Sublinear-Time Maintenance of Breadth-First Spanning Tree in Partially Dynamic Networks

Henzinger, Monika; Krinninger, Sebastian; Nanongkai, Danupon

doi:10.1007/978-3-642-39212-2_53

Cited by 12 publications

(16 citation statements)

References 30 publications

(9 reference statements)

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“…This is the key insight that allows us to check whether v's distance increases by Ω(δ). We can thus obtain the same result as we have obtained in the case of the incremental model [17], i.e., O(n 1.8 ) total update time when m = O(n).…”

Section: Review Of Even-shiloach Tree (Es-treesupporting

confidence: 78%

“…no previous algorithm achieves o(n 2 ) total update time). Our general approach to attack this special case is inspired by our recent technique called lazyupdate Even-Shiloach tree introduced in [17]. In [17], we used this technique to obtain a (1+ )-approximation O(n 1.8 )-time algorithm for incremental SSSP (allowing only edge insertions) when m = O(n)…”

Section: Review Of Even-shiloach Tree (Es-treementioning

confidence: 99%

“…However, it also leads to a distance error. In [17], we showed that by regularly recomputing the BFS tree from scratch, we can keep this distance error small. One crucial subroutine of this approach is checking, after an edge deletion, whether a node's distance (from the source) increases by at least δ.…”

Section: Review Of Even-shiloach Tree (Es-treementioning

confidence: 99%

“…Besides being interesting in its own right, it also arises as a subproblem in many other dynamic algorithms and can be adapted to other settings (e.g. [5,6,7,16,17,19,21,22]). The first non-trivial result is from 1981 by Even and Shiloach [14].…”

Section: Related Workmentioning

confidence: 99%

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A Subquadratic-Time Algorithm for Decremental Single-Source Shortest Paths

Henzinger¹,

Krinninger²,

Nanongkai³

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study dynamic (1 + )-approximation algorithms for the single-source shortest paths problem in an unweighted undirected n-node m-edge graph under edge deletions. The fastest algorithm for this problem is an algorithm with O(n 2+o(1) ) total update time and constant query time by Bernstein and Roditty (SODA 2011). In this paper, we improve the total update time to O(n 1.8+o(1) + m 1+o (1) ) while keeping the query time constant. This running time is essentially tight when m = Ω(n 1.8 ) since we need Ω(m) time even in the static setting. For smaller values of m, the running time of our algorithm is subquadratic, and is the first that breaks through the quadratic time barrier.In obtaining this result, we develop a fast algorithm for what we call center cover data structure. We also make non-trivial extensions to our previous techniques called lazyupdate and monotone Even-Shiloach trees (ICALP 2013 and FOCS 2013). As by-products of our new techniques, we obtain two new results for the decremental all-pairs shortestpaths problem. Our first result is the first approximation algorithm whose total update time is faster than O(mn) for all values of m. Our second result is a new trade-off between the total update time and the additive approximation guarantee.

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Section: Review Of Even-shiloach Tree (Es-treesupporting

confidence: 78%

Section: Review Of Even-shiloach Tree (Es-treementioning

confidence: 99%

Section: Review Of Even-shiloach Tree (Es-treementioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

A Subquadratic-Time Algorithm for Decremental Single-Source Shortest Paths

Henzinger¹,

Krinninger²,

Nanongkai³

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…For the dynamic setting, Henzinger et al [25] presented a model that has a preprocessing stage followed by an alternating sequence of non-overlapping stages for update and recovery (see Section 6.2 for details). We use this model with an additional constraint of space restriction of O(n) size at each node.…”

Section: Preliminariesmentioning

confidence: 99%

Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs

Khan

2017

Proceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures

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Depth first search (DFS) tree is a fundamental data structure for solving various graph problems. The classical algorithm [47] for building a DFS tree requires O(m + n) time for a given undirected graph G having n vertices and m edges. Recently, Baswana et al.[6] presented a simple algorithm for updating the DFS tree of an undirected graph after an edge/vertex update inÕ(n) 1 time. However, their algorithm is strictly sequential. We present an algorithm achieving similar bounds that can be easily adopted to the parallel environment.In the parallel environment, a DFS tree can be computed from scratch using O(m) processors in expectedÕ(1) time [2] on an EREW PRAM, whereas the best deterministic algorithm takes O( √ n) time [2, 22] on a CRCW PRAM. Our algorithm can be used to develop optimal time (upto poly log n factors) deterministic parallel algorithms for maintaining fully dynamic DFS and fault tolerant DFS of an undirected graph. Parallel Fully Dynamic DFS:Given an arbitrary online sequence of vertex or edge updates, we can maintain a DFS tree of an undirected graph inÕ(1) time per update using m processors on an EREW PRAM. Parallel Fault tolerant DFS:An undirected graph can be preprocessed to build a data structure of size O(m), such that for any set of k updates (where k is constant) in the graph, a DFS tree of the updated graph can be computed inÕ(1) time using n processors on an EREW PRAM. For constant k, this is also work optimal (upto poly log n factors).Moreover, our fully dynamic DFS algorithm provides, in a seamless manner, nearly optimal (upto poly log n factors) algorithms for maintaining a DFS tree in the semi-streaming environment and a restricted distributed model. These are the first parallel, semi-streaming and distributed algorithms for maintaining a DFS tree in the dynamic setting.

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Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

Brand

Nanongkai

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Consider the following distance query for an n-node graph G undergoing edge insertions and deletions: given two sets of nodes I and J, return the distances between every pair of nodes in I × J. This query is rather general and captures several versions of the dynamic shortest paths problem. In this paper, we develop an efficient (1 + ǫ)-approximation algorithm for this query using fast matrix multiplication. Our algorithm leads to answers for some open problems for Single-Source and All-Pairs Shortest Paths (SSSP and APSP), as well as for Diameter, Radius, and Eccentricities. Below are some highlights. Note that all our algorithms guarantee worstcase update time and are randomized (Monte Carlo), but do not need the oblivious adversary assumption.Subquadratic update time for SSSP, Diameter, Centralities, ect.: When we want to maintain distances from a single node explicitly (without queries), a fundamental question is to beat trivially calling Dijkstra's static algorithm after each update, taking Θ(n 2 ) update time on dense graphs. A better time complexity was not known even with amortization. It was known to be improbable for exact algorithms and for combinatorial any-approximation algorithms to polynomially beat the Ω(n 2 ) bound (under some conjectures) [Roditty, Zwick, ESA'04; Abboud, V. Williams, FOCS'14]. 1 Our algorithm with I = {s} and J = V (G) implies a (1+ǫ)-approximation algorithm for this, guaranteeingÕ(n 1.823 /ǫ 2 ) worst-case update time for directed graphs with positive real weights in [1, W ]. 2 With ideas from [Roditty, V. Williams, STOC'13], we also obtain the first subquadratic worst-case update time for (5/3 + ǫ)-approximating the eccentricities and (1.5 + ǫ)-approximating the diameter and radius for unweighted graphs (with small additive errors). We also obtain the first subquadratic worst-case update time for (1 + ǫ)-approximating the closeness centralities for undirected unweighted graphs.Worst-case update time for APSP: When we want to maintain distances between all-pairs of nodes explicitly, theÕ(n 2 ) amortized update time by Demetrescu and Italiano [STOC'03] already matches the trivial Ω(n 2 ) lower bound. A fundamental question is whether it can be made worst-case. The state-of-the-art algorithm takesÕ(n 2+2/3 ) worst-case update time to maintain the distances exactly [Abraham, Chechik, Krinninger, SODA'17; Thorup STOC'05]. When it comes to (1 + ǫ) approximation, this bound is still higher than calling theÕ(n ω /ǫ)-time static algorithm of Zwick [FOCS'98], where ω ≈ 2.373. Our algorithm with I = J = V (G) implies nearly tight bounds for this, namelyÕ(n 2 /ǫ 1+ω ) for undirected unweighted graphs and O(n 2.045 /ǫ 2 ) for directed graphs with positive real weights. Besides this, we also obtain the first dynamic APSP algorithm with subquadratic update time and sublinear query time.

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Sublinear-Time Maintenance of Breadth-First Spanning Tree in Partially Dynamic Networks

Cited by 12 publications

References 30 publications

A Subquadratic-Time Algorithm for Decremental Single-Source Shortest Paths

A Subquadratic-Time Algorithm for Decremental Single-Source Shortest Paths

Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs

Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

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