[. . . ] The two variants—the classical one and the streaming technique (cf. Section 2.3.1)—of the algorithm by Overmars and Yap (1991) have the same run time. Actually the same operations are done, though in different order. The authors de- scribe the analysis for the variant which completely builds the orthogonal partition tree before the sweep. The pre-processive sorting requires O(mlogm) time. It is shown that a cuboid is stored in O(m(d−2)/2) leaves of the partition tree since the
partitioning ensures that this is an upper bound for the number of partially covered regions.
The proof of the upper bound of the runtime is based on the number of partially covered regions per cuboid. Here, the explanation that this number does not exceed
O(√md−2)is given. Recall that a cuboid partially covers a region if ani-boundary cuts through the region. This characteristic is illustrated in Figure 2.13. A region that is generated by a splitting through dimensioniis termed ani-partition. There are O(√mi−1) (i−1)-partitions. The i-boundary of a cuboid intersects an (i− 1)-partition at most once and thereby cuts through one of its i-partition. Thus an i-boundary intersects O(√mi−1) i-partitions. When the partitioning is done concerning the remaining (d− 1−i) dimensions, each i-partition is subdivided into O(√md−1−i) (d − 1)-partitions. The cuboid’s i-boundary cuts O(√mi−1)· O(√md−1−i) =O(√md−2)(d−1)-partitions, which corresponds to the number of leafs that contain the cuboid.
The contained volume within these leaves has to be updated when the cuboid is inserted or removed from the orthogonal partition tree during the sweep. Thus, over all cuboids there are O(m(d−2)/2 ·m) = O(md/2) updates. Updating means
computing the measure in the trellis for each step of the sweeping. This originally takes timeO(logm)with the help of the segment trees, but the adapted algorithm (Algorithm 2.2) computes an update in constant time. The number of inner nodes influenced by a cuboid is bounded by the number of influenced leaves times the depth of the tree, so updating these results in a run time of O(md/2logm).
To get a notion of the actual performance of the presented algorithm (termed HOY), it is applied to common test sets provided by While et al. (2006) and its
Fig. 2.13:Illustration of the number of intersected regions. The dashed lines adum-
brate the induced weakly dominated hypercuboid. The columns show (i−1)-
partitions with their contained i-partitions. The bold i-partitions are inter-
sected by thei-boundary of the hypercuboid.
runtime is considered in comparison to algorithms by Fonseca et al. (2006) (termed FPL). The question we want to pose is which algorithm is the fastest on these test functions.
HOY has been implemented in C++ and the FPL algorithms in C. The programs were compiled with G++-4.2, GCC-4.2 respectively, with optimization level 3 and run under Ubuntu 8.04.1 on a single core Intel Pentium IV 3.4GHz processor with 1MB of cache, which is almost the same hardware setup as used in the study by Fonseca et al. (2006). To allow comparability we choose the test family DTLZ- Sphere (corresponding to DTLZ2,see Deb et al. (2002b)) and Random which were also invoked in the previous study and additionally DTLZLinear (DTLZ1). The data sets are random samples of these Pareto fronts of different size with three to nine objectives. The reference point r for the hypervolume calculation is cho- sen as(2, . . . ,2)>,(11, . . . ,11)>,(1, . . . ,1)>, respectively, so that all Pareto optimal points dominate it. For each setting (dimension and size fixed) 10 sample sets are processed. Source code of the algorithms are available at the authors’ homepages. The variants FPL3 and FPL4 have been considered since they outperform variants 1 and 2 in the earlier study. FPL4 shows a slightly better performance than FPL3 with lower variance, so only the results of FPL4 are shown here. They resemble the results in Fonseca et al. (2006) where objectives have been read in reverse order to comply with the preceding study by While et al. (2006). Since FPL4 outperforms the HSO in that study and no implementation of the HSO with the permutation heuristics by While et al. (2005) is available, HSO is not considered here.
The experimental results for the sets withd∈ {3,4,5,6}are shown using Box-and- Whisker plots in Fig. 2.14. Each set is processed twice: in original order and with objectives in reverse order, whereas the longer runtime corresponds to the latter. On the sets, both algorithm perform similar ford <5but FPL4 is a bit faster. For
Fig. 2.14:Results of FPL 4 (left) and HOY (right) on the test set DTLZSphere (top),
DTLZLinear (middle), and Random (bottom), with the number of points on
the horizontal axis and the computation time in seconds on the logarithmic vertical axis. The objectives in the input set are in the original order for the
lower runtimes and in reverse order for the higher ones, respectively. The
DTLZSphere data set with 800 6-dimensional points is corrupt so the results have been left out.
Fig. 2.15:Results of HOY on the test set DTLZSphere divided by different functions as potential upper bounds of the runtime.
the spherical and the linear Pareto fronts (Fig. 2.14, top, middle) there is partly a hundredfold magnification of the runtime. HOY’s runtime does not deviate that much due to permutations of the input so that its runtime lies between FPL4’s slow and its fast runs. This might be due to the different techniques of dividing the space into sub-problems. On the randomly generated fronts no significant difference can be observed for different input orders. Further studies on different permutations emphasize these observations.
Although HOY has a lower worst case runtime, this advantage seem not to strike through for low dimensional sets of the considered size. Larger sets seem to be required to reveal the lower magnitude of the runtime resembling the asymptotic behavior. For the 5- and 6-dimensional data HOY performs better than the slow runs of FPL4 also for small sets.
The indifference of permuting the random sets might be due to the lack of structure so that no adversely constellation is formed. The questions arise how worst case instances might be structured and if they occur at all in the practical application scenario of the algorithms. To get insight to this, other sets featuring different properties shall be considered in future studies. Both algorithms might be improved by the heuristics of While et al. (2005), where it is expected that FPL gains more since there seemed to be easy constellations for it. However, it is unclear how much speed-up can be gained. The HOY serves as a robust method regarding the order of the input.
To visualize the order of the measured runtime, we consider the longest observed runtime on DTLZSphere (sets in reverse order) and divide it by an asymptotic upper bound of md/2 (denoted as ub) as shown in Fig. 2.15 (red symbols). The
graph is not a constant but decreases alluding that the concrete runtime is lower. The blue curve with an additional factor of log(m)/√m seems to be an adequate estimate for this instance. The concrete constant of the runtime is irrelevant here and just reflects processor speed. Small 3-dimensional sets seemed to fall out of the schema, so it is assumed that their runtime is dominated by constants that lose influence on the magnitude for larger sets.