2020 [ nach oben ]

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  Bläsius, Thomas; Friedrich, Tobias; Schirneck, MartinThe Minimization of Random Hypergraphs. European Symposium on Algorithms (ESA) 2020: 21:1-21:15
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  We investigate the maximum-entropy model \(\mathcal{B}_{n,m,p}\) for random \(n\)-vertex, \(m\)-edge multi-hypergraphs with expected edge size \(pn\). We show that the expected size of the minimization \(\min(\mathcal{B}_{n,m,p})\), i.e., the number of inclusion-wise minimal edges of \(\mathcal{B}_{n,m,p}\), undergoes a phase transition with respect to \(m\). If \(m\) is at most \(1/(1-p)^(1-p)n}\), then \(\mathrm{E[|\min(\mathcal{B}_{n,m,p})|]\) is of order \(\Theta(m)\), while for \(m ge 1/(1-p)^(1-p+\varepsilon)n}\) for any \(\varepsilon > 0\), it is \(\Theta( 2^(\mathrm{H(\alpha) + (1-\alpha) \log_2 p) n/ \sqrt{n})\). Here, \(\mathrm{H}\) denotes the binary entropy function and \(alpha = - (\log_{1-p m)/n\). The result implies that the maximum expected number of minimal edges over all \(m\) is \(\Theta((1+p)^n/\sqrt{n})\). Our structural findings have algorithmic implications for minimizing an input hypergraph. This has applications in the profiling of relational databases as well as for the Orthogonal Vectors problem studied in fine-grained complexity. We make several technical contributions that are of independent interest in probability. First, we improve the Chernoff--Hoeffding theorem on the tail of the binomial distribution. In detail, we show that for a binomial variable \(Y sim \mathrm{Bin(n,p)\) and any \(0 < x < p\), it holds that \(\mathrm{P[Y le xn] = \Theta( 2^-\!\mathrm{D(x \,\|}\, p) n}/\sqrt{n})\), where \(\mathrm{D}\) is the binary Kullback--Leibler divergence between Bernoulli distributions. We give explicit upper and lower bounds on the constants hidden in the big-O notation that hold for all \(n\). Secondly, we establish the fact that the probability of a set of cardinality \(i\) being minimal after \(m\) i.i.d. maximum-entropy trials exhibits a sharp threshold behavior at \(i^* = n + \log_{1-p m\).
  @inproceedings{Blaesius20MinimizationESA, abstract = {We investigate the maximum-entropy model \(\mathcal{B}_{n,m,p}\) for random \(n\)-vertex, \(m\)-edge multi-hypergraphs with expected edge size \(pn\). We show that the expected size of the minimization \(\min(\mathcal{B}_{n,m,p})\), i.e., the number of inclusion-wise minimal edges of \(\mathcal{B}_{n,m,p}\), undergoes a phase transition with respect to \(m\). If \(m\) is at most \(1/(1-p)^{(1-p)n}\), then \(\mathrm{E}[|\min(\mathcal{B}_{n,m,p})|]\) is of order \(\Theta(m)\), while for \(m \ge 1/(1-p)^{(1-p+\varepsilon)n}\) for any \(\varepsilon > 0\), it is \(\Theta( 2^{(\mathrm{H}(\alpha) + (1-\alpha) \log_2 p) n}/ \sqrt{n})\). Here, \(\mathrm{H}\) denotes the binary entropy function and \(\alpha = - (\log_{1-p} m)/n\). The result implies that the maximum expected number of minimal edges over all \(m\) is \(\Theta((1+p)^n/\sqrt{n})\). Our structural findings have algorithmic implications for minimizing an input hypergraph. This has applications in the profiling of relational databases as well as for the Orthogonal Vectors problem studied in fine-grained complexity. We make several technical contributions that are of independent interest in probability. First, we improve the Chernoff--Hoeffding theorem on the tail of the binomial distribution. In detail, we show that for a binomial variable \(Y \sim \mathrm{Bin}(n,p)\) and any \(0 < x < p\), it holds that \(\mathrm{P}[Y \le xn] = \Theta( 2^{-\!\mathrm{D}(x \,{\|}\, p) n}/\sqrt{n})\), where \(\mathrm{D}\) is the binary Kullback--Leibler divergence between Bernoulli distributions. We give explicit upper and lower bounds on the constants hidden in the big-O notation that hold for all \(n\). Secondly, we establish the fact that the probability of a set of cardinality \(i\) being minimal after \(m\) i.i.d. maximum-entropy trials exhibits a sharp threshold behavior at \(i^* = n + \log_{1-p} m\).}, author = {Bläsius, Thomas and Friedrich, Tobias and Schirneck, Martin}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {esa martinschirneck thomasblaesius tobiasfriedrich year2020}, pages = {21:1-21:15}, title = {The Minimization of Random Hypergraphs}, year = 2020 }

2019 [ nach oben ]

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  Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian; Meyer, Ulrich; Penschuck, Manuel; Weyand, ChristopherEfficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs. European Symposium on Algorithms (ESA) 2019: 21:2-21:14

  EATCS Best Paper Award
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a power-law degree distribution with controllable exponent \(\beta\), and high clustering that can be controlled via the temperature \(T\). We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to \(T = 0\). We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, i.e., they involve no approximation. Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify \(\bar{d}\) as input and obtain a graph with expected average degree \(\bar{d}\). Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straight-forward inclusion does not hold in practice. However, the difference is negligible for most use cases.
  @inproceedings{blaesius2019efficiently, abstract = {Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a power-law degree distribution with controllable exponent \(\beta\), and high clustering that can be controlled via the temperature \(T\). We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to \(T = 0\). We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, i.e., they involve no approximation. Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify \(\bar{d}\) as input and obtain a graph with expected average degree \(\bar{d}\). Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straight-forward inclusion does not hold in practice. However, the difference is negligible for most use cases.}, author = {Bläsius, Thomas and Friedrich, Tobias and Katzmann, Maximilian and Meyer, Ulrich and Penschuck, Manuel and Weyand, Christopher}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {christopherweyand esa hyperbolic_generation manuelpenschuck maximiliankatzmann thomasblaesius tobiasfriedrich ulrichmeyer year2019}, note = {EATCS Best Paper Award}, pages = {21:2-21:14}, title = {Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs}, year = 2019 }

2017 [ nach oben ]

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  Friedrich, Tobias; Krohmer, Anton; Rothenberger, Ralf; Sauerwald, Thomas; Sutton, Andrew M.Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT. European Symposium on Algorithms (ESA) 2017: 37:1-37:15
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The average-case analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random \(k\)-SAT on \(n\) variables, \(m=\Theta(n)\) clauses, and a power law distribution on the variable occurrences with exponent \(\beta\). We observe a satisfiability threshold at \(\beta=(2k-1)/(k-1)\). This threshold is tight in the sense that instances with \( \beta\) \(< (2k-1)/(k-1)-\varepsilon\) for any constant \(\varepsilon>0\) are unsatisfiable with high probability (w.h.p.). For \(\beta\ge(2k-1)/(k-1)+\varepsilon\), the picture is reminiscent of the uniform case: instances are satisfiable w.h.p. for sufficiently small constant clause-variable ratios \(m/n\); they are unsatisfiable above a ratio \(m/n\) that depends on \(\beta\).
  @inproceedings{friedrich2017bounds, abstract = {Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The average-case analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random \(k\)-SAT on \(n\) variables, \(m=\Theta(n)\) clauses, and a power law distribution on the variable occurrences with exponent \(\beta\). We observe a satisfiability threshold at \(\beta=(2k-1)/(k-1)\). This threshold is tight in the sense that instances with \( \beta\) \(< (2k-1)/(k-1)-\varepsilon\) for any constant \(\varepsilon>0\) are unsatisfiable with high probability (w.h.p.). For \(\beta\ge(2k-1)/(k-1)+\varepsilon\), the picture is reminiscent of the uniform case: instances are satisfiable w.h.p. for sufficiently small constant clause-variable ratios \(m/n\); they are unsatisfiable above a ratio \(m/n\) that depends on \(\beta\).}, author = {Friedrich, Tobias and Krohmer, Anton and Rothenberger, Ralf and Sauerwald, Thomas and Sutton, Andrew M.}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {andrewmsutton antonkrohmer esa ralfrothenberger scalefree_sat tobiasfriedrich year2017}, month = {sep}, pages = {37:1-37:15}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik}, series = {LIPIcs}, title = {Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT}, volume = 87, year = 2017 }

2016 [ nach oben ]

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  Arndt, Tobias; Hafner, Danijar; Kellermeier, Thomas; Krogmann, Simon; Razmjou, Armin; Krejca, Martin S.; Rothenberger, Ralf; Friedrich, TobiasProbabilistic Routing for On-Street Parking Search. European Symposium on Algorithms (ESA) 2016: 6:1-6:13
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  An estimated \(30\%\) of urban traffic is caused by search for parking spots. Traffic could be reduced by suggesting effective routes leading along potential parking spots. In this paper, we formalize parking search as a probabilistic problem on a road graph and show that it is NP-complete. We explore heuristics that optimize for the driving duration and the walking distance to the destination. Routes are constrained to reach a certain probability threshold of finding a spot. Empirically estimated probabilities of successful parking attempts are provided by TomTom on a per-street basis. We release these probabilities as a dataset of about 80,000 roads covering the Berlin area. This allows to evaluate parking search algorithms on a real road network with realistic probabilities for the first time. However, for many other areas, parking probabilities are not openly available. Because they are effortful to collect, we propose an algorithm that relies on conventional road attributes only. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. This leads to the conclusion that conventional road attributes may be sufficient to compute reasonably good parking search routes.
  @inproceedings{DBLP:conf/esa/ArndtHKKRKRF16, abstract = {An estimated \(30\%\) of urban traffic is caused by search for parking spots. Traffic could be reduced by suggesting effective routes leading along potential parking spots. In this paper, we formalize parking search as a probabilistic problem on a road graph and show that it is NP-complete. We explore heuristics that optimize for the driving duration and the walking distance to the destination. Routes are constrained to reach a certain probability threshold of finding a spot. Empirically estimated probabilities of successful parking attempts are provided by TomTom on a per-street basis. We release these probabilities as a dataset of about 80,000 roads covering the Berlin area. This allows to evaluate parking search algorithms on a real road network with realistic probabilities for the first time. However, for many other areas, parking probabilities are not openly available. Because they are effortful to collect, we propose an algorithm that relies on conventional road attributes only. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. This leads to the conclusion that conventional road attributes may be sufficient to compute reasonably good parking search routes.}, author = {Arndt, Tobias and Hafner, Danijar and Kellermeier, Thomas and Krogmann, Simon and Razmjou, Armin and Krejca, Martin S. and Rothenberger, Ralf and Friedrich, Tobias}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {ESA arminrazmjou danijarhafner martinskrejca ralfrothenberger simonkrogmann thomaskellermeier tobiasarndt tobiasfriedrich year2016}, pages = {6:1-6:13}, title = {Probabilistic Routing for On-Street Parking Search}, year = 2016 }
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  Baum, Moritz; Bläsius, Thomas; Gemsa, Andreas; Rutter, Ignaz; Wegner, FranziskaScalable Exact Visualization of Isocontours in Road Networks via Minimum-Link Paths. European Symposium on Algorithms (ESA) 2016: 7:1-7:18
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, large-scale networks. We propose isocontours represented by polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that run in (almost) linear time and are simple enough to be implemented in practice. A key ingredient is a new practical linear-time algorithm for minimum-link paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, computing near-optimal solutions in a few milliseconds on average, even for long ranges.
  @inproceedings{DBLP:conf/esa/BaumBGRW16, abstract = {Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, large-scale networks. We propose isocontours represented by polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that run in (almost) linear time and are simple enough to be implemented in practice. A key ingredient is a new practical linear-time algorithm for minimum-link paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, computing near-optimal solutions in a few milliseconds on average, even for long ranges.}, author = {Baum, Moritz and Bläsius, Thomas and Gemsa, Andreas and Rutter, Ignaz and Wegner, Franziska}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {ESA thomasblaesius year2016}, pages = {7:1-7:18}, title = {Scalable Exact Visualization of Isocontours in Road Networks via Minimum-Link Paths}, year = 2016 }
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  Bläsius, Thomas; Friedrich, Tobias; Krohmer, AntonHyperbolic Random Graphs: Separators and Treewidth. European Symposium on Algorithms (ESA) 2016: 15:1-15:16
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  When designing and analyzing algorithms, one can obtain better and more realistic results for practical instances by assuming a certain probability distribution on the input. The worst-case run-time is then replaced by the expected run-time or by bounds that hold with high probability (whp), i.e., with probability \(1 - O(1/n)\), on the random input. Hyperbolic random graphs can be used to model complex real-world networks as they share many important properties such as a small diameter, a large clustering coefficient, and a power-law degree-distribution. Divide and conquer is an important algorithmic design principle that works particularly well if the instance admits small separators. We show that hyperbolic random graphs in fact have comparatively small separators. More precisely, we show that a hyperbolic random graph can be expected to have a balanced separator hierarchy with separators of size \(O(\sqrt{n^{(3-\beta)}})\), \(O(\log n)\), and \(O(1)\) if \(2 < \beta < 3\), \(\beta = 3\) and \(3 < \beta\), respectively (\(\beta\) is the power-law exponent). We infer that these graphs have whp a treewidth of \(O(\sqrt{n^{(3 - \beta)}})\), \(O(\log^{2}n)\), and \(O(\log n)\), respectively. For \(2 < \beta < 3\), this matches a known lower bound. For the more realistic (but harder to analyze) binomial model, we still prove a sublinear bound on the treewidth. To demonstrate the usefulness of our results, we apply them to obtain fast matching algorithms and an approximation scheme for Independent Set.
  @inproceedings{DBLP:conf/esa/BlasiusFK16, abstract = {When designing and analyzing algorithms, one can obtain better and more realistic results for practical instances by assuming a certain probability distribution on the input. The worst-case run-time is then replaced by the expected run-time or by bounds that hold with high probability (whp), i.e., with probability \(1 - O(1/n)\), on the random input. Hyperbolic random graphs can be used to model complex real-world networks as they share many important properties such as a small diameter, a large clustering coefficient, and a power-law degree-distribution. Divide and conquer is an important algorithmic design principle that works particularly well if the instance admits small separators. We show that hyperbolic random graphs in fact have comparatively small separators. More precisely, we show that a hyperbolic random graph can be expected to have a balanced separator hierarchy with separators of size \(O(\sqrt{n^{(3-\beta)}})\), \(O(\log n)\), and \(O(1)\) if \(2 < \beta < 3\), \(\beta = 3\) and \(3 < \beta\), respectively (\(\beta\) is the power-law exponent). We infer that these graphs have whp a treewidth of \(O(\sqrt{n^{(3 - \beta)}})\), \(O(\log^{2}n)\), and \(O(\log n)\), respectively. For \(2 < \beta < 3\), this matches a known lower bound. For the more realistic (but harder to analyze) binomial model, we still prove a sublinear bound on the treewidth. To demonstrate the usefulness of our results, we apply them to obtain fast matching algorithms and an approximation scheme for Independent Set.}, author = {Bläsius, Thomas and Friedrich, Tobias and Krohmer, Anton}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {ESA antonkrohmer hyperbolic_analysis thomasblaesius tobiasfriedrich year2016}, pages = {15:1-15:16}, title = {Hyperbolic Random Graphs: Separators and Treewidth}, year = 2016 }
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  Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton; Laue, SörenEfficient Embedding of Scale-Free Graphs in the Hyperbolic Plane. European Symposium on Algorithms (ESA) 2016: 16:1-16:18

  EATCS Best Paper Award
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require a runtime of \(\Omega(n^{2})\). Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Log-likelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.
  @inproceedings{DBLP:conf/esa/BlasiusFKL16, abstract = {Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require a runtime of \(\Omega(n^{2})\). Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Log-likelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.}, author = {Bläsius, Thomas and Friedrich, Tobias and Krohmer, Anton and Laue, Sören}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {ESA antonkrohmer bestpaper hyperbolic_embedding thomasblaesius tobiasfriedrich year2016}, note = {EATCS Best Paper Award}, pages = {16:1-16:18}, title = {Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane}, year = 2016 }

2014 [ nach oben ]

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  Bringmann, Karl; Friedrich, Tobias; Krohmer, AntonDe-anonymization of Heterogeneous Random Graphs in Quasilinear Time. European Symposium on Algorithms (ESA) 2014: 197-208
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  There are hundreds of online social networks with billions of users in total. Many such networks publicly release structural information, with all personal information removed. Empirical studies have shown, however, that this provides a false sense of privacy - it is possible to identify almost all users that appear in two such anonymized network as long as a few initial mappings are known. We analyze this problem theoretically by reconciling two versions of an artificial power-law network arising from independent subsampling of vertices and edges. We present a new algorithm that identifies most vertices and makes no wrong identifications with high probability. The number of vertices matched is shown to be asymptotically optimal. For an \(n\)-vertex graph, our algorithm uses \(n^\epsilon\) seed nodes (for an arbitrarily small \(\epsilon\)) and runs in quasilinear time. This improves previous theoretical results which need \(\Theta(n)\) seed nodes and have runtimes of order \(n^{1 + \Omega(1)}\). Additionally, the applicability of our algorithm is studied experimentally on different networks.
  @inproceedings{DBLP:conf/esa/BringmannFK14, abstract = {There are hundreds of online social networks with billions of users in total. Many such networks publicly release structural information, with all personal information removed. Empirical studies have shown, however, that this provides a false sense of privacy - it is possible to identify almost all users that appear in two such anonymized network as long as a few initial mappings are known. We analyze this problem theoretically by reconciling two versions of an artificial power-law network arising from independent subsampling of vertices and edges. We present a new algorithm that identifies most vertices and makes no wrong identifications with high probability. The number of vertices matched is shown to be asymptotically optimal. For an \(n\)-vertex graph, our algorithm uses \(n^\epsilon\) seed nodes (for an arbitrarily small \(\epsilon\)) and runs in quasilinear time. This improves previous theoretical results which need \(\Theta(n)\) seed nodes and have runtimes of order \(n^{1 + \Omega(1)}\). Additionally, the applicability of our algorithm is studied experimentally on different networks.}, author = {Bringmann, Karl and Friedrich, Tobias and Krohmer, Anton}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {antonkrohmer esa imported tobiasfriedrich year2014}, pages = {197-208}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {De-anonymization of Heterogeneous Random Graphs in Quasilinear Time}, volume = 8737, year = 2014 }