Fast Shortest Path Distance Estimation in Large Networks

We study the problem of preprocessing a large graph so that point-to-point shortest-path queries can be answered very fast. Computing shortest paths is a well studied problem, but exact algorithms do not scale to huge graphs encountered on the web, social networks, and other applications. In this paper we focus on approximate methods for distance estimation, in particular using landmark-based distance indexing. This approach involves selecting a subset of nodes as landmarks and computing (offline) the distances from each node in the graph to those landmarks. At runtime, when the distance between a pair of nodes is needed, we can estimate it quickly by combining the precomputed distances of the two nodes to the landmarks. We prove that selecting the optimal set of landmarks is an NP-hard problem, and thus heuristic solutions need to be employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the suggested techniques is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach in the literature which considers selecting landmarks at random. Finally, we study applications of our method in two problems arising naturally in large-scale networks, namely, social search and community detection.

Exact Geosedics and Shortest Paths on Polyhedral Surface

We present two algorithms for computing distances along a non-convex polyhedral surface. The first algorithm computes exact minimal-geodesic distances and the second algorithm combines these distances to compute exact shortest-path distances along the surface. Both algorithms have been extended to compute the exact minimalgeodesic paths and shortest paths. These algorithms have been implemented and validated on surfaces for which the correct solutions are known, in order to verify the accuracy and to measure the run-time performance, which is cubic or less for each algorithm. The exact-distance computations carried out by these algorithms are feasible for large-scale surfaces containing tens of thousands of vertices, and are a necessary component of near-isometric surface flattening methods that accurately transform curved manifolds into flat representations.

Small-World Internet Topologies: Possible Causes and Implications on Scalability of End-System Multicast

Recent work has shown the prevalence of small-world phenomena [28] in many networks. Small-world graphs exhibit a high degree of clustering, yet have typically short path lengths between arbitrary vertices. Internet AS-level graphs have been shown to exhibit small-world behaviors [9]. In this paper, we show that both Internet AS-level and router-level graphs exhibit small-world behavior. We attribute such behavior to two possible causes–namely the high variability of vertex degree distributions (which were found to follow approximately a power law [15]) and the preference of vertices to have local connections. We show that both factors contribute with different relative degrees to the small-world behavior of AS-level and router-level topologies. Our findings underscore the inefficacy of the Barabasi-Albert model [6] in explaining the growth process of the Internet, and provide a basis for more promising approaches to the development of Internet topology generators. We present such a generator and show the resemblance of the synthetic graphs it generates to real Internet AS-level and router-level graphs. Using these graphs, we have examined how small-world behaviors affect the scalability of end-system multicast. Our findings indicate that lower variability of vertex degree and stronger preference for local connectivity in small-world graphs results in slower network neighborhood expansion, and in longer average path length between two arbitrary vertices, which in turn results in better scaling of end system multicast.

On the Scalability-Performance Tradeoffs in MPLS and IP Routing

MPLS (Multi-Protocol Label Switching) has recently emerged to facilitate the engineering of network traffic. This can be achieved by directing packet flows over paths that satisfy multiple requirements. MPLS has been regarded as an enhancement to traditional IP routing, which has the following problems: (1) all packets with the same IP destination address have to follow the same path through the network; and (2) paths have often been computed based on static and single link metrics. These problems may cause traffic concentration, and thus degradation in quality of service. In this paper, we investigate by simulations a range of routing solutions and examine the tradeoff between scalability and performance. At one extreme, IP packet routing using dynamic link metrics provides a stateless solution but may lead to routing oscillations. At the other extreme, we consider a recently proposed Profile-based Routing (PBR), which uses knowledge of potential ingress-egress pairs as well as the traffic profile among them. Minimum Interference Routing (MIRA) is another recently proposed MPLS-based scheme, which only exploits knowledge of potential ingress-egress pairs but not their traffic profile. MIRA and the more conventional widest-shortest path (WSP) routing represent alternative MPLS-based approaches on the spectrum of routing solutions. We compare these solutions in terms of utility, bandwidth acceptance ratio as well as their scalability (routing state and computational overhead) and load balancing capability. While the simplest of the per-flow algorithms we consider, the performance of WSP is close to dynamic per-packet routing, without the potential instabilities of dynamic routing.

Indexing Distances in Large Graphs and Applications in Search Tasks

This thesis elaborates on the problem of preprocessing a large graph so that single-pair shortest-path queries can be answered quickly at runtime. Computing shortest paths is a well studied problem, but exact algorithms do not scale well to real-world huge graphs in applications that require very short response time. The focus is on approximate methods for distance estimation, in particular in landmarks-based distance indexing. This approach involves choosing some nodes as landmarks and computing (offline), for each node in the graph its embedding, i.e., the vector of its distances from all the landmarks. At runtime, when the distance between a pair of nodes is queried, it can be quickly estimated by combining the embeddings of the two nodes. Choosing optimal landmarks is shown to be hard and thus heuristic solutions are employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the techniques presented in this thesis is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach which considers selecting landmarks at random. Finally, they are applied in two important problems arising naturally in large-scale graphs, namely social search and community detection.

Search Space Reduction in QoS Routing

To provide real-time service or engineer constrained-based paths, networks require the underlying routing algorithm to be able to find low-cost paths that satisfy given Quality-of-Service (QoS) constraints. However, the problem of constrained shortest (least-cost) path routing is known to be NP-hard, and some heuristics have been proposed to find a near-optimal solution. However, these heuristics either impose relationships among the link metrics to reduce the complexity of the problem which may limit the general applicability of the heuristic, or are too costly in terms of execution time to be applicable to large networks. In this paper, we focus on solving the delay-constrained minimum-cost path problem, and present a fast algorithm to find a near-optimal solution. This algorithm, called DCCR (for Delay-Cost-Constrained Routing), is a variant of the k-shortest path algorithm. DCCR uses a new adaptive path weight function together with an additional constraint imposed on the path cost, to restrict the search space. Thus, DCCR can return a near-optimal solution in a very short time. Furthermore, we use the method proposed by Blokh and Gutin to further reduce the search space by using a tighter bound on path cost. This makes our algorithm more accurate and even faster. We call this improved algorithm SSR+DCCR (for Search Space Reduction+DCCR). Through extensive simulations, we confirm that SSR+DCCR performs very well compared to the optimal but very expensive solution.

On the Origin of Power Laws in Internet Topologies

Recent empirical studies have shown that Internet topologies exhibit power laws of the form for the following relationships: (P1) outdegree of node (domain or router) versus rank; (P2) number of nodes versus outdegree; (P3) number of node pairs y = x^α within a neighborhood versus neighborhood size (in hops); and (P4) eigenvalues of the adjacency matrix versus rank. However, causes for the appearance of such power laws have not been convincingly given. In this paper, we examine four factors in the formation of Internet topologies. These factors are (F1) preferential connectivity of a new node to existing nodes; (F2) incremental growth of the network; (F3) distribution of nodes in space; and (F4) locality of edge connections. In synthetically generated network topologies, we study the relevance of each factor in causing the aforementioned power laws as well as other properties, namely diameter, average path length and clustering coefficient. Different kinds of network topologies are generated: (T1) topologies generated using our parametrized generator, we call BRITE; (T2) random topologies generated using the well-known Waxman model; (T3) Transit-Stub topologies generated using GT-ITM tool; and (T4) regular grid topologies. We observe that some generated topologies may not obey power laws P1 and P2. Thus, the existence of these power laws can be used to validate the accuracy of a given tool in generating representative Internet topologies. Power laws P3 and P4 were observed in nearly all considered topologies, but different topologies showed different values of the power exponent α. Thus, while the presence of power laws P3 and P4 do not give strong evidence for the representativeness of a generated topology, the value of α in P3 and P4 can be used as a litmus test for the representativeness of a generated topology. We also find that factors F1 and F2 are the key contributors in our study which provide the resemblance of our generated topologies to that of the Internet.