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Dynamic Minimum Spanning Forest with Subpolynomial Worst-case Update Time
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. 950-961Conference paper, Published paper (Refereed)
##### Abstract [en]

We present a Las Vegas algorithm for dynamically maintaining a minimum spanning forest of an nnode graph undergoing edge insertions and deletions. Our algorithm guarantees an O(n(o(1))) worst-case update time with high probability. This significantly improves the two recent Las Vegas algorithms by Wulff-Nilsen [2] with update time O(n(0.5-epsilon)) for some constant epsilon > 0 and, independently, by Nanongkai and Saranurak [3] with update time O(n(0.494)) (the latter works only for maintaining a spanning forest). Our result is obtained by identifying the common framework that both two previous algorithms rely on, and then improve and combine the ideas from both works. There are two main algorithmic components of the framework that are newly improved and critical for obtaining our result. First, we improve the update time from O(n(0.5-epsilon)) in [2] to O(n(o(1))) for decrementally removing all low-conductance cuts in an expander undergoing edge deletions. Second, by revisiting the "contraction technique" by Henzinger and King [4] and Holm et al. [5], we show a new approach for maintaining a minimum spanning forest in connected graphs with very few (at most (1 + o(1))n) edges. This significantly improves the previous approach in [2], [3] which is based on Frederickson's 2-dimensional topology tree [6] and illustrates a new application to this old technique.

##### Place, publisher, year, edition, pages
IEEE, 2017. p. 950-961
##### Series
Annual IEEE Symposium on Foundations of Computer Science, ISSN 0272-5428
##### National Category
Electrical Engineering, Electronic Engineering, Information Engineering Computer Sciences
##### Identifiers
ISI: 000417425300083Scopus ID: 2-s2.0-85041099602ISBN: 978-1-5386-3464-6 (print)OAI: oai:DiVA.org:kth-220661DiVA, id: diva2:1171866
##### Conference
58th IEEE Annual Symposium on Foundations of Computer Science (FOCS), OCT 15-17, 2017, Berkeley, CA
##### Note

QC 20180108

Available from: 2018-01-08 Created: 2018-01-08 Last updated: 2018-07-25Bibliographically 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
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)
##### Note

QC 20180725

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

#### Open Access in DiVA

No full text in DiVA

Publisher's full textScopus

#### Authority records BETA

Na Nongkai, DanuponSaranurak, Thatchaphol

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Cite
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