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Improved dynamic algorithms for maintaining approximate shortest paths under deletions
 In 22nd ACM Symp. on Discrete Algorithms (SODA
"... We present the first dynamic shortest paths algorithms that make any progress beyond a longstanding O(n) update time barrier (while maintaining a reasonable query time), although it is only progress for nottoosparse graphs. In particular, we obtain new decremental algorithms for two approximate sh ..."
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We present the first dynamic shortest paths algorithms that make any progress beyond a longstanding O(n) update time barrier (while maintaining a reasonable query time), although it is only progress for nottoosparse graphs. In particular, we obtain new decremental algorithms for two approximate shortestpath problems in unweighted, undirected graphs. Both algorithms are randomized (Las Vegas). • Given a source s, we present an algorithm that maintains (1 + ɛ)approximate shortest paths from s with an expected total update time of Õ(n 2+O(1/ √ log n)) over all deletions (so the amortized time is about Õ(n2 /m)). The worstcase
Averagecase analysis of online topological ordering
 of Lecture Notes in Computer Science
, 2007
"... Many applications like pointer analysis and incremental compilation require maintaining a topological ordering of the nodes of a directed acyclic graph (DAG) under dynamic updates. All known algorithms for this problem are either only analyzed for worstcase insertion sequences or only evaluated exp ..."
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Many applications like pointer analysis and incremental compilation require maintaining a topological ordering of the nodes of a directed acyclic graph (DAG) under dynamic updates. All known algorithms for this problem are either only analyzed for worstcase insertion sequences or only evaluated experimentally on random DAGs. We present the first averagecase analysis of online topological ordering algorithms. We prove an expected runtime of O(n 2 polylog(n)) under insertion of the edges of a complete DAG in a random order for the algorithms of Alpern et
On Pairwise Spanners
, 2013
"... Given an undirected nnode unweighted graph G = (V, E), a spanner with stretch function f(·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparses ..."
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Given an undirected nnode unweighted graph G = (V, E), a spanner with stretch function f(·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the uv distance only for pairs (u, v) in a given set P ⊆ V × V. Such Pspanners were studied before [Coppersmith,Elkin’05] only in the special case that f(·) is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same P) and of the best known spanners (with the same f(·)). In more detail, for arbitrary P, we show that there exists a Pspanner of size O(n(P  log n) 1/4) with f(d) = d+4 log n. Alternatively, for any ε> 0, there exists a Pspanner of size O(nP  1/4 log n ε) with f(d) = (1 + ε)d + 4. We also consider the relevant special case that there is a critical set of nodes S ⊆ V, and we wish to approximate either the distances within nodes in S or from nodes in S to any other node. We show that there exists an (S × S)spanner of size O(n √ S) with f(d) = d + 2, and an (S × V)spanner of size O(n √ S  log n) with f(d) = d + 2 log n. All the mentioned pairwise spanners can be constructed in polynomial time.
Small stretch spanners on dynamic graphs
 IN PROCEEDINGS OF 13TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS, VOLUME 3669 OF LNCS
, 2005
"... We present fully dynamic algorithms for maintaining 3 and 5spanners of undirected graphs. For unweighted graphs we maintain a 3or 5spanner under insertions and deletions of edges in O(n) amortized time per operation over a sequence of Ω(n) updates. The maintained 3spanner (resp., 5spanner) has ..."
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We present fully dynamic algorithms for maintaining 3 and 5spanners of undirected graphs. For unweighted graphs we maintain a 3or 5spanner under insertions and deletions of edges in O(n) amortized time per operation over a sequence of Ω(n) updates. The maintained 3spanner (resp., 5spanner) has O(n 3/2) edges (resp., O(n 4/3)edges), which is known to be optimal. On weighted graphs with d different edge cost values, we maintain a 3 or 5spanner in O(n) amortized time per operation over a sequence of Ω(d·n) updates. The maintained 3spanner (resp., 5spanner) has O(d·n 3/2) edges (resp., O(d·n 4/3)edges). The same approach can be extended to graphs with realvalued edge costs in the range [1,C]. All our algorithms are deterministic and are substantially faster than recomputing a spanner from scratch after each update.
Finding Topk Shortest Path Distance Changes in an Evolutionary Network
"... Abstract. Networks can be represented as evolutionary graphs in a variety of spatiotemporal applications. Changes in the nodes and edges over time may also result in corresponding changes in structural garph properties such as shortest path distances. In this paper, we study the problem of detectin ..."
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Abstract. Networks can be represented as evolutionary graphs in a variety of spatiotemporal applications. Changes in the nodes and edges over time may also result in corresponding changes in structural garph properties such as shortest path distances. In this paper, we study the problem of detecting the topk most significant shortestpath distance changes between two snapshots of an evolving graph. While the problem is solvable with two applications of the allpairs shortest path algorithm, such a solution would be extremely slow and impractical for very large graphs. This is because when a graph may contain millions of nodes, even the storage of distances between all node pairs can become inefficient in practice. Therefore, it is desirable to design algorithms which can directly determine the significant changes in shortest path distances, without materializing the distances in individual snapshots. We present algorithms that are up to two orders of magnitude faster than such a solution, while retaining comparable accuracy. 1
Fully Dynamic Randomized Algorithms for Graph Spanners
, 2008
"... Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. ..."
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Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a tspanner of itself, the research as well as applications of spanners invariably deal with a tspanner which has as small number of edges as possible. We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner. Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a tspanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a tspanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.
On the Complexity of TimeDependent Shortest Paths
"... We investigate the complexity of shortest paths in timedependent graphs, in which the costs of edges vary as a function of time, and as a result the shortest path between two nodes s and d can change over time. Our main result is that when the edge cost functions are (polynomialsize) piecewise line ..."
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We investigate the complexity of shortest paths in timedependent graphs, in which the costs of edges vary as a function of time, and as a result the shortest path between two nodes s and d can change over time. Our main result is that when the edge cost functions are (polynomialsize) piecewise linear, the shortest path from s to d can change Θ(log n) n times, settling a severalyearold conjecture of Dean [Technical Reports, 1999, 2004]. We also show that the complexity is polynomial if the slopes of the linear function come from a restricted class, present an outputsensitive algorithm for the general case, and describe a scheme for a (1 + ɛ)approximation of the travel time function in nearquadratic space. Finally, despite the fact that the arrival time function may have superpolynomial complexity, we show that a minimum delay path for any departure time interval can be computed in polynomial time. 1
Average update times for fullydynamic allpairs shortest paths
 Proceedings of the 19th International Symposium on Algorithms and Computation (ISAAC 2008), Gold Coast, Australia, 2008, LNCS 5369
"... Abstract We study the fullydynamic all pairs shortest path problem for graphs with arbitrary nonnegative edge weights. It is known for digraphs that an update of the distance matrix costs Õ(n2.75) 1 worstcase time [Thorup, STOC ’05] and Õ(n2) amortized time [Demetrescu and Italiano, J.ACM ’04] wh ..."
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Abstract We study the fullydynamic all pairs shortest path problem for graphs with arbitrary nonnegative edge weights. It is known for digraphs that an update of the distance matrix costs Õ(n2.75) 1 worstcase time [Thorup, STOC ’05] and Õ(n2) amortized time [Demetrescu and Italiano, J.ACM ’04] where n is the number of vertices. We present the first averagecase analysis of the undirected problem. For a random update we show that the expected time per update is bounded by O(n 4/3+ε) for all ε> 0.
New data structures for subgraph connectivity
 In Proc. 37th International Colloquium on Automata, Languages and Programming (ICALP),pages 201–212
, 2010
"... Abstract. We study the “subgraph connectivity ” problem for undirected graphs with sublinear vertex update time. In this problem, we can make vertices active or inactive in a graph G, and answer the connectivity between two vertices in the subgraph of G induced by the active vertices. In this paper, ..."
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Abstract. We study the “subgraph connectivity ” problem for undirected graphs with sublinear vertex update time. In this problem, we can make vertices active or inactive in a graph G, and answer the connectivity between two vertices in the subgraph of G induced by the active vertices. In this paper, we solve two open problems in subgraph connectivity. We give the first subgraph connectivity structure with worstcase sublinear time bounds for both updates and queries. Our worstcase subgraph connectivity structure supports Õ(m4/5) update time, Õ(m1/5) query time and occupies Õ(m) space, where m is the number of edges in the whole graph G. In the second part of our paper, we describe another dynamic subgraph connectivity structure with amortized Õ(m2/3) update time, Õ(m1/3) query time and linear space, which improves the structure introduced by [Chan, Pǎtra¸scu, Roditty, FOCS’08] that takes Õ(m4/3) space. 1