As a ﬁrst attempt to exploit symmetries in continuous con- straint problems, we focus on permutations of the variables consisting of one single cycle. We propose a procedure that takes advantage of these symmetries by interacting with a Branch-and-Prune algorithm without interfering with it. A key concept in this procedure are the classes of symmetric boxes formed by bisecting a n-dimensional cube at the same point in all dimensions at the same time. We quantify these classes as a function of n. Moreover, we propose a simple algorithm to generate the representatives of all these classes for any number of variables at very high rates. A problem example from the chemical ﬁeld and a kinematics solver are used to show the performance of the approach in practice.Peer Reviewe

Ruiz de Angulo García, Vicente

Torras, Carme

English

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Exploiting single-cycle symmetriesin Branch-and-Prune algorithmsVicente Ruiz de Angulo and Carme TorrasInstitut de Robo`tica i Informa`tica Industrial (CSIC-UPC)Llorens i Artigas 4-6, 08028-Barcelona, Spain.{ruiz, torras}@iri.upc.eduAbstract. As a first attempt to exploit symmetries in continuous con-straint problems, we focus on permutations of the variables consisting ofone single cycle. We propose a procedure that takes advantage of thesesymmetries by interacting with a Branch-and-Prune algorithm withoutinterfering with it. A key concept in this procedure are the classes ofsymmetric boxes formed by bisecting a n-dimensional cube at the samepoint in all dimensions at the same time. We quantify these classes as afunction of n. Moreover, we propose a simple algorithm to generate therepresentatives of all these classes for any number of variables at veryhigh rates. A problem example from the chemical field and a kinematicssolver are used to show the performance of the approach in practice.1 Symmetry in Continuous Constraints ProblemsSymmetry exploitation in discrete constraint problems has received a great dealof attention lately [6, 4, 5, 11]. On the contrary, symmetries have been largelydisregarded in continuous constraint solving, despite the important growth inboth theory and applications that this field has recently experienced [12, 1, 9].Continuous (or numerical) constraint solving is often tackled using Branch-and-Prune algorithms [13], which iteratively locate solutions inside an initial do-main box, by alternating box subdivision (branching) and box reduction (prun-ing) steps. Motivated by a molecular conformation problem, in this paper wedeal with the most simple type of box symmetry, namely that in which domainvariables (i.e., box dimensions) undergo a single-cycle permutation leaving theconstraints invariant. This can be seen, thus, as a form of constraint symmetryin the terminology introduced in [3].We are interested in solving the following general Continuous ConstraintSatisfaction Problem (CCSP): Find all points x = (x1, . . . , xn) lying in an initialbox of Rn satisfying the constraints f1(x) ∈ C1 , . . . , fm(x) ∈ Cm, where fi is afunction fi : Rn → R, and Ci is an interval in R.We assume the problem is tackled using a Branch-and-Prune (B&P) algo-rithm. The only particular feature that we require of this algorithm is that ithas to work with boxes in Rn.We say that a function s : Rn → Rn is a point symmetry of the problemif there exists an associated permutation σ ∈ Σm such that fi(x) = fσ(i)(s(x))IIand Ci = Cσ(i), ∀i = 1, . . . ,m. We consider symmetry as a property that relatespoints that are equivalent as regards to a CCSP. Concretely, from the abovedefinition one can conclude that x is a solution to the problem iff s(x) is asolution to the problem. Let s and t be two symmetries of a CCSP with associatedpermutations σs and σt. It is easy to see that the composition of symmetriess(t(·)) is also a symmetry with associated permutation σs(σt(·)).An interesting type of symmetries are permutations of the components of x.Let D be a finite set. A cycle of length k is a permutation ψ such that there existdistinct elements a1, . . . ak ∈ D such that ψ(ai) = ψ(a(i+1)mod k) and ψ(z) = zfor any other element z ∈ D. Such a cycle is represented as (a1, . . . ak). Everypermutation can be expressed as a composition of disjoint cycles. In this paper wefocus on a particular type of permutations, namely those constituted by a singlecycle. In its simplest form, this is s(x1, x2, . . . xn) = (xθ(1), xθ(2), . . . xθ(n)) =(x2, x3...xn, x1), where θ(i) = (i+ 1) mod n.Example: n = 3,m = 4,x = (x1, x2, x3) ∈ [−1, 1]× [−1, 1]× [−1, 1],f1(x) : x21 + x22 + x23 ∈ [5, 5] ≡ x21 + x22 + x23 = 5f2(x) : 2x1 − x2 ∈ [0,∞] ≡ 2x1 − x2 > 0f3(x) : 2x2 − x3 ∈ [0,∞] ≡ 2x2 − x3 > 0f4(x) : 2x3 − x1 ∈ [0,∞] ≡ 2x3 − x1 > 0There exists a symmetry s(x1, x2, x3) = (x2, x3, x1). The constraint permu-tation associated to s is σ(1) = 1, σ(2) = 3, σ(3) = 4, and σ(4) = 2.For n > 3 there is never a unique symmetry for a given problem. If thereexists a symmetry s, then for example s2(x) = s(s(x)) is another symmetry. Ingeneral, using the convention of denoting s0(x) the identity mapping, {si(x), i =0 . . . n− 1} is the set of different symmetries that can be obtained by composings(x) with itself, while for i > n we have that si(x) = si mod n(x). Thus, asingle-cycle symmetry generates by composition n−1 symmetries, excluding thetrivial identity mapping. Some of them may have different numbers of cycles.The algorithm presented in this paper deals with all the compositions of a single-cycle symmetry, even if some of them are not single-cycle symmetries. The gainobtained with the proposed algorithm will be shown to be generally proportionalto the number of different compositions of the selected symmetry. Therefore,when several single-cycle symmetries exist in a CCSP problem, the algorithmshould be used with that generating the most symmetries by composition, i.e.,with that having the longest cycle.2 Box symmetrySince B&P algorithms work with boxes, we turn our attention now to the set ofpoints symmetric to those belonging to a box B ⊆ Rn.Let s be a single-cycle symmetry corresponding to the circular variable shift-ing θ introduced in the preceding section, and B = [x1, x1] × . . . × [xn, xn] abox in Rn. The box symmetry function S is defined as S(B) = {s(x) s.t. x ∈IIIB} = [xθ(1), xθ(1)] × . . . × [xθ(n), xθ(n)] = [x2, x2] × . . . × [xn, xn] × [x1, x1]. Thebox symmetry function has also an associated constraint permutation σ, whichis the same associated to s. Si will denote S composed i times. We say, then,that Si(B) and Sj(B) are symmetric boxes, 0 6 i, j < n, i 6= j.As in the case of point symmetry, box symmetry has implications for theCCSP. If there is no solution inside a box B, there is no solution inside any ofits symmetric boxes either. A box B∫ ⊆ B is a solution iff Si(B∫ ) ⊆ Si(B) is asolution box for any i ∈ {1 . . . n− 1}.Box symmetry is an equivalence relation defining symmetry equivalence classes.Let R(B) be the set of different boxes in the symmetry class of B, R(B) ={Si(B), i ∈ {0, . . . , n − 1}}. We define the period P (B) of a box B as P (B) =|R(B)|. Therefore R(B) = {Si(B), i ∈ {0, . . . , P (B) − 1}}. For example, for boxB′ = [0, 4]× [2, 5]× [2, 5]× [0, 4]× [2, 5]× [2, 5], P (B′) = 3.In the following sections we will show that the implications of box symmetryfor a CCSP can be exploited to save much computing time in a meta-algorithmthat uses the B&P algorithm as a tool without interfering with it.3 Algorithm to exploit box symmetry classesThe algorithm we will propose to exploit box symmetry makes much use of thesymmetry classes formed by bisecting a n-dimensional cube In (i.e., of period1) in all dimensions at the same time and at the same point, resulting in 2nboxes. We will denote L and H the two subintervals into which the originalrange I is divided. For example, for n = 2, we have the following set of boxes{L × L,L × H,H × L,H × H} whose periods are 1, 2, 2 and 1, respectively.And their symmetry classes are: {L × L}, {L × H,H × L}, and {H × H}.Representing the two intervals L and H as 0 and 1, respectively, and droppingthe × symbol, the sub-boxes can be coded as binary numbers. Let SRn be the setof representatives, formed by choosing the smallest box in binary order from eachclass. For example, SR2 = {00, 01, 11}. Note that the cube In to be partitionedcan be thought of as the the set of binary numbers of length n, and that SRn isnothing more than a subset whose elements are different under circular shift.Section 4 will show how many components SRn has, how they are distributedand, more importantly, how can they be generated. The symmetry exploitationalgorithm we propose below uses the B&P algorithm as an external routine.Thus, the internals of the B&P algorithm do not need to be modified.The idea is to first divide the initial box into a number of symmetry classes.Next, one needs to process only a representative of each class with the B&Palgorithm. At the end, by applying box symmetries to the solution boxes ob-tained in this way, one would get all the solutions lying in the space coveredby the whole classes, i.e., the initial box. The advantage of this procedure isthat the B&P algorithm would have to process only a fraction of the initial box.Assuming that the initial box is a n-cube covering the same interval [xl, xh] inall dimensions, we can directly apply the classes associated to SRn. A procedureto exploit single-cycle symmetries in this way is presented in Algorithm 1.IVAlgorithm 1: CSym algorithm.Input: A n-cube, [xl, xh]× · · · × [xl, xh]; S: a single-cycle box symmetry;B&P : a B&P algorithm.Output: A set of boxes covering all solutions.SolutionBoxSet← EmptySet1x∗ ← SelectBisectionPoint(xl, xh)2foreach b ∈ SRn do3B ← GenerateSubBox(b, xl, xh, x∗)4SolutionBoxSet← SolutionBoxSet ∪ ProcessRepresentative(B)5return SolutionBoxSet6Algorithm 2: The ProcessRepresentative function.Input: A box B; S: a single-cycle box symmetry; B&P : a B&P algorithm.Output: The set of solution boxes contained in B and its symmetric boxes.SolSet← B&P (B)1SymSolSet← SolSet2for i=1: P (B)− 1 do3SymSolSet← SymSolSet ∪ApplySymmetry(SolSet, Si)4return SymSolSet5The operator GenerateSubBox(b, xl, xh, x∗) returns the box correspond-ing to binary code b when [xl, xh] is the range of the initial box in all dimensionsand x∗ is the point in which this interval is bisected. The iterations over line 4of Algorithm 1 generate a set of representative boxes such that, together withtheir symmetries, cover the initial n-cube.ProcessRepresentative(B) returns all the solution boxes associated to B,that is, the solutions inside B and inside its symmetric boxes. Since the numberof symmetries of B is P (B), the benefits of exploiting the symmetries of a classrepresentative is proportional to its period.4 An illustrative exampleMolecules can be modeled as mechanical chains by making some reasonable ap-proximations, such as constant bond length and constant angle between consec-utive bonds. Finding all valid conformations of a molecule can be formulated asa distance-geometry [2] problem in which some distances between points (atoms)are fixed and known, and one must find the set of values of unknown (variable)distances by solving a set of constraints consisting of equalities or inequalities ofdeterminants formed with subsets of the fixed and variable distances [2].The problem can be solved using a B&P algorithm [9, 8]. Figure 1(a) displaysthe known and unknown distances of the cycloheptane, a molecule composed ofa ring of seven carbon atoms. The distance between two consecutive atoms ofVd1d 6d 3d4d5d7d2(a) (b)Fig. 1. (a) Cycloheptane. Disks represent carbon atoms. Constant and variable dis-tances between atoms are represented with continuous and dashed lines, respectively.(b) Three-dimensional projection of the cycloheptane solutions. The lightest (yellow)boxes are the solutions found inside the representatives using the B&P algorithm (line1 in Algorithm 2). The other colored boxes are the solutions obtained by applyingsymmetries to the yellow boxes (line 4 in Algorithm 2).the ring is constant and equal everywhere. The distance between two atomsconnected to a same atom is also known and constant no matter the atoms. Theproblem has several symmetries. One is s(d1, . . . , d7) = (dθ(1), dθ(2), . . . , dθ(7)) =(d2, d3 . . . , d7, d1). When this symmetry is exploited with the CSym algorithmthe problem is solved in 4.64 minutes, which compares very favorably with the31.6 minutes spent when using the algorithm in [8] alone. Thus, a reduction bya factor close to n = 7 (i.e., the length of the symmetry cycle) in computingtime is obtained, which suggests that the handling of box symmetries doesn’tintroduce a significant overhead. Figure 1(b) shows a projection into d1 d2 andd3 of the solutions obtained using CSym.5 Counting and generating box symmetry classesLet us define some quantities of interest:-Nn: Number of elements of SRn.-FPn: Number of elements of SRn that correspond to full-period boxes , i.e.,boxes of period n.-N pn : Number of elements of SRn having period p.Polya’s theorem [7] could be used to determine some of these quantitiesfor a given n by building a possibly huge polynomial and elucidating some ofits coefficients. We present a simpler way of calculating them by means of theformulae below. A demonstration of their validity as well as expressions forsimilar quantities distinguishing the number of 1’s can be found in [10]. Webegin with FPn:FPn = 2nn−∑p∈div(n), p<npnFPp. (1)VI2 4 6 8 10 12box dimensionality (i.e., number of variables)050100150200250300350number of classes of symmetric boxes(a)2 4 6 8 10 12 14 16 18 20box dimensionality (i.e., number of variables)020406080100percentage of full-period classes(b)Fig. 2. (a) Number of elements of SRn as a function of n. (b) Percentage of full-periodelements in SRn as a function of n.This recurrence has a simple baseline condition: FP1 = 2. Nn is calculatedwithNn = 2nn+∑p∈div(n), p<nn− pnFPp. (2)This formula is valid for n > 1. The remaining case is N1 = 2. Finally,N pn ={0 if p /∈ div(n)FPp otherwise(3)Figure 2(a) displays the number of classes (Nn) as a function of n. The curveindicates an exponential-like behavior. Figure 2(b) shows the percentage of full-period classes in SRn (100 Nnn /Nn). Note that the percentage of classes withperiod different from n is significant for low n, but approaches quickly 0 as ngrows.The naive procedure to generate SRn involves a huge number of operations.Here we suggest an algorithm capable of calculating SRn on the fly. We usea compact coding of the binary numbers representing the boxes consisting ofchains of numbers. The first number in the code is the number of 0’s appearingbefore the first 1. The i-th number in the code for i > 1 is the number of 0’sbetween the (i− 1)-th and the i-th 1’s. For example, the number 0100010111 iscodified as 13100. The length of this numerical codification is the number of 1’sof the codified binary number, which has been denoted by m.Algorithm 3, explained in [10], generates SRnnm, the subset of the elementsof SRn having m 1’s and period n, from which the whole SRn can be obtained,as showed also in [10]. The algorithm outputs a list of codes in decreasing nu-merical order. For instance, the output obtained when requesting SR993 withClassGen(6, 1, 1, 3, A) is: {600, 510, 501, 420, 411, 402, 330, 321, 312}.6 ConclusionsWe have approached the problem of exploiting symmetries in continuous con-straint satisfaction problems using B&P algorithms. Our approach is generalVIIAlgorithm 3: ClassGen algorithm.Input: sum is the sum of the digits that remain to be written on the right(from position pos to m); pos: the index of the next position to bewritten; ctrol: the index of the current control element, whose valuecannot be surpassed in the next position; m: the length of the code;A: array where class codes are being generated.Output: A set of codes representing classes, SR.SR← EmptySet1if pos = m then2if sum < A[ctrol] then /* otherwise, SR will remain EmptySet */3A[m]← sum4SR← {A};5else6if pos 6= 1 then7LowerLimit = 08UpperLimit←Minimum(A[ctrol], sum)9else10LowerLimit = dsum/me11UpperLimit← sum12for i = UpperLimit to LowerLimit do13A[pos]← i14if i = A[ctrol] then /* i = A[ctrol] = UpperLimit */15SR← SRSClassGen(sum− i, pos+ 1, ctrol + 1,m,A)16else SR← SRSClassGen(sum− i, pos+1, 1,m,A); /* i < A[ctrol] */17return SR18and could be used also with other box-oriented algorithms, such as Branch-and-Bound for nonlinear optimization. The particular symmetries we have tackledare single-cycle permutations of the problem variables.The suggested strategy is to bisect the domain, the initial n-cube, simulta-neously in all dimensions at the same point. This forms a set of boxes that canbe grouped in box symmetry classes. A representative of each class is selectedto be processed by the B&P algorithm and all the symmetries of the representa-tive are applied to the resulting solutions. In this way, the solutions within thewhole initial domain are found, while having processed only a fraction of it –theset of representatives– with the B&P algorithm. The time savings tend to beproportional to the number of symmetric boxes of the representative. Therefore,symmetry exploitation is complete for full-period representatives.We have also developed a method for generating the classes resulting frombisecting a n-cube. The numerical analysis of the classes revealed that the aver-age number of symmetries of the class representatives tends quickly to n as thenumber of variables, n, grows. These are good news, since n is the maximumnumber of symmetries attainable with single-cycle symmetries of n variables.VIIIHowever, for small n there is still a significant fraction of the representatives nothaving the maximum number of symmetries. Another weakness of the proposedstrategy is the exponential growth in the number of classes as a function of n.The problems with small and large n should be tackled with a more refinedsubdivision of the initial domain in box symmetry classes, which is left for nearfuture work. We are also currently approaching the extension of this work todeal with permutations of the problem variables composed of several cycles.AcknowledgmentsThis work has been partially supported by the Spanish Ministry of Educationand Science through the contract DPI2004-07358 and by the “Comunitat deTreball dels Pirineus” under contract 2006ITT-10004.References1. Benhamou F., Goulard F.: Universally Quantified Interval Constraints. Proc. 6thCP (2000) 67-822. Blumenthal, L.: Theory and aplications of distance geometry. Oxford UniversityPress, 1953.3. Cohen D., Jeavons P., Jefferson Ch., Petrie K.E., Smith B.M.: Symmetry Definitionsfor Constraint Satisfaction Problems. Constraints 11(2-3) (2006) 115-1374. Flener P., Frisch A.M., Hnich B., Kiziltan Z., Miguel I., Pearson J., Walsh, T.:Breaking Row and Column Symmetries in Matrix Models. Proc. 8th CP (2002)462-4765. Gent I.P., Harvey W., Kelsey T.: Groups and Constraints: Symmetry BreakingDuring Search. Proc. 8th CP (2002) 415-4306. Meseguer P., Torras C.: Exploiting symmetries within constraint satisfaction search.Artificial Intelligence 129 (2001) 133-1637. Polya, G. and Read, R.C.: Combinatorial enumeration of groups, graphs and chem-ical compounds. Springer-Verlag, New York, 1987.8. Porta, J.M., Ros Ll., Thomas F., Corcho F., Canto J. and Perez J.J.: Complete mapsof molecular loop conformational spaces. Journal of Computational Chemistry (toappear)9. Porta J.M., Ros Ll., Thomas F., Torras C.: A Branch-and Prune Solver for DistanceConstraints. IEEE Trans. on Robotics 21(2) (2005) 176-18710. Ruiz de Angulo V., Torras C.: Exploiting single-cycle symmetries incontinuous constraint satisfaction problems. IRI DT 2007/02, June 2007,http://www.iri.upc.edu/people/ruiz/articulos/symmetriesreport1.pdf.11. Puget J-F.: Symmetry Breaking Revisited. Constraints 10(1) (2005) 23-4612. Sam-Haroud D., Faltings B.: Consistency Techniques for Continuous Constraints.Constraints 1(1-2) (1996) 85-11813. Vu X-H., Silaghi M., Sam-Haroud D., Faltings B.: Branch-and-Prune Search Strate-gies for Numerical Constraint Solving. Swiss Federal Institute of Technology (EPFL)LIA-REPORT 7 (2006)

Exploiting single-cycle symmetries in branch-and-prune algorithms

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Exploiting single-cycle symmetries in branch-and-prune algorithms

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