Computing hypervolume contributions in low dimensions: asymptotically optimal algorithm and complexity results

Transcription

1 EMO 2011, 5-8 April, Ouro Preto, Brazil Computing hypervolume contributions in low dimensions: asymptotically optimal algorithm and complexity results Michael T.M. Emmerich 1,2,3 and Carlos M. Fonseca 1,3,4 1 Faculty of Science and Technology, Universidade do Algarve, Campus de Gambelas, Faro, Portugal 2 Algorithms and Natural Computing Group, Leiden University, NL 3 CEG-IST, Instituto Superior Técnico, Technical University of Lisbon Av. Rovisco Pais 1, Lisboa, Portugal 4 Department of Informatics Engineering, University of Coimbra, Pólo II, Coimbra, Portugal (cmfonsecdei.uc.pt, emmerichliacs.nl)

2 EMO 2011, 5-8 April, Ouro Preto, Brazil Computing hypervolume contributions in low dimensions: asymptotically optimal algorithm and complexity results Michael T.M. Emmerich 1,2,3 and Carlos M. Fonseca 1,3,4 1 Faculty of Science and Technology, Universidade do Algarve, Campus de Gambelas, Faro, Portugal 2 Algorithms and Natural Computing Group, Leiden University, NL 3 CEG-IST, Instituto Superior Técnico, Technical University of Lisbon Av. Rovisco Pais 1, Lisboa, Portugal 4 Department of Informatics Engineering, University of Coimbra, Pólo II, Coimbra, Portugal (cmfonsecdei.uc.pt, emmerichliacs.nl)

3 Synopsis Hypervolume contributions are often computed for fitness assignment in EMOA, e.g. SMS-EMOA, MOO-CMA, or IBEA. This work presents: new results on the (time) complexity of computing all hypervolume contributions. It is proved that: for d = 2, 3 the problem has time complexity Θ(n log n). for d > 3, the time complexity is bounded below by Ω(n log n). a new dimensions sweep algorithm with asymptotically optimal time complexity O(n log n) for three dimensions.

4 Table of Contents Introduction Complexity results Dimensions Sweep Algorithm Speed-Test Summary/Outlook

5 Hypervolume Indicator Definition (Hypervolume (indicator)) Given a finite set Y of mutually non-dominated vectors in R d, the hypervolume indicator measures the volume (Lebesgue measure) of the subspace simultaneously dominated by Y and bounded by a reference point. hyp(y) = Vol [y r, y]. y Y where Vol() denotes the Lebesgue measure in d dimensions.

6 In 2-D and 3-D the time complexity for computing the hypervolume indicator is Θ(n log n) [Fonseca et al., 2006],[Beume et al., 2009]

7 The hypervolume indicator (or S-metric, Lebesgue measure) was introduced by Zitzler and Thiele [Zitzler & Thiele, 1998] to measure the quality of Pareto front approximations. The hypervolume is a unary set-indicator for performance assessment. it requires no a-priori knowledge of Pareto front. the hypervolume indicator is often used in guiding selection in indicator-based metaheuristics [Emmerich et al., 2005, Fleischer, 2003, Huband et al., 2003, Igel et al., 2007, Mostaghim et al., 2007, Zitzler & Künzli, 2004,?] and archivers [Knowles et al., 2003]. in this context, the problem of computing hypervolume contributions and/or the minimal hypervolume contributor of a set of points often arises.

8 Definition (hypervolume contribution) hyp(y, Y) = hyp(y) hyp(y \ {y}) In two dimensions: for d = 2 a hypervolume contribution depends on 3 points all contributions can be obtained by sorting of Y in O(n log n)

9 Hypervolume contribution in 3-D 3-D contributions can depend on up to n points. their computation requires at least Ω(n).

10 Problem definitions Let A d denote all d-dimensional sets of mutually non-dominated points. Problem (AllContributions) Given Y A d as an input, compute the hypervolume contributions of all points y Y. Problem (OneContribution) Given Y A d and y Y as an input, compute the hypervolume contribution S(y, Y). Problem (MinimalContribution) Given Y A d as an input, compute the minimal hypervolume contribution, i.e., min y Y S(y, Y). Problem (MinimalContributor) Given Y A d as an input, find a point with minimal hypervolume contribution, i.e., y arg min y Y S(y, Y).

11 Lower bound for AllContributions Problem (AllContributions) Given Y A d as an input, compute the hypervolume contributions of all points y Y. Theorem Any algorithm that solves AllContributions in d > 1 dimensions requires Ω(n log n) time in the algebraic decision tree model of computation. Proof. Reduction to uniform gap problem.

12 Reduction proof, uniform gap Problem (UniformGap) The problem of deciding whether a set of n points on the real line is equidistant is called UniformGap. Lemma The complexity of UniformGap is Ω(n log n) in the algebraic decision tree model of computation. ([Preparata & Shamos, 1985], p. 260) Reduction proof: Show that, if AllContributions could be computed in less than O(n log n) time, then also UniformGap could be decided in less than O(n log n) time. This would contradict the lemma.

13 Reduction of UniformGap to AllContributions Reduction of uniform gap: Step 1 Augment uniform gap input coordinates by equal y-coordinates (see figure left). Step 2 Solve AllContributions for augmented problem. Step 3 If contributions all equal to ((x max x min )/(n 1)) 2, uniform gap is positive, otherwise not. Step 1 and 3 would require only linear time. Lemma: For odd number of input points on the diagonal, hyp(y, Y ) c = ((x max x min )/(n 1)) 2 is obtained for all points y Y, iff gaps are uniform.

14 Even vs. odd number of points ( ) 2(xmax x min ) maximize c(δ) = δ δ δ = 2(x max x min ) = κ n 1 n 1 For even number of points, there is no such unique optimum: Discard maximum from set X, and solve problem O(n). Check distance of maximum to 2nd smallest solution O(n).

15 Complexity results OneContribution Problem (OneContribution) Given Y A d and y Y as an input, compute the hypervolume contribution hyp(y, Y). Theorem The time complexity of OneContribution is Θ(n) for d = 2, Θ(n log n) for d = 3, and for d > 3 it is bounded below by Ω(n log n). Proof. For d = 2 proof is elementary. d = 3: first upper bound then lower bound.

16 Lower bound Ω(n log n) for d = 3: The problem of computing the hypervolume of a set in two dimensions Hypervolume2d can be reduced in linear time to OneContribution in three dimensions (see picture). The complexity of Hypervolume2D is bounded below by Ω(n log n) [Beume et al., 2009]. Upper bound O(n log n) for d = 3: Compute hyp(y) hyp(y \ {y}); hyp requires O(n log n) [Fonseca et al., 2006].

17 Fast dimensions sweep algorithm for d = 3 General idea of algorithm: Sweep in f 3 -direction; update hypervolume slice by slice. Yielded asymptotically optimal algorithm for 3-D hypervolume computation.

18 Single layer, update step There are three events that trigger box creations/completions: a) all boxes owned by point s dominated in xy-plane by p[i] are discarded/completed; per dominated point a new box is created b) all boxes owned by p[t] that are partly dominated by p[i] are discarded, a single new box is added to the right position of p[t] s boxlist c) All boxes owned by p[r] that are partly dominated by p[i] are discarded, a single box is added to the left of p[r] s boxlist

19 Computational cost analysis each box is created once and discarded once; z-levels are only updated at create and discard time; finding p[r], p[t] requires O(log(n)) (AVL tree); each such box appears either at tail or head of box list processing time is constant per box; the number of boxes is given by: a) n + n boxes are created/discarded; each box is caused by one, unique, point, excepting the rightmost box (one such box per step). b, c) step adds at most two boxes, in total: 2n boxes. Total cost for all steps together: O(n log n)

20 Sharper complexity bounds Based on the new O(n log n) algorithm for AllContributionssharper complexity bounds can be stated: Theorem The following statements hold for d = 3 and an input set Y A d of size n: 1. AllContributions has time complexity Θ(n log n), i.e. the algorithm HYCON3D is asymptotically optimal. 2. The time complexity of MinimalContribution is bounded by O(n log n). 3. The time complexity MinimalContributor is bounded by O(n log n).

21 Tested code EF: The dimension sweep algorithm discussed in this paper. 1 FPLI: iterated application of total hypervolume computation with dimension sweep algorithm by Fonseca, Paquete and López-Ibáñez 2, cf. [Fonseca et al., 2006]. WFG: IHSO Algorithm 3 by Walking Fish Group (WFG) e.g. [Bradstreet et al., 2008]. 1 C++ source code is available on natcomp.liacs.nl (

22 Experimental results Problem (convexspherical) Find all contributions of Y A d, where y are generated independently, and y i = 10 v i / v, v i Normal(0, 1), i = 1,..., d. Problem (concavespherical) Find all contributions of Y A d, where y Y are generated independently, and y i = v i / v, v i Normal(0, 1), i = 1,..., d. Problem (cliff3d) Find all contributions for Y A d, where y Y are generated independently, and y i = 10 v i / v, v i Normal(0, 1), i = 1, 2, y 3 Uniform(0, 10).

23 Results of the empirical speed-test

24 Conclusions Sharper complexity bounds for 2D and 3D, lower bound for N dimensions. Interesting finding: OneContribution is Θ(n) in 2-D, while Θ(n log n) in 3-D. AllContributions is Θ(n log n) in 2-D and in 3-D. Asymptotically optimal algorithm for computing one and all contributions in 3-D Improves minimal contributor algorithm [Bringmann & Friedrich, 2010] by factor n. Again, as in [Fonseca et al., 2006] and [Kung et al., 1975] the particular structure of 3-D Pareto fronts could be exploited using dimensions sweep and amortization. Practical performance in tests up to 300 times faster than existing implementations.

25 Outlook generalization to higher dimensions incremental updates like in IHSO heuristics for sorting objectives application in algorithms such as SMS-EMOA, CMA-MOO, IBEA, or Hypervolume-based archivers will yield much larger Pareto front approximations

26 I gratefully acknowledge funding by Fundaçao para a Ciência e a Technologia (FCT), Portugal. Grant: Set-Indicator Based Multiobjective Optimization (SIMO) Thanks for organizing EMO2011 and for your attention!!

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