Mehlhorn, K., Data Structures and Algorithms, Vol. 1, Springer Verlag, Berlin, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....X with A = t there exists an element h # H such that h restricted to A is one to one. When H = # we refer to the triple (X, Y, H) as a PHF (#; n, t, t) Perfect hash families are interesting combinatorial objects that have found numerious applications to cryptography (see, for example, [25, 28, 29, 30]) Given a PHF (#; n, t, t) X, Y, H) we can construct a generalised cumulative array for the (t, n) threhold structure on in the following way. Let h 1 , h # . For each h i we associate a t set A i = a 1 , a and define a function f i from to A i ....

....1 elements in A i for all 1 #. Note that each participant will get only one element from each A i ) Thus (f 1 , f # , A 1 , A # ) is a generalised cumulative array for the (t, n) threshold structure defined P. If # = 1 then we know that # . However, from [25] we have the following result: Result 4.2 For any integer #, n, t, there exists a PHF (#; n, t, t) provided # # #te log n#. Thus, from Result 4.2, when t is small (compared to n) and fixed the value of can be reduced from to O(log n) 11 Many questions arise from this ....

K. Mehlhorn. Data Structures and Algorithms, Volume 1, Springer-Verlag, 1984.

....is given. Computational results are presented in section 4. 2. Preliminaries 2. 1 Planarity test In the sequel we will consider a depth first search tree of the graph G = V; T; B) where V is the set of DFS numbers of the vertices, T is the set of tree edges and B the set of back edges [M]. G is assumed to be biconnected. The idea is the following: Suppose we identify a cycle, in the sequel called spine cycle, starting in the root (node 1) of the DFS tree consisting of tree edges followed by one back edge leading to node 1 again. Such a back edge must exist, because of the ....

....of S(e i ) are A(e i ) f3; 5; 6g and low (e) f3g. A segment S(e) is called strongly planar if there exists a planar embedding of S(e) in which the stem of the daughter cycle C(e) borders the outerface. The correctness of the following lemma is obvious if you consider Figure 2 (see [M]) Let e be an edge emanating from a spine cycle C. Then C S(e) is planar if and only if S(e) is strongly planar. Task (A) is solved if we will have an algorithm which tests strong planarity. Now, suppose that all of the k segments S(e i ) are strongly planar. Under what conditions can they ....

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Mehlhorn, K., Graph Algorithms and NP-Completeness, Data Structures and Algorithms, vol. 2, pp. 93-122, Springer-Verlag, 1984.

....structures trade error of approximation for faster operation, leading to theoretical and practical speedups for a wide variety of algorithms. We give approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure [28, 20], except that answers to queries are approximate. The variants support all operations in constant time provided the error of approximation is 1 polylog(n) and in O(loglog n) time provided the error is 1 polynomial(n) for n elements in the data struc ture. We consider the tolerance of ....

....output with the error of approximation tending to zero, Primes algorithm requires only linear time, Dijkstras algorithm requires O(m log log n) time, and the on line variant of Grahams algorithm requires constant amortized time per operation. 1 Introduction The van Emde Boas data structure (VEB) [28, 20] represents an ordered multiset of integers. The data AT T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974. Emaih matias research.att.com. tDepartment of Computer Science, Duke University, Box 90129, Durham, N.C. 27708 0129. art of this research was done while the author was at ....

K. Mehlhorn. Data Structures and Algorithms. Springer-Verlag, Berlin, Heidelberg, 1984.

....these components. Remark 2.15 Recall that a subset W of the nodes of a directed graph is called a strongly connected component of the graph, if for all w; w 2 W there is a path from w to w. The set of all strongly connected components of a given directed graph can be computed in linear time (Mehlhorn (1984)) Intuitively it is plausible that the sequence of box coverings B k converges to the chain recurrent set of f . Indeed, under mild assumptions on the box coverings one can prove convergence, see Eidenschink (1995) Osipenko (1999) 10 10 5 0 5 10 2 1 0 1 2 20 10 0 10 20 (a) k = 8 10 5 ....

K. Mehlhorn. Data Structures and Algorithms. Springer, 1984.

.... a strongly connected component of G, if for all w; w 2 W there is a path from w to w (i.e. if there is a sequence (w i ; w i 1 ) 2 E, i = 0; m 1, such that w = w 0 and w = wm ) The set of all strongly connected components of a given directed graph can be computed in linear time [5]. Intuitively it is plausible that the sequences of box coverings B k converge to the chain recurrent set of f . Indeed, under mild assumptions on the box coverings one can prove convergence, see [4, 8] 5 Numerical Example In this section we apply the algorithm described above to a speci c ....

K. Mehlhorn. Data Structures and Algorithms. Springer, 1984.

....can build a call graph. The nodes of the call graph are the procedures and functions of the program. If procedure A calls B, there is an edge from node A to B in the call graph. If there are no recursive procedures and functions, the call graph is acyclic. By topologically sorting (cf. Knu73a, Meh84] an acyclic call graph, symbolic evaluation can be applied to the graph in such a way that a symbolic formula for the WCET of a procedure P is available before another procedure Q, which calls P , is analyzed. We have the following theorem: Theorem 5.4. If the call graph of a program is ....

Kurt Mehlhorn, Graph algorithms and NP-completeness, Data Structures and Algorithms, vol. 2, Springer-Verlag, Berlin, 1984. 28

....Geometry and Graphics binary space partition tree, chain tree, trapezoid tree, range tree, segment tree, interval tree, priority search tree, hull tree, quad tree, R tree, grid file, metablock tree. 3 Further Information Many textbooks and monographs have been written on data structures, e.g. [1, 3, 5, 6, 7, 8, 9, 10, 13, 14, 16, 17, 15, 19]. Recent papers surveying the state of the art in data structures include [2, 4, 12, 18] The LEDA project [11] aims at developing a C library of efficient and reliable implementations of sophisticated data structures. 4 ....

K. Mehlhorn. Data Structures and Algorithms. Springer-Verlag, 1984. Volumes 1--3.

....we apply to each of the three problems, is as follows. First, in Section 2, we characterize proofs that an algorithm has obtained the correct answer. Then, in Section 3, we see how to best encode proofs in binary, the idea being 1 Aspects of this idea are explored for the case of two sets in [10]. that easy instances have succinctly encodable proofs. In Section 4, we extend lower bounds beyond the most basic information theoretic argument. In Section 5, we develop an algorithm to find a proof in time matching our lower bound. Finally, in Section 6, we extend these algorithms to produce ....

....other words, given a B tree T and elements x 1 ; xm , we would like to be able to construct a new B tree with contents T Gamma fx 1 ; xm g, without modifying T but by reusing nodes of T . This can be done using a standard persistence trick: perform the standard Btree multidelete [10], but whenever a node is modified, first make a copy of the node and then modify the copy instead. This proves the following theorem: Theorem 6.1. The difference of k sorted sets stored in read only B trees can be computed as another B tree in O(kG) time. 6.2 Computing Unions. The situation for ....

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K. Mehlhorn. Data Structures and Algorithms, vol. 1, pp. 240--241. Springer-Verlag, 1984.

....a pattern t called the subject, find the set MPTM (P,t) p, q ] p P, q sub (t) p q of all patterns in P matching subpatterns of t. 3 This paper is concerned with linear pattern matching and with solutions to the Multi Pattern Matching Problem on a uniform cost sequential RAM [1, 28]. More complex kinds of pattern matching can be solved by extensions to our algorithms. However, even for linear pattern matching, solving MPTM (P,t) efficiently seems to be extremely difficult. The current best space efficient top down algorithm to solve MPTM (P,t) where P contains a single ....

Mehlhorn, K., Sorting and Searching, Data Structures and Algorithms, 1, Springer-Verlag, 1984.

....operations. Traditional heap algorithms for insertion and deletion have a complexity of O(log 2 V ) for V virtual channels. There are a number of ways for reducing this complexity for ATM networks where timestamps take integer values in a finite range. A recursive algorithm was proposed in [14, 64, 69] for implementing add and delete operations in such a priority queue with O(log log V ) time complexity, where V is the number of elements in the queue. These algorithms were further refined by Johnson [49] who presented a non recursive algorithm with O(log log D) complexity for the add and delete ....

K. Mehlhorn, Data structures and algorithms. Springer-Verlag, 1984.

....these components. Remark 2.8: Recall that a subset W of the nodes of a directed graph is called a strongly connected component of the graph, if for all w; w 2 W there is a path from w to w. The set of all strongly connected components of a given directed graph can be computed in linear time [24]. Intuitively it is plausible that the sequences of box coverings B k converge to the chain recurrent set of T . Indeed, under mild assumptions on the box coverings one can prove convergence, see [11, 25] 2.5 Example: Chain Recurrent Set of the Knotted Flow We return to the previous example ....

K. Mehlhorn. Data Structures and Algorithms. Springer, 1984.

....in time proportional to log K. The array Z 1 ; Zm is called a (max) heap if it is partially ordered so as to satisfy (2) Z i max(Z 2i ; Z 2i 1 ) a min (heap) reverses the inequality. A convenient reference for heaps and other data structures is the text of Sara Baase (1988) or the one of Mehlhorn (1984). There it is shown that a heap of size m may be constructed in at most 4m steps and that a new item may be inserted so as to preserve the heap property in at most log m steps. It is clear that after an item is deleted, the heap property can be restored in at most 2 log m steps. The complexity of ....

Mehlhorn, K. (1984). Data Structures and Algorithms. Vol.1: Sorting and Searching.

....A memoryless algorithm maintains no state information besides the current tree. The proposed memoryless heuristics are: 1. Move to root by rotation [7] 2. Single rotation [7] 3. Splaying [65] The state based algorithms are 1. Dynamic monotone trees [17] 2. WPL trees [21] 3. D trees [57,58]. 3.3 State based algorithms Bitner [17] proposed and analyzed dynamic monotone trees. Dynamic monotone trees are a dynamic version of a data structure suggested by Knuth [45] for approximating optimal binary search trees given a distribution D. The element with maximum probability of access is ....

....the weighted path length of the tree can be computed efficiently. The WPL tree can be no better than Omega (log n) competitive (once again using repeated sequential access) but it is unknown if this bound is achieved. It is also unknown whether WPL trees are O(1) static competitive. Mehlhorn [57,58] introduced the D tree. The basic idea behind a D tree is that each time an element is accessed the binary search tree is extended by adding a dummy node as a leaf in the subtree rooted at accessed element. The extended tree is then maintained using a weight balanced or height balanced binary tree ....

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K. Mehlhorn. Data Structures and Algorithms. Springer-Verlag, New York, 1984. (3 volumes).

....e.g. 15] Examples of fundamental data structures used in three major application domains are mentioned below. Graphs and Networks adjacency matrix, adjacency lists, link cut tree [33] dynamic expression tree [5] topology tree [14] SPQR tree [8] sparsification tree [11] See also, e.g. [12, 22, 34]. Text Processing string, suffix tree, Patricia tree. See, e.g. 16] Geometry and Graphics binary space partition tree, chain tree, trapezoid tree, range tree, segment tree, interval tree, priority search tree, hull tree, quad tree, R tree, grid file, metablock tree. See, e.g. 4, 10, 13, 22, ....

....[12, 22, 34] Text Processing string, suffix tree, Patricia tree. See, e.g. 16] Geometry and Graphics binary space partition tree, chain tree, trapezoid tree, range tree, segment tree, interval tree, priority search tree, hull tree, quad tree, R tree, grid file, metablock tree. See, e.g. [4, 10, 13, 22, 26, 27, 29]. 1.5 Organization of the Chapter The rest of this chapter focuses on three fundamental abstract data types: sequences, priority queues, and dictionaries. Examples of efficient data structures and algorithms for implementing them are presented in detail in Sections 2, 3 and 4, respectively. ....

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K. Mehlhorn. Data Structures and Algorithms. Springer-Verlag, 1984. Volumes 1--3.

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Mehlhorn, K., Data Structures and Algorithms, Vol. 1, Springer Verlag, Berlin, 1984.

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K. Mehlhorn: "Data Structures and Algorithms I", Springer-Verlag, 1984

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Mehlhorn, K., Data Structures and Algorithms, Vol. 1, Springer Verlag, Berlin, 1984.

....the implementation of both versions of HT PLANAR and a demo, and report on our experimental experience. Procedure HT PLANAR is based on the Hopcroft and Tarjan linear time planarity testing algorithm as described in [Meh84, section IV.10] For the sequel we assume knowledge of section IV.10 of [Meh84]. Our procedure HT PLANAR differs from [Meh84, section IV.10] in two respects: Firstly, it works for arbitrary directed graphs and not only for biconnected undirected graphs. To this end we augment the input graph by additional edges to make it biconnected and bidirected. The augmentation does not ....

....it1; assert(H.ismap( edge e; foralledges(e,G) assert(offset[e] offset[G.reversal(e) foralledges(e,G) assert(G.source(e) G.target(e) einH[e] nil) foralledges(e,G) assert(offset[e] 0) 4 The Planarity Test We are now ready for the planarity test proper. We follow [Meh84, page 95]. We first compute dfsnumber s and parents, we delete all forward edges and all reversals of tree edges, and we reorder the adjaceny lists as described in [Meh84, page 101] We then test the strong 7 planarity. The array alpha is needed for the embedding process. It records the placement of the ....

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K. Mehlhorn. Data Structures and Algorithms. Springer Verlag, 1984.

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K. Mehlhorn. Data Structures and Algorithms. Springer, 1984.

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K. Mehlhorn. Data Structures And Algorithms. Springer Verlag, 1990.

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K. Mehlhorn. Sorting and Searching, Data Structures and Algorithms, volume 1. Springer-Verlag, 1984.

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Meh84c. Kurt Mehlhorn, Sorting and searching, Data Structures and Algorithms, vol. 1, SpringerVerlag, Berlin, 1984.

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Meh84a. Kurt Mehlhorn, Graph algorithms and NP-completeness, Data Structures and Algorithms, vol. 2, Springer-Verlag, Berlin, 1984.

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K. Mehlhorn. Sorting and Searching, Data Structures and Algorithms, volume 1. Springer-Verlag, 1984.

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K. Mehlhorn, Data Structures and Algorithms, (Springer-Verlag, Berlin, 1984), Volumes 1--3.

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