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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We study the connections between sorting and the binary search tree (BST) model, with an aim towards showing that the fields are connected more deeply than is currently appreciated. While any BST can be used to sort by inserting the keys one-by-one, this is a very limited relationship and importantly says nothing about parallel sorting. We show what we believe to be the first formal relationship between the BST model and sorting. Namely, we show that a large class of sorting algorithms, which includes mergesort, quicksort, insertion sort, and almost every instance-optimal sorting algorithm, are equivalent in cost to offline BST algorithms. Our main theoretical tool is the geometric interpretation of the BST model introduced by Demaine et al. [Demaine et al., 2009], which finds an equivalence between searches on a BST and point sets in the plane satisfying a certain property. To give an example of the utility of our approach, we introduce the log-interleave bound, a measure of the information-theoretic complexity of a permutation π, which is within a lg lg n multiplicative factor of a known lower bound in the BST model; we also devise a parallel sorting algorithm with polylogarithmic span that sorts a permutation π using comparisons proportional to its log-interleave bound. Our aforementioned result on sorting and offline BST algorithms can be used to show existence of an offline BST algorithm whose cost is within a constant factor of the log-interleave bound of any permutation π.

Guy E. Blelloch and Magdalen Dobson. The Geometry of Tree-Based Sorting. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blelloch_et_al:LIPIcs.ICALP.2023.26, author = {Blelloch, Guy E. and Dobson, Magdalen}, title = {{The Geometry of Tree-Based Sorting}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {26:1--26:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.26}, URN = {urn:nbn:de:0030-drops-180780}, doi = {10.4230/LIPIcs.ICALP.2023.26}, annote = {Keywords: binary search trees, sorting, dynamic optimality, parallelism} }

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**Published in:** LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

We present a general technique for garbage collecting old versions for multiversion concurrency control that simultaneously achieves good time and space complexity. Our technique takes only O(1) time on average to reclaim each version and maintains only a constant factor more versions than needed (plus an additive term). It is designed for multiversion schemes using version lists, which are the most common.
Our approach uses two components that are of independent interest. First, we define a novel range-tracking data structure which stores a set of old versions and efficiently finds those that are no longer needed. We provide a wait-free implementation in which all operations take amortized constant time. Second, we represent version lists using a new lock-free doubly-linked list algorithm that supports efficient (amortized constant time) removals given a pointer to any node in the list. These two components naturally fit together to solve the multiversion garbage collection problem - the range-tracker identifies which versions to remove and our list algorithm can then be used to remove them from their version lists. We apply our garbage collection technique to generate end-to-end time and space bounds for the multiversioning system of Wei et al. (PPoPP 2021).

Naama Ben-David, Guy E. Blelloch, Panagiota Fatourou, Eric Ruppert, Yihan Sun, and Yuanhao Wei. Space and Time Bounded Multiversion Garbage Collection. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bendavid_et_al:LIPIcs.DISC.2021.12, author = {Ben-David, Naama and Blelloch, Guy E. and Fatourou, Panagiota and Ruppert, Eric and Sun, Yihan and Wei, Yuanhao}, title = {{Space and Time Bounded Multiversion Garbage Collection}}, booktitle = {35th International Symposium on Distributed Computing (DISC 2021)}, pages = {12:1--12:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-210-5}, ISSN = {1868-8969}, year = {2021}, volume = {209}, editor = {Gilbert, Seth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.12}, URN = {urn:nbn:de:0030-drops-148143}, doi = {10.4230/LIPIcs.DISC.2021.12}, annote = {Keywords: Lock-free, data structures, memory management, snapshot, version lists} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

When designing concurrent algorithms, Load-Link/Store-Conditional (LL/SC) is a very useful primitive since it avoids ABA problems. The full semantics of LL/SC are not supported in hardware by any modern architecture, so there has been a significant amount of work on simulations of LL/SC using CAS. However, all previous algorithms that are constant time either use unbounded sequence numbers (and thus base objects of unbounded size), or require Ω(MP) space to implement M LL/SC objects for P processes.
We present the first constant time implementation of LL/SC from bounded-sized CAS objects using only constant space overhead per LL/SC variable. In particular, our implementation uses Θ(M+kP²) space, where k is the number of outstanding LL operations per process, and only requires pointer-width CAS operations. In most algorithms that use LL/SC, k is a small constant which reduces our additive space overhead to Θ(P²). Our algorithm can also be extended to implement L word LL/SC objects in Θ(L) time for LL and SC, O(1) time for VL, and Θ((M+kP²)L) space.
To achieve these bounds, our main technical contribution is implementing a new primitive called Single-Writer Copy which takes a pointer to a word sized memory location and atomically copies its contents into another object. The restriction is that only one process is allowed to write/copy into the destination object at a time. The ability to read from one memory location and write to another atomically, and in constant-time, is very powerful and we believe this primitive will be useful in designing other algorithms.

Guy E. Blelloch and Yuanhao Wei. LL/SC and Atomic Copy: Constant Time, Space Efficient Implementations Using Only Pointer-Width CAS. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blelloch_et_al:LIPIcs.DISC.2020.5, author = {Blelloch, Guy E. and Wei, Yuanhao}, title = {{LL/SC and Atomic Copy: Constant Time, Space Efficient Implementations Using Only Pointer-Width CAS}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {5:1--5:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.5}, URN = {urn:nbn:de:0030-drops-130831}, doi = {10.4230/LIPIcs.DISC.2020.5}, annote = {Keywords: LL/SC, Atomic Copy, CAS, Constant Time} }

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Brief Announcement

**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

We describe an algorithm for supporting allocation and free for fixed-sized blocks, for p asynchronous processors, with O(1) worst-case time per operation, Θ(p²) additive space overhead, and using only single-word read, write, and CAS. While many algorithms rely on having constant-time fixed-size allocate and free, we present the first implementation of these two operations that is constant time with reasonable space overhead.

Guy E. Blelloch and Yuanhao Wei. Brief Announcement: Concurrent Fixed-Size Allocation and Free in Constant Time. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 51:1-51:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blelloch_et_al:LIPIcs.DISC.2020.51, author = {Blelloch, Guy E. and Wei, Yuanhao}, title = {{Brief Announcement: Concurrent Fixed-Size Allocation and Free in Constant Time}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {51:1--51:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.51}, URN = {urn:nbn:de:0030-drops-131291}, doi = {10.4230/LIPIcs.DISC.2020.51}, annote = {Keywords: malloc, free, fixed-size, concurrent, constant time} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

The dynamic trees problem is to maintain a forest subject to edge insertions and deletions while facilitating queries such as connectivity, path weights, and subtree weights. Dynamic trees are a fundamental building block of a large number of graph algorithms. Although traditionally studied in the single-update setting, dynamic algorithms capable of supporting batches of updates are increasingly relevant today due to the emergence of rapidly evolving dynamic datasets. Since processing updates on a single processor is often unrealistic for large batches of updates, designing parallel batch-dynamic algorithms that achieve provably low span is important for many applications.
In this work, we design the first work-efficient parallel batch-dynamic algorithm for dynamic trees that is capable of supporting both path queries and subtree queries, as well as a variety of nonlocal queries. Previous work-efficient dynamic trees of Tseng et al. were only capable of handling subtree queries [ALENEX'19, (2019), pp. 92 - 106]. To achieve this, we propose a framework for algorithmically dynamizing static round-synchronous algorithms to obtain parallel batch-dynamic algorithms. In our framework, the algorithm designer can apply the technique to any suitably defined static algorithm. We then obtain theoretical guarantees for algorithms in our framework by defining the notion of a computation distance between two executions of the underlying algorithm.
Our dynamic trees algorithm is obtained by applying our dynamization framework to the parallel tree contraction algorithm of Miller and Reif [FOCS'85, (1985), pp. 478 - 489], and then performing a novel analysis of the computation distance of this algorithm under batch updates. We show that k updates can be performed in O(klog(1+n/k)) work in expectation, which matches the algorithm of Tseng et al. while providing support for a substantially larger number of queries and applications.

Umut A. Acar, Daniel Anderson, Guy E. Blelloch, Laxman Dhulipala, and Sam Westrick. Parallel Batch-Dynamic Trees via Change Propagation. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{acar_et_al:LIPIcs.ESA.2020.2, author = {Acar, Umut A. and Anderson, Daniel and Blelloch, Guy E. and Dhulipala, Laxman and Westrick, Sam}, title = {{Parallel Batch-Dynamic Trees via Change Propagation}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {2:1--2:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.2}, URN = {urn:nbn:de:0030-drops-128686}, doi = {10.4230/LIPIcs.ESA.2020.2}, annote = {Keywords: Dynamic trees, Graph algorithms, Parallel algorithms, Dynamic algorithms} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

The future of main memory appears to lie in the direction of new non-volatile memory technologies that provide strong capacity-to-performance ratios, but have write operations that are much more expensive than reads in terms of energy, bandwidth, and latency. This asymmetry can have a significant effect on algorithm design, and in many cases it is possible to reduce writes at the cost of more reads. This paper studies which algorithmic techniques are useful in designing practical write-efficient algorithms. We focus on several fundamental algorithmic building blocks including unordered set/map implemented using hash tables, comparison sort, and graph traversal algorithms including breadth-first search and Dijkstra's algorithm. We introduce new algorithms and implementations that can reduce writes, and analyze the performance experimentally using a software simulator. Finally, we summarize interesting lessons and directions in designing write-efficient algorithms that can be valuable to share.

Yan Gu, Yihan Sun, and Guy E. Blelloch. Algorithmic Building Blocks for Asymmetric Memories. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gu_et_al:LIPIcs.ESA.2018.44, author = {Gu, Yan and Sun, Yihan and Blelloch, Guy E.}, title = {{Algorithmic Building Blocks for Asymmetric Memories}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.44}, URN = {urn:nbn:de:0030-drops-95070}, doi = {10.4230/LIPIcs.ESA.2018.44}, annote = {Keywords: Asymmetric Memory, I/O Cost, Write-Efficient Algorithms, Hash Tables, Graph-Traversal Algorithms} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

In this paper we describe an algorithm that embeds a graph metric (V,d_G) on an undirected weighted graph G=(V,E) into a distribution of tree metrics (T,D_T) such that for every pair u,v in V, d_G(u,v)<=d_T(u,v) and E_T[d_T(u,v)]<=O(log n)d_G(u,v). Such embeddings have proved highly useful in designing fast approximation algorithms, as many hard problems on graphs are easy to solve on tree instances. For a graph with n vertices and m edges, our algorithm runs in O(m log n) time with high probability, which improves the previous upper bound of O(m log^3 n) shown by Mendel et al. in 2009.
The key component of our algorithm is a new approximate single-source shortest-path algorithm, which implements the priority queue with a new data structure, the bucket-tree structure. The algorithm has three properties: it only requires linear time in terms of the number of edges in the input graph; the computed distances have the distance preserving property; and when computing the shortest-paths to the k-nearest vertices from the source, it only requires to visit these vertices and their edge lists. These properties are essential to guarantee the correctness and the stated work bound.
Using this shortest-path algorithm, we show how to generate an intermediate structure, the approximate dominance sequences of the input graph, in O(m log n) time, and further propose a simple yet efficient algorithm to converted this sequence to a tree embedding in O(n log n) time, both with high probability. Combining the three subroutines gives the stated work bound of the algorithm.
We also show a new application of probabilistic tree embeddings: they can be used to accelerate the construction of a series of approximate distance oracles.

Guy E. Blelloch, Yan Gu, and Yihan Sun. Efficient Construction of Probabilistic Tree Embeddings. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blelloch_et_al:LIPIcs.ICALP.2017.26, author = {Blelloch, Guy E. and Gu, Yan and Sun, Yihan}, title = {{Efficient Construction of Probabilistic Tree Embeddings}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {26:1--26:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.26}, URN = {urn:nbn:de:0030-drops-75034}, doi = {10.4230/LIPIcs.ICALP.2017.26}, annote = {Keywords: Graph Algorithm, Metric Embeddings, Probabilistic Tree Embeddings, Single-source Shortest-paths} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

In several emerging technologies for computer memory (main memory), the cost of reading is significantly cheaper than the cost of writing. Such asymmetry in memory costs poses a fundamentally different model from the RAM for algorithm design. In this paper we study lower and upper bounds for various problems under such asymmetric read and write costs. We consider both the case in which all but O(1) memory has asymmetric cost, and the case of a small cache of symmetric memory. We model both cases using the (M,omega)-ARAM, in which there is a small (symmetric) memory of size M and a large unbounded (asymmetric) memory, both random access, and where reading from the large memory has unit cost, but writing has cost omega >> 1.
For FFT and sorting networks we show a lower bound cost of Omega(omega*n*log_{omega*M}(n)), which indicates that it is not possible to achieve asymptotic improvements with cheaper reads when omega is bounded by a polynomial in M. Moreover, there is an asymptotic gap (of min(omega,log(n)/log(omega*M)) between the cost of sorting networks and comparison sorting in the model. This contrasts with the RAM, and most other models, in which the asymptotic costs are the same. We also show a lower bound for computations on an n*n diamond DAG of Omega(omega*n^2/M) cost, which indicates no asymptotic improvement is achievable with fast reads. However, we show that for the minimum edit distance problem (and related problems), which would seem to be a diamond DAG, we can beat this lower bound with an algorithm with only O(omega*n^2/(M*min(omega^{1/3},M^{1/2}))) cost. To achieve this we make use of a "path sketch" technique that is forbidden in a strict DAG computation. Finally, we show several interesting upper bounds for shortest path problems, minimum spanning trees, and other problems. A common theme in many of the upper bounds is that they require redundant computation and a tradeoff between reads and writes.

Guy E. Blelloch, Jeremy T. Fineman, Phillip B. Gibbons, Yan Gu, and Julian Shun. Efficient Algorithms with Asymmetric Read and Write Costs. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{blelloch_et_al:LIPIcs.ESA.2016.14, author = {Blelloch, Guy E. and Fineman, Jeremy T. and Gibbons, Phillip B. and Gu, Yan and Shun, Julian}, title = {{Efficient Algorithms with Asymmetric Read and Write Costs}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {14:1--14:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.14}, URN = {urn:nbn:de:0030-drops-63656}, doi = {10.4230/LIPIcs.ESA.2016.14}, annote = {Keywords: Computational Model, Lower Bounds, Shortest-paths, Non-Volatile Memory, Sorting Networks, Fast Fourier Transform, Diamond DAG, Minimum Spanning Tree} }

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**Published in:** LIPIcs, Volume 32, 1st Summit on Advances in Programming Languages (SNAPL 2015)

We articulate the need for managing (data) locality automatically rather than leaving it to the programmer, especially in parallel programming systems. To this end, we propose techniques for coupling tightly the computation (including the thread scheduler) and the memory manager so that data and computation can be positioned closely in hardware. Such tight coupling of computation and memory management is in sharp contrast with the prevailing practice of considering each in isolation. For example, memory-management techniques usually abstract the computation as an unknown "mutator", which is treated as a "black box". As an example of the approach, in this paper we consider a specific class of parallel computations, nested-parallel computations. Such computations dynamically create a nesting of parallel tasks. We propose a method for organizing memory as a tree of heaps reflecting the structure of the nesting. More specifically, our approach creates a heap for a task if it is separately scheduled on a processor. This allows us to couple garbage collection with the structure of the computation and the way in which it is dynamically scheduled on the processors. This coupling enables taking advantage of locality in the program by mapping it to the locality of the hardware. For example for improved locality a heap can be garbage collected immediately after its task finishes when the heap contents is likely in cache.

Umut A. Acar, Guy Blelloch, Matthew Fluet, Stefan K. Muller, and Ram Raghunathan. Coupling Memory and Computation for Locality Management. In 1st Summit on Advances in Programming Languages (SNAPL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 32, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{acar_et_al:LIPIcs.SNAPL.2015.1, author = {Acar, Umut A. and Blelloch, Guy and Fluet, Matthew and Muller, Stefan K. and Raghunathan, Ram}, title = {{Coupling Memory and Computation for Locality Management}}, booktitle = {1st Summit on Advances in Programming Languages (SNAPL 2015)}, pages = {1--14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-80-4}, ISSN = {1868-8969}, year = {2015}, volume = {32}, editor = {Ball, Thomas and Bodík, Rastislav and Krishnamurthi, Shriram and Lerner, Benjamin S. and Morriset, Greg}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SNAPL.2015.1}, URN = {urn:nbn:de:0030-drops-50121}, doi = {10.4230/LIPIcs.SNAPL.2015.1}, annote = {Keywords: Parallel computing, locality, memory management, parallel garbage collection, functional programming, nested parallelism, thread scheduling} }

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