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Don’t Rush into a Union: Take Time to Find Your Roots
, 2011
"... We present a new threshold phenomenon in data structure lower bounds where slightly reduced update times lead to exploding query times. Consider incremental connectivity, letting tU be the time to insert an edge and tq be the query time. For tU = Ω(tq), the problem is equivalent to the wellundersto ..."
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We present a new threshold phenomenon in data structure lower bounds where slightly reduced update times lead to exploding query times. Consider incremental connectivity, letting tU be the time to insert an edge and tq be the query time. For tU = Ω(tq), the problem is equivalent to the wellunderstood union–find problem: INSERTEDGE(s, t) can be implemented by UNION(FIND(s), FIND(t)). This gives worstcase time tU = tq = O(lg n / lg lg n) and amortized tU = tq = O(α(n)). By contrast, we show that if tU = o(lg n / lg lg n), the query time explodes to tq ≥ n 1−o(1). In other words, if the data structure doesn’t have time to find the roots of each disjoint set (tree) during edge insertion, there is no effective way to organize the information! For amortized complexity, we demonstrate a new inverseAckermann type tradeoff in the regime tU = o(tq). A similar lower bound is given for fully dynamic connectivity, where an update time of o(lg n) forces the query time to be n 1−o(1). This lower bound allows for amortization and Las Vegas randomization, and comes close to the known O(lg n · (lg lg n) O(1) ) upper bound. 1
MinMaxProfiles: A unifying view of common intervals, nested common intervals and conserved intervals of K permutations
"... Common intervals of K permutations over the same set of n elements were firstly investigated by T. Uno and M.Yagiura (Algorithmica, 26:290:309, 2000), who proposed an efficient algorithm to find common intervals when K = 2. Several particular classes of intervals have been defined since then, e.g. c ..."
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Common intervals of K permutations over the same set of n elements were firstly investigated by T. Uno and M.Yagiura (Algorithmica, 26:290:309, 2000), who proposed an efficient algorithm to find common intervals when K = 2. Several particular classes of intervals have been defined since then, e.g. conserved intervals and nested common intervals, with applications mainly in genome comparison. Each such class, including common intervals, led to the development of a specific algorithmic approach for K = 2, and except for nested common intervals for its extension to an arbitrary K. In this paper, we propose a common and efficient algorithmic framework for finding different types of common intervals in a set P of K permutations, with arbitrary K. Our generic algorithm is based on a global representation of the information stored in P, called the MinMaxprofile of P, and an efficient data structure, called an LRstack, that we introduce here. We show that common intervals (and their subclasses of irreducible common intervals and samesign common intervals), nested common intervals (and their subclass of maximal nested common intervals) as well as conserved intervals (and their subclass of irreducible conserved intervals) may be obtained by appropriately setting the parameters of our algorithm in each case. All the resulting algorithms run in O(Kn + N)time and need O(n) additional space, where N is the number of solutions. The algorithms for nested common intervals and maximal nested common intervals are new for K> 2, in the sense that no other algorithm has been given so far to solve the problem with the same complexity, or better. The other algorithms are as efficient as the best known algorithms.
Supervised by:
, 2007
"... The Universe type system allows a programmer to control aliasing and dependencies in objectoriented programs by applying an ownership relation to structure the object store. In previous projects the Universe type system was extended by static checks for Uniqueness and ownership transfer [[5], [6]] a ..."
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The Universe type system allows a programmer to control aliasing and dependencies in objectoriented programs by applying an ownership relation to structure the object store. In previous projects the Universe type system was extended by static checks for Uniqueness and ownership transfer [[5], [6]] and for generic types in the Universe Type System [[1], [2]]. The main part of this master project is to design and implement the runtime support for Uniqueness (and ownership transfer) and for generic types. From Java 5 generic types are introduced. Java does not store the runtime type of the type arguments. But the Universe type system needs this information for its runtime model. Based on GUT [[2], [3]] which defines the runtime model for generic types in the Universe type system the current runtime implementation in the multijava [11] compiler is now extended by an implementation for this runtime model. The runtime support for Uniqueness is an extension of the basic Universe runtime implementation to fulfill the requirements of Uniqueness
A Simple and Efficient UnionFindDelete Algorithm
, 2010
"... The UnionFind data structure for maintaning disjoint sets is one of the best known and widespread data structures, in particular the version with constanttime Union and efficient Find. Recently, the question of how to handle deletions from the structure in an efficient manner has been taken up, fi ..."
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The UnionFind data structure for maintaning disjoint sets is one of the best known and widespread data structures, in particular the version with constanttime Union and efficient Find. Recently, the question of how to handle deletions from the structure in an efficient manner has been taken up, first by Kaplan, Shafrir and Tarjan (2002) and subsequently by Alstrup et al. (2005). The latter work shows that it is possible to implement deletions in constant time, without affecting adversely the asymptotic complexity of other operations, even when this complexity is calculated as a function of the current size of the set. In this note we present a conceptual and technical simplification of the algorithm, which has the same theoretical efficiency, and is probably more attractive in practice. Keywords: Data structures, Disjoint sets, UnionFind 1
Disjoint Set Union with Randomized Linking
, 2014
"... A classic result in the analysis of data structures is that path compression with linking by rank solves the disjoint set union problem in almostconstant amortized time per operation. Recent experiments suggest that in practice, a naıve linking method works just as well if not better than linking ..."
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A classic result in the analysis of data structures is that path compression with linking by rank solves the disjoint set union problem in almostconstant amortized time per operation. Recent experiments suggest that in practice, a naıve linking method works just as well if not better than linking by rank, in spite of being theoretically inferior. How can this be? We prove that randomized linking is asymptotically as efficient as linking by rank. This result provides theory that matches the experiments, which implicitly do randomized linking as a result of the way the input instances are generated.