Abstract
In dynamic data structures, one is interested in efficiently facilitating queries to a data set, while being able to efficiently perform updates as the data set undergoes changes. Often, relaxing the efficiency measure to the amortised setting allows for simpler algorithms. A wellknown example of a data structure with amortised guarantees is the splay tree by Sleator and Tarjan [Daniel D. Sleator and Robert E. Tarjan, 1985].
Similarly, in data structures for dynamic graphs, one is interested in efficiently maintaining some information about the graph, or facilitating queries, as the graph undergoes changes in the form of insertion and deletion of edges. Examples of such information include connectivity, planarity, and approximate sparsity of the graph: is the graph presently connected? Is it planar? Has its arboricity grossly exceeded some specified number α̃? The related queries could be: is a connected to b? Are the edges uv and uw consecutive in the ordering around u in its current planar embedding? Or, report the O(α) outedges of vertex x.
In this talk, we will see Brodal and Fagerberg’s amortised algorithm for orienting sparse graphs (i.e. of arboricity ≤ α), so that each vertex has O(α) outedges [Gerth Stølting Brodal and Rolf Fagerberg, 1999]. The algorithm itself is extremely simple, and uses an elegant amortised argument in its analysis. Then, we will visit the problem of dynamic planarity testing: is the graph presently planar? Here, we will see an elegant amortised reduction to the seemingly easier problem, where planarityviolating edges may be detected and rejected [Eppstein et al., 1996]. We will see a sketch of how the current stateoftheart algorithm for efficient planarity testing [Jacob Holm and Eva Rotenberg, 2020] uses ideas similar to those in [Gerth Stølting Brodal and Rolf Fagerberg, 1999] to analyse the behaviour of a greedy algorithm via a possibly inefficient algorithm with provably low recourse [Jacob Holm and Eva Rotenberg, 2020]. If time permits, we will touch upon a recent simple amortised data structure for maintaining information in dynamic forests [Jacob Holm et al., 2023], which builds on ideas from splay trees.
The talk concludes with some open questions in the area.
BibTeX  Entry
@InProceedings{rotenberg:LIPIcs.STACS.2023.2,
author = {Rotenberg, Eva},
title = {{Amortised Analysis of Dynamic Data Structures}},
booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
pages = {2:12:2},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772662},
ISSN = {18688969},
year = {2023},
volume = {254},
editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17654},
URN = {urn:nbn:de:0030drops176547},
doi = {10.4230/LIPIcs.STACS.2023.2},
annote = {Keywords: Amortised analysis, splaying, dynamic graphs, planarity testing}
}
Keywords: 

Amortised analysis, splaying, dynamic graphs, planarity testing 
Collection: 

40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023) 
Issue Date: 

2023 
Date of publication: 

03.03.2023 