Comparison of Maximal Upward Planar Subgraph Computation Algorithms

Transcription

1 01 10th International Conference on Frontiers of Information Technology Comparison of Maximal Upward Planar Subgraph Computation Algorithms Aimal Tariq Rextin Department of Computing and Technology, Abasyn University, Pakistan Abstract A digraph G =(V,E) is upward planar if it has a planar drawing with all edges pointing upward. A subgraph G of a digraph G with an upward planar drawing is called a maximal upward planar subgraph of G if the addition of any edge in G G to G causes non-upward planarity. Binucci et al. showed that finding even the maximum upward planar subgraph of an embedded digraph G φ is NP-Complete [1]. In this paper, we compare different algorithms to find maximal upward planar subgraph of an embedded digraph. We also use a large test suite of embedded digraphs to gain a deeper understanding of upward planarity and see how the different heuristics perform in practice. Index Terms Graph Drawing, Upward Planarity, Algorithms I. INTRODUCTION A graph G =(V,E) consists of a set V, whose elements are called vertices, and a set E of pairs of vertices called edges. It is used to represent relationships between discrete objects. The objects are represented by vertices and the edges represent the relationship between different objects. If the edges has an associated direction then it is a directed graph or digraph. It is generally believed that graphs were first used by Leonhard Eular to study the famous Seven Bridges of Königsberg Problem. Graph Drawing is the area in computer science that is concerned with constructing drawings of graphs that are easily comprehended by humans. In recent years, a lot of research has been done in graph drawing and an overview of the field can be found in [], [3]. Techniques from graph drawing have found applications in a wide variety of other areas, some examples are : VLSI design [4], web searching [5], visualization and analysis of large and complex networks such as social and biological networks [6]. A graph is planar if it has a drawing without any edge crossings, and a drawing of a graph with no edge crossing is called a planar drawing. Let G be a planar graph. There are infinite number of possible planar drawings of G, however all possible planar drawings of G can be be divided into a finite number of topologically equivalent classes. A topologically equivalent class of planar drawings is called an embedding. A particular planar drawing of G realizes φ if it belongs to the equivalence class φ. We called a graph G an embedded graph of φ and denote it by G φ, if we restrict ourselves to planar drawings from a particular embedding φ. We would like to minimize the number of crossings in a drawing of a graph. Although we would like to have no edge crossing in a drawing, but not all graphs admit such a drawing. Moreover, in the case of a digraph, we would want to maximize the edges pointing in the same direction. A digraph G =(V,E) is upward planar if it has a planar drawing with all edges pointing upward []. It is reasonable to say that a non-upward planar digraph is more readable if we can draw the largest possible subgraph in an upward planar fashion. A graph H =(V,E ) is a subgraph of a graph G =(V,E) if V V and E E. A graph H =(V,E ) is a spanning subgraph of a graph G = (V,E) if V = V and E E. Inthis paper, we will only consider spanning subgraphs, because a digraph G is upward planar if and only if all its components are upward planar. This is simple to see because each connected component of an upward planar graph can be independently drawn in an upward planar fashion. A subgraph G of a digraph G with an upward planar drawing is called a maximal upward planar subgraph of G if the addition of any edge in G G to G causes non-upward planarity. Binucci et al. showed that finding even the maximum upward planar subgraph of an embedded digraph G φ is NP- Complete [1]. Binucci et al. [1] studied two heuristics to find the maximal upward planar subgraph for an embedded digraph and compared these heuristics with an exact algorithm. In this paper, we compare their algorithms with some wider heuristics including more general optimization heuristics like Genetic Algorithms and Simulated Annealing. We also use a larger test suite with different types of digraphs to gain a deeper understanding of upward planarity. Another aim is to gain a deeper understanding of how the different heuristics perform in practice. A. Preliminaries First, we briefly review some of the relevant results in the area of upward planarity. An st-digraph is an acyclic digraph G with a single source s, a single sink t and an edge (s, t). Di Battista et al. and Kelly independently showed that a digraph is upward planar if and only if it is a spanning subgraph of a planar st-graph [7], [8]. Garg and Tamassia [9] showed that upward planarity testing is NP-Complete for general digraphs. Bertolazzi and Di Battista designed a quadratic time algorithm to test upward planarity of embedded graphs [10]. Hutton and Lubiw [14] presented an O(n ) upward planarity testing algorithm for single source digraphs, and it was optimized to O(n) by Bertolazzi et al. [11]. Healy and Lynch [1] designed /1 $ IEEE DOI /FIT

2 an algorithm that determines if a digraph is upward planar by checking if its biconnected components have upward planar drawings with certain properties. In a digraph, a source vertex has only outgoing edges and a sink vertex has only incoming edges. We denote the set of sources and sinks in a graph G by S and T respectively; the set of nodes that are neither in S nor in T are represented by I. A path p in a directed graph G is a sequence of vertices, such that every two consecutive vertices in p has an edge between them in G. Every drawing of an embedded graph G φ has a fixed circular ordering of edges φ(v) =e 1,e e k around each vertex v G. Moreover, every drawing of G φ have the same set of facial boundaries which we denote by F. An embedded graph G φ is bimodal if φ(v) can be partitioned into consecutive incoming and outgoing edges for every node v G. Let G φ be a subgraph of G φ, G φ is compatible with G φ if for each vertex v G φ, φ (v) is obtained by deleting zero or more edges from φ(v). In an embedded digraph G φ, the triplet e 1,v,e is an angle if e 1 precedes e in φ(v), where e 1 may equal e. There is a correspondence between each angle e 1,v,e of G φ and the geometrical angle between e 1 and e in a drawing of G φ. An angle is labeled large if the corresponding geometrical angle is larger than π and small otherwise. An angle e 1,v,e is a switch if both e 1 and e point in the same direction. It is a sink-switch if both e 1 and e point toward v, and it is a source-switch if both e 1 and e point away from v. In an acyclic digraph, the number of switches inafacef is always even and is denoted by n f. Lemma 1 (Bertolazzi et al. [10]): Let F be the set of faces in a bimodal embedded digraph G φ. Then (n f 1) + = S + T. f F The O(n )-time embedded upward planarity testing algorithm by Bertolazzi et al. [10] first ensures that G φ is planar, bimodal and has no directed cycles and then checks if G φ satisfies the following theorem. Theorem 1 (Bertolazzi et al. [10]): Let G φ be a be planar acyclic, and bimodal embedded DAG. G φ is upward planar if and only if we can find a mapping from sources and sinks of G φ to its faces, such that the following properties are satisfied. A face h has n h +1 sources or sinks mapped to it; Every other face f has n f 1 sources or sinks mapped to it. The mapping defined in Theorem 1 is called as a consistent assignment. Note that in the final upward planar drawing of G φ, the face h with n h +1 sources or sinks assigned to it is the external face, while all other faces are internal. Given an embedded digraph G φ with a given external face, we define capacity of face f (cap(f)) as following: cap(f) =n f +1 if f is the external face. cap(f) =n f 1 if f is one of the internal faces. II. MAXIMAL UPWARD PLANAR COMPUTATION ALGORITHMS In this section, we introduce the different algorithms to find the maximal upward planar subgraph of an embedded digraph G φ. In the next section, we compare the practical performance of these algorithms. Note that the Edge Insert Algorithm and Bend Algorithm are also discussed in [1], while the other are new algorithms. A. Incremental Algorithms We first propose a set of incremental algorithms, each of which computes a maximal upward planar subgraph for an embedded digraph G φ. Each algorithm maintains an upward planar digraph G φ called the current upward planar subgraph, such that G φ is compatible with G φ. We start with an initial upward planar subgraph and one by one add the removed edges to it. The digraph after the addition of a new edge is denoted by G φ. We retain the new edge if G φ is upward planar, else we discard it. We denote the face g G φ in which the new edge e is restored as the insert-face. We will check if the resulting embedded digraph remains to be bimodal and acyclic after each edge insertion. Bimodality can be checked in O(1) time, while acyclicity testing requires O(n) time. The insertion of an edge e = (u, v) bisects an angle α u = e 1,u,e at vertex u into two new angles e 1,u,e and e, u, e. Similarly, an angle is bisected at v. 1) Edge Insertion Algorithm: In this algorithm, we start with an initial upward planar digraph of all vertices of G φ but none of its edges. We then restore the edges one by one, each time checking if the resulting embedded digraph G φ is upward planar or not. We first check if G φ is bimodal and acyclic, we then use Bertolazzi s algorithm to check if G φ is upward planar. Since G φ is planar so Bertolazzi s algorithm runs O(n) times. The time complexity of Bertolazzi s algorithm is O(n ) [10] and so the worst case time complexity of Algorithm 1 is O(n 3 ). Algorithm 1 Edge Insert G φ =(V, ) for all e/ G φ do restore e in G φ if G φ is not bimodal or not acyclic then if G φ return G φ ) Improved Edge Insertion Algorithm : In this section, we introduce an algorithm which bypasses running Bertolazzi s algorithm at some edge insertions. This is done by efficiently 361

3 constructing a consistent assignment for G φ from the consistent assignment of G φ for some cases. We now discuss the cases where Bertolazzi s algorithm can be bypassed. An edge insertion always changes the facial boundary of the insert face g, and might also change its capacity. We let SW (g) denote the set of switches in the face g G φ and we let SW (g) denote its set of switches in G φ. Moreover, we let ΔSW(g) = SW (g) SW (g). The following lemma relates ΔSW(g) to Δcap(g), which is the change in capacity of g. Lemma : If cap (g) is the capacity of face g G φ and cap (g) is its capacity in G φ, then cap (g) =cap (g) + ΔSW(g). Proof: From the definition of capacity, cap (g) = SW (g) ± 1 Where we take + or depending on if g is an external or an internal face of an embedded digraph G φ with a given external face. Moreover, SW (g) = SW (g) +ΔSW(g). Hence cap (g) = SW (g) +ΔSW(g) ± 1 = SW (g) ± 1+ ΔSW(g) = cap (g)+ ΔSW(g) The following is possible when an edge e is restored: at least one of the end vertices of e is an isolated vertex; e is connects two previously disconnected components of G φ ; and, finally, the restored edge e has both its ends in the same connected component of G φ. We first discuss the case when at least one of the end vertices of e is an isolated vertex. Since G φ will be upward planar when both end vertices of e are isolated vertices, we only discuss the case when there is one isolated vertex. We denote the non-isolated vertex by v and the isolated vertex by v. Let α v = e 1,v,e be the angle that is bisected by the edge insertion, resulting in two new angles α 1 and α. When the α v is not a switch then e will have the same direction as either e 1 or e and hence either α 1 or α will be a switch and the other will not be a switch. Similarly, when e 1,v,e is a switch, then e can either have the same direction as e 1 and e or it can point in the opposite direction. Hence, we can reduce the original 3 =8possible types of edge insertions into the following 3 classes. The following three properties are easy to see from Lemma. Property 1: When an edge e is inserted between node v and a disconnected node v in g G φ such that it divides a non-switch angle e 1,v,e, then ΔSW(g) =and hence Δcap(g) =1. Property : When an edge e is inserted between v and a disconnected node v in g G φ such that e 1,v,e is a switch in G φ and the direction of e is opposite to e 1 and e, then ΔSW(g) =0and hence Δcap(g) =0. Property 3: When an edge e is inserted between v and a disconnected node v in g G φ such that e 1,v,e is a switch in G φ and the direction of e is same as e 1 and e, then ΔSW(g) =and hence Δcap(g) =1. Lemma 3: Let an edge e be inserted between at least one isolated vertex in a embedded upward planar graph G φ with a consistent assignment A. Then, G φ will always be upward planar if both end vertices of e are isolated. Moreover, G φ will be upward planar with the same external face as G φ if G has a non isolated end vertex v and one of the following is true: 1) v is a source/sink in both G and G ; ) v is a source/sink in G but not in G and v was assigned to face g G φ ; 3) v is not a source/sink in G. Proof: If both end vertices of e are isolated in G then we will get a new connected component in G φ with one source and one sink. We will get a consistent assignment for this component by assigning both the source and the sink to its unique face. Hence, G φ is upward planar because all it connected components are upward planar. When one end vertex of e is isolated and the other is not isolated then let v denote the isolated vertex and v the non isolated vertex. We show that G φ always has a consistent assignment under the given conditions. Let a = e 1,v,e be the angle incident on v in face g G φ. When v is a source/sink in both G and G then α is a switch in G φ. Moreover the direction of e in G φ is the same as e 1 and e. Hence we know from Property 3 that Δcap(g) =1.We satisfy the capacity of G φ by assigning all sources/sinks of G to their old faces in G φ and assigning the new source/sink v to g to satisfy the increase in capacity of g. The second case is when v was a source/sink in G that was assigned to g but it is not a source/sink in G. We know that α was a switch in g in G φ but not in G φ, hence from Property Δcap(g) =0. In this case, we assign v to g to compensate for the destruction of v as a source/sink and the remaining sources/sinks of G to their old faces in G φ to get a consistent assignment for G φ. When v is not a source/sink in G, then α can be a switch or a non switch angle. If α is a switch, then e will be inserted in the direction of e 1 and e because otherwise G φ will not be bimodal. In this case Property 3 shows that Δcap(g) = 1. Similarly when α is a non-switch angle then Property 1 shows that Δcap(g) =1. In both these cases we assign all sources/sinks of G to their old faces in G φ and assign the new source/sink v to g to satisfy the increase in capacity of g. When the conditions of Lemma 3 are met, we get a consistent assignment for G φ by retaining the assignment for all sources/sinks of G present in G and assigning the previously isolated vertex to g The following lemma tells us when we can bypass running Bertolazzi s algorithm when an edge is inserted between two connected components of G φ, such that each component has 36

4 at least one edge. Note that the insert-face of both components will merge to form a new face in the resulting graph G φ. Moreover, the insert-face of both components can be an internal or the external face in their respective connected components. Lemma 4: Let G φ be an embedded upward planar graph with at least two connected components Cφ 1 and C φ, each containing of at least one edge. Moreover, let A be the consistent assignment of G φ. Let an edge e be inserted between node v Cφ 1 and v Cφ. We assume that e is inserted in face g 1 of Cφ 1 and g of Cφ, such that at least one of g 1 and g is the external face their respective connected components. The resulting embedded graph G φ will be upward planar if one of the following is true for both v and v. 1) It is a source/sink in both G and G ; or it is a source/sink in G but not in G and it was assigned to its respective insert-face (g 1 or g ); or it is not a source/sink in G. Proof: If g 1 and g are external faces of their respective connected components in G φ, then let g be the external face of G φ. If one of g 1 or g is the external face of its connected component and h (h = g 1 and h = g ) is the external face of the other connected component, then we make h the external face of G φ and g its internal face. We assign all sources/sinks not assigned to g in G φ to the faces they were assigned to in G φ. This will satisfy the capacity of all faces except g in G φ and now all unassigned sources and sinks are on the facial boundary of g. Weknow from Lemma 1 that the sum of unsatisfied capacity equals the number unassigned sources/sinks, therefore we get a consistent assignment for G φ by assigning all unassigned sources/sinks to g. When the above conditions are met, we can get a consistent assignment for G φ by assigning the sources and sinks previously assigned to g 1 or g to the new face g and assigning all other sources and sinks to their previous faces. An edge e in an embedded graph is incident to at most faces, therefore a single edge insertion can at most split a face, resulting in one extra face in G φ. Let us restore an edge e between nodes of the same connected component in the insert-face g, this will split g into face f 1 and face f. The sources and sinks in G φ that were previously assigned to face g can be divided into the following three sets: S f1 is the set of sources/sinks now incident to f 1 but not f, S f is the set of sources/sinks now they are incident to f but not f 1, and S s is the set of sources/sinks now incident to both f 1 and f. We have the following lemma for this case. Lemma 5: Let G φ be an embedded upward planar graph with a consistent assignment A. Let an edge e be added to G φ such that the insert-face g is divided into f 1 and f. Then G φ is upward planar if S f 1 cap(f 1 ), S f cap(f ), and the edge insertion does not destroy a source/sink assigned to a face other than g. Proof: We satisfy the capacity of all faces in G φ except f 1 and f by assigning them the sources and sinks that they were previously assigned to by A. The only unassigned sources/sinks now are S f1, S f and S s.ifg was an internal face in G φ then we assign f 1 and f the capacity of an internal face. However, if g was the external face of G φ then we assign f 1 the capacity of an external face and we assign f the capacity of an internal face. We then assign all sources and sinks of S f1 to face f 1, which is possible since every v S f1 is incident to f 1 and S f1 cap(f 1 ). We also assign all sources and sinks of S f to f, which is possible since every v S f is incident to f and S f cap(f ). The only faces that may have unsatisfied capacity are f 1 and f and the only unassigned sources/sinks are in S s. Since S s is incident to both f 1 and f, we assign S s to f 1 and f to get a consistent assignment for G φ from Lemma 1. When the above conditions are met, then we get a consistent assignment for G φ in the following manner: assign S f 1 to f 1 and assign S f to f ; assign S s such that the capacity of f 1 and f gets completed; and assign all other sources and sinks to the faces they were assign to in G φ These results lead to Algorithm. Algorithm Improved Edge Insert G φ =(V, ) for all edges e/ G φ do restore e in G φ if G φ is not bimodal or not acyclic then if edge is inserted such that the conditions of Lemmas 3, 4 or 5 are satisfied then according to the respec- Assign sources/sinks of G φ tive lemma else run the Bertolazzi algorithm on G φ if G φ return G φ 3) Algorithm: This algorithm is based on the fact that any directed graph whose underlying undirected graph is a tree is always upward planar. In the Tree Insert Algorithm, we first calculated a subgraph G φ of the input embedded digraph G φ, such that the underlying undirected graph of G φ is a tree. We then one by one, insert the remaining edges of G φ, and check if the resulting embedded digraph is upward planar. We discard an edge if its insertion makes G φ non-upward planar, otherwise we insert the next 363

5 edge. Algorithm 3 Tree Insert calculate a spanning tree of G φ let G φ denote the spanning tree for all edges e/ G φ do restore e in G φ if G φ is not bimodal or not acyclic then run the Bertolazzi algorithm on G φ if G φ return G φ B. Genetic Algorithm and Simulated Annealing Based Algorithm Genetic Algorithms and Simulated Annealing are generic methods for optimization that are inspired from biological evaluation and a physical process. Both simulated annealing and genetic algorithms engine require a fitness function, that returns a score that indicates how good or bad a given solution is. The objective of genetic algorithms and simulated annealing is to maximize or minimize the value returned by the fitness function. For our purposes, we used a fitness function that returns 0 if a given instance is non-upward planar, and if a given instance is upward planar then the value returned by the fitness function is the number of edges in that instance. We use the following fitness function: If G φ is not upward planar and U are the sources and sinks that are not assigned to any face, then return 10 U. Otherwise if G φ is upward planar and E be the number of edges in G φ, then the fitness function returns E. We then used MATLAB genetic algorithm engine and its simulated annealing engine to find upward planar subgraphs for each instance of the test suite. We discuss their results in more detail later in the paper. C. Bend Algorithm Quasi-upward planarity is a generalization of upward planarity. In a quasi-upward planar drawing Γ of a digraph G, all outgoing edges of a vertex v leave it from above and all incoming edges of v enter it from below. Bertolazzi et al. [13] discusses an algorithm to compute a quasi-upward planar drawing of G φ with minimum number of bends. Binucci et al. present a heuristic called the Bend Algorithm to find the maximal upward planar subgraph of an embedded digraph G φ [1]. This heuristic uses the concept of quasi-upward planarity. Binucci et al. show that V Edge Density GA(sec) SA (sec) TABLE I SAMPLE OF TIME TAKEN BY GENETIC ALGORITHM AND SIMULATED ANNEALING BASED ALGORITHMS it generally performs better than the Simple Algorithm. The algorithm for bend algorithm is shown in Algorithm 4. Algorithm 4 Bend Algorithm find maximal bimodal subgraph of G φ and denote it by G φ find the quasi-upward planar drawing of G φ remove all edges with non zero bends, let R denote the removed edges for all e R do restore e in G φ if G φ return G φ III. EXPERIMENTAL ANALYSIS AND CONCLUSIONS In this section we discuss the experimental analysis of our algorithms. The algorithms were implemented in C++ using LEDA and AGD libraries and they were executed on a 1.6 GHz processor with 104 MB of memory. We generated a test suite of 5837 random planar digraphs. The number of nodes in a random graph, n, were between 10 and 00. For each n we generate digraphs with edge density 0.9 till.4 with a step of 0.05 Each maximal upward planar subgraph algorithm first generate a planar embedding for each randomly generated digraph and then we proceed with the rest of the algorithm as described in Section II. In the figures we show the results of our experimental analysis of the different algorithms. Our main interest is the retention rate of the different algorithms, which is the percentage of the original edges retained in the final upward planar subgraph of the input embedded digraph. We also measure the time taken the different algorithms. We measure these two quantities with respect to the number of nodes of the input graph and the edge density. Note that, the edge density for a graph G =(V,E) is defined as the following: edge density = E V. We conclude that Bend Algorithm performs very well both in terms of time and retention rate. Moreover, The retention rate depends on the edge density. Further, Our genetic algorithm performs very poorly in terms of retention rate compared to all other algorithms. It retention rate drops 364

6 Retained Edges With Respect to Number of Nodes 7 6 Improved Edge Insert Bend Algorithm Time With Respect to Edge Density for Edge Insert Algos Percentage of Retained Edges Time (secs) Genetic Algorithm Simulated Anealing Bend Algo Number of Nodes Number of Nodes Fig. 1. Retention rate of algorithms with respect to the number of nodes Fig. 4. Time taken by algorithms with respect to edge density Percentage of Retained Edges Time (secs) Retained Edges With Respect to Edge Density 60 Genetic Algorithm Simulated Anealing Bend Algorithm Edge Density Fig Retention rate of algorithms with respect to edge density Improved Edge Insert Bend Algorithm Time With Respect to Number of Nodes Edge Insert Algos Number of Nodes Fig. 3. Time taken algorithms with respect to number of nodes significantly below the other algorithms for graphs with more than 100 nodes. We also observe that the retention rate for all algorithms (except genetic algorithms) are very close. As expected, Genetic algorithms and simulated annealing based algorithms take significantly more time as shown in Table I. We do not recommend these two approaches given their retention rate and time taken. REFERENCES [1] C. Binucci, W. Didimo and F. Giordano, On The Complexity of Finding Maximum Upward Planar Subgraph of an Embedded Planar Digraph. Technical Report RT001-01, University of Perugia, 007. [] Giuseppe Di Battista and Peter Eades and Roberto Tamassia and Ioannis G. Tollis, Graph Drawing: Algorithms for the Visualization of Graphs.Prentice Hall, [3] Michael Kaufmann and Dorothea Wagner, Drawing Graphs, Methods and Models. Springer, 001. [4] MFranco P. Preparata, Optimal Three-Dimensional VLSI Layouts. Mathematical Systems Theory, [5] Emilio Di Giacomo, Walter Didimo, Luca Grilli and Giuseppe Liotta, WhatsOnWeb: Using Graph Drawing to Search the Web. GD 005, 005. [6] Adel Ahmed, Tim Dwyer, Michael Forster, Xiaoyan Fu, Joshua Wing Kei Ho, Seok-Hee Hong, Dirk Koschützki, Colin Murray, Nikola S. Nikolov, Ronnie Taib, Alexandre Tarassov and Kai Xu, GEOMI: GEOmetry for Maximum Insight. GD 005, 005. [7] Giuseppe Di Battista and Roberto Tamassia, Algorithms for Plane Representations of Acyclic Digraphs. Theor. Comput. Sci., [8] D. Kelly, Fundamentals of planar ordered sets. Discrete Mathematics, [9] Ashim Garg and Roberto Tamassia, On the Computational Complexity of Upward and Rectilinear Planarity Testing, SIAM Journal Computing, 001. [10] Paola Bertolazzi, Giuseppe Di Battista, Giuseppe Liotta and Carlo Mannino, Upward Drawings of Triconnected Digraphs. Algorithmica, [11] Paola Bertolazzi, Giuseppe Di Battista, Carlo Mannino and Roberto Tamassia, Optimal Upward Planarity Testing of Single-Source Digraphs. SIAM Journal Computing, [1] Patrick Healy and Karol Lynch, Building Blocks of Upward Planar Digraphs, Proceedings of Graph Drawing, 004. [13] MPaola Bertolazzi, Giuseppe Di Battista and Walter Didimo, Quasi- Upward Planarity, Algorithmica, 00. [14] Michael D. Hutton and Anna Lubiw, Upward Planar Drawing of Single- Source Acyclic Digraphs, SIAM Journal Computing,