Inspired by a challenge during Graph Drawing 2010 "Find an orthogonal drawing whose longest edge is as short as possible", we investigate techniques to incorporate this goal into the "standard" topology-shape-metrics approach at moderate extra computational complexity.

Experiments indicate that this project is worth pursuing

Gronemann, Martin

Jünger, Michael

Mallach, Sven

Schmidt, Daniel R.

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Towards shortest longest edges in orthogonalgraph drawingMartin Gronemann, Michael Ju¨nger, Sven Mallach, and Daniel R. SchmidtInstitut fu¨r Informatik, Universita¨t zu Ko¨ln, GermanyAbstract. Inspired by a challenge during Graph Drawing 2010 ‘Find anorthogonal drawing whose longest edge is as short as possible’, we inves-tigate techniques to incorporate this goal into the ‘standard’ topology-shape-metrics approach at moderate extra computational complexity.Experiments indicate that this project is worth pursuing.1 IntroductionIn an orthogonal drawing, all nodes lie on a grid and all edge segments are parallelto the axes. We assume familiarity with the ‘standard’ basic technique to producesuch drawings, the topology-shape-metrics (TSM), that has been introduced inthe 1980s by Batini, Di Battista, Nardelli and Tamassia [3, 2]. Its topology stepyields a planar(ized) embedding, the shape step an orthogonal representation(OR) and the metric step edge lengths from a compaction procedure.Optimization goals include the minimization of the area of the drawing, thenumber of its bends and the total and maximum edge lengths. A bend-minimalOR of a four-planar embedding can be computed in time O(n74 log n) [11].However, even given a fixed OR, the exact solution of the other optimizationproblems is NP-hard [16]. Even worse, finding a globally optimal solution cannotbe reduced to the compaction step, but relies on all the three TSM steps. To thebest of our knowledge, currently no such integrated algorithm exists.Orthogonal compaction is strongly related to VLSI layout problems anddrawings of relational diagrams. Hence, research has mainly been devoted toimprovements of the area and total edge length. Concerning total edge length,Klau and Mutzel [14] provided an ILP-based exact compaction algorithm. Itruns in polynomial time for restrictive kinds of ORs which are called completeor uniquely completable. Experimental studies [13, 8] show that heuristics basedon turn-regularity [5] and network flows [7] typically perform best for practicaland hard instances. There also exist heuristics that try to refine an already givenorthogonal drawing like, e.g., [12] and [10]. Focusing on relational diagrams, in [9,6] heuristics for compacting orthogonal drawings with fixed node sizes are pre-sented, and in [4] an algorithm that tries to preserve a given initial sketch of agraph while computing a drawing with few bends.In this paper we focus on keeping the longest edge as short as possible whichhas not yet received much attention in the literature. We restrict our attentionto graphs whose nodes have a maximum degree of four and assume that the2 Martin Gronemann, Michael Ju¨nger, Sven Mallach, and Daniel R. Schmidttopology phase has already taken place. Since our ideas rely on the establishedflow-based compaction heuristics, Sect. 2 deals with the network flow problemsaddressed. In Sect. 3, we propose our new heuristic techniques to improve ORsand the metric step. Finally, we present an experimental analysis in Sect. 4.2 Computing Minimum Maximum FlowsGiven a network, i.e. a directed graph (V,E) with a source/sink s, t ∈ V , acapacity function u : E → Z≥0∪{∞} and lower bounds l : E → Z≥0, an s-t-flowis a function f : E → R≥0 with l(e) ≤ f(e) ≤ u(e) on all edges e ∈ E that hasthe flow conservation property, i.e.∑e=(v,u)∈E f(e) −∑e=(u,v)∈E f(e) = 0 forall v ∈ V \ {s, t}. We say that f is integral iff f(e) ∈ Z≥0 holds for all e ∈ E.The Minimum Cost Flow Problem (MinCostFlow) in a network with costsc : E → Z asks for an s-t-flow that minimizes the overall cost∑e∈E c(e)f(e). Forour implementation, we use the (non-polynomial but practically fast) networksimplex algorithm [1]. There is also an algorithm by Garg and Tamassia [11]that was developed for a similar graph drawing problem.Given an algorithm for the MinCostFlow problem, we can solve the IntegralMinimum Maximum s-t-Flow Problem with lower bounds (MinMaxFlow): Ina network (V,E, s, t, u) with u ≡ ∞, find an integral s-t-flow such that themaximum flow on any arc is minimum while the lower bounds are respected.The integrality constraint is important as a fractional solution can have a lowervalue than any integral MinMaxFlow. We can compute an integral MinMaxFlowby calling a MinCostFlow algorithm in a binary search on the capacities.Algorithm 1: Computing a MinMaxFlow by binary search.Input: A network N = (V,E, s, t,∞), lower bounds l : E → Z≥0Result: An optimal MinMaxFlow on N given lSolve MinCostFlow on N , l, c ≡ 1 and u ≡ ∞, obtain fM ;Set fmax = max{fM (e) | e ∈ E}, low = 0 and high = fmax;while low < high domid = b(low + high)/2c;Solve MinCostFlow on N , l, c ≡ 1 and u ≡ mid, obtain fM ;if fM feasible then high = mid else low = mid+ 1Solve MinCostFlow on N , l, c ≡ 1 and u ≡ high, obtain and return fM ;The algorithm runs in time O(Tmcf · log fmax) where Tmcf is the time neededby the MinCostFlow algorithm. In particular, a polynomial time MinCostFlowalgorithm [1, Chapter 9], yields a polynomial time algorithm for the integralMinMaxFlow problem. For the correctness of Alg. 1, observe that every Min-CostFlow is a feasible MinMaxFlow. Thus, the condition that fM is feasible, ismonotonic regarding u on {0, . . . , fmax} and thus, binary search can be applied.3 Minimizing the Longest EdgeIn principle, the TSM approach is not well-suited for minimizing the total ormaximum edge length of a drawing, since these properties depend on all threeTowards shortest longest edges in orthogonal graph drawing 3steps; yet, until the last step, there is no notion of edge lengths. However, it isstate-of-the-art and widely used, so in this section we present heuristic strategiesfor the shape and metrics steps that can help to keep the longest edge lengthsmall. They can easily be incorporated into TSM frameworks.3.1 MinMax Flow Compaction in the Metrics StepHeuristic compaction is typically done in two steps. First, after making the ORturn-regular and dissecting all faces to rectangular shapes, a constructive heuris-tic is used to obtain a first feasible drawing. Then improvement heuristics arerepeatedly applied until local optima are reached. As evaluated in [13], flow-basedheuristics are a good choice. Here, constraint graphs that model adjacencies andvisibilities are built for each compaction direction. Their dual graphs can beinterpreted as networks N where each edge corresponds to exactly one edge ofthe constraint graph. After a MinCostFlow computation on N with l ≡ 1 andu ≡ ∞, the edge length can be set to dual edge flows which are of minimumtotal value. Instead, we use MinMaxFlow to find edge lengths so that the longestedge is short. This procedure is referred to by ML from now on. We also considera variant CML, in which the length of the longest edge is reduced subject to theconstraint that the total edge length be kept minimum. This requires an easyadaption of Alg. 1 in which the binary search direction depends on whether thecurrent total flow exceeds its minimum value.3.2 Balancing Bends in the Shape StepThe standard bend-minimization (TB) relies on a MinCostFlow problem in a dualnetwork N . On those edges in N that correspond to primal edges, each flow unitcorresponds to a bend in that primal edge. Minimizing the total number ofbends can be counterproductive for our goal, as a better solution can sometimesbe found by introducing further bends. Nevertheless, experiments [8] showedthat bend-minimal ORs serve at least as a good basis for the compaction phase.However, we believe that a concentration of bends on single edges has severaldrawbacks. First of all, it is not considered to be aesthetically pleasant. Evenmore, in many cases, edges with a lot of bends will be long, have complex routesor block a broad area. Heuristically, a more balanced distribution of bends couldhelp to escape from local minima. Our results indicate that the shape step hasa large influence on the achievable improvements of the compaction step. Weevaluate this by applying Alg. 1 in two variations: MB minimizes the maximumnumber of bends on an edge as far as possible and CMB does the same as long asthe total number of bends does not increase.4 ResultsWe experimentally evaluate whether our balancing approach yields drawingswith shorter longest edges. Our code is a modification of the TSM frameworkimplementation in the Open Graph Drawing Framework (OGDF) [15].4 Martin Gronemann, Michael Ju¨nger, Sven Mallach, and Daniel R. SchmidtFor a general comparison of the different drawing strategies, we randomlygenerate instances1 using rudy [17] and use realistic instances from the romelibrary [8]. From these instances, we first remove loops and isolated nodes, thenall nodes with a degree of more than four in the order of their indices. If thegraph becomes disconnected, we select the largest connected component thatcontains the smallest node index. Non-planar instances are planarized by theOGDF code in the topology step. The results are depicted in Tab. 1: On the rudyinstances, minimizing the maximum number of bends per edge (MB) in the shapestep yields the same solutions as the conventional total bend-minimization TB.However, using the constrained variant CMB in the shape step can lead to largeimprovements that are of the metric step approach. Yet, CMB can perform worsethan the conventional TB+TL. Striving for short longest edges in the metric step(ML) is never worse than total edge length minimization (TL) and can lead tosmall improvements. CML performs roughly like ML. We observe the same on therome library instances.Finally, we analyze the influence of a graph’s shape and density on the per-formance of the different drawing methods. To that end, we use rudy to generatecomplete grid graphs with integer edge weights distributed uniformly at randomfrom 0 to 1002. The density of our instance is controlled by deleting all edgeswhose weight is at least ∆ = 65, 70, . . . , 95. Disconnected instances are rejected.No planarization is needed on these instances. In Tab. 2(a), each entry depictshow many times a given strategy yielded the best result on one of the 100 in-stances. Here, none of the bend-minimization strategies is clearly superior, butfor the non-quadratic instances, CMB is slightly better in total. This is in contrastto the performance of MB which is not better than the conventional TB method.For the metric step, ML consistently beats the other strategies; only for the verynarrow 40×5 grid not much is gained. When CMB is used, ML has the highest im-pact for densities from 70% to 85%. Overall, CMB+ML seems to be a good choicehere. A more detailed look at the performance (see Tab. 2(b)) reveals, however,that CMB+ML still can perform much worse than TL+TB.5 ConclusionThe experiments show that one can – with little extra computational effort – putemphasis on short longest edges in the TSM framework. However, in most cases,concentrating on the metric step is not enough. For significant progress one hasto integrate the shape step into the optimization process. In some cases, buildingthe OR with the CMB strategy is a good idea; in other cases, it can lead to badresults. Thus, it is still unclear how to find the best OR, but the experimentsshow that if the right OR is found, one can expect drastic improvements.1 We call rudy -rnd graph i (200 / (i-1)) (4711 + i) for i = 175, . . . , 200. Eachcall yields an instance with i initial nodes that have average degree four.2 We call rudy -grid 2D height width -random 0 100 (seed+i) for i = 1, . . . , 100with an initial seed of 215986 (#inhabitants of Eindhoven according to Wikipedia).Towards shortest longest edges in orthogonal graph drawing 5TB CMB MBi TL CML ML TL CML ML TL CML ML175 29 28 28 29 28 28 29 28 28176 12 12 12 12 12 12 12 12 12177 21 20 20 24 22 22 21 20 20178 12 12 12 11 11 11 12 12 12179 33 33 31 27 27 27 33 33 31180 22 16 16 18 18 18 22 16 16181 16 16 15 15 15 15 16 16 15182 16 16 16 16 16 16 16 16 16183 21 21 15 18 18 15 21 21 15184 26 26 26 27 27 27 26 26 26185 20 20 19 20 20 19 20 20 19186 8 8 8 8 8 8 8 8 8187 19 19 13 24 24 24 19 19 13188 23 23 23 24 24 24 23 23 23189 48 49 49 49 49 49 48 49 49190 28 28 28 32 32 32 28 28 28191 17 17 17 21 21 21 17 17 17192 28 28 28 24 24 24 28 28 28193 16 16 16 6 6 6 16 16 16194 39 39 39 30 26 26 39 39 39195 10 10 10 17 17 17 10 10 10196 34 34 34 30 30 30 34 34 34197 29 29 29 25 25 25 29 29 29198 64 64 64 35 30 30 64 64 64199 4 4 4 4 4 4 4 4 4200 27 27 27 24 24 24 27 27 27(a) Random rudy graphs.TB CMB MBTL CML ML TL CML ML TL CML MLug10.10 6 6 6 6 6 6 6 6 6ug20.10 2 2 2 2 2 2 2 2 2ug30.10 8 8 8 4 4 4 8 8 8ug40.10 4 4 4 2 2 2 4 4 4ug50.10 4 4 4 2 2 2 4 4 4ug10.30 12 12 11 7 7 7 12 12 11ug20.30 7 7 7 7 7 7 7 7 7ug30.30 12 12 12 12 12 12 12 12 12ug40.30 10 10 10 8 8 8 10 10 10ug50.30 6 6 6 4 4 4 6 6 6ug10.50 18 18 18 13 11 12 18 18 18ug20.50 10 10 10 10 10 10 10 10 10ug30.50 20 20 20 15 15 15 20 20 20ug40.50 10 10 10 9 9 9 10 10 10ug50.50 20 20 20 10 10 10 20 20 20ug20.70 16 16 16 17 17 17 16 16 16ug30.70 38 38 38 38 38 38 38 38 38ug40.70 31 31 31 33 33 33 31 31 31ug50.70 16 16 16 17 17 17 16 16 16ug60.70 38 38 38 24 23 23 38 38 38ug10.90 23 23 23 23 23 23 23 23 23ug20.90 25 25 25 28 28 28 25 25 25ug30.90 45 45 45 45 45 45 45 45 45ug40.90 25 25 25 21 21 21 25 25 25ug50.90 22 22 22 19 16 16 22 22 22(b) 10 to 90 node graphs from the rome-lib.Table 1. Length of the longest edge in the computed drawing.#con. TB CMB MB∆ inst. TL CML ML TL CML ML TL CML ML14 × 14 grid, seed 21598665 21 10 13 15 6 8 12 10 13 1570 46 14 14 25 14 18 31 14 14 2575 74 24 29 46 18 25 45 24 29 4680 87 31 35 57 28 35 54 31 35 5785 95 46 55 67 42 49 66 46 55 6790 96 63 66 70 66 66 73 63 66 7095 100 71 74 79 77 79 81 71 74 7920 × 10 grid, seed 21618865 14 3 4 8 1 4 8 3 4 870 35 11 16 22 15 16 23 11 16 2275 66 21 26 37 28 33 45 21 26 3780 83 30 38 51 28 41 59 30 38 5185 96 45 51 65 54 61 74 45 51 6590 98 61 65 71 70 73 77 61 65 7195 100 71 73 76 61 65 70 71 73 7640 × 5 grid, seed 21628965 8 6 6 6 6 6 7 6 6 670 27 20 21 22 23 24 24 20 21 2275 57 39 39 40 50 52 53 39 39 4080 79 62 63 66 59 60 62 62 63 6685 93 70 71 76 75 76 78 70 71 7690 99 61 65 69 71 71 75 61 65 6995 100 73 75 76 70 73 75 73 75 76(a) Number of times that a given strategy yieldsbest results. Larger values are better. The thirdcolumn gives the total number of connected in-stances (out of 100). For ∆ ≤ 60, only very fewinstances are connected.TB CMB MBi TL CML ML TL CML ML TL CML ML4 26 26 23 18 18 14 26 26 236 30 30 30 18 14 14 30 30 308 17 17 17 23 23 23 17 17 1712 28 28 28 34 31 31 28 28 2814 25 25 13 32 32 32 25 25 1317 23 23 23 23 23 23 23 23 2320 31 31 31 31 31 31 31 31 3121 24 24 13 28 28 28 24 24 1323 19 19 19 19 19 10 19 19 1924 27 27 27 27 27 27 27 27 2725 17 17 11 17 17 11 17 17 1126 34 34 34 30 30 29 34 34 3427 16 16 10 37 37 37 16 16 1029 25 25 23 20 20 12 25 25 2331 21 21 21 11 11 11 21 21 2132 17 17 17 16 16 8 17 17 1733 31 31 31 13 13 13 31 31 3134 33 33 33 12 12 12 33 33 3338 38 38 38 24 24 16 38 38 3839 17 13 13 16 16 9 17 13 1342 13 13 13 17 12 12 13 13 1343 25 25 16 14 13 13 25 25 1644 34 34 34 32 32 32 34 34 3445 9 9 9 9 9 9 9 9 9(b) Length of the longest edge. Smaller val-ues are better. First 24 connected instancesfrom the 14x14 grid, ∆ = 70. The newstrategies work especially well on this set.Table 2. Results on grid graphs of different shapes and densities.6 Martin Gronemann, Michael Ju¨nger, Sven Mallach, and Daniel R. Schmidt5 333333Fig. 1. Example graph compacted using TB with TL (left) and ML (right).References1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, andApplications. Prentice Hall (1993)2. Batini, C., Di Battista, G., Tamassia, R.: Automatic graph drawing and readabilityof diagrams. IEEE Trans. Syst. Man Cybern. 18, 61–79 (1988)3. Batini, C., Nardelli, E., Tamassia, R.: A layout algorithm for data flow diagrams.IEEE Trans. Softw. Eng. 12, 538–546 (1986)4. Brandes, U., Eiglsperger, M., Kaufmann, M., Wagner, D.: Sketch-driven orthogonalgraph drawing. In: Proc. GD 2002. pp. 1–11. LNCS, Springer (2002)5. Bridgeman, S.S., Di Battista, G., Didimo, W., Liotta, G., Tamassia, R., Vismara,L.: Turn-regularity and optimal area drawings of orthogonal representations. Com-put. Geom. Theory Appl. 16, 53–93 (2000)6. 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Towards shortest longest edges in orthogonal graph drawing

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