2005년 대한 생화학·분자생물학회 하계학술대회를 다녀와서

Despite the success of traditional network analysis, standard networks provide a limited representation of complex systems, which often include different types of relationships (or ‘multiplexity’) between their components. Such structural complexity has a significant effect on both dynamics and function. Throwing away or aggregating available structural information can generate misleading results and be a major obstacle towards attempts to understand complex systems. The recent multilayer approach for modelling networked systems explicitly allows the incorporation of multiplexity and other features of realistic systems. It allows one to couple different structural relationships by encoding them in a convenient mathematical object. It also allows one to couple different dynamical processes on top of such interconnected structures. The resulting framework plays a crucial role in helping to achieve a thorough, accurate understanding of complex systems. The study of multilayer networks has also revealed new physical phenomena that remain hidden when using ordinary graphs, the traditional network representation. Here we survey progress towards attaining a deeper understanding of spreading processes on multilayer networks, and we highlight some of the physical phenomena related to spreading processes that emerge from multilayer structure. Reshaping network theory to describe the multilayered structures of the real world has formed a focus in complex networks research in recent years. Progress in our understanding of dynamical processes is but one of the fruits of this labour.

The physics of spreading processes in multilayer networks

We consider several well-studied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 
1. Is the 3SUM problem on $n$ numbers in $O(n^{2-\epsilon})$ time for some $\epsilon>0$? 
2. Can one determine the satisfiability of a CNF formula on $n$ variables in $O((2-\epsilon)^n poly n)$ time for some $\epsilon>0$? 
3. Is the All Pairs Shortest Paths problem for graphs on $n$ vertices in $O(n^{3-\epsilon})$ time for some $\epsilon>0$? 
4. Is there a linear time algorithm that detects whether a given graph contains a triangle? 
5. Is there an $O(n^{3-\epsilon})$ time combinatorial algorithm for $n\times n$ Boolean matrix multiplication? 
The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems such as Pagh's problem defined in a recent paper by Patrascu [STOC 2010].

Popular conjectures imply strong lower bounds for dynamic problems

An investigation is made of relays whose reliability can be described in simple terms by means of probabilities. It is shown that by using a sufficiently large number of these relays in the proper manner, circuits can be built which are arbitrarily reliable, regardless of how unreliable the original relays are. Various properties of these circuits are elucidated.

Reliable Circuits Using Less Reliable Relays

We consider the point-to-point (approximate) shortest-path query problem, which is the following generalization of the classical single-source (SSSP) and all-pairs shortest-path (APSP) problems: we are first presented with a network (graph). A so-called preprocessing algorithm may compute certain information (a data structure or index) to prepare for the next phase. After this preprocessing step, applications may ask shortest-path or distance queries, which should be answered as fast as possible.Due to its many applications in areas such as transportation, networking, and social science, this problem has been considered by researchers from various communities (sometimes under different names): algorithm engineers construct fast route planning methods; database and information systems researchers investigate materialization tradeoffs, query processing on spatial networks, and reachability queries; and theoretical computer scientists analyze distance oracles and sparse spanners. Related problems are considered for compact routing and distance labeling schemes in networking and distributed computing and for metric embeddings in geometry as well.In this survey, we review selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time. We survey methods for general graphs as well as specialized methods for restricted graph classes, in particular for those classes with arguable practical significance such as planar graphs and complex networks.

/pdf/shortest-path-queries-in-static-networks-t3dlcej7at.pdf

Shortest-path queries in static networks

The diameter is a fundamental graph parameter and its computation is necessary in many applications. The fastest known way to compute the diameter exactly is to solve the All-Pairs Shortest Paths (APSP) problem.In the absence of fast algorithms, attempts were made to seek fast algorithms that approximate the diameter. In a seminal result Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] designed an algorithm that computes in O (n2 + m√n) time an estimate D for the diameter D in directed graphs with nonnegative edge weights, such that [EQUATION], where M is the maximum edge weight in the graph. In recent work, Roditty and Vassilevska W. [STOC 13] gave a Las Vegas algorithm that has the same approximation guarantee but improves the (expected) runtime to O (m√n). Roditty and Vassilevska W. also showed that unless the Strong Exponential Time Hypothesis fails, no O (n2-e) time algorithm for sparse unweighted undirected graphs can achieve an approximation ratio better than 3/2. Thus their algorithm is essentially tight for sparse unweighted graphs. For weighted graphs however, the approximation guarantee can be meaningless, as M can be arbitrarily large.In this paper we exhibit two algorithms that achieve a genuine 3/2-approximation for the diameter, one running in O (m3/2) time, and one running in O (mn2/3). time. Furthermore, our algorithms are deterministic, and thus we present the first deterministic (2 -- e)-approximation algorithm for the diameter that takes subquadratic time in sparse graphs.In addition, we address the question of obtaining an additive c-approximation for the diameter, i.e. an estimate D such that D -- c ≤ D ≤ D. An extremely simple O (mn1-e) time algorithm achieves an additive ne-approximation; no better results are known. We show that for any e > 0, getting an additive ne-approximation algorithm for the diameter running in O (n2-e) time for any δ > 2e would falsify the Strong Exponential Time Hypothesis. Thus the simple algorithm is probably essentially tight for sparse graphs, and moreover, obtaining a subquadratic time additive c-approximation for any constant c is unlikely.Finally, we consider the problem of computing the eccentricities of all vertices in an undirected graph, i.e. the largest distance from each vertex. Roditty and Vassilevska W. [STOC 13] show that in O (m√n) time, one can compute for each v e V in an undirected graph, an estimate e(v) for the eccentricity e (v) such that max{R, 2/3 · e(v)} ≤ e (v) ≤ min {D, 3/2 · e(v)} where R = minv e (v) is the radius of the graph. Here we improve the approximation guarantee by showing that a variant of the same algorithm can achieve estimates e' (v) with 3/5 · e (v) ≤ e' (v) ≤ e (v).

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Better approximation algorithms for the graph diameter

This paper considers additive and purely additive spanners. We present a new purely additive spanner of size O(n7/5) with additive stretch 4. This construction fills in the gap between the two existing constructions for purely additive spanners, one for 2-additive spanner of size O(n3/2) and the other for 6-additive spanner of size O(n4/3), and thus answers a main open question in this area. In addition, we present a construction for additive spanners with O(n1+δ) edges and additive stretch of O(n1/2−3δ/2) for any 3/17 ≤ δ

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973105.36

New additive spanners

Compact Routing Schemes.

The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected n-vertex graph G=(V,E) and an integer k ≥ 1, the subgraph H=(V,E'), E'⊆ E, is a spanner of stretch k (or, a k-spanner) of G if δH(u,v) ≤ k· δG(u,v) for every u,v ∈ V, where δG'(u,v) denotes the distance between u and v in G'. Graph spanners were extensively studied since their introduction over two decades ago. It is known how to efficiently construct a (2k-1)-spanner of size O(n1+1/k), and this size-stretch tradeoff is conjectured to be tight.The notion of fault tolerant spanners was introduced a decade ago in the geometric setting [Levcopoulos et al., STOC'98]. A subgraph H is an f-vertex fault tolerant k-spanner of the graph G if for any set F⊆ V of size at most f and any pair of vertices u,v ∈ V \ F, the distances in H satisfy δH \ F(u,v) ≤ k· δG \ F(u,v). Levcopoulos et al. presented an efficient algorithm that given a set S of n points in Rd, constructs an f-vertex fault tolerant geometric (1+e)-spanner for S, that is, a sparse graph H such that for every set F⊆ S of size f and any pair of points u,v ∈ S \ F, δH \ F(u,v) ≤ (1+e) |uv|, where |uv| is the Euclidean distance between u and v. A fault tolerant geometric spanner with optimal maximum degree and total weight was presented in [Czumaj & Zhao, SoCG'03]. This paper also raised as an open problem the question whether it is possible to obtain a fault tolerant spanner for an arbitrary undirected weighted graph.The current paper answers this question in the affirmative, presenting an f-vertex fault tolerant (2k-1)-spanner of size O(f2 kf+1 · n1+1/klog1-1/kn). Interestingly, the stretch of the spanner remains unchanged while the size of the spanner only increases by a factor that depends on the stretch k, on the number of potential faults f, and on logarithmic terms in n. In addition, we consider the simpler setting of f-edge fault tolerant spanners (defined analogously). We present an f-edge fault tolerant 2k-1 spanner with edge set of size O(f· n1+1/k) (only f times larger than standard spanners). For both edge and vertex faults, our results are shown to hold when the given graph G is weighted.

/pdf/fault-tolerant-spanners-for-general-graphs-1oupcaqejl.pdf

Fault-tolerant spanners for general graphs

An approximate distance oracle is a succinct data structure that provides fast answers to distance queries between any two nodes of a given graph. In this paper we consider approximate distance oracles for general undirected graphs with non-negative edge weights with constant query time. We present a distance oracle of size O(kn1+1/k), with 2k --- 1 stretch and O(1) query time. This improves the O(log k) query time of Wulff-Nilsen's distance oracle [SODA '13], which in turn improved the O(k) query time of Thorup and Zwick's distance oracle [J. ACM '05].

Shiri Chechik

Papers

Better approximation algorithms for the graph diameter

New additive spanners

Compact Routing Schemes.

Fault-tolerant spanners for general graphs

Approximate distance oracles with constant query time