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29
LinearTime PointerMachine Algorithms for Least Common Ancestors, MST Verification, and Dominators
 IN PROCEEDINGS OF THE THIRTIETH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1998
"... We present two new data structure toolsdisjoint set union with bottomup linking, and pointerbased radix sortand combine them with bottomlevel microtrees to devise the first lineartime pointermachine algorithms for offline least common ancestors, minimum spanning tree (MST) verification, ..."
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Cited by 27 (4 self)
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We present two new data structure toolsdisjoint set union with bottomup linking, and pointerbased radix sortand combine them with bottomlevel microtrees to devise the first lineartime pointermachine algorithms for offline least common ancestors, minimum spanning tree (MST) verification, randomized MST construction, and computing dominators in a flowgraph.
Dynamic Optimality–Almost
 Proc. 45th Annu. IEEE Sympos. Foundations Comput. Sci
"... We present an O(lg lg n)competitive online binary search tree, improving upon the best previous (trivial) competitive ratio of O(lg n). This is the first major progress on Sleator and Tarjan’s dynamic optimality conjecture of 1985 that O(1)competitive binary search trees exist. 1. ..."
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Cited by 11 (1 self)
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We present an O(lg lg n)competitive online binary search tree, improving upon the best previous (trivial) competitive ratio of O(lg n). This is the first major progress on Sleator and Tarjan’s dynamic optimality conjecture of 1985 that O(1)competitive binary search trees exist. 1.
UNIFYING THE LANDSCAPE OF CELLPROBE LOWER BOUNDS
, 2008
"... We show that a large fraction of the datastructure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: • highdimensional problems, where the goal is to show large space lower bounds. • co ..."
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Cited by 11 (0 self)
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We show that a large fraction of the datastructure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: • highdimensional problems, where the goal is to show large space lower bounds. • constantdimensional geometric problems, where the goal is to bound the query time for space O(n·polylogn). • dynamic problems, where we are looking for a tradeoff between query and update time. (In this case, our bounds are slightly weaker than the originals, losing a lglgn factor.) Our reductions also imply the following new results: • an Ω(lgn/lglgn) bound for 4dimensional range reporting, given space O(n · polylogn). This is quite timely, since a recent result [39] solved 3D reporting in O(lg 2 lgn) time, raising the prospect that higher dimensions could also be easy. • a tight space lower bound for the partial match problem, for constant query time. • the first lower bound for reachability oracles. In the process, we prove optimal randomized lower bounds for lopsided set disjointness.
Planning for fast connectivity updates
 In Proceedings 48th IEEE Symposium on Foundations of Computer Science (FOCS
, 2007
"... Understanding how a single edge deletion can affect the connectivity of a graph amounts to finding the graph bridges. But when faced with d> 1 deletions, can we establish as easily how the connectivity changes? When planning for an emergency, we want to understand the structure of our network ahead ..."
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Cited by 10 (0 self)
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Understanding how a single edge deletion can affect the connectivity of a graph amounts to finding the graph bridges. But when faced with d> 1 deletions, can we establish as easily how the connectivity changes? When planning for an emergency, we want to understand the structure of our network ahead of time, and respond swiftly when an emergency actually happens. We describe a linearspace representation of graphs which enables us to determine how a batch of edge updates can impact the graph. Given a set of d edge updates, in time O(d polylg n) we can obtain the number of connected components, the size of each component, and a fast oracle for answering connectivity queries in the updated graph. The initial representation is polynomialtime constructible. 1.
Counting Inversions, Offline Orthogonal Range Counting, and Related Problems
"... We give an O(n √ lg n)time algorithm for counting the number of inversions in a permutation on n elements. This improves a longstanding previous bound of O(n lg n / lg lg n) that followed from Dietz’s data structure [WADS’89], and answers a question of Andersson and Petersson [SODA’95]. As Dietz’s ..."
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Cited by 9 (3 self)
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We give an O(n √ lg n)time algorithm for counting the number of inversions in a permutation on n elements. This improves a longstanding previous bound of O(n lg n / lg lg n) that followed from Dietz’s data structure [WADS’89], and answers a question of Andersson and Petersson [SODA’95]. As Dietz’s result is known to be optimal for the related dynamic rank problem, our result demonstrates a significant improvement in the offline setting. Our new technique is quite simple: we perform a “vertical partitioning ” of a trie (akin to van Emde Boas trees), and use ideas from external memory. However, the technique finds numerous applications: for example, we obtain • in d dimensions, an algorithm to answer n offline orthogonal range counting queries in time O(n lg d−2+1/d n); • an improved construction time for online data structures for orthogonal range counting; • an improved update time for the partial sums problem; • faster Word RAM algorithms for finding the maximum depth in an arrangement of axisaligned rectangles, and for the slope selection problem. As a bonus, we also give a simple (1 + ε)approximation algorithm for counting inversions that runs in linear time, improving the previous O(n lg lg n) bound by Andersson and Petersson.
Design of data structures for mergeable trees
 In Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2006
"... Motivated by an application in computational topology, we consider a novel variant of the problem of efficiently maintaining dynamic rooted trees. This variant requires merging two paths in a single operation. In contrast to the standard problem, in which only one tree arc changes at a time, a singl ..."
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Cited by 5 (1 self)
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Motivated by an application in computational topology, we consider a novel variant of the problem of efficiently maintaining dynamic rooted trees. This variant requires merging two paths in a single operation. In contrast to the standard problem, in which only one tree arc changes at a time, a single merge operation can change many arcs. In spite of this, we develop a data structure that supports merges on an nnode forest in O(log 2 n) amortized time and all other standard tree operations in O(log n) time (amortized, worstcase, or randomized depending on the underlying data structure). For the special case that occurs in the motivating application, in which arbitrary arc deletions (cuts) are not allowed, we give a data structure with an O(log n) time bound per operation. This is asymptotically optimal under certain assumptions. For the evenmore special case in which both cuts and parent queries are disallowed, we give an alternative O(log n)time solution that uses standard dynamic trees as a black box. This solution also applies to the motivating application. Our methods use previous work on dynamic trees in various ways, but the analysis of each algorithm requires novel ideas. We also investigate lower bounds for the problem under various assumptions. 1
A Framework for Dynamizing Succinct Data Structures ⋆
"... Abstract. We present a framework to dynamize succinct data structures, to encourage their use over nonsuccinct versions in a wide variety of important application areas. Our framework can dynamize most stateoftheart succinct data structures for dictionaries, ordinal trees, labeled trees, and text ..."
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Cited by 5 (1 self)
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Abstract. We present a framework to dynamize succinct data structures, to encourage their use over nonsuccinct versions in a wide variety of important application areas. Our framework can dynamize most stateoftheart succinct data structures for dictionaries, ordinal trees, labeled trees, and text collections. Of particular note is its direct application to XML indexing structures that answer subpath queries [2]. Our framework focuses on achieving informationtheoretically optimal space along with nearoptimal update/query bounds. As the main part of our work, we consider the following problem central to text indexing: Given a text T over an alphabet Σ, construct a compressed data structure answering the queries char(i), rank s(i), and select s(i) for a symbol s ∈ Σ. Many data structures consider these queries for static text T [5, 3, 16, 4]. We build on these results and give the best known query bounds for the dynamic version of this problem, supporting arbitrary insertions and deletions of symbols in T. Specifically, with an amortized update time of O(n ɛ), any static succinct data structure D for T, taking t(n) time for queries, can be converted by our framework into a dynamic succinct data structure that supports rank s(i), select s(i), and char(i) queries in O(t(n) + log log n) time, for any constant ɛ> 0. When Σ  = polylog(n), we achieve O(1) query times. Our update/query bounds are nearoptimal with respect to the lower bounds from [13]. 1
On range searching in the group model and combinatorial discrepancy
 In Proc. 52nd IEEE Symposium on Foundations of Computer Science
, 2011
"... Abstract. In this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that tutq = Ω(disc2) where tu is ..."
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Cited by 5 (2 self)
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Abstract. In this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that tutq = Ω(disc2) where tu is the worst case update time, tq the worst case query time and disc is the combinatorial discrepancy of the range searching problem in question. This relation immediately implies a whole range of exceptionally high and neartight lower bounds for all of the basic range searching problems. We list a few of them in the following: • For ddimensional halfspace range searching, we get a lower bound of tutq = Ω(n1−1/d). This comes within a lg lg n factor of the best known upper bound. • For orthogonal range searching, we get a lower bound of tutq = Ω(lgd−1 n). • For ball range searching, we get a lower bound of tutq = Ω(n1−1/d). We note that the previous highest lower bound for any explicit problem, due to Pǎtra¸scu [STOC’07], states that tq = Ω((lg n / lg(lg n + tu)) 2), which does however hold for a less restrictive class of data structures. Our result also has implications for the field of combinatorial discrepancy. Using textbook range searching solutions, we improve on the best known discrepancy upper bound for axisaligned rectangles in all dimensions d ≥ 3.
Dynamic connectivity: Connecting to networks and geometry
 In Proceedings 49th FOCS
, 2008
"... Dynamic connectivity is a wellstudied problem, but so far the most compelling progress has been confined to the edgeupdate model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fu ..."
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Cited by 4 (0 self)
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Dynamic connectivity is a wellstudied problem, but so far the most compelling progress has been confined to the edgeupdate model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems: Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by on vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in Õ(m2/3) amortized time, wheremdenotes the number of edges in the graph. This greatly improves over the previous result [Chan, STOC’02], which required fast matrix multiplication and had an update time of O(m 0.94). The new data structure is also simpler. Geometric connectivity asks to maintain a dynamic set of n geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axisparallel line segments and rectangles. We provide similarly improved update times, Õ(n2/3), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublineartime range queries. In particular, we obtain the first sublinear update time for arbitrary 2D line segments: O ∗ (n9/10); for ddimensional simplices: O ∗ 1 1− (n d(2d+1)); and for ddimensional balls: O ∗ (n 1 − 1
Dynamic rank/select dictionaries with applications to XML indexing
, 2006
"... \lie consider a central problem in text indexing: Given a text T over an alphabet C, construct a conlpressed data structure answering the queries char(i), rank,(i); and select,(i) for a synlbol s E C. Wlany data structures consider these queries for static text T [GGVOS; FI\/IOl, SGOG, GMROG]. We co ..."
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Cited by 4 (0 self)
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\lie consider a central problem in text indexing: Given a text T over an alphabet C, construct a conlpressed data structure answering the queries char(i), rank,(i); and select,(i) for a synlbol s E C. Wlany data structures consider these queries for static text T [GGVOS; FI\/IOl, SGOG, GMROG]. We consider the dynainic version of the problem, where we are allowed to insert and delete symbols at arbitrary positions of T. This problenl is a key challenge in compressed text illdexing and has direct applicatioil to dynaillic XI\/IL iildexing structures that answer subpath queries [FLMM05]. We build on the results of [RRROZ, GMROG] and give the best known query bounds for the dynanlic version of this problem, supporting arbitrary insertions and deletions of sylllbols in T. Specifically, with an amortized update time of O((l/e)ne), we suggest how to support rank,(i), select,(i): and char(i) queries in O((~/E) loglogn) time, for ally e < 1. The best previous query tinles for this problem were O(logn1og ICI): given by [MNOG]. Our bounds are conlpetitive with stateoftheart static structures [GhlROG]. Sonle applicable lower bounds for the partial sunls probleln [PD06] show that our update/query tradeoff is also nearly optimal. In addition, our space bound is conlpetitive with the corresponding static structures. For the special case of bitvectors (i.e., 1x1 = 2); we also show the best tradeoffs for query/update time, inlproving upoil the results of [MNOG, HSSO3; RRR021. Finally, our focus on fast query/slower update is wellsuited for a queryintensive XhlIL indexing ellvironment. Using the XBW transform [FLhllM05], we also present a dynanlic data structure that succinctly maintains an ordered labeled tree T and supports a powerf~~l set of queries on T.