P. v. Emde Boas, R. Kaa.s and E. Zijlstra, Design and Implementation of an Efficient Priority Queue, Math. Systems Theory 10 (1977), 99-127.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in advance) do not have a linear solution on a pointer machin, and that although the Split Find problem can be solved in linear time on a RAM (cf; 6] this is not possible on a pointer machine. Problem UNION FIND Table 1: Complexity on Pointer Mchines SPLIT FIND General model O(loglogn) [3] 2 O(n rn.a(rn, n) new Separation condition e( 15] amortized e(loglogn) 12] O(n m.a(m,n) new O(logn) 12] O(n m.ct(rn, n) new Recently, in [4] a lower bound was proved for the Union Find problem on the Cell Probe Machine with word size log n, where n is the size of the ....

P. v. Emde Boas, R. Kaa.s and E. Zijlstra, Design and Implementation of an Efficient Priority Queue, Math. Systems Theory 10 (1977), 99-127.

....was sorting. After sorting the method runs in O(n) time. Hence, applying the efficient sorting method on a grid we obtain: Theorem 7.1 Given a set of n points on a u x u grid, the convez hull can be computed in time O(nloglog.u) 26 A second basic tool is searching. Van Emde Boas [27] see also [28], has shown that one can store n values between 0 and u 1 using O(u) storage such that values can be inserted, deleted and searched for in time O(loglog u) Also 1 dimensional range queries can be answered this efficienty. Now reconsider the axis parallel line segment intersection problem where ....

va Emde Boas, P., Kay, tl.., Zijlstra, E., Design and implementation of an efficient priority queue. Math. Systems Theory I0, 99-127 (1977)

....and delete are amortized and randomized,i.e. the averaging involved in the analysis is over choices made by the algorithm and not over the input sequence. q.e.d. This result extends and unifies the work of v. Emde Boas et al. v. Emde Boas, D. Johnson, R. Karlsson, and D. Willard. In [E77] [EKZ77], and [J82] it is shown that the time bounds can be achieved using O(N) space and that a space bound O(N ffl ) can be achieved at the cost of multiplying the time bounds by 1=ffl for any ffl 0. In [W83] the space bound O(jdom Dj) is achieved for static dictionaries, and in [K84] W84] the ....

P. v. Emde Boas, R. Kaas, E. Zijlstra: "Design and Implementation of an Efficient Priority Queue", Math. Systems Theory 10, 99--127, 1977

....5 Epilogue The main question that we have not dealt with in this paper is the determination of a lower bound for the T P problem. Techniques for proving lower bounds on pointer machines used by other researchers (e.g. 14, 11] and results achieved by other researchers on similar problems [4, 15, 16, 7] do not seem to be suitable for the T P problem. Since this paper was submitted, using a different technique, we have been able to prove that the T P problem has a lower bound time complexity of Omega Gammaf lg n) 12] This result proves that the data structures proposed in this paper are ....

P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and Implementation of an Efficient Priority Queue. Mathematical Systems Theory, 10, 1977. 7

....results, we present several successively better schemes, finally ending with the best result. The data structure we design and use to obtain the optimal upper bound of O(lg lg n) has some similarities with van Emde Boas stratified tree structure designed to efficiently implement priority queues [18] and used to provide efficient solutions to various set manipulation problems (e.g. 18, 9, 10] Both these data structures use the idea of a nested tree structure, where the same logical organization is repeated at different levels of nesting. In our case the levels of nesting are represented ....

.... The data structure we design and use to obtain the optimal upper bound of O(lg lg n) has some similarities with van Emde Boas stratified tree structure designed to efficiently implement priority queues [18] and used to provide efficient solutions to various set manipulation problems (e.g. [18, 9, 10]) Both these data structures use the idea of a nested tree structure, where the same logical organization is repeated at different levels of nesting. In our case the levels of nesting are represented by different colored trees, while in the stratified tree structured the levels are represented ....

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van Emde Boas, P., Kaas, R., and Zijlstra, E. Design and Implementation of an Efficient Priority Queue. Mathematical Systems Theory 10 (1977). 15

....RAM memory cells can contain (log n) bit integers that may be compared, added, subtracted, multiplied and divided (with rest) Furthermore, the integers can be also used as pointers to other memory cells (indirect addressing) All these operations take constant time. A variant of the RAM model [81, 45, 44, 51] allows also bitwise logical operations on the integers in constant time. In the real RAM model memory cells also can store real numbers. Since a real number can contain an infinite amount of information in its binary expansion, the set of valid operations for real numbers must be carefully ....

P. van Emde Boas, R. Kaas, and E. Ziljstra. Design and implementation of an efficient priority queue. Math. Systems Theory, 10:99--127, 1977. 98

....Sciences and Engineering Research Council of Canada and Communications and Information Technology Ontario. Part of the research was conducted while visiting Laboratoire de Recherche en Informatique, Universite de Paris Sud, Orsay, France. 1 amounts of space. For example, van Emde Boas trees [41, 40] can be used to perform predecessor queries on any set of integers from a universe of size N in O(loglogN) time, but they require W(N) space. However, there have been important algorithmic breakthroughs showing that such techniques have more general applicability. For example, with two level ....

....updates in time O(n e ) for any constant e 0. All of these data structures use O(n) space. Although hashing provides an optimal solution to the static dictionary problem, it is not directly applicable to the predecessor problem. Another data structure, van Emde Boas (or stratified) trees [41, 40], is useful for both static and dynamic versions of the predecessor problem. These trees support membership queries, predecessor queries, and updates in O(loglogN) time. The set is stored in a binary trie and binary search is performed on the logN bits used to represent individual elements. The ....

P. Van Emde Boas, R Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10:99--127, 1977.

....the Search step visits a constant number of nodes by Lemma 6.2. For 6.1. Removing a site from the Delaunay triangulation 135 each node, we must locate the creator of the node in Reinsert, which can be done in O(log log n) worst case deterministic time, by using a bounded ordered dictionary [vEBKZ77]. The universe for this dictionary is the insertion age of the points, or in other words the number of the sites. The required finiteness of the universe can be circumvented using standard dynamization techniques, see for example [Ove83, section5. 2] To preserve the simplicity of the ....

P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Sysyems Theory, 10:99--127, 1977.

.... this efficiently, we need to use a priority queue with the following operations: Insert, Delete, and Successor (i.e. given some value i in the priority queue, determine j, the next larger value that is currently in the priority queue) The integer priority queue originally 6 due to van Emde Boas [VKZ77, Joh82] has just the properties we need. Each site can be assigned one of the integers 1 through n based on its order when sorted by x coordinate (i.e. the data point with least x coordinate is 1, the one with next smallest x coordinate is 2, etc. These integer values can be used in an integer ....

P. van Emde Boas, R. Kaas, and E. Zijlstra, Design and Implementation of an Efficient Priority Queue, Math. Systems Theory, 10 (1977), 99--127.

....we perform a reporting step again. We repeat this until either there are no red points left or there are no blue points left. At the end of the algorithm, we will have reported all red blue dominance pairs. Since the points have coordinates from a finite universe, we can use van Emde Boas trees [vEB77a, vEB77b] in order to search among them in O(log log n) time rather than O(log n) time. In this way, the merge step of the divide and conquer algorithm takes 2 O O O O O O O O O O O O p q Figure 1: The maximal layer of a planar point set S forms a contour. The point p is inside the contour, ....

....points and a set B of blue points in U 3 , and we have to find all dominance pairs (r; b) where r 2 R and b 2 B. The points in both sets R and B are sorted by their third coordinates. 5 In the final algorithm, we first construct an empty van Emde Boas tree (vEB tree) on the universe U . See [vEB77a, vEB77b]. During the entire algorithm, elements will be inserted and deleted in this tree and we will perform queries on it. Its construction time is O(u) its query and update times are O(log log u) and it uses O(u) space. In the rest of this section, we assume that we have this tree available. First ....

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P. van Emde Boas, R. Kaas and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10:99--127, 1977.

....of elements on the queue as heaps. The fastest implementations of heaps are described in [3, 10, 12] Alternative implementations of priority queues use buckets (e.g. 1, 5, 7, 8] Operation times for bucket based implementations depend on the maximum event duration C, defined in Section 2. See [2] for a related data structure. Heaps are particularly efficient when the number of elements on the heap is small. Bucket based priority queues are particularly efficient when the maximum event duration C is small. Furthermore, some of the work done in bucket based implementations can be amortized ....

P. Van Emde Boas, R. Kaas, and E. Zijlstra. Design and Implementation of an Efficient Priority Queue. Math. Systems Theory, 10:99--127, 1977.

....operations on words in constant time. Miltersen [10] refers to this model as a Practical RAM. We assume the elements are integers in the range 0: 2 w Gamma 1. A tradeoff similar to the one for the comparison model [6] is not known for a Practical RAM. A data structure of van Emde Boas et al. [15, 16] supports the operations Insert, Delete, Pred, FindMin and FindMax on a Practical RAM in worst case O(log w) time. For word size log O(1) n this implies an O(log log n) time implementation. Thorup [14] recently presented a priority queue supporting Insert and ExtractMin in worst case O(log log ....

....The data structure is the first allowing predecessor queries in O(logn= log log n) time while having O(log log n) update time. If f(n) p log n, we achieve time bounds matching those of Andersson [2] The basic idea of our construction is to apply the data structure of van Emde Boas et al. [15, 16] for O(f(n) levels and then switch to a packed search tree of height O(logn=f(n) This is very similar to the data structure of Andersson [2] But where Andersson uses O(logn=f(n) time to update his packed B tree, we only need O(f(n) time. The idea we apply to achieve this speedup is to add ....

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Peter van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10:99--127, 1977. This article was processed using the L a T E X macro package with LLNCS style

....j] return the maximum value in A[i] A[i 1] A[j] can be processed in O(1) time and work. Lemma 2.4 ( 6] Given an array, A, of n numbers, we can compute, for each location i, the nearest position j such that j i and A[j] A[i] in O(log log n) time with O(n) work. Lemma 2. 5 ([26]) A subset of numbers from the universe 1 : N can be maintained under insert, delete, extract maximum or minimum and find predecessor or successor queries in O(log log N ) time using O(s) space where s is the size of the subset. Lemma 2.6 ( 7] Given a suffix tree of a string T , it can be ....

....Each query takes time O(1) Both these bounds follow from Lemma 2.6. In what follows, we show how the preprocessing work can be reduced to O( P all real skeleton trees R c jR c j) O(n C) while taking O(log log n) time for each query. To achieve this, we use the van Emde Boas result [26]. For any c 2 C, associate with each node in the naive skeleton tree T c , the group of out leaves which are its descendant but which are not the descendant of any other internal node in T c . The key observation is that these groups form a partition of the out leaves of T into sets of disjoint ....

P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Math. Systems Theory, 10:99--127, 1977.

....as representing integers, is order preserving. Hence we can feed floating points to an integer priority queue, if we just don t tell it that it is floats. Similarly, van Emde Boas data Most of this work was done while the author visited the Max Planck Institut fur Informatik. structure [vBKZ77] works in time O(log ) where is the word length, no matter whether the words represent integers or floats. Up to recently, all theoretical developments in the single source shortest paths problem (SSSP) were based in Dijkstra s algorithm [Dij59] where vertices are visited in increasing order ....

P. van Emde Boas, R. Kaas, and E. Zijlstra, Design and implementation of an efficient priority queue, Math. Syst. Th. 10 (1977), 99--127.

....based data structures for representing sets, such as trees and heaps of all kinds. It is also sufficiently powerful to implement several fundamental algorithms and data structures where indirect addressing and or bit manipulation is essential, such as for example tries, van Emde Boas trees [11], Gabow and Tarjan s special case Union Find algorithm [15] Chazelle s M structure [7] Dietz list indexing structure [8] the O(n log log n) sorting algorithm of Andersson et al. [2] Andersson s O( p log n) search structure [1] and Thorup s O(log log n) priority queue [16] For some of these ....

P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10:99--127, 1977.

.... [76] but one representative example is (a; b) trees [64] which support all the mentioned operations in time O(log n) The first data structure showing that the Omega Gammae 2 n) lower bound for comparison based implementations does not hold for bounded universes is due to van Emde Boas et al. [104, 106]. The data structure of van Emde Boas et al. supports the operations Insert, Delete, Pred, FindMin and FindMax on a Practical RAM in worst case O(log w) time. For word size log O(1) n this implies a time O(log log n) implementation. Thorup [102] has presented a priority queue supporting Insert ....

....we achieve the result of Thorup but in the worst case sense. Furthermore we support Pred queries in worst case O(log n= log log n) time. If f(n) p log n, we achieve time bounds matching those of Andersson [5] Our construction is obtained by combining the data structure of van Emde Boas et al. [104, 106] with packed search trees similar to those of Andersson [5] but where we add buffers of delayed insertions and deletions to the nodes of the packed search tree. The idea of adding buffers to a search tree is inspired by Arge [7] who designs efficient external memory data structures. The data ....

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Peter van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10:99--127, 1977.

....can be implemented on a random access computer so that all operations have worst case complexity O(log log n) Furthermore, if we allow the complexities of union and split operations to be n ffl , then find can be performed in constant time. This is due to Van Emde Boas, Kaas and Zijlstra [22]. Imai and Asano [13] prove that SPLIT FIND(n) can be implemented on a random access computer so that the amortized complexity of the operations is O(1) i.e. for any m, m operations can be performed in time O(m) On the pointer machine model, some more bounds were known. The pointer machine ....

P. Van Emde Boas, R. Kaas, E. Zijlstra, Design and implementation of an efficient priority queue, Math. Systems Theory 10 (1977) 99-127.

....n) bound in deterministic linear space using standard AC 0 instructions only [Ram96] There has also been a substantial development based on the maximal edge weight C, again assuming integer edge weights, each fitting in one word. First note that using van Emde Boas s general search structure [MN90, vBoa77, vBKZ77], and bucketing according to bD(v) nc, we get an O(m log log C) algorithm for SSSP. Ahuja, Melhorn, Orlin, and Tarjan have found a priority queue for SSSP giving a running time of O(n p log C m) AMOT90] Recently, this has been improved by Cherkassky, Goldberg, and Silverstein to O(n 3 p ....

P. van Emde Boas, R. Kaas, and E. Zijlstra, Design and implementation of an efficient priority queue, Math. Syst. Th. 10 (1977), 99--127.

....of Fredman, Koml os and Szemer edi [7] Furthermore, for the first half of this paper, we require that the rectangle corners and query points lie in a fixed size integer grid [0; U Gamma 1] d . Stratified trees, a data structure introduced by van Emde Boas [19] and extended by him and others [10, 14, 21, 22], exploit the power of a RAM on a fixed universe. They have been used for log logarithmic time queries in one dimensional point location, more commonly known as searching a list for the successor of a query point. Muller [15] used a type of stratified tree as a twodimensional point location ....

....tree and T nodes of the interval tree. The height of L is h = Theta(log log U= log k) For each L node , define the T depth( as the number of the deepest level of T in the subtree rooted at . Figure 2: A stratified tree storing two intervals To store an interval I in a van Emde Boas tree [19, 21], one would record the bounds of I in each T node that I cuts. We use the level search tree as a guide and store I only cut nodes certain levels of T as follows. Store interval I in level I of T , and, for each L node on the path in L from I to the L root, store I at level 1 T depth( ....

[Article contains additional citation context not shown here]

P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Math. Systems Theory, 10:99--127, 1977.

....point standard is made such that interpreting the bit string representation of floating points as representing integers, is order preserving. Hence we can feed floating points to an integer priority queue, if we just don t tell it that it is floats. Similarly, van Emde Boas data structure [vBKZ77] works in time O(log ) where the the word length, no matter whether the words represent integers or floats. Up to recently, all theoretical developments in the single source shortest paths problem (SSSP) were based in Dijkstra s algorithm [Dij59] where vertices are visited in increasing ....

P. van Emde Boas, R. Kaas, and E. Zijlstra, Design and implementation of an efficient priority queue, Math. Syst. Th. 10 (1977), 99--127.

....recently introduced in [MVY94] the (word based) radix tree. This data structure is qualitatively different than traditional priority queues in that its time per operation does not increase as the number of keys grows (in fact it can decrease) Instead, as in the van Emde Boas data structure [vKZ77], the running time depends on the size of the universe the set of possible keys which is assumed to be f0; 1; Ug, for some integer U 0. Most notably, the data structure is designed to take advantage of any tolerance the application has for working with approximate, rather than ....

P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Math. Systems Theory, 10:99--127, 1977.

.... (code indx[i] blacks) g Decode Algorithm 1: Linear decoding algorithm. 4 Faster Decoding In Algorithm 1 we search for the level of the symbol linearly from the top and in this section we improve the solution by using binary search on the levels of the mutated Huffman tree (cf. [3]) for a toy example see Figure 3. The search, when probing on some level, shifts in or out the appropriate number of bits from the input stream and checks if the computed code corresponds to the black, gray or white node. If the corresponding node is: down 7 10 11 5 6 9 12 13 8 1 2 3 4 0 2 1 ....

P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10(1):99--127, 1977.

....Instead, notice that it suffices to use a data structure that solves the static predecessor problem on this list of numbers. That gives a number of tradeoffs for our problem based on those for the static predecessor problem. For example, placing the elements of each list in a van Emde Boas tree [vKZ77] allows such queries to be answered in O(log log n) time. The upshot is that we get a O(n log n) preprocessing algorithm with O(log log n) time queries. To reduce the preprocessing, we can partition T into subtrees of size Theta(log n) which we collapse into nodes. Now the tree is of size ....

P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Math. Systems Theory, 10:99--127, 1977.

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Emde Boaz P.v., Kaas R. and Zijlstra E., Design and Implementation of an efficient Priority Queue, Mathematical Systems Theory 10, (1977), 99-127.

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P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10:99--127, 1977.

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