the system must be equal to the number of unknown values, the complexity depends on the structure of Pn, its dimension, and on the size of its denominator. Moreover, ne initial values of the enumerator have to be counted. Example 4. The previous example could have been resolved by this method. The pseudo-polynomial could have been determined by resolving a system of 5 rational linear equations, since the coecients of n and n are periodic, and the coecient of n is constant. Moreover, the period of the periodic coecients is equal to the denominator of P , i.e. 2. ut Computing the coecients in the Ehrhart polynomial can be done through other approaches. For example, Barvinok in [5] proposes a complexity study of the computation of these coecients in terms of computation of the volumes of faces, since any coecient can be related to volumes or surface areas of Pn . But since the problem of volume computation is NP-hard [10], results given by Barvinok steps on the strong assumption of a Volume Computation Oracle. Barvinok also has proposed for the rst time in [4], a polynomial-time algorithm for counting integral points in non-homothetic polyhedra, i.e. when the dimension is xed. The results presented in this section are limited to homothetic polyhedra of only one parameter n. Ehrhart in [14, 15] started a limited study of bipartite systems with two parameters and wrote in introduction : La determination du 1

1

1

0

2

1

compteur d'un systeme diophantien lineaire a deux parametres est un probleme interessant, mais vaste et dicile. Talking about systems with several parame-

ters, Ehrhart ends his study with the following conjecture.

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Conjecture 1. For any signi cant diophantine linear system of any dimension,

linearly dependent of several positive integral parameters, the enumerator depending on these parameters is expressed at dierent intervals by dierent pseudo-polynomials. Hence, it is an opened problem since no other related results are known by us at this moment. But it gets more and more necessary to have new results in this research area, since the concept of parametric domains constitutes one of the major foundations in the theory of automatic parallelization of programs [17, 23]. In particular, it seems to be opportune to introduce the concept of multi-dimensional homothetic systems and multi-dimensional periodic numbers. Our geometrical model of a linear multiprocessor mapping is recalled in the next section. Associated with the above results, it allows the determination of useful values in the parallelization of nested loops or systems of ane recurrence equations.

4. Geometrical analysis of a linear allocation As it was said in introduction, any linear allocation of the iteration space Pn consists in a projection along a unimodular vector p called the allocation direction. Without loss of generality, the vector p is chosen such that 
p > 0. All the points of Pn resulting in the same virtual processor by this projection de ne a line-segment. Two successive points z and z 0 of such a line-segment, z 0 ? z = p, characterize successive computations on the virtual processor. Moreover, these two computations are time-spaced by 
p steps. Hence, any virtual processor is active one step over 
p steps. By considering all the so de ned line-segments, their extremities de ne facets of Pn. De nition 21. Let Ai be the normal vector to any facet Fi of Pn. The inner product Ai
p determines three sets of facets of Pn : { If Ai
p  1, Fi is a rst facet denoted Fi , { if Ai
p  1, Fi is a last facet denoted Fi?, { if Ai
p = 0, Fi is a null facet denoted Fi . Our model will then only consider the rst and last facets. Since only integral points are considered, the extreme integral points of the line-segments de ne either facets or inside facets of Pn according to the following de nition. This case occurs for any facet such that jAi
pj > 1. De nition 22. (inside facets) Let Fi be a facet of Pn contained in the supporting hyperplane de ned by the equation Ai
z = bi. The intersection, if it is not empty, between an hyperplane of equation Ai
z = bi 
and Pn is called an inside facet of Pn denoted Fi;
,
2 Z. For any rst or last facet, we consider a number of inside facets equal to jAi
pj ? 1, contained each in the hyperplane of equation Ai
z = bi 
, ? . From now
2 [1::jAi
pj ? 1]. They are denoted by Fi;q and Fi;q on, but only ? ? in useful cases, Fi and Fi will be denoted by Fi; and Fi; , and in the case of ? , the notation F  will be used. common notions for Fi;q and Fi;q i;q +

0

+

+

+

0

+

0

12

Ph. Clauss

All the points of the rst (or last) facets and of the rst (or last) inside facets characterize the rst (or last) computations of the virtual processors. As it was said in introduction, the schedule is de ned by an ane function t(z) = 
z + . The vector  is the normal vector to any hyperplane de ned by Ht0 = fz 2 Zd j 
z +  = t g, t 2 Z. All the points of the iteration space Pn belonging to such an hyperplane Ht0 characterize simultaneous computations at step t . It comes that the intersection between all the rst (or last) facets, rst (or last) inside facets and any hyperplane Ht0 characterizes all the rst (or last) simultaneous computations at step t . The projection of this intersection along the allocation direction p characterizes the virtual processors whose rst (or last) computations occur simultaneously at step t . By considering the intersection between Ht0 and one of the rst (or last) facet or rst (or last) inside facet, such an intersection is either the rst (or last) facet or rst (or last) inside facet, or a lattice polytope with dimension less or equal than d ? 2. Since 
p 6= 0, the projection of this intersection along p is of the same dimension.  is sensible only on a The projection of such an intersection Ht0 \ Fi;q subset of virtual processors whose rst (or last) computations are de ned by  . By considering the projection along p of all the couples the points of Fi;q ? (Ht0?
p \ Fi ;q ; Ht0 \ Fi;q ), we characterize three sets of virtual processors : the dead ones, the active ones of step t , and the previously unused ones. If  results in Fi;q  , its projection results in the whole subany intersection Ht0 \ Fi;q set of virtual processors for which it is de ned. Otherwise, the projection of any couple (Ht0?
p \ Fi?;q ; Ht0 \ Fi;q ) results in a diedron according to the following de nition. 0

0

0

0

0

0

+

0

0

0

0

+

De nition 23. (diedron) Let H and H 0 be two half-hyperplanes of0 Zd, bounded by the same ane span D of dimension d ? 2. The couple (H; H ) is called a 0

diedron of facets H and H . The ane span D is called the edge of the diedron.

All the virtual processors situated between the facets of the diedra at step t are processors which are active at t . 0

0

Proposition 2. (translation of the diedron) Any diedron (Ht0?
p \ Fi?;q ; Ht0 \ Fi;q ) can be deduced from the diedron of the previous time step (Ht0?
p? \ Fi?;q ; Ht0? \ Fi;q ) by a translation of the edge of the diedron along a vector d = D=
D where D is such that Ai
D = 0 and Ai
D = 0. Proof. The edge of any diedron (Ht0 ?
p \ Fi?;q ; Ht0 \ Fi;q ) de nes particular virtual processors whose rst computation occurs at step t and last computation occurs at step t ? 
p. Therefore, these virtual processors are characterized by a negative activity duration of ?
p steps. It comes that the edges of two 0

0

+

1

0

0

+

1

0

0

0

+

0

0

successive diedra are spaced by vectors allowing to generate the virtual processors characterized by the same activity duration. By reasoning at the level of the iteration space Pn, two processors of same activity duration are de ned by two allocation line-segments containing the same number of integral points. Therefore, these two line-segments de ne a parallelogram. So ?their extreme points belonging to Fi;q and their extreme points belonging to Fi ;q are spaced by the +

0

0

The Volume of a Lattice Polyhedron

13

:::

same vector D, i.e. Ai
D = 0 and Ai
D = 0. Moreover, in order to translate by a one time step, this vector is normalized. ut This linear multiprocessor schedule modeling allows the determination of several useful values by means of combinatorial computations on lattice polyhedra. In the next section, a direct application of Ehrhart's method is developped. 0

5. Computation of indicators Now these great results are used for the computation of { the processor count in the virtual grid allocated from any linear allocation, { the potential parallelism associated with any linear time schedule. In fact, any indicator resulting from lattice points enumeration in lattice polyhedra could have been considered. We just limit ourselves to these two items which seem to be representative of this kind of computations.

 Processor count Our geometrical model of a linear allocation presented above characterizes the rst (or last) computations of all the virtual processors as being points of rst (or last) facets and rst (or last) inside facets. Since it is obvious that any virtual processor has only one rst (or last) computation, the processor count is given by the number of integral points in all the rst (or last) facets and rst (or last) inside facets. Proposition 3. (processor count) Since jP j denotes the number of integral points in P, the processor count pn resulting from any linear allocation of Pn is given by,

j pn = jF j = j [fi Fi j = j [fi [jqA
pj? Fi;q i

=1

=1

1

=0

where f denotes the number of rst (or last) facets.

 is a (k ? 1)-dimensional polytope since Pn Any facet or inside facet Fi;q is a k-polytope. Moreover, it is a rational polytope : its vertices have rational  can be seen as the intersection between an hyperplane coordinates since any Fi;q and Pn. Hence, for any xed i, the set Fi is a k-dimensional rational polytope de ned by Fi = fz 2 Zd j A
z  Bn + C; Ai
z  bi n + ci  (jAi
pj ? 1)g In the general case f > 1, the union of all these rational polytopes is a nonconvex rational polyhedron and Ehrhart's results can apply. Such a polyhedron belongs to the set of normal polyhedra according to the de nition of Ehrhart in [12, 14, 15] and to the one given in section 2. The determination of the Ehrhart polynomial for such a rational polyhedron yields rst the computation of the rational coordinates of its vertices. Finding all the vertices of a convex polyhedron is a well-known problem. A survey and comparison of methods can be found in [21]. For example, Chernikova's algorithm [21] or more recently Avis and Fukuda's pivoting algorithm [2] can be used. But since our considered polyhedron F  is non-convex in a general way,

14

Ph. Clauss

the following heuristic procedure is needed to compute all the vertices of F  for n=1: { Compute all the vertices of each rational polytope Fi, for n = 1 and i 2 [1::f]. { For each found vertex v, if v is a vertex of a unique polytope Fi and does not belong to any other polytope Fj, then v is a vertex of F  . If v is a vertex of two distinct polytopes Fi and Fj, then v is a vertex of F . If v is a vertex of a unique polytope Fi and belongs to any other polytope Fj , then v is not a vertex of F . { Compute all the vertices of each intersection Fi \ Fj , for n = 1, i 6= j and i; j 2 [1::f]. { For each found vertex v, if v is not a vertex of any polytope Fi, then v is a  vertex of F . Example 5. A simple 3-dimensional example is shown on gure 1 : { v is a vertex of F  and does not belong to F . Hence, v is a vertex of F. { v is a vertex of F and is also a vertex of F . Hence, v is a vertex of F . { v is a vertex of F  and belongs to F without being one of its vertices. Hence, v is not a vertex of F  . { v is a vertex of F \ F  and is not a vertex, neither of F  nor of F . Hence, v is a vertex of F . ut 1

1

2

1

3

1

2

2

2

1

2

3

4

1

2

1

2

4

v3

v2

v1

F* 2

F*1 v4

Fig. 1. A 3D illustration for the vertex nding procedure of non-convex sets

F



The Ehrhart polynomial of F , giving the processor count in the grid allocated from any linear allocation, is then determined by using directly the results presented in section 3.

 Potential parallelism Through our geometrical model, the potential parallelism of any time step t , i.e. the number of simultaneous computations at t , is equal to the number of integral points in Pn;t0 = Ht0 \ Pn , i.e. pp(t ) = jPn;t0 j. It is de ned by 0

0

0

Pn;t0 = fz 2 Zd j A
z  Bn + C; 
z +  = t g 0

The Volume of a Lattice Polyhedron

15

:::

This set Pn;t0 is a (k ? 1)-dimensional rational polytope. Moreover, it is a parametric polytope with two parameters n and t . But since Ehrhart's method applies only with one parameter, we choose to express t as a polynomial in n of degree 1, i.e. t = xn + y, where x and y are rational. Pn;xn y = fz 2 Zd j A
z  Bn + C; 
z +  = xn + yg By computing the vertices of Pn;x y , the Ehrhart polynomial, giving the potential parallelism pp(t ) of any step t = xn + y, is directly determined. An opportune choice of the coecients x and y is made by computing the latency  of the parallel algorithm, i.e.  = MAXz2P 
z + . At least one vertex of Pn , say v, is such that t(v) = . The parametric coordinates of this vertex, depending on n, allows to get a polynomial expression of  in n of degree 1. Hence, we let the coecient of n of this expression be the numerator of x, and the constant term be the numerator of y. For example, with such coecients x and y, the potential parallelism occurring at mid-time can be expressed as a pseudo-polynomial in n. Moreover, such a parametric expression of t allows to nd the step where pp(t ) is maximum by enumeration of some initial values and by the knowledge of the following theorem found in [16, p. 87], since the potential parallelism associated to a given linear time-schedule, i.e. the maximum number of simultaneous computations, is de ned by pp = MAXt0 pp(t ). Theorem 10. (Brunn's theorem) The volume variation of parallel hyperplane 0

0

0

+

+

0

0

n

0

0

0

sections of a convex set occurs in at most three phases : increasing, constant, decreasing.

The whole method is now applied to the Gauss elimination algorithm.

6. Example : The Gaussian elimination algorithm This well-known algorithm can be expressed as the following nested loops : FOR k := 1 TO N ? 1 DO FOR i := k + 1 TO N DO FOR j := k + 1 TO N + 1 DO a[i; j; k] := a[i; j; k ? 1] ? a[i; k; k ? 1]  a[k; j; k ? 1]=a[k; k; k ? 1] ENDFOR ENDFOR ENDFOR The associated iteration space PN is de ned by 0 ?1 0 1 1 0 0 1 0 ?1 1 B 1 0 0 CC BB 1 CC BB 0 CC PN = fz 2 Z j B B@ 0 ?1 1 CA
z  B@ 0 CA N + B@ 0 CAg 0 1 0 1 1 0 0 ?1 0 ?1 Since P and P are empty, and P does not show all the vertices, this domain is signi cant from N = 3. Hence, in order to de ne the iteration space as an 3

0

1

2

16

Ph. Clauss

homothetic polyhedron Pn , we let N = n + 3 and de ne it as follows. It is represented on gure 2 where the parametric coordinates of the vertices depending on n are given. (n+3,n+2,n+2)

(n+3,n+4,n+2)

k i

j

(2,1,1)

(n+3,1,1)

(2,n+4,1)

(n+3,n+4,1)

Fig. 2. Gauss iteration space Pn = fz 2 Z j A
z  Bn + C g 3

0 ?1 0 B 1 0 = fz 2 Z j B B @ 00 ?11

0 0 1 0 ?1 1 11 0C B1C B 3C 1C CA
z  BB@ 0 CCA n + BB@ 0 CCAg 0 1 4 0 0 ?1 0 ?1 Let us consider the linear allocation de ned by the projection vector p = (1; ?1; 1) and the time schedule de ned by t(z) = (1; 1; 1)
z ? 3. Such a choice leads to the following parallel program obtained through the Fourier-Motzkin Elimination Method [3], where the rst coordinate denotes the calculation step and the second and third coordinates denote the processor where the item is computed. 3

FOR t := 1 TO 3N ? 3 DO FORALL (x = MAX(t ? N + 4; t ) : min(2N + 1; t + 2)) FORALL (y = MAX(t ? x ? N + 3; 2t ? 3x + 6) : min(t ? 2x + N + 4; ?1)) a[t + 3; x; y] := a[t + 2; x; y ? 1] ?a[3t ? 3x ? y + 8; 2t ? 2x ? y + 6; y ? 1]  a[t + y + 2; x + y; ?1] =a[3t ? 3x + 8; 2t ? 2x + 6; ?1] ENDFOR In order to implement such a program, many related informations can be useful. In particular, the number of processors (x; y) used by the program and the maximum parallelism, i.e. the maximum number of processors (x; y) simultaneously active at a given step t, play an important role. The normal vector to any facet Fi of Pn is given by any row of A. But in order to classify the facets in rst, last and null facets, the normal vectors have to be oriented to the inside of Pn. We get the following normal vectors : A = (1; 0; ?1) A = (?1; 0; 0) A = (0; 1; ?1) A = (0; ?1; 0) A = (0; 0; 1) The inner product Ai
p indicates that 2 +7 3

1

2

3

4

5

The Volume of a Lattice Polyhedron

17

:::

{ F is a null facet F . { F and F are last facets F ? and F ?. Moreover, F ? is associated to one 0 1

1

inside facet F ?; . Hence, we consider F ? , F ?; and F ?; represented on gure 3. { F and F are rst facets F and F represented on gure 3. 2

3

2

3

3 1

+

4

3

2

5

3 0

3 1

+

4

5

F+ 5

F+ 4

F -3,0

F -3,1

F -2

Fig. 3. First and last facets de ned by p

 Processor count In order to put our method to the test, we determine the processor count resulting from the projection along p by rst considering the rst facets, and then by considering the last facets.

 Considering the rst facets The set F [ F is a normal integral polyhedron, union of two homothetic +

+

2-polyhedra. Theorem 6 can be applied : P jn = q Cnq (j ? j )q + j = n(j ? j ) + n n? (j ? 2j + j ) + j 4

5

2

0

=1

(

1

0

0

1)

2

2

1

0

0

By counting the number of points in F [ F for n = 0, n = 1 and n = 2, we get j = 9, j = 18 and j = 30. Hence, the processor count is determined and given by 2 pN = N N ? pn = n n +

4

0

1

+

5

2

3(

+5 2

+6)

3

(

1)

2

 Considering the last facets In this case, an inside facet F ?; has to be considered. Hence, the set F ? = ? ; [ F ; is a rational 3-polytope de ned by

F?

3 0

3 1

3 1

3

18

Ph. Clauss

F ? = fz 2 Z j A 0
?z 1 Bn0+ C;1A1
z  b0n 0+ 1c ? (jA0
?p1j ?11)g BB 1 0 0 C BB 1 CC BB 3 CC C 0 ? 1 1 B C = fz 2 Z j B
z  B BB 01 CCC n + BBB 04 CCCg B@ 0 1 0 C C @ 0 A @ ?1 A 0 0 ?1 A 0 1 ?1 0 1 Its vertices for n = 1 have to be computed. We obtain (2; 1; 1) (4; 1; 1) (4; 3; 3) (2; 2; 1) (4; 2; 1) (4; 4; 3) 3

3

3

3

3

3

3

All these vertices have integral coordinates and hence the set F ? [ F ?; [ F ?; is an integral 3-polyhedron. Theorem 6 can also be applied : P jn = q Cnq (j ? j )q + j = n(j ? j ) + n n? (j ? 2j + j ) + n n? n? (j ? 3j + 3j ? j ) + j 2

3 0

3 1

3

0

=1

0

(

1

0

(

1)

2

2

1

0

1)(

2)

3

6

2

1

0

By counting the number of points in F ? [ F ?; [ F ?; for n = 0, n = 1, n = 2 and n = 3, we get j = 9, j = 18, j = 30 and j = 45. Hence, we successfully get the same result as by considering the rst facets. 2

0

1

2

3 0

3 1

3

 Potential parallelism The latency of the parallel algorithm is given by  = MAXz2P (1; 1; 1)
z ? 3 = 3n+6. The potential parallelism occurring at mid-time is de ned by Pn; 3 2+6 if n mod 2 = 0 and by Pn; 3 2+7 if n mod 2 = 1. n

n

n

 n mod 2 = 0

The set Pn; 3 2+6 is de ned by n

Pn; 3 2+6 = fz 2 Z j A
z  Bn + C; (1; 1; 1)
z ? 3 = 3n 2+ 6 g 3

n

This set is signi cant only if n mod 2 = 0. Hence, let n = 2m and de ne Pn; 3 2+6 as being, Pm; m = fz 2 Z j A
z  2m  B + C; (1; 1; 1)
z ? 3 = 3m + 3g This set is a rational 2-polytope. The vertices of P ; are computed and represented on gure 4. (2; 6; 1) (5; 3; 1) (5; 2; 2) ( ; ; ) The denominator of P ; is equal to the denominator of the rational vertex, i.e. 3. Hence, the Ehrhart polynomial is of period 3 and of degree 2. By applying theorem 8, we observe that there is one ane span of a 0-dimensional face, i.e. a vertex, which is not an integral point. But for any q > 0, the ane span of any q-dimensional face contains integral points. Hence, the coecients of m and m are constant. It comes that the ve coecients c ; c ; c ; c ; c of the n

3

+3

3

1 6

11

8

8

3

3

3

1 6

2

1

1

2

3

4

5

0

The Volume of a Lattice Polyhedron

19

:::

(5,4,4)

(5,6,4)

(11/3,8/3,8/3)

(5,2,2) (2,1,1)

(2,6,1)

(5,1,1)

(5,6,1) (5,3,1)

Fig. 4.

P1;6

and its vertices

pseudo-polynomial c m + c m + [c ; c ; c ] have to be determined by resolving the following linear system of 5 equations : 8 c =j > > < c +c +c =j 4c + 2c + c = j > 9c > : 16c ++3c4c ++c c ==j j By counting the number of points in P ; , P ; , P ; , P ; and P ; , we get j = 2, j = 6, j = 12, j = 20 and j = 30. Hence, the resolution of the above system gives the following results : c =1 c =3 c =2 c =2 c =2 Hence, the potential parallelism occurring at mid-time when n mod 2 = 0 is a polynomial. pp(3m + 3) = m + 3m + [2; 2; 2] = m + 3m + 2 2

1

2

3

4

5

0

1

2

5

3

1

1

2

4

2

1

2

5

3

1

2

3

4

0 3

0

1

2

3

1 6

2 9

3 12

4 15

4

1

2

3

4

5

2

2

pp( n ) = n + n + 2 3

+6

1

2

2

4

3

2

pp( N ? ) = N ? 3

3

2

1

4

2

1 4

 n mod 2 = 1 The set Pn; 3 2+7 is de ned by n

Pn; 3 2+7 = fz 2 Z j A
z  Bn + C; (1; 1; 1)
z ? 3 = 3n 2+ 7 g 3

n

This set is signi cant only if n mod 2 = 1. Hence, let n = 2m + 1 and de ne Pn; 3 2+7 as being, n

Pm; m = fz 2 Z j A
z  (2m + 1)  B + C; (1; 1; 1)
z ? 3 = 3m + 5g 3

+5

3

20

Ph. Clauss

This set is a rational 2-polytope. The vertices of P ; are computed. ( ; 7; ) (6; 4; 1) (6; ; ) ( ; ; ) The denominator of P ; is equal to the lowest common multiple of the denominators of the rational vertices, i.e. 6. Hence, the Ehrhart polynomial is of period 6 and of degree 2. By applying theorem 8, we observe that there is at least one ane span of a 1-dimensional face, i.e. the edge containing vertices ( ; 7; ) and ( ; ; ), which contains no integral point. Hence, only the coecient of m is constant. It comes that the thirteen coecients of the pseudo-polynomial c m + [c ; c ; c ; c ; c ; c ]m + [c ; c ; c ; c ; c ; c ] have to be determined by resolving the following linear system of 13 equations : 8 c =j > > c +c +c = j > > 4c + 2c + c = j > > 9c + 3c + c = j > > 16c + 4c + c = j > < 25c + 5c + c = j 36c + 6c + c = j > 49c + 7c + c = j > > 64c + 8c + c = j > > 81c + 9c + c = j > > 100c + 10c + c = j > > + 11c + c = j : 121c 144c + 12c + c = j By counting the number of points in P ; , P ; , P ; , P ; , P ; , P ; , P ; , P ; , P ; , P ; , P ; , P ; and P ; , we get j = 4, j = 9, j = 16, j = 25, j = 36, j = 49, j = 64, j = 81, j = 100, j = 121, j = 144, j = 169 and j = 196. Hence, the resolution of the above system gives the following results : c =1 c =c =c =c =c =c =c =c =c =c =c =c =4 Hence, the potential parallelism occurring at mid-time when n mod 2 = 1 is a polynomial. pp(3m + 5) = m + [4; 4; 4; 4;4; 4]m+ [4; 4; 4; 4;4; 4] = (m + 2) 1 8

5

3

5

5

13

10

10

2

2

2

2

3

3

3

1 8

2

1

2

13

10

10

3

3

3

2

3

4

5

6

7

8

9

13

3

8 29

9 32

4

10 35

2

8

12

13

1

1

3

9

1

4

10

2

3

1

5

11

4

1

6

12

5

1

7

13

1

2

8

7

1

3

9

8

1

4

10

6

9

1

5

11

10

1

6

12

11

1

7

13

12

0 5

1 8

2 11

11 38

5

11

11

3

2

0

1

7 26

10

5

2

12 41

6

3 14

0

7

8

4 17

5 20

1

6 23

2

9

10

12

1

2

3

4

5

6

7

8

9

10

11

12

13

2

2

? pp( n ) = n


?
pp( N ? ) = N 3

+7

+3

2

3

2

2

2

2

2

2

The above results can be summerized by the following pseudo-polynomial since the potential parallelism at mid-time consists in considering step d  e for any value of n : pp(d 2 e) = 41 n + 23 n + [ 94 ; 2] = 14 N + [? 41 ; 0] In order to determine the potential parallelism pp of the time-schedule by using Brunn's theorem, we determine in the same way some other polynomials : 2

2

2

The Volume of a Lattice Polyhedron

21

:::

 n mod 2 = 0

pp(3m + 2) = m + 3m + 1 pp(3m + 3) = m + 3m + 2 pp(3m + 4) = m + 3m + 2 pp(3m + 5) = m + 3m + 2 pp(3m + 6) = m + 3m + 1 These results show clearly that when n mod 2 = 0, the potential parallelism increases until time-step 3m+3, keeps constant from time-step 3m+3 to time-step 3m + 5, and then decreases. Hence, according to Brunn's theorem, the potential parallelism of time-step 3m + 3 previously determined is maximum and is equal to the potential parallelism pp of the time-schedule. 2

2

2

2

2

 n mod 2 = 1

pp(3m + 4) = m + 4m + 3 pp(3m + 5) = m + 4m + 4 pp(3m + 6) = m + 4m + 4 pp(3m + 7) = m + 4m + 3 These results show clearly that when n mod 2 = 1, the potential parallelism increases until time-step 3m + 5, keeps constant from time-step 3m + 5 to timestep 3m + 6, and then decreases. Hence, according to Brunn's theorem, the potential parallelism of time-step 3m + 5 previously determined is maximum and is equal to the potential parallelism pp of the time-schedule. pp = pp(d 2 e) = 41 n + 32 n + [ 94 ; 2] = 41 N + [? 41 ; 0] 2

2

2

2

2

2

7. Conclusion It has been shown that Ehrhart's powerful results allow to resolve eciently the problem of counting the number of integral points in integral and rational homothetic polyhedra, in their linear projections, and in their intersections with cutting hyperplanes. In our context of automatic parallelization of programs, these results yields an easy determination of the number of processors resulting from linear allocations and of the potential parallelism resulting from ane time-schedules. These two quantities are representative examples for such kind of computations. But the same method can be used to compute, for example, the number of calculations (or processors) using a same data. But it is not generalized at this time to parametric polyhedra of any number of size parameters. Hence, this dicult problem needs some further developments by introducing multi-dimensional homothetic systems and multi-dimensional periodic numbers. Our own method previously presented in [7, 9, 8], also applied on the Gaussian elimination, can apply for any number of size parameters, but needs the parametric coordinates of the vertices and is limited to integral polyhedra. Hence, it yields more complex computations. Acknowledgement. I am grateful to Eugene Ehrhart for having accepted to meet me and to answer my rudimentary questions.

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Ph. Clauss

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