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Thatchaphol Saranurak - One of the best experts on this subject based on the ideXlab platform.
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multi finger Binary Search trees
arXiv: Data Structures and Algorithms, 2018Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:We study multi-finger Binary Search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied $k$-server problem. Finger Search is a popular technique for speeding up BST operations when a query sequence has locality of reference. BSTs with multiple fingers can exploit more general regularities in the input. In this paper we consider the cost of serving a sequence of queries in an optimal (offline) BST with $k$ fingers, a powerful benchmark against which other algorithms can be measured. We show that the $k$-finger optimum can be matched by a standard dynamic BST (having a single root-finger) with an $O(\log{k})$ factor overhead. This result is tight for all $k$, improving the $O(k)$ factor implicit in earlier work. Furthermore, we describe new online BSTs that match this bound up to a $(\log{k})^{O(1)}$ factor. Previously only the "one-finger" special case was known to hold for an online BST (Iacono, Langerman, 2016; Cole et al., 2000). Splay trees, assuming their conjectured optimality (Sleator and Tarjan, 1983), would have to match our bounds for all $k$. Our online algorithms are randomized and combine techniques developed for the $k$-server problem with a multiplicative-weights scheme for learning tree metrics. To our knowledge, this is the first time when tools developed for the $k$-server problem are used in BSTs. As an application of our $k$-finger results, we show that BSTs can efficiently serve queries that are close to some recently accessed item. This is a (restricted) form of the unified property (Iacono, 2001) that was previously not known to hold for any BST algorithm, online or offline.
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multi finger Binary Search trees
International Symposium on Algorithms and Computation, 2018Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:We study multi-finger Binary Search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied k-server problem. Finger Search is a popular technique fo ...
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pattern avoiding access in Binary Search trees
Foundations of Computer Science, 2015Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:The dynamic optimality conjecture is perhaps the most fundamental open question about Binary Search trees (BST). It postulates the existence of an asymptotically optimal online BST, i.e. One that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and Greedy [Lucas, 1988, Munro, 2000, Demaine et al. 2009]. Despite BSTs being among the simplest data structures in computer science, and despite extensive effort over the past three decades, the conjecture remains elusive. Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-n sequences is Theta(n log n), achievable by any balanced BST. Thus, the obvious missing step towards the conjecture is an understanding of the "easy" access sequences, and indeed the most fruitful reSearch direction so far has been the study of specific sequences, whose "easiness" is captured by a parameter of interest. For instance, splay provably achieves the bound of O(nd) when d roughly measures the distances between consecutive accesses (dynamic finger), the average entropy (static optimality), or the delays between multiple accesses of an element(working set). The difficulty of proving dynamic optimality is witnessed by other highly restricted special cases that remain unresolved, one prominent example is the traversal conjecture [Sleator and Tarjan, 1985], which states that preorder sequences (whose optimum is linear) are linear-time accessed by splay trees, no online BST is known to satisfy this conjecture. In this paper, we prove two different relaxations of the traversal conjecture for Greedy: (i) Greedy is almost linear for preorder traversal, (ii) if a linear-time preprocessing is allowed, Greedy is in fact linear. These statements are corollaries of our more general results that express the complexity of access sequences in terms of a pattern avoidance parameter k. Pattern avoidance is a well-established concept in combinatorics, and the classes of input sequences thus defined are rich, e.g. The k = 3 case includes preorder sequences. For any sequence X with parameter k, our most general result shows that Greedy achieves the cost n*2(a#x0391;(n))O(k) where a#x0391; is the inverse Ackermann function. Furthermore, a broad subclass of parameter-k sequences has a natural combinatorial interpretation as k-decomposable sequences. For this class of inputs, we obtain an n*2O(k) bound for Greedy when preprocessing is allowed. For k = 3, these results imply (i) and (ii). To our knowledge, these are the first upper bounds for Greedy that are not known to hold for any other online BST. To obtain these results we identify an input-revealing property of Greedy. Informally, this means that the execution log partially reveals the structure of the access sequence. This property facilitates the use of rich technical tools from forbidden sub matrix theory. Further studying the intrinsic complexity of k-decomposable sequences, we make several observations. First, in order to obtain an offline optimal BST, it is enough to bound Greedy on non-decomposable access sequences. Furthermore, we show that the optimal cost for k-decomposable sequences is Theta(n log k), which is well below the proven performance of all known BST algorithms. Hence, sequences in this class can be seen as a "candidate counterexample" to dynamic optimality.
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pattern avoiding access in Binary Search trees
arXiv: Data Structures and Algorithms, 2015Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:The dynamic optimality conjecture is perhaps the most fundamental open question about Binary Search trees (BST). It postulates the existence of an asymptotically optimal online BST, i.e. one that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and Greedy [Lucas, 1988; Munro, 2000; Demaine et al. 2009] [..] Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-n sequences is Theta(n log n), achievable by any balanced BST. Thus, the obvious missing step towards the conjecture is an understanding of the "easy" access sequences. [..] The difficulty of proving dynamic optimality is witnessed by highly restricted special cases that remain unresolved; one prominent example is the traversal conjecture [Sleator and Tarjan, 1985], which states that preorder sequences (whose optimum is linear) are linear-time accessed by splay trees; no online BST is known to satisfy this conjecture. In this paper, we prove two different relaxations of the traversal conjecture for Greedy: (i) Greedy is almost linear for preorder traversal, (ii) if a linear-time preprocessing is allowed, Greedy is in fact linear. These statements are corollaries of our more general results that express the complexity of access sequences in terms of a pattern avoidance parameter k. [..] To our knowledge, these are the first upper bounds for Greedy that are not known to hold for any other online BST. To obtain these results we identify an input-revealing property of Greedy. Informally, this means that the execution log partially reveals the structure of the access sequence. This property facilitates the use of rich technical tools from forbidden submatrix theory. [Abridged]
Laszlo Kozma - One of the best experts on this subject based on the ideXlab platform.
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multi finger Binary Search trees
arXiv: Data Structures and Algorithms, 2018Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:We study multi-finger Binary Search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied $k$-server problem. Finger Search is a popular technique for speeding up BST operations when a query sequence has locality of reference. BSTs with multiple fingers can exploit more general regularities in the input. In this paper we consider the cost of serving a sequence of queries in an optimal (offline) BST with $k$ fingers, a powerful benchmark against which other algorithms can be measured. We show that the $k$-finger optimum can be matched by a standard dynamic BST (having a single root-finger) with an $O(\log{k})$ factor overhead. This result is tight for all $k$, improving the $O(k)$ factor implicit in earlier work. Furthermore, we describe new online BSTs that match this bound up to a $(\log{k})^{O(1)}$ factor. Previously only the "one-finger" special case was known to hold for an online BST (Iacono, Langerman, 2016; Cole et al., 2000). Splay trees, assuming their conjectured optimality (Sleator and Tarjan, 1983), would have to match our bounds for all $k$. Our online algorithms are randomized and combine techniques developed for the $k$-server problem with a multiplicative-weights scheme for learning tree metrics. To our knowledge, this is the first time when tools developed for the $k$-server problem are used in BSTs. As an application of our $k$-finger results, we show that BSTs can efficiently serve queries that are close to some recently accessed item. This is a (restricted) form of the unified property (Iacono, 2001) that was previously not known to hold for any BST algorithm, online or offline.
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improved bounds for multipass pairing heaps and path balanced Binary Search trees
European Symposium on Algorithms, 2018Co-Authors: Dani Dorfman, Laszlo Kozma, Haim Kaplan, Seth Pettie, Uri ZwickAbstract:We revisit multipass pairing heaps and path-balanced Binary Search trees (BSTs), two classical algorithms for data structure maintenance. The pairing heap is a simple and efficient "self-adjusting" heap, introduced in 1986 by Fredman, Sedgewick, Sleator, and Tarjan. In the multipass variant (one of the original pairing heap variants described by Fredman et al.) the minimum item is extracted via repeated pairing rounds in which neighboring siblings are linked. Path-balanced BSTs, proposed by Sleator (cf. Subramanian, 1996), are a natural alternative to Splay trees (Sleator and Tarjan, 1983). In a path-balanced BST, whenever an item is accessed, the Search path leading to that item is re-arranged into a balanced tree. Despite their simplicity, both algorithms turned out to be difficult to analyse. Fredman et al. showed that operations in multipass pairing heaps take amortized O(log n · log log n/log log log n) time. For Searching in path-balanced BSTs, Balasubramanian and Raman showed in 1995 the same amortized time bound of O(log n · log log n/log log log n), using a different argument. In this paper we show an explicit connection between the two algorithms and improve both bounds to O(log n · 2log∗n · log∗ n), respectively O (log n · 2log∗n · (log∗ n)2), where log∗(·) denotes the slowly growing iterated logarithm function. These are the first improvements in more than three, resp. two decades, approaching the information-theoretic lower bound of Ω(log n).
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improved bounds for multipass pairing heaps and path balanced Binary Search trees
arXiv: Data Structures and Algorithms, 2018Co-Authors: Dani Dorfman, Laszlo Kozma, Haim Kaplan, Seth Pettie, Uri ZwickAbstract:We revisit multipass pairing heaps and path-balanced Binary Search trees (BSTs), two classical algorithms for data structure maintenance. The pairing heap is a simple and efficient "self-adjusting" heap, introduced in 1986 by Fredman, Sedgewick, Sleator, and Tarjan. In the multipass variant (one of the original pairing heap variants described by Fredman et al.) the minimum item is extracted via repeated pairing rounds in which neighboring siblings are linked. Path-balanced BSTs, proposed by Sleator (Subramanian, 1996), are a natural alternative to Splay trees (Sleator and Tarjan, 1983). In a path-balanced BST, whenever an item is accessed, the Search path leading to that item is re-arranged into a balanced tree. Despite their simplicity, both algorithms turned out to be difficult to analyse. Fredman et al. showed that operations in multipass pairing heaps take amortized $O(\log{n} \cdot \log\log{n} / \log\log\log{n})$ time. For Searching in path-balanced BSTs, Balasubramanian and Raman showed in 1995 the same amortized time bound of $O(\log{n} \cdot \log\log{n} / \log\log\log{n})$, using a different argument. In this paper we show an explicit connection between the two algorithms and improve the two bounds to $O\left(\log{n} \cdot 2^{\log^{\ast}{n}} \cdot \log^{\ast}{n}\right)$, respectively $O\left(\log{n} \cdot 2^{\log^{\ast}{n}} \cdot (\log^{\ast}{n})^2 \right)$, where $\log^{\ast}(\cdot)$ denotes the very slowly growing iterated logarithm function. These are the first improvements in more than three, resp. two decades, approaching in both cases the information-theoretic lower bound of $\Omega(\log{n})$.
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multi finger Binary Search trees
International Symposium on Algorithms and Computation, 2018Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:We study multi-finger Binary Search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied k-server problem. Finger Search is a popular technique fo ...
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pattern avoiding access in Binary Search trees
Foundations of Computer Science, 2015Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:The dynamic optimality conjecture is perhaps the most fundamental open question about Binary Search trees (BST). It postulates the existence of an asymptotically optimal online BST, i.e. One that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and Greedy [Lucas, 1988, Munro, 2000, Demaine et al. 2009]. Despite BSTs being among the simplest data structures in computer science, and despite extensive effort over the past three decades, the conjecture remains elusive. Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-n sequences is Theta(n log n), achievable by any balanced BST. Thus, the obvious missing step towards the conjecture is an understanding of the "easy" access sequences, and indeed the most fruitful reSearch direction so far has been the study of specific sequences, whose "easiness" is captured by a parameter of interest. For instance, splay provably achieves the bound of O(nd) when d roughly measures the distances between consecutive accesses (dynamic finger), the average entropy (static optimality), or the delays between multiple accesses of an element(working set). The difficulty of proving dynamic optimality is witnessed by other highly restricted special cases that remain unresolved, one prominent example is the traversal conjecture [Sleator and Tarjan, 1985], which states that preorder sequences (whose optimum is linear) are linear-time accessed by splay trees, no online BST is known to satisfy this conjecture. In this paper, we prove two different relaxations of the traversal conjecture for Greedy: (i) Greedy is almost linear for preorder traversal, (ii) if a linear-time preprocessing is allowed, Greedy is in fact linear. These statements are corollaries of our more general results that express the complexity of access sequences in terms of a pattern avoidance parameter k. Pattern avoidance is a well-established concept in combinatorics, and the classes of input sequences thus defined are rich, e.g. The k = 3 case includes preorder sequences. For any sequence X with parameter k, our most general result shows that Greedy achieves the cost n*2(a#x0391;(n))O(k) where a#x0391; is the inverse Ackermann function. Furthermore, a broad subclass of parameter-k sequences has a natural combinatorial interpretation as k-decomposable sequences. For this class of inputs, we obtain an n*2O(k) bound for Greedy when preprocessing is allowed. For k = 3, these results imply (i) and (ii). To our knowledge, these are the first upper bounds for Greedy that are not known to hold for any other online BST. To obtain these results we identify an input-revealing property of Greedy. Informally, this means that the execution log partially reveals the structure of the access sequence. This property facilitates the use of rich technical tools from forbidden sub matrix theory. Further studying the intrinsic complexity of k-decomposable sequences, we make several observations. First, in order to obtain an offline optimal BST, it is enough to bound Greedy on non-decomposable access sequences. Furthermore, we show that the optimal cost for k-decomposable sequences is Theta(n log k), which is well below the proven performance of all known BST algorithms. Hence, sequences in this class can be seen as a "candidate counterexample" to dynamic optimality.
Parinya Chalermsook - One of the best experts on this subject based on the ideXlab platform.
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multi finger Binary Search trees
arXiv: Data Structures and Algorithms, 2018Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:We study multi-finger Binary Search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied $k$-server problem. Finger Search is a popular technique for speeding up BST operations when a query sequence has locality of reference. BSTs with multiple fingers can exploit more general regularities in the input. In this paper we consider the cost of serving a sequence of queries in an optimal (offline) BST with $k$ fingers, a powerful benchmark against which other algorithms can be measured. We show that the $k$-finger optimum can be matched by a standard dynamic BST (having a single root-finger) with an $O(\log{k})$ factor overhead. This result is tight for all $k$, improving the $O(k)$ factor implicit in earlier work. Furthermore, we describe new online BSTs that match this bound up to a $(\log{k})^{O(1)}$ factor. Previously only the "one-finger" special case was known to hold for an online BST (Iacono, Langerman, 2016; Cole et al., 2000). Splay trees, assuming their conjectured optimality (Sleator and Tarjan, 1983), would have to match our bounds for all $k$. Our online algorithms are randomized and combine techniques developed for the $k$-server problem with a multiplicative-weights scheme for learning tree metrics. To our knowledge, this is the first time when tools developed for the $k$-server problem are used in BSTs. As an application of our $k$-finger results, we show that BSTs can efficiently serve queries that are close to some recently accessed item. This is a (restricted) form of the unified property (Iacono, 2001) that was previously not known to hold for any BST algorithm, online or offline.
-
multi finger Binary Search trees
International Symposium on Algorithms and Computation, 2018Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:We study multi-finger Binary Search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied k-server problem. Finger Search is a popular technique fo ...
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pattern avoiding access in Binary Search trees
Foundations of Computer Science, 2015Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:The dynamic optimality conjecture is perhaps the most fundamental open question about Binary Search trees (BST). It postulates the existence of an asymptotically optimal online BST, i.e. One that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and Greedy [Lucas, 1988, Munro, 2000, Demaine et al. 2009]. Despite BSTs being among the simplest data structures in computer science, and despite extensive effort over the past three decades, the conjecture remains elusive. Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-n sequences is Theta(n log n), achievable by any balanced BST. Thus, the obvious missing step towards the conjecture is an understanding of the "easy" access sequences, and indeed the most fruitful reSearch direction so far has been the study of specific sequences, whose "easiness" is captured by a parameter of interest. For instance, splay provably achieves the bound of O(nd) when d roughly measures the distances between consecutive accesses (dynamic finger), the average entropy (static optimality), or the delays between multiple accesses of an element(working set). The difficulty of proving dynamic optimality is witnessed by other highly restricted special cases that remain unresolved, one prominent example is the traversal conjecture [Sleator and Tarjan, 1985], which states that preorder sequences (whose optimum is linear) are linear-time accessed by splay trees, no online BST is known to satisfy this conjecture. In this paper, we prove two different relaxations of the traversal conjecture for Greedy: (i) Greedy is almost linear for preorder traversal, (ii) if a linear-time preprocessing is allowed, Greedy is in fact linear. These statements are corollaries of our more general results that express the complexity of access sequences in terms of a pattern avoidance parameter k. Pattern avoidance is a well-established concept in combinatorics, and the classes of input sequences thus defined are rich, e.g. The k = 3 case includes preorder sequences. For any sequence X with parameter k, our most general result shows that Greedy achieves the cost n*2(a#x0391;(n))O(k) where a#x0391; is the inverse Ackermann function. Furthermore, a broad subclass of parameter-k sequences has a natural combinatorial interpretation as k-decomposable sequences. For this class of inputs, we obtain an n*2O(k) bound for Greedy when preprocessing is allowed. For k = 3, these results imply (i) and (ii). To our knowledge, these are the first upper bounds for Greedy that are not known to hold for any other online BST. To obtain these results we identify an input-revealing property of Greedy. Informally, this means that the execution log partially reveals the structure of the access sequence. This property facilitates the use of rich technical tools from forbidden sub matrix theory. Further studying the intrinsic complexity of k-decomposable sequences, we make several observations. First, in order to obtain an offline optimal BST, it is enough to bound Greedy on non-decomposable access sequences. Furthermore, we show that the optimal cost for k-decomposable sequences is Theta(n log k), which is well below the proven performance of all known BST algorithms. Hence, sequences in this class can be seen as a "candidate counterexample" to dynamic optimality.
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pattern avoiding access in Binary Search trees
arXiv: Data Structures and Algorithms, 2015Co-Authors: Parinya Chalermsook, Mayank Goswami, Laszlo Kozma, Kurt Mehlhorn, Thatchaphol SaranurakAbstract:The dynamic optimality conjecture is perhaps the most fundamental open question about Binary Search trees (BST). It postulates the existence of an asymptotically optimal online BST, i.e. one that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and Greedy [Lucas, 1988; Munro, 2000; Demaine et al. 2009] [..] Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-n sequences is Theta(n log n), achievable by any balanced BST. Thus, the obvious missing step towards the conjecture is an understanding of the "easy" access sequences. [..] The difficulty of proving dynamic optimality is witnessed by highly restricted special cases that remain unresolved; one prominent example is the traversal conjecture [Sleator and Tarjan, 1985], which states that preorder sequences (whose optimum is linear) are linear-time accessed by splay trees; no online BST is known to satisfy this conjecture. In this paper, we prove two different relaxations of the traversal conjecture for Greedy: (i) Greedy is almost linear for preorder traversal, (ii) if a linear-time preprocessing is allowed, Greedy is in fact linear. These statements are corollaries of our more general results that express the complexity of access sequences in terms of a pattern avoidance parameter k. [..] To our knowledge, these are the first upper bounds for Greedy that are not known to hold for any other online BST. To obtain these results we identify an input-revealing property of Greedy. Informally, this means that the execution log partially reveals the structure of the access sequence. This property facilitates the use of rich technical tools from forbidden submatrix theory. [Abridged]
Eran Yahav - One of the best experts on this subject based on the ideXlab platform.
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practical concurrent Binary Search trees via logical ordering
ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, 2014Co-Authors: Dana Drachsler, Martin Vechev, Eran YahavAbstract:We present practical, concurrent Binary Search tree (BST) algorithms that explicitly maintain logical ordering information in the data structure, permitting clean separation from its physical tree layout. We capture logical ordering using intervals, with the property that an item belongs to the tree if and only if the item is an endpoint of some interval. We are thus able to construct efficient, synchronization-free and intuitive lookup operations. We present (i) a concurrent non-balanced BST with a lock-free lookup, and (ii) a concurrent AVL tree with a lock-free lookup that requires no synchronization with any mutating operations, including balancing operations. Our algorithms apply on-time deletion; that is, every request for removal of a node, results in its immediate removal from the tree. This new feature did not exist in previous concurrent internal tree algorithms.We implemented our concurrent BST algorithms and evaluated them against several state-of-the-art concurrent tree algorithms. Our experimental results show that our algorithms with lock-free contains and on-time deletion are practical and often comparable to the state-of-the-art.
Neeraj Mittal - One of the best experts on this subject based on the ideXlab platform.
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fast concurrent lock free Binary Search trees
ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, 2014Co-Authors: Aravind Natarajan, Neeraj MittalAbstract:We present a new lock-free algorithm for concurrent manipulation of a Binary Search tree in an asynchronous shared memory system that supports Search, insert and delete operations. In addition to read and write instructions, our algorithm uses (single-word) compare-and-swap (CAS) and bit-test-and-set (SETB) atomic instructions, both of which are commonly supported by many modern processors including Intel~64 and AMD64.In contrast to existing lock-free algorithms for a Binary Search tree, our algorithm is based on marking edges rather than nodes. As a result, when compared to other lock-free algorithms, modify (insert and delete) operations in our algorithm work on a smaller portion of the tree, thereby reducing conflicts, and execute fewer atomic instructions (one for insert and three for delete). Our experiments indicate that our lock-free algorithm significantly outperforms all other algorithms for a concurrent Binary Search tree in many cases, especially when contention is high, by as much as 100%.
