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diff --git a/thesis/Justfile b/thesis/Justfile index 4e71af3..bd5a60d 100644 --- a/thesis/Justfile +++ b/thesis/Justfile @@ -1,10 +1,10 @@ default: build build: - latexmk -bibtex -pdf + latexmk -pdf watch: - latexmk -bibtex -pdf -pvc + latexmk -pdf -pvc clean: latexmk -c diff --git a/thesis/biblio.bib b/thesis/biblio.bib index bea3669..5237df9 100644 --- a/thesis/biblio.bib +++ b/thesis/biblio.bib @@ -1,36 +1,25 @@ - @inproceedings{jung_brainy_2011, - location = {New York, {NY}, {USA}}, title = {Brainy: Effective Selection of Data Structures}, - isbn = {978-1-4503-0663-8}, url = {https://doi.org/10.1145/1993498.1993509}, doi = {10.1145/1993498.1993509}, series = {{PLDI} '11}, - abstract = {Data structure selection is one of the most critical aspects of developing effective applications. By analyzing data structures' behavior and their interaction with the rest of the application on the underlying architecture, tools can make suggestions for alternative data structures better suited for the program input on which the application runs. Consequently, developers can optimize their data structure usage to make the application conscious of an underlying architecture and a particular program input.This paper presents the design and evaluation of Brainy, a new program analysis tool that automatically selects the best data structure for a given program and its input on a specific microarchitecture. The data structure's interface functions are instrumented to dynamically monitor how the data structure interacts with the application for a given input. The instrumentation records traces of various runtime characteristics including underlying architecture-specific events. These generated traces are analyzed and fed into an offline model, constructed using machine learning, to select the best data structure. That is, Brainy exploits runtime feedback of data structures to model the situation an application runs on, and selects the best data structure for a given application/input/architecture combination based on the constructed model. The empirical evaluation shows that this technique is highly accurate across several real-world applications with various program input sets on two different state-of-the-art microarchitectures. Consequently, Brainy achieved an average performance improvement of 27\% and 33\% on both microarchitectures, respectively.}, pages = {86--97}, booktitle = {Proceedings of the 32nd {ACM} {SIGPLAN} Conference on Programming Language Design and Implementation}, publisher = {Association for Computing Machinery}, author = {Jung, Changhee and Rus, Silvius and Railing, Brian P. and Clark, Nathan and Pande, Santosh}, date = {2011}, - note = {event-place: San Jose, California, {USA}}, - keywords = {application generator, data structure selection, performance counters, training framework}, } @inproceedings{thomas_framework_2005, - location = {New York, {NY}, {USA}}, title = {A Framework for Adaptive Algorithm Selection in {STAPL}}, - isbn = {1-59593-080-9}, url = {https://doi.org/10.1145/1065944.1065981}, doi = {10.1145/1065944.1065981}, series = {{PPoPP} '05}, - abstract = {Writing portable programs that perform well on multiple platforms or for varying input sizes and types can be very difficult because performance is often sensitive to the system architecture, the run-time environment, and input data characteristics. This is even more challenging on parallel and distributed systems due to the wide variety of system architectures. One way to address this problem is to adaptively select the best parallel algorithm for the current input data and system from a set of functionally equivalent algorithmic options. Toward this goal, we have developed a general framework for adaptive algorithm selection for use in the Standard Template Adaptive Parallel Library ({STAPL}). Our framework uses machine learning techniques to analyze data collected by {STAPL} installation benchmarks and to determine tests that will select among algorithmic options at run-time. We apply a prototype implementation of our framework to two important parallel operations, sorting and matrix multiplication, on multiple platforms and show that the framework determines run-time tests that correctly select the best performing algorithm from among several competing algorithmic options in 86-100\% of the cases studied, depending on the operation and the system.}, pages = {277--288}, booktitle = {Proceedings of the Tenth {ACM} {SIGPLAN} Symposium on Principles and Practice of Parallel Programming}, publisher = {Association for Computing Machinery}, author = {Thomas, Nathan and Tanase, Gabriel and Tkachyshyn, Olga and Perdue, Jack and Amato, Nancy M. and Rauchwerger, Lawrence}, date = {2005}, - note = {event-place: Chicago, {IL}, {USA}}, - keywords = {ml, read}, } @inproceedings{osterlund_dynamically_2013, @@ -40,50 +29,36 @@ booktitle = {2013 28th {IEEE}/{ACM} International Conference on Automated Software Engineering ({ASE})}, author = {Österlund, Erik and Löwe, Welf}, date = {2013}, - keywords = {read, rules-based}, } @inproceedings{franke_collection_2022, - location = {New York, {NY}, {USA}}, title = {Collection Skeletons: Declarative Abstractions for Data Collections}, - isbn = {978-1-4503-9919-7}, url = {https://doi.org/10.1145/3567512.3567528}, doi = {10.1145/3567512.3567528}, series = {{SLE} 2022}, - abstract = {Modern programming languages provide programmers with rich abstractions for data collections as part of their standard libraries, e.g. Containers in the C++ {STL}, the Java Collections Framework, or the Scala Collections {API}. Typically, these collections frameworks are organised as hierarchies that provide programmers with common abstract data types ({ADTs}) like lists, queues, and stacks. While convenient, this approach introduces problems which ultimately affect application performance due to users over-specifying collection data types limiting implementation flexibility. In this paper, we develop Collection Skeletons which provide a novel, declarative approach to data collections. Using our framework, programmers explicitly select properties for their collections, thereby truly decoupling specification from implementation. By making collection properties explicit immediate benefits materialise in form of reduced risk of over-specification and increased implementation flexibility. We have prototyped our declarative abstractions for collections as a C++ library, and demonstrate that benchmark applications rewritten to use Collection Skeletons incur little or no overhead. In fact, for several benchmarks, we observe performance speedups (on average between 2.57 to 2.93, and up to 16.37) and also enhanced performance portability across three different hardware platforms.}, pages = {189--201}, booktitle = {Proceedings of the 15th {ACM} {SIGPLAN} International Conference on Software Language Engineering}, publisher = {Association for Computing Machinery}, author = {Franke, Björn and Li, Zhibo and Morton, Magnus and Steuwer, Michel}, date = {2022}, - note = {event-place: Auckland, New Zealand}, - keywords = {read, functional requirements}, - file = {Accepted Version:/home/aria/Zotero/storage/TJ3AGL2S/Franke et al. - 2022 - Collection Skeletons Declarative Abstractions for.pdf:application/pdf}, } @article{qin_primrose_2023, title = {Primrose: Selecting Container Data Types by Their Properties}, volume = {7}, - issn = {2473-7321}, url = {http://arxiv.org/abs/2205.09655}, doi = {10.22152/programming-journal.org/2023/7/11}, shorttitle = {Primrose}, - abstract = {Context: Container data types are ubiquitous in computer programming, enabling developers to efficiently store and process collections of data with an easy-to-use programming interface. Many programming languages offer a variety of container implementations in their standard libraries based on data structures offering different capabilities and performance characteristics. Inquiry: Choosing the *best* container for an application is not always straightforward, as performance characteristics can change drastically in different scenarios, and as real-world performance is not always correlated to theoretical complexity. Approach: We present Primrose, a language-agnostic tool for selecting the best performing valid container implementation from a set of container data types that satisfy *properties* given by application developers. Primrose automatically selects the set of valid container implementations for which the *library specifications*, written by the developers of container libraries, satisfies the specified properties. Finally, Primrose ranks the valid library implementations based on their runtime performance. Knowledge: With Primrose, application developers can specify the expected behaviour of a container as a type refinement with *semantic properties*, e.g., if the container should only contain unique values (such as a `set`) or should satisfy the {LIFO} property of a `stack`. Semantic properties nicely complement *syntactic properties* (i.e., traits, interfaces, or type classes), together allowing developers to specify a container's programming interface *and* behaviour without committing to a concrete implementation. Grounding: We present our prototype implementation of Primrose that preprocesses annotated Rust code, selects valid container implementations and ranks them on their performance. The design of Primrose is, however, language-agnostic, and is easy to integrate into other programming languages that support container data types and traits, interfaces, or type classes. Our implementation encodes properties and library specifications into verification conditions in Rosette, an interface for {SMT} solvers, which determines the set of valid container implementations. We evaluate Primrose by specifying several container implementations, and measuring the time taken to select valid implementations for various combinations of properties with the solver. We automatically validate that container implementations conform to their library specifications via property-based testing. Importance: This work provides a novel approach to bring abstract modelling and specification of container types directly into the programmer's workflow. Instead of selecting concrete container implementations, application programmers can now work on the level of specification, merely stating the behaviours they require from their container types, and the best implementation can be selected automatically.}, - pages = {11}, number = {3}, journaltitle = {The Art, Science, and Engineering of Programming}, - shortjournal = {Programming}, author = {Qin, Xueying and O'Connor, Liam and Steuwer, Michel}, urldate = {2023-09-25}, date = {2023-02-15}, eprinttype = {arxiv}, eprint = {2205.09655 [cs]}, - keywords = {read, functional requirements}, - file = {arXiv Fulltext PDF:/home/aria/Zotero/storage/IL59NESA/Qin et al. - 2023 - Primrose Selecting Container Data Types by Their .pdf:application/pdf;arXiv.org Snapshot:/home/aria/Zotero/storage/DCIW4XE4/2205.html:text/html}, } @inproceedings{costa_collectionswitch_2018, - location = {Vienna Austria}, title = {{CollectionSwitch}: a framework for efficient and dynamic collection selection}, isbn = {978-1-4503-5617-6}, url = {https://dl.acm.org/doi/10.1145/3168825}, @@ -96,15 +71,10 @@ author = {Costa, Diego and Andrzejak, Artur}, urldate = {2023-09-21}, date = {2018-02-24}, - langid = {english}, - keywords = {read, estimate-based}, - file = {Costa and Andrzejak - 2018 - CollectionSwitch a framework for efficient and dy:/home/aria/Zotero/storage/7B8QMVRU/Costa and Andrzejak - 2018 - CollectionSwitch a framework for efficient and dy:application/pdf}, } @inproceedings{shacham_chameleon_2009, - location = {Dublin Ireland}, title = {Chameleon: adaptive selection of collections}, - isbn = {978-1-60558-392-1}, url = {https://dl.acm.org/doi/10.1145/1542476.1542522}, doi = {10.1145/1542476.1542522}, shorttitle = {Chameleon}, @@ -115,30 +85,20 @@ author = {Shacham, Ohad and Vechev, Martin and Yahav, Eran}, urldate = {2023-09-21}, date = {2009-06-15}, - langid = {english}, - keywords = {read, rules-based}, - file = {Shacham et al. - 2009 - Chameleon adaptive selection of collections.pdf:/home/aria/Zotero/storage/75CS9CWY/Shacham et al. - 2009 - Chameleon adaptive selection of collections.pdf:application/pdf}, } @incollection{hutchison_coco_2013, - location = {Berlin, Heidelberg}, title = {{CoCo}: Sound and Adaptive Replacement of Java Collections}, volume = {7920}, - isbn = {978-3-642-39037-1 978-3-642-39038-8}, - url = {http://link.springer.com/10.1007/978-3-642-39038-8_1}, + url = "http://link.springer.com/10.1007/978-3-642-39038-8_1", shorttitle = {{CoCo}}, pages = {1--26}, booktitle = {{ECOOP} 2013 – Object-Oriented Programming}, publisher = {Springer Berlin Heidelberg}, author = {Xu, Guoqing}, - editor = {Castagna, Giuseppe}, - editorb = {Hutchison, David and Kanade, Takeo and Kittler, Josef and Kleinberg, Jon M. and Mattern, Friedemann and Mitchell, John C. and Naor, Moni and Nierstrasz, Oscar and Pandu Rangan, C. and Steffen, Bernhard and Sudan, Madhu and Terzopoulos, Demetri and Tygar, Doug and Vardi, Moshe Y. and Weikum, Gerhard}, - editorbtype = {redactor}, urldate = {2023-10-17}, date = {2013}, - doi = {10.1007/978-3-642-39038-8_1}, - note = {Series Title: Lecture Notes in Computer Science}, - keywords = {read, rules-based}, + doi = "10.1007/978-3-642-39038-8_1", } @inproceedings{l_liu_perflint_2009, @@ -149,48 +109,33 @@ booktitle = {2009 International Symposium on Code Generation and Optimization}, author = {{L. Liu} and {S. Rus}}, date = {2009-03-22}, - note = {Journal Abbreviation: 2009 International Symposium on Code Generation and Optimization}, - keywords = {read, rules-based}, - file = {Full Text:/home/aria/Zotero/storage/KTJNYCES/L. Liu and S. Rus - 2009 - Perflint A Context Sensitive Performance Advisor .pdf:application/pdf}, } @article{jung_brainy_2011-1, title = {Brainy: effective selection of data structures}, volume = {46}, - issn = {0362-1340, 1558-1160}, url = {https://dl.acm.org/doi/10.1145/1993316.1993509}, doi = {10.1145/1993316.1993509}, shorttitle = {Brainy}, - abstract = {Data structure selection is one of the most critical aspects of developing effective applications. By analyzing data structures' behavior and their interaction with the rest of the application on the underlying architecture, tools can make suggestions for alternative data structures better suited for the program input on which the application runs. Consequently, developers can optimize their data structure usage to make the application conscious of an underlying architecture and a particular program input. - This paper presents the design and evaluation of Brainy, a new program analysis tool that automatically selects the best data structure for a given program and its input on a specific microarchitecture. The data structure's interface functions are instrumented to dynamically monitor how the data structure interacts with the application for a given input. The instrumentation records traces of various runtime characteristics including underlying architecture-specific events. These generated traces are analyzed and fed into an offline model, constructed using machine learning, to select the best data structure. That is, Brainy exploits runtime feedback of data structures to model the situation an application runs on, and selects the best data structure for a given application/input/architecture combination based on the constructed model. The empirical evaluation shows that this technique is highly accurate across several real-world applications with various program input sets on two different state-of-the-art microarchitectures. Consequently, Brainy achieved an average performance improvement of 27\% and 33\% on both microarchitectures, respectively.}, pages = {86--97}, number = {6}, journaltitle = {{ACM} {SIGPLAN} Notices}, - shortjournal = {{SIGPLAN} Not.}, + shortjournal = {{SIGPLAN} Notices}, author = {Jung, Changhee and Rus, Silvius and Railing, Brian P. and Clark, Nathan and Pande, Santosh}, urldate = {2023-09-21}, date = {2011-06-04}, - langid = {english}, - keywords = {ml, read}, - file = {Jung et al. - 2011 - Brainy effective selection of data structures.pdf:/home/aria/Zotero/storage/DPJPURT8/Jung et al. - 2011 - Brainy effective selection of data structures.pdf:application/pdf}, } @inproceedings{costa_empirical_2017, - location = {New York, {NY}, {USA}}, title = {Empirical Study of Usage and Performance of Java Collections}, - isbn = {978-1-4503-4404-3}, url = {https://doi.org/10.1145/3030207.3030221}, doi = {10.1145/3030207.3030221}, series = {{ICPE} '17}, - abstract = {Collection data structures have a major impact on the performance of applications, especially in languages such as Java, C\#, or C++. This requires a developer to select an appropriate collection from a large set of possibilities, including different abstractions (e.g. list, map, set, queue), and multiple implementations. In Java, the default implementation of collections is provided by the standard Java Collection Framework ({JCF}). However, there exist a large variety of less known third-party collection libraries which can provide substantial performance benefits with minimal code changes.In this paper, we first study the popularity and usage patterns of collection implementations by mining a code corpus comprised of 10,986 Java projects. We use the results to evaluate and compare the performance of the six most popular alternative collection libraries in a large variety of scenarios. We found that for almost every scenario and {JCF} collection type there is an alternative implementation that greatly decreases memory consumption while offering comparable or even better execution time. Memory savings range from 60\% to 88\% thanks to reduced overhead and some operations execute 1.5x to 50x faster.We present our results as a comprehensive guideline to help developers in identifying the scenarios in which an alternative implementation can provide a substantial performance improvement. Finally, we discuss how some coding patterns result in substantial performance differences of collections.}, pages = {389--400}, booktitle = {Proceedings of the 8th {ACM}/{SPEC} on International Conference on Performance Engineering}, publisher = {Association for Computing Machinery}, author = {Costa, Diego and Andrzejak, Artur and Seboek, Janos and Lo, David}, date = {2017}, - note = {event-place: L'Aquila, Italy}, - keywords = {collections, empirical study, execution time, java, memory, performance}, - file = {Full Text:/home/aria/Zotero/storage/DLA43MW4/Costa et al. - 2017 - Empirical Study of Usage and Performance of Java C.pdf:application/pdf}, } @online{wastl_advent_2015, @@ -221,6 +166,4 @@ author = {Bayer, R. and {McCreight}, E.}, urldate = {2024-03-08}, date = {1970}, - langid = {english}, - file = {Full Text:/home/aria/Zotero/storage/84VSCDAG/Bayer and McCreight - 1970 - Organization and maintenance of large ordered indi.pdf:application/pdf}, } diff --git a/thesis/main.tex b/thesis/main.tex index a3c71c3..9bf6e6f 100644 --- a/thesis/main.tex +++ b/thesis/main.tex @@ -10,8 +10,9 @@ \usepackage{microtype} \usepackage{calc} -\usepackage[style=numeric]{biblatex} -\addbibresource{biblio.bib} +\usepackage{url} +\usepackage{natbib} +\bibliographystyle{unsrtnat} %% Convenience macros \newcommand{\code}[1]{\lstinline$#1$} @@ -98,7 +99,7 @@ from the Informatics Research Ethics committee. \input{parts/conclusion} -\printbibliography +\bibliography{biblio} %% \appendix diff --git a/thesis/parts/background.tex b/thesis/parts/background.tex index a4383bc..f705aad 100644 --- a/thesis/parts/background.tex +++ b/thesis/parts/background.tex @@ -77,7 +77,7 @@ This means that developers are forced to guess based on their knowledge of the u \subsection{Rules-based approaches} One approach to the container selection problem is to allow the developer to make the choice initially, but use some tool to detect poor choices. -Chameleon\parencite{shacham_chameleon_2009} uses this approach. +Chameleon\citep{shacham_chameleon_2009} uses this approach. It first collects statistics from program benchmarks using a ``semantic profiler''. This includes the space used by collections over time and the counts of each operation performed. @@ -94,13 +94,13 @@ This results in selection rules being more restricted than they otherwise could For instance, a rule cannot suggest a \code{HashSet} instead of a \code{LinkedList} as the two are not semantically identical. Chameleon has no way of knowing if doing so will break the program's functionality and so it does not make the suggestion. -CoCo \parencite{hutchison_coco_2013} and work by \"{O}sterlund \parencite{osterlund_dynamically_2013} use similar techniques, but work as the program runs. +CoCo \citep{hutchison_coco_2013} and work by \"{O}sterlund \citep{osterlund_dynamically_2013} use similar techniques, but work as the program runs. This works well for programs with different phases of execution, such as loading and then working on data. However, the overhead from profiling and from checking rules may not be worth the improvements in other programs, where access patterns are roughly the same throughout. \subsection{ML-based approaches} -Brainy\parencite{jung_brainy_2011} gathers statistics similarly, however it uses machine learning (ML) for selection instead of programmed rules. +Brainy\citep{jung_brainy_2011} gathers statistics similarly, however it uses machine learning (ML) for selection instead of programmed rules. ML has the advantage of being able to detect patterns a human may not be aware of. For example, Brainy takes into account statistics from hardware counters, which are difficult for a human to reason about. @@ -108,7 +108,7 @@ This also makes it easier to add new collection implementations, as rules do not \subsection{Estimate-based approaches} -CollectionSwitch\parencite{costa_collectionswitch_2018} is an online solution which adapts as the program runs and new information becomes available. +CollectionSwitch\citep{costa_collectionswitch_2018} is an online solution which adapts as the program runs and new information becomes available. First, a performance model is built for each container implementation. This gives an estimate of some cost for each operation at a given collection size. @@ -131,7 +131,7 @@ However, it does not take the collection size into account. Most of the approaches we have highlighted focus on non-functional requirements, and use programming language features to enforce functional requirements. We will now examine tools which focus on container selection based on functional requirements. -Primrose \parencite{qin_primrose_2023} is one such tool, which uses a model-based approach. +Primrose \citep{qin_primrose_2023} is one such tool, which uses a model-based approach. It allows the application developer to specify semantic requirements using a Domain-Specific Language (DSL), and syntactic requirements using Rust's traits. The semantic requirements are expressed as a list of predicates, each representing a semantic property. diff --git a/thesis/parts/design.tex b/thesis/parts/design.tex index 18efa9c..6d9c482 100644 --- a/thesis/parts/design.tex +++ b/thesis/parts/design.tex @@ -87,7 +87,7 @@ We now go into more detail on how each step works, although we leave some specif %% Explain role in entire process As described in Chapter \ref{chap:background}, any implementation we pick must satisfy the program's functional requirements. -To do this, we integrate Primrose \parencite{qin_primrose_2023} as a first step. +To do this, we integrate Primrose \citep{qin_primrose_2023} as a first step. Primrose allows users to specify both the traits they require in an implementation (essentially the API and methods available), and what properties must be satisfied. @@ -126,7 +126,7 @@ Although we use primrose in our implementation, the rest of our system isn't dep \section{Cost Models} Now that we have a list of possible implementations, we need to understand the performance characteristics of each of them. -We use an approach similar to CollectionSwitch\parencite{costa_collectionswitch_2018}, which assumes that the main factor in how long an operation takes is the current size of the collection. +We use an approach similar to CollectionSwitch\citep{costa_collectionswitch_2018}, which assumes that the main factor in how long an operation takes is the current size of the collection. %% Benchmarks An implementation has a seperate cost model for each operation, which we obtain by executing the operation repeatedly on collections of various sizes. @@ -199,7 +199,7 @@ But when the size of the container grows, the cost of doing \code{contains} may Adaptive containers attempt to address this need, by starting off with one implementation (the low or before implementation), and switching to a new implemenation (the high or after implementation) once the size of the container passes a certain threshold. -This is similar to systems such as CoCo\parencite{hutchison_coco_2013} and in work by \"{O}sterlund\parencite{osterlund_dynamically_2013}. +This is similar to systems such as CoCo\citep{hutchison_coco_2013} and in work by \"{O}sterlund\citep{osterlund_dynamically_2013}. However, we decide when to switch container implementation before the program is run, rather than as it is running. We also do so in a way that requires no knowledge of the implementation internals. diff --git a/thesis/parts/implementation.tex b/thesis/parts/implementation.tex index cd7b4b7..706dbd5 100644 --- a/thesis/parts/implementation.tex +++ b/thesis/parts/implementation.tex @@ -32,7 +32,7 @@ The library source can be found in \code{src/crates/library}. \code{SortedUniqueVec} & Vec kept in sorted order, with no duplicates \\ \code{HashMap} & Hash map with quadratic probing \\ \code{HashSet} & Hash map with empty values \\ - \code{BTreeMap} & B-Tree\parencite{bayer_organization_1970} map with linear search. \\ + \code{BTreeMap} & B-Tree\citep{bayer_organization_1970} map with linear search. \\ \code{BTreeSet} & B-Tree map with empty values \\ \end{tabular} \caption{Implementations in our library} diff --git a/thesis/parts/results.tex b/thesis/parts/results.tex index 747631d..dbc8b14 100644 --- a/thesis/parts/results.tex +++ b/thesis/parts/results.tex @@ -41,7 +41,7 @@ It's unsurprising that these two implementations are the cheapest, as they have This is likely due to hash collisions being more likely as the size of the collection increases. \code{BTreeSet} insertions are also expensive, however the cost appears to level out as the collection size goes up (a logarithmic curve). -It's important to note that Rust's \code{BTreeSet}s are not based on binary tree search, but instead a more general tree search originally proposed by R Bayer and E McCreight\parencite{bayer_organization_1970}, where each node contains $B-1$ to $2B-1$ elements in an array. +It's important to note that Rust's \code{BTreeSet}s are not based on binary tree search, but instead a more general tree search originally proposed by R Bayer and E McCreight\citep{bayer_organization_1970}, where each node contains $B-1$ to $2B-1$ elements in an array. Our two mapping types, \code{BTreeMap} and \code{HashMap}, mimic the behaviour of their set counterparts. @@ -92,7 +92,7 @@ This is possibly a case of overfitting, as the observations for both implementat \code{HashSet} appears roughly linear as expected, with only a slow logarithmic rise, probably due to an increasing amount of collisions. \code{BTreeSet} is consistently above it, with a slightly higher logarithmic rise. -The standard library documentation states that searches are expected to take $B\log(n)$ comparisons on average\parencite{rust_documentation_team_btreemap_2024}, which is in line with observations. +The standard library documentation states that searches are expected to take $B\log(n)$ comparisons on average\citep{rust_documentation_team_btreemap_2024}, which is in line with observations. \code{BTreeMap} and \code{HashMap} both mimic their set counterparts, though are more expensive in most places. This is probably due to the increased size more quickly exhausting CPU cache. @@ -120,7 +120,7 @@ Future improvements could address the overfitting problems some operations had, Our test cases broadly fall into two categories: Example cases, which just repeat a few operations many times, and our 'real' cases, which are implementations of common algorithms and solutions to programming puzles. We expect the results from our example cases to be relatively unsurprising, while our real cases are more complex and harder to predict. -Most of our real cases are solutions to puzzles from Advent of Code\parencite{wastl_advent_2015}, a popular collection of programming puzzles. +Most of our real cases are solutions to puzzles from Advent of Code\citep{wastl_advent_2015}, a popular collection of programming puzzles. Table \ref{table:test_cases} lists and briefly describes our test cases. \begin{table}[h!] |