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Distributed Exact Weighted All-Pairs Shortest Paths in (O)over-tilde(n(5/4)) Rounds
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0003-4468-2675
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0003-3694-740X
2017 (English)In: 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), IEEE, 2017, p. 168-179Conference paper, Published paper (Refereed)
Abstract [en]

We study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1+ o(1))-approximation (O) over tilde (n)-time algorithms [2], [3], which are matched with (Omega) over tilde (n)-time lower bounds [3], [4], [5](1). No omega(n) lower bound or o(m) upper bound were known for exact computation. In this paper, we present an (O) over tilde (n(5/4))-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive O(m)-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are asymmetric (a. k. a. the directed case where communication is bidirectional). Our techniques also yield an (O) over tilde (n(3/4) k(1/2) + n)-time algorithm for the k-source shortest paths problem where we want every node to know distances from k sources; this improves Elkin's recent bound [6] when k = (omega) over tilde (n(1/4)). We achieve the above results by developing distributed algorithms on top of the classic scaling technique, which we believe is used for the first time for distributed shortest paths computation. One new algorithm which might be of an independent interest is for the reversed r-sink shortest paths problem, where we want every of r sinks to know its distances from all other nodes, given that every node already knows its distance to every sink. We show an (O) over tilde (n root r)-time algorithm for this problem. Another new algorithm is called short range extension, where we show that in (O) over tilde (n root h) time the knowledge about distances can be "extended" for additional h hops. For this, we use weight rounding to introduce small additive errors which can be later fixed. Remark: Independently from our result, Elkin recently observed in [6] that the same techniques from an earlier version of the same paper (https://arxiv.org/abs/1703.01939v1) led to an O(n(5/3) log(2/3) n)-time algorithm.

Place, publisher, year, edition, pages
IEEE, 2017. p. 168-179
Series
Annual IEEE Symposium on Foundations of Computer Science, ISSN 0272-5428
Keywords [en]
distributed graph algorithms, all-pairs shortest paths, exact distributed algorithms
National Category
Electrical Engineering, Electronic Engineering, Information Engineering
Identifiers
URN: urn:nbn:se:kth:diva-220659DOI: 10.1109/FOCS.2017.24ISI: 000417425300015Scopus ID: 2-s2.0-85041116469ISBN: 978-1-5386-3464-6 OAI: oai:DiVA.org:kth-220659DiVA, id: diva2:1172111
Conference
58th IEEE Annual Symposium on Foundations of Computer Science (FOCS), OCT 15-17, 2017, Berkeley, CA
Funder
EU, Horizon 2020, 715672Swedish Research Council, 2015-04659
Note

QC 20170109

Available from: 2018-01-09 Created: 2018-01-09 Last updated: 2018-07-24Bibliographically approved
In thesis
1. Dynamic algorithms: new worst-case and instance-optimal bounds via new connections
Open this publication in new window or tab >>Dynamic algorithms: new worst-case and instance-optimal bounds via new connections
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis studies a series of questions about dynamic algorithms which are algorithms for quickly maintaining some information of an input data undergoing a sequence of updates. The first question asks \emph{how small the update time for handling each update can be} for each dynamic problem. To obtain fast algorithms, several relaxations are often used including allowing amortized update time, allowing randomization, or even assuming an oblivious adversary. Hence, the second question asks \emph{whether these relaxations and assumptions can be removed} without sacrificing the speed. Some dynamic problems are successfully solved by fast dynamic algorithms without any relaxation. The guarantee of such algorithms, however, is for a worst-case scenario. This leads to the last question which asks for \emph{an algorithm whose cost is nearly optimal for every scenario}, namely an instance-optimal algorithm. This thesis shows new progress on all three questions.

For the first question, we give two frameworks for showing the inherent limitations of fast dynamic algorithms. First, we propose a conjecture called the Online Boolean Matrix-vector Multiplication Conjecture (OMv). Assuming this conjecture, we obtain new \emph{tight} conditional lower bounds of update time for more than ten dynamic problems even when algorithms are allowed to have large polynomial preprocessing time. Second, we establish the first analogue of ``NP-completeness'' for dynamic problems, and show that many natural problems are ``NP-hard'' in the dynamic setting. This hardness result is based on the hardness of all problems in a huge class that includes a number of natural and hard dynamic problems. All previous conditional lower bounds for dynamic problems are based on hardness of specific problems/conjectures.

For the second question, we give an algorithm for maintaining a minimum spanning forest in an $n$-node graph undergoing edge insertions and deletions using $n^{o(1)}$ worst-case update time with high probability. This significantly improves the long-standing $O(\sqrt{n})$ bound by {[}Frederickson STOC'83, Eppstein, Galil, Italiano and Nissenzweig FOCS'92{]}. Previously, a spanning forest (possibly not minimum) can be maintained in polylogarithmic update time if either amortized update is allowed or an oblivious adversary is assumed. Therefore, our work shows how to eliminate these relaxations without slowing down updates too much.

For the last question, we show two main contributions on the theory of instance-optimal dynamic algorithms. First, we use the forbidden submatrix theory from combinatorics to show that a binary search tree (BST) algorithm called \emph{Greedy} has almost optimal cost when its input \emph{avoids a pattern}. This is a significant progress towards the Traversal Conjecture {[}Sleator and Tarjan JACM'85{]} and its generalization. Second, we initialize the theory of instance optimality of heaps by showing a general transformation between BSTs and heaps and then transferring the rich analogous theory of BSTs to heaps. Via the connection, we discover a new heap, called the \emph{smooth heap}, which is very simple to implement, yet inherits most guarantees from BST literature on being instance-optimal on various kinds of inputs. The common approach behind all our results is about making new connections between dynamic algorithms and other fields including fine-grained and classical complexity theory, approximation algorithms for graph partitioning, local clustering algorithms, and forbidden submatrix theory.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2018. p. 51
Series
TRITA-EECS-AVL ; 2018:51
National Category
Computer Sciences
Research subject
Computer Science
Identifiers
urn:nbn:se:kth:diva-232471 (URN)978-91-7729-865-6 (ISBN)
Public defence
2018-08-27, F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20180725

Available from: 2018-07-25 Created: 2018-07-24 Last updated: 2018-07-25Bibliographically approved

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Na Nongkai, DanuponSaranurak, Thatchaphol

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