An Experimental Comparison of Fast Algorithms for Drawing General Large Graphs Stefan Hachul and Michael Jünger Universität zu Köln, Institut für Informatik, Pohligstraße 1, 50969 Köln, Germany {hachul, mjuenger}@informatik.uni-koeln.de Abstract. In the last decade several algorithms that generate straightline drawings of general large graphs have been invented. In this paper we investigate some of these methods that are based on force-directed or algebraic approaches in terms of running time and drawing quality on a big variety of artificial and real-world graphs. Our experiments indicate that there exist significant differences in drawing qualities and running times depending on the classes of tested graphs and algorithms. 1 Introduction Force-directed graph drawing methods generate drawings of a given general graph G =(V,E) in the plane in which each edge is represented by a straight line connecting its two adjacent nodes. The computation of the drawings is based on associating G with a physical model. Then, an iterative algorithm tries to find a placement of the nodes so that the total energy of the physical system is minimal. Important esthetic criteria are uniformity of edge length, few edge crossings, non-overlapping nodes, and the display of symmetries if some exist. Classical force-directed algorithms like [5, 15, 7, 4, 6] are used successfully in practice (see e.g. [2]) for drawing general graphs containing few hundreds of vertices. However, in order to generate drawings of graphs that contain thousands or hundreds of thousands of vertices more efficient force-directed techniques have been developed [19, 18, 9, 8, 12, 21, 11, 10]. Besides fast force-directed algorithms other very fast methods for drawing large graphs (see e.g. [13, 16]) have been invented. These methods are based on techniques of linear algebra instead of physical analogies. But they strive for the same esthetic drawing criteria. Previous experimental tests of these methods are mainly restricted to regular graphs with grid-like structures (see e.g. [13, 16, 9, 21, 12]). Since general graphs share these properties quite seldom, and since the test environments of these experiments are different, a standardized comparison of the methods on a wider range of graphs is needed. In this study we experimentally compare some of the fastest state-of-the-art algorithms for straight-line drawing of general graphs on a big variety of graph classes. In particular, we investigate the force-directed algorithm GRIP of Gajer and Kobourov [9] and Gajer et al. [8], the Fast Multi-scale Method (FMS) of P. Healy and N.S. Nikolov (Eds.): GD 2005, LNCS 3843, pp. 235 250, 2005. c Springer-Verlag Berlin Heidelberg 2005
236 S. Hachul and M. Jünger Harel and Koren [12], and the Fast Multipole Multilevel Method (FM 3 )ofhachul and Jünger [11, 10]. The examined algebraic methods are the algebraic multigrid method ACE of Koren et al. [16] and the high-dimensional embedding approach (HDE) by Harel and Koren [13]. Additionally, one of the faster classical forcedirected algorithms, namely the grid-variant algorithm (GVA) of Fruchterman and Reingold [7], is tested as a benchmark. After a short description of the tested algorithms in Section 2 and of the experimental framework in Section 3, our results are presented in Section 4. 2 The Algorithms 2.1 The Grid-Variant Algorithm (GVA) The grid-variant algorithm of Fruchterman and Reingold [7] is based on a model of pairwise repelling charged particles (the nodes) and attracting springs (the edges), similar to the model of the Spring Embedder of Eades [5]. Since a naive exact calculation of the repulsive forces acting between all pairs of charges needs Θ( V 2 ) time per iteration, GVA does only calculate the repulsive forces acting between nodes that are placed relatively near to each other. Therefore, the rectangular drawing area is subdivided into a regular square grid. The repulsive forces that act on a node v that is contained in a grid box B are approximated by summing up only the repulsive forces that are induced by the nodes contained in B and the nodes in the grid boxes that are neighbors of B. Ifthenumberof iterations is assumed to be constant, the best-case running time of the GVA is Θ( V + E ). The worst-case running time, however, remains Θ( V 2 + E ). 2.2 The Method GRIP Gajer et al. [8] and Gajer and Kobourov [9] developed the force-directed multilevel algorithm GRIP. In general, multilevel algorithms are based on two phases. A coarsening phase, in which a sequence of coarse graphs with decreasing sizes is computed and a refinement phase in which successively drawings of finer graphs are computed, using the drawings of the next coarser graphs and a variant of a suitable force-directed single-level algorithm. The coarsening phase of GRIP is based on the construction of a maximum independent set filtration or MIS filtration ofthenodesetv.amisfiltration is a family of sets {V =: V 0,V 1,...,V k } with V k V k 1... V 0 so that each V i with i {1,...,k} is a maximal subset of V i 1 for which the graphtheoretic distance between any pair of its elements is at least 2 i 1 +1. Gajer and Kobourov [9] use a Spring Embedder-like method as single-level algorithm at each level. The used force vector is similar to that used in the Kamada-Kawai method [15], but is restricted to a suitable chosen subset of V i. Other notable specifics of GRIP are that it computes the MIS filtration only and no edge sets of the coarse graphs G 0,...,G k that are induced by the filtrations. Furthermore, it is designed to place the nodes in an n-dimensional space
An Experimental Comparison of Fast Algorithms 237 (n 2), to draw the graph in this space, and to project it into two or three dimensions. The asymptotic running time of the algorithm, excluding the time that is needed to construct the MIS filtration, is Θ( V (log diam(g) 2 )) for graphs with bounded maximum node degree, where diam(g) denotes the diameter of G. 2.3 The Fast Multi-scale Method (FMS) In order to create the sequence of coarse graphs in the force-directed multilevel method FMS, Harel and Koren [12] use an O(k V ) algorithm that finds a 2- approximative solution of the NP-hard k-center problem. The node set V i of a graph G i in the sequence G 0,...,G k is determined by the approximative solution of the k i -center problem on G with k i >k i+1 for all i {0,...,k 1}. The authors use a variation of the algorithm of Kamada and Kawai [15] as force-directed single-level algorithm. In order to speed up the computation of this method, they modify the energy function of Kamada and Kawai [15] that is associated with a graph G i with i {0,...,k 1}. The difference to the original energy of Kamada and Kawai [15] is that only some of the V (G i ) 1springs that are connected with a node v V (G i ) are considered. The asymptotic running time of the FMS is Θ( V E ). Additionally, Θ( V 2 ) memory is needed to store the distances between all pairs of nodes. 2.4 The Fast Multipole Multilevel Method (FM 3 ) The force-directed multilevel algorithm FM 3 has been introduced by Hachul and Jünger [11, 10]. It is based on a combination of an efficient multilevel technique with an O( V log V ) approximation algorithm to obtain the repulsive forces between all pairs of nodes. In the coarsening step subgraphs with a small diameter (called solar systems) are collapsed to obtain a multilevel representation of the graph. In the used single-level algorithm, the bottleneck of calculating the repulsive forces acting between all pairs of charged particles in the Spring Embedder-like force model is overcome by rapidly evaluating potential fields using a novel multipole-based tree-code. The worst-case running time of FM 3 is O( V log V + E ) with linear memory requirements. 2.5 The Algebraic Multigrid Method ACE In the description of their method ACE, Koren et al. [16] define the quadratic optimization problem (P ) min x T Lx so that x T x = 1 in the subspace x T 1 n =0. Here n = V and L is the Laplacian matrix of G. The minimum of (P) is obtained by the eigenvector that corresponds to the smallest positive eigenvalue of L. The problem of drawing the graph G in two dimensions is reduced to the problem of finding the two eigenvectors of L that are associated with the two smallest eigenvalues.
238 S. Hachul and M. Jünger Instead of calculating the eigenvectors directly, an algebraic multigrid algorithm is used. Similar to the force-directed multilevel ideas, the idea is to express the originally high-dimensional problem in lower and lower dimensions, solving the problem at the lowest dimension, and progressively solving a highdimensional problem by using the solutions of the low-dimensional problems. The authors do not give an upper bound on the asymptotic running time of ACE in the number of nodes and edges. 2.6 High-Dimensional Embedding (HDE) The method HDE of Harel and Koren [13] is based on a two phase approach that first generates an embedding of the graph in a very high-dimensional vector space and then projects this drawing into the plane. The high-dimensional embedding of the graph is generated by first using a linear time algorithm for approximatively solving the k-center problem. A fixed value of k = 50 is chosen, and k is also the dimension of the high-dimensional vector space. Then, breadth-first search starting from each of the k center nodes is performed resulting in k V -dimensional vectors that store the graph-theoretic distances of each v V to each of the k centers. These vectors are interpreted as a k-dimensional embedding of the graph. In order to project the high-dimensional embedding of the graph into the plane, the k vectors are used to define a covariance matrix S. The x- and y- coordinates of the two-dimensional drawing are obtained by calculating the two eigenvectors of S that are associated with its two largest eigenvalues. HDE runs in O( V + E ) time. 3 The Experiments 3.1 Test-Environment, Implementations, and Parameter Settings All experiments were performed on a 2.8 GHz Intel Pentium 4 PC with one gigabyte of memory. We tested a version of GVA that has been implemented in the framework of AGD [14] by S. Näher andd. Alberts, animplementationof GRIP by R. Yusufov that is available from [22], and implementations of FMS, ACE, HDE by Y. Koren that are available from [17]. Finally, we tested our own implementation of FM 3. In order to obtain a fair comparison, we ran each algorithm with the same set of standard-parameter settings (given by the authors) on each tested graph. However, we are aware that in some cases it might be possible to obtain better results by spending a considerable amount of time with trial-and-error searching for an optimal set of parameters for each algorithm and graph. 3.2 The Set of Test Graphs Since only few implementations can handle disconnected and weighted graphs, we restrict to connected unweighted graphs, here.
An Experimental Comparison of Fast Algorithms 239 We generated several classes of artificial graphs to examine the scaling of the algorithms on graphs with predefined structures but different sizes. These are random grid graphs that were obtained by first creating regular square grid graphs and then randomly deleting 3% of the nodes. The sierpinski graphs were created by associating the Sierpinski Triangles with graphs. Furthermore, we generated complete 6-nary trees. The next two classes of artificial graphs were designed to test how well the algorithms can handle highly non-uniform distributions of the nodes and high node degrees. Therefore, we created these graphs in a way so that one can expect that an energy-minimal configuration of the nodes in a drawing that relies on a Spring Embedder-like force model induces a tiny subregion of the drawing area which contains Θ( V ) nodes. In particular, we constructed trees that contain a root node r with V /4 neighbors. The other nodes were subdivided into six subtrees of equal size rooted at r. We called these graphs snowflake graphs. Additionally we created spider graphs by constructing a circle C containing 25% of the nodes. Each node of C is also adjacent to 12 other nodes of the circle. The remaining nodes were distributed on 8 paths of equal length that were rooted at one node of C. In contrast to the snowflake graphs is that the spider graphs have bounded maximum degree. The last kind of artificial graphs are graphs with a relatively high edge density E / V 14. We called them flower graphs. They are constructed by joining 6 circles of equal length at a single node before replacing each of the nodes by a complete subgraph with 30 nodes (K 30 ). The rest of the test graphs are taken from real-world applications. In particular, we selected graphs from the AT&T graph library [1], from C. Walshaw s graph collection [20], and a graph that describes a social network of 2113 people that we obtained from C. Lipp. We partitioned the artificial and real-world graphs into two sets. The first set are graphs that consist of few biconnected components, have a constant maximum node degree, and have a low edge density. Furthermore, one can expect that an energy-minimal configuration of the nodes in a Spring Embedder drawing of such a graph does not contain Θ( V ) nodes in an extremely tiny subregion of the drawing area. Since one can anticipate from previous experiments [13, 16, 9, 12] that the graphs contained in this set do not cause problems for many of the tested algorithms, we call the set of these graphs kind. The second set is the complement of the first one, and we call the set of these graphs challenging. 3.3 The Criteria of Evaluation The natural criteria to evaluate a graph-drawing algorithm in practice are the needed running times and the quality of the drawings. Unlike evaluating the first criterion, evaluating the quality of a drawing is a difficult task. Possible ways are the calculation of the total energy in the underlying force models or the measurement of relevant esthetic criteria (e.g. crossing number, uniformity of the edge lengths). However, one of the most important goals is that an individual user is satisfied with a drawing. Hence,
240 S. Hachul and M. Jünger we decided to print the drawings and to comment how well they display the structure of each graph by keeping the modeled esthetic criteria in mind. 4 The Results 4.1 Comparison of the Running Times Table 1 presents the running times of the methods GVA, FM 3, GRIP, FMS, ACE, and HDE for the tested graphs. Table 1. The test graphs and the running times that are needed by the tested algorithms to draw them. Explanations: (E) No drawing was generated due to an error in the executable. (M) No drawing was generated because the memory is restricted to graphs with 10, 000 nodes. (T ) No drawing was generated within 10 hours of CPU time. B denotes the set of biconnected components of the graphs. Graph Information Type Name V E B Algorithm Information E max. CPU Time in Seconds V degree GVA FM 3 GRIP FMS ACE HDE rnd grid 032 985 1834 2 1.8 4 12.5 1.9 0.3 1.0 < 0.1 < 0.1 rnd grid 100 9497 17849 6 1.8 4 203.4 19.1 4.4 32.0 0.5 0.1 Kind rnd grid 320 97359 184532 2 1.9 4 6316.1 215.4 (E) (M) 4.1 1.3 ficial Artisierpinski 06 1095 2187 1 2.0 4 13.1 1.8 0.3 1.0 < 0.1 < 0.1 sierpinski 08 9843 19683 1 2.0 4 171.7 16.8 4.8 33.0 1.0 0.1 sierpinski 10 88575 177147 1 2.0 4 3606.4 162.0 (E) (M) 23.4 1.0 crack 10240 30380 1 2.9 9 317.5 23.0 6.8 (M) 0.4 0.2 Kind fe pwt 36463 144794 55 3.9 15 1869.1 69.0 (E) (M) (T ) 0.5 Real finan 512 74752 261120 1 3.4 54 6319.8 158.2 (E) (M) 7.5 1.0 World fe ocean 143437 409593 39 2.8 6 19247.0 355.9 (E) (M) 4.0 3.4 tree d 4 1555 1554 1554 1.0 7 14.3 2.6 0.3 2.0 < 0.1 < 0.1 tree d 5 9331 9330 9330 1.0 7 130.3 17.7 2.4 43.0 0.5 < 0.1 tree d 6 55987 55986 55986 1.0 7 1769.2 121.3 (E) (M) 4.5 0.5 snowflake A 971 970 970 1.0 256 8.0 1.6 0.4 73.0 0.4 < 0.1 Chal- snowflake B 9701 9700 9700 1.0 2506 143.2 17.4 6.1 3320.0 (T ) < 0.1 lenging snowflake C 97001 97000 97000 1.0 25006 14685.7 166.5 (E) (M) (T ) 0.8 Arti- spider A 1000 2200 801 2.2 18 17.6 1.9 0.4 1.0 1.1 < 0.1 ficial spider B 10000 22000 8001 2.2 18 189.0 17.7 7.2 47.0 8.9 0.1 spider C 100000 220000 80001 2.2 18 4568.3 177.2 (E) (M) 280.7 1.3 flower A 930 13521 1 14.5 30 61.7 1.2 0.7 1.0 < 0.1 < 0.1 flower B 9030 131241 1 14.5 30 595.1 11.9 19.3 46.0 1.4 0.2 flower C 90030 1308441 1 14.5 30 11841.5 121.4 (E) (M) (T ) 1.4 ug 380 1104 3231 27 2.9 856 23.1 2.1 0.4 1.0 < 0.1 < 0.1 esslingen 2075 5530 867 2.6 97 43.8 4.0 0.5 404.0 1.0 < 0.1 Chaladd 32 4960 9462 951 1.9 31 80.6 12.1 1.6 17.0 0.5 < 0.1 lenging dg 1087 7602 7601 7601 1.0 6566 624.8 18.1 3.6 5402.0 108.4 < 0.1 Real bcsstk 33 8738 291583 1 33.3 140 1494.6 23.8 29.1 6636.0 0.4 0.3 World bcsstk 31 35586 572913 48 16.1 188 4338.4 83.6 (E) (M) 1.9 0.7 bcsstk 32 44609 985046 3 22.0 215 6387.1 110.9 (E) (M) 3.6 0.9
An Experimental Comparison of Fast Algorithms 241 As expected, in most cases GVA is the slowest method among the forcedirected algorithms. The largest graph fe ocean is drawn by GVA in 5 hours and 20 minutes. (a) GVA (b) FM 3 (c) GRIP (d) FMS (e) ACE (f) HDE (g) GVA (h) FM 3 (i) GRIP (j) FMS (k) ACE (l) HDE Fig. 1. (a)-(f) Drawings of rnd grid 100 and (g)-(l) sierpinski 08 generated by different algorithms
242 S. Hachul and M. Jünger The method FM 3 is significantly faster than GVA for all tested graphs. The running times range from less than 2 seconds for the smallest graphs to less than 6 minutes for the largest graph fe ocean. The subquadratic scaling of FM 3 can be experimentally confirmed for all classes of tested graphs. (a) GVA (b) FM 3 (c) GRIP (d) ACE (e) HDE (f) GVA (g) FM 3 (h) HDE (i) GVA (j) FM 3 (k) ACE (l) HDE Fig. 2. (a)-(e) Drawings of crack, (f)-(h) fe pwt, and (i)-(l) finan 512 generated by different algorithms
An Experimental Comparison of Fast Algorithms 243 Except for the dense graphs flower B and bcsstk 33 GRIP is faster than FM3 (up to a factor 9). Unfortunately, we could not examine the scaling of GRIP for the largest graphs due to an error in the executable. Since the memory requirement of FMS is quadratic in the size of the graph, the implementation of FMS is restricted to graphs that contain at most 10, 000 nodes. The running time of FMS is comparable with that of FM3 for the smallest and the medium sized kind graphs. In contrast to this, the CPU time of FMS increases (b) FM3 (a) GVA (c) ACE 2861 2866 6887 7207 7208 7211 7239 6888 7220 6897 7210 7240 9246 2878 2868 6895 477 6916 6884 6910 6907 2863 479 7238 6892 6889 7229 7214 7215 2886 2049 2851 6891 2873 2855 475 2874 2870 2869 480 2881 478 6883 1147 (d) HDE 2871 2885 2854 2050 7223 2859 476 2852 2876 2865 7217 9233 2883 2858 2875 6900 6885 7241 7212 1201 2860 2857 2867 2879 6899 6908 6886 7209 7221 2862 2864 6914 6912 6911 7222 9230 9231 9232 6894 2452 2853 2872 2453 2880 9229 2048 2943 2884 2877 9249 7224 6913 4680 2056 6915 7232 7233 20382882 9234 1152 2944 7237 2053 9245 4678 2454 2856 341 6896 9241 6898 9252 1148 9237 20472033 7231 2422 4675 92517216 9228 6902 9225 29322942 1203 2039 1538 2419 7219 2433 6909 6917 2428 7242 2030 24512450 1149 7235 9248 4677 6893 2035 2423 2036 1151 2945 7225 9240 2052 2952 2424 408 1206 9247 9238 7213 2031 1202 7534 2948 3754 7230 4682 2951 9224 9150 9243 46614679 7543 2057 2430 7234 1541 7531 4659 4676 4683 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1469 8731 4801 8807 4328 8824 1472 4353 8751 8803 8860 4355 4816 8819 1475 8734 8835 8852 4332 8857 8837 1466 8735 8736 8798 8800 4351 8733 8804 8799 4813 8858 8820 1471 8817 8829 8797 1473 8845 8831 8838 8847 8832 8836 8818 8862 8802 8834 8816 8830 8828 1474 8827 8849 8850 8844 8853 8848 6180 6195 6166 6168 6193 6164 6177 6175 6165 6167 6197 6196 1029 61866182 6198 6178 6189 6190 6169 6194 6187 1032 6188 6181 6170 1027 6192 6174 6184 1030 6171 4666 1953 2064 1956 1978 1973 5906 1954 2068 1976 1972 2069 4603 1980 4592 1982 4428 2070 2084 2073 329 328 2065 2066 4611 1968 1952 348 2062 5909 58782092 2076 1979 4595 4596 4593 764 4585 5892 5877 5888 2091 1970 1975 4582 2085 1986 1985 1961 767 4584 6163 2093 2094 1031 1951 1969 6185 1971 1977 6172 4608 4606 4605 4586 5889 5910 6183 6191 1028 6173 4590 4604 4588 4589 4579 325 4414 1984 171 4424330 345 4426 344 19742067 1965 2072 1966 2074 4607 4417 1981 2083 4587 4580 1983 4406 7654581 763 4601 4591 4594 2218 1967 2236 766 46124609 2203 2207 4614 2228 2063 4106 4418 4401 8635 1964 369 327 4610 2217 2220 4403 2205 8640 2209 2204 4613 4410 4420 2215 2216 8639 2233 4598 4602 768 4583 4600 4597 4422 4421 4080 4076 4400 2087 8636 5243 4079 8643 4416 1963 4402 4429 2086 4407 8627 4409 5239 4095 736 2554 8846 88418842 8840 8839 8843 (e) GVA (h) FMS (f) FM3 (i) ACE (g) GRIP (j) HDE Fig. 3. (a)-(d) Drawings of fe ocean and (e)-(j) tree 06 05 generated by different algorithms
244 S. Hachul and M. Jünger drastically for several challenging graphs, in particular for graphs that either contain nodes with a very high degree or have a high edge density. The algorithm ACE is much faster than the force-directed algorithms for nearly all kind graphs. However, like for FMS, the running times grow extremely when ACE is used to draw several of the challenging graphs. The linear time method HDE is by far the fastest algorithm. It needs less than 3.4 seconds for drawing even the largest tested graph. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 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509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 (a) GVA (b) FM 3 (c) GRIP (d) FMS (e) ACE (f) HDE 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 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553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 (g) GVA (h) FM 3 (i) GRIP (j) FMS (k) ACE (l) HDE Fig. 4. (a)-(f) Drawings of snowflake A and (g)-(l) spider A generated by different algorithms
295 291 317 292 294 297 318 15 957 10 11 9 13 296 321 315 26 172 319 941 1065 293 271 6 322 16 36 12 2 31 3 25 375 1067 1 32 861 890 301 305 268 316 304 299 267 269 258 270 30 27 21 8 411 35 1045 309 310 314 917 1049 1050 670 28 29 298 262 0 158 264 939 680 668 694 174 320 7 683 303 154 678 18 300 261 37 163 20 682 1043 688 709 24 1048 706 671 257 373 676 679 1069 307 289 662 155 159 312 256 308 302 681 669 1044 156 162 161 313 266 160 43 866 1064 674 673 710 685 284 171 860 340 858 412 642 672 1046 677 675 260 661 684 157 41 282 286 287 311 283 323 17 42 22 1047 695 958 38 241 144 145 265 689 687 686 663 711 691708 693 705 285 263 899 651 690 4 168 169 281 259 712 141 280 288 306 374 633 702 697 940 290 636 641 900 660 278 248 272 275 823 865 921 343 713 733 147 664 643 692 667 699 170 47 44 849 898 732 696 698 45 19 649 173 273 637 349 632 714 700 92 276 896 747 648 246 734 822 707 137 640 344 279 634 23 635 249 625 715 274 277 244 653 665 666 718 701 821 1066 922 902 717 1101 716 704 722 726 725 864 980 920 850 639 631 750 735 638 647 729 721 91 400 626 240 767 48 345 624 445 247 135 138 49 89 39 346 630 430 628 255 703 719 752 655 769 749 751 879 855 895 848 768 746 736 645 1100 753 819 851 901 817 731 650 46 748 1040 629 409 242 659 720 1041 724 139 723 90 969 852 815 813 410 399 623 443 780 252 376 627 250 408 243 608 609 14 40 251 406 444763 764 622 208 239 404 254 398 253 787 782 189 617 88 245 134 191 348 407 136 744 148 607 87 402 33 2071103 131 132 133 517 621 620 658 1029 657 1042 785 728 727 818 925 897 652 730 737 766 1030 770 1084 1012 847 856 978 781 784 745 1102 924 814 859 644 1039 738 1013 786 952 846 790 610 754 772 618 802 739 857 832 853 812 953 923 979 762 872 1085 998 1068 755 646 743 820 765972 1031 742 511 806 954 1051 1011 122 518 825 589 741 740 1083 942 606 837 140 1075 783 221 146 223 233 586 84 362371 442 656 1028 997 761 827 462 977 788 463 397 1032 512 403 507 401 234 5 1036 1037 441 235 513 118 516 200 435 142 117 364 366 368 370 757 86 85 966 229 106 120 119 121 453 508 128 461 886 152 143 238 369 372 347 361 654 1038 583 759 190 1099 592 616 587 801 845 854 619 756 829 771 805 816 996 844 867 884 826 779 611 590 760 842 831 990 894 841 843 935 1096 1027 834 824 584 448 605 791 604 614 758 876 883 976 789 124 184 615 877 773 1080 1033 800 1014 778 206 593 907 450 180 111 803 104 107 330 79 509 510 110 514 936 909 113 836 519 840 830 363 365 1095 971 974 439 794 1098 580 613 503 828 114 449 451 776 774 1089 995 1026 918 888 798 209 123 360 434 151 115 149188 881 331 777 165 217 595 186 581222 329 575 968 908 577 878 504 838 413 405 452 1090 862 1035 967 1002 987 429 433 573 127 460 150 199 103 83 563 204201 202 75 105 505 116 350 599 231 338 1097 793 1034 175 880 839 1022 893 797 833 988 989 955 795 396 77 571 491591 506 956 546 915 601 612 775 596 804 835 994 916 1074 598 498 490 108 112 129 339 1025 914 179 76 588 882 100 432 578 568 570 874 483 98 454 1021 457 420 975 459 34 328 78 177 792 951 1061 130 125 326 873 437 929 562 192 97 63 178 796 74 164 70 808205 109 61 181 96 60 71 73 153811 214 167 185 57 182 72 810 187 567 64 212 50 53 99 94 95 126 183 102 458 82 55 54 213 51 218 219 220 576 807 166 198 203 582 600 891 579 565 230 1079 1062 419 367 934 341 1024 868 887 602 569 585 594 431 1009 499 1086 992 327 1094 436 572 910 885 991 377 500 545 799 337 232 352 354 597 332 481 455 938 1087 973 421 937 863 359 561 560 965603 496 945 959 986 456 809 497 905 574 911 479 1008 1063 1088 1070 422 1093 482 906 447 912 515 414 386 358 334 501 415 960 930 502 478 477 236 380 1073 557 889 559 964 892 970 943 993 1020 981 385 438 423 351 1019 544 1091 983 52 984 1010 216 549 101 1060 566 378 425 446 387 93 58 59 228 68 81 357 355 985 335 196 550 547 480 417 962 1057 982 1092 80 353 1018 56543 1023 556 215 564 237 197 210 495 933 416 537 552 333 1001 473 194 356 224 388 424 62 533 69 913 532 553 176 476 211 944 1071 1017 66 226 875 65 193 379 418 927 539 542 919 541 525 475 225 227 535 427 336 486 342 324 195 381 426 67 325 382 383 389 390 391 540 470 440 526 523 521 474 472 494 484 522 469 394 488 464 467 471 524 520 465 384 428 551 527 531 554 528 1016 1007 1015 548 558 1059 1055 1082 871 932 1076 961 926 928 870 1072 529 999 536 1000 555 931 1058 538 869 1003 946 1053 949 903 904 534 1006 468 1077 493 530 1081 489 466 485 1056 392 947 492 950 393 963 395 1052 487 1054 1004 1078 1005 948 4.2 Comparison of the Drawings An Experimental Comparison of Fast Algorithms 245 For all kind graphs the classical method GVA does not untangle the drawings that were induced by the random initial placements. In contrast to this nearly all algorithms generated comparable pleasing drawings of the kind graphs (see Figure 1, Figure 2, and Figure 3(a)-(d)). None of the drawings of the complete 6-nary trees (see Figure 3(e)-(j)) is really convincing, since the force-directed algorithms produce many unnecessary (a) GVA (b) FM 3 (c) GRIP (d) FMS (e) ACE (f) HDE (g) GVA (h) FM 3 (i) GRIP (j) FMS (k) ACE (l) HDE Fig. 5. (a)-(f) Drawings of flower B and (g)-(l) ug 380 generated by different algorithms
246 S. Hachul and M. Ju nger edge crossings. However, the drawings generated by FM3 and FMS display parts of the regularity of these graphs. The algebraic methods ACE and HDE place many nodes at the same coordinates. In general, this behavior of the algebraic methods can be observed for graphs that consist of many biconnected components. Explanations of the theoretical reasons can be found in [3, 16] and [10]. Except FM3 none of the tested algorithms displays the global structure of the snowflake graphs. Even the drawings of the smallest snowflake graph (see (b) FM3 (a) GVA (d) FMS (e) ACE (c) GRIP (f) HDE 1177 1181 1179 1015 1735 1182 1180 1379 1380 971 790 1178 399 788 1088 970 160 1388 1383 969 1298 794 1208 1160 1207 460 533 1247 1243 1211 968 1245 746 1143 1264 1736 678 1446 1520 1521 1035 1518 516 885 520 534 180 561 2018 176 1256 170 524 1671 184 625 1668 1424 1527 198 494 191 1957 495 493 1921 282 185 491 1474 1072 551 398 316 507 154 1320 1890 1651 1663 1786 1163 1660 1900 672 589 1625 1808 1937 1309 1385 1773 1304 116 541 546 543 544 1255 204 545 217 220 934 1227 1270 1286 690 222 935 540 119 375 123 120 212 59 58 931 576 219 377 2038 113 122 112 223 1425 114 586 109 928 581 209 216 110 929 575 205 966 930 1121 1120 121 1774 225 583 1744 1535 967 221 590 965 932 933 601 239 600 1756 1755 591 358 940 1750 1377 1758 1751 1391 594 598 592 1376 1743 1759 1455 1757 362 1579 366 363 599 238 2031 213 1413 580 603 1944 1985 926 927 207 224 593 1754 584 588 1393 1781 1934 215 964 214 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446 450 1482 915 1280 1469 608 189 720 1395 1447 1490 1259 635 723 272 269 719 712 710 721 1191 1140 1981 1431 24 1980 8 523 670 724 1285 1519 626 380 1522 463 884 514 713 722 949 1858 1405 1020 1026 1171 667 61 1226 634 711 715 1433 693 1030 941 1967 889 888 692 9 716 911 942 1416 525 1907688 240 2002 18 16 1483 1215 1622 379 924 1507 374 793 1438 998 1639 1762 2025 1516 1905 1197 1971 624 733 461 890 1412 1092 1619 1485 1577 12 23 887 1016 1621 1620 1441 1618 1468 1523 990 796 787 1729 1515 662 45 1970 1863 802 800 855 973 1841 1200 1739 1086 1095 382 2522 1241 1984 1234 1198 1500 1249 1152 13 959 1435 227 1190 1027 1233 27 11 17 1645 789 1503 1502 1501 237 892 1432 2024 1908 2029 1362 1281 1631 1440 1237 727 1238 1369 1888 1368 1017 674 1634 1240 1442 511 1429 1371 891 1370 2052 1196 1176 241 1011 1235 1244 1210 249 1265 1450 1195 1040 1041 1248 1253 1443 1266 1261 1453 137 1202 44 1038 1229 141 515 1124 1263 248 1199 1204 408 1288 1498 1252 1797 1887 1161 60 1212 1232 1236 513 512 684 1293 1242 744 1297 1451 529 1262 1260 797 1156 140 1499 745 528 680 676 791 792 1206 1014 1159 1158 747 682 679 681 992 1201 383 2075 400 1470 1382 677 1203 1183 883 972 1372 1734 685 683 136 1157 372 373 1731 880 1753 1752 364 361 1792 365 1317 1799 1800 367 1935 1318 2014 1291 1848 368 369 1278 359 772 360 1311 1847 2032 (g) GVA (j) FMS (h) FM3 (k) ACE (i) GRIP (l) HDE Fig. 6. (a)-(f) Drawings of dg 1087 and (g)-(l) esslingen generated by different algorithms
An Experimental Comparison of Fast Algorithms 247 Figure 4(a)-(f)) leave room for improvement. However, GVA and GRIP visualize parts of its structure in an appropriate way. The drawings of the spider A graph (see Figure 4(g)-(l)) that are generated by GRIP, FMS, andhde are not as symmetric as that generated by FM 3.Butthey display the global structure of the graph. The drawing generated by GVA shows the dense subregion, but GVA does not untangle the 8 paths. The paths in the (a) GVA (b) FM 3 (c) GRIP (d) FMS (e) ACE (f) HDE (g) GVA (h) FM 3 (i) GRIP (j) FMS (k) ACE (l) HDE Fig. 7. (a)-(f) Drawings of add 32 and (g)-(l) bcsstk 33 generated by different algorithms
248 S. Hachul and M. Jünger drawing of ACE are not displayed in the same length. The drawings of the larger spider graphs are of comparable quality. The drawings of the flower B graph (see Figure 5(a)-(f)) that are generated by FMS and HDE display the global structure of the graph but the symmetries are not as clear as in the drawing generated by FM 3. The drawings of the other flower graphs are of comparable quality. We concentrate on the challenging real-world graphs now. The graphs ug 380 and dg 1087 both contain one node with a very high degree. Furthermore, dg 1087 has many biconnected components, since it is a tree. Only the drawings that are generated by GVA, FM 3,andGRIP (see Figure 5(g)-(l) and Figure 6(a)- (f)) clearly display the central regions of these graphs. It can be observed that (a) GVA (b) FM 3 (c) ACE (d) HDE (e) GVA (f) FM 3 (g) ACE (h) HDE Fig. 8. (a)-(d) Drawings of bcsstk 31 con and (e)-(h) bcsstk 32 generated by different algorithms