Chazelle B.,Guibas L., Fractional Cascading: a data structuring technique with Geometric Applications, 12 ICALP 1985, pp. 90-100.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....containing p. The entire search from top to bottom in the multi level data structure takes therefore O(log n) time, resulting in O(log n) candidate answers. The minimum among these candidates is the final answer to the query. The query time can be improved by applying fractional cascading [8] to the two lower levels of the data structure. This is possible because the bottom level structures are ordered lists (a sequence of intervals) Fractional cascading improves the query time in a 2 level data structure consisting of a segment tree with the one dimensional ordered lists as ....

B. Chazelle and L. Guibas, Fractional cascading I: A data structuring technique, Algorithmica 1 (1986), pp. 133-162.

....here, we are given a monotone planar decomposition having JV vertices, and a series of query points. For each query point, we are to return the identifier of the region in which it lies. In main memory, this problem can be solved in optimal time O( iv If)log iV) using fractional cascading [7,8]; O(iv log iv) is spent on preprocessing and O(If log iv) is needed to perform the queries. We can apply the technique of Lemma 4.1 to the main tree, but the bridge pointers connecting the catalogs make the dag non planar. To get around this, we note that as queries traverse the edges between ....

B. Chazelle and L. J. Guibas, "Fractional Cascading: I. A Data Structuring Technique," Algorithmica 1 (1986), 133 162. II. Applications," Algorithmlea 1 (1986), 163 191.

....here, we are given a monotone planar decomposition having N vertices, and a series of K query points. For each query point, we are to return the identifier of the region in which it lies. In main memory, this problem can be solved in optimal time O( N I )log N) using fractional cascading [7,8]; O(N log N) is spent on preprocessing and O(I log N) is needed to perform the queries. Tamassia and Vitter [35] have demonstrated a technique by which the fractional cascading used to solve this problem can be implemented in parallel. Their technique can solve our problem in O( N p K) logp N) ....

B. Chazelle & L. J. Guibas, "Fractional Cascading: I. A Data Structuring Technique," Algorithmlea 1 (1986), 133 162.

....Query Problem In order to construct the dominance tree efficiently our method relies on the solution to a query problem involving line segments and rays. The solution utilizes both fractional cascading and segment trees. We now review the fractional cascading technique of Chazelle and Guibas [24]. We are given a directed bounded degree graph G = V; E) where each node v in G contains a sorted list C(v) The problem is to construct a data structure so that given a walk (v 1 ; v 2 ; v s ) and an arbitrary element x, one processor can locate x in all of the C(v i ) s quickly. An ....

Chazelle, B., and Guibas, L. J., "Fractional Cascading: I. A Data Structuring Technique," Algorithmica Vol. 1, No. 2, pp. 133-162.

....t) d Gamma1 log log n) query time, where t can be any integer with 2 t m. For instance, taking t = 2 we get a structure with O(m log d Gamma1 m) space and O(log d Gamma1 m log log n) query time; in this case the query time can be improved to O(log d Gamma1 m) using fractional cascading [CG86]. We can also take t = m ffl (where ffl can be any fixed positive constant) to obtain a structure with O(m 1 ffl ) space and O(log log n) query time. In the latter result, we achieve the same query time as the best known bound for the classical case d = 1, while the space is still close to ....

B. Chazelle andL. J. Guibas. Fractional Cascading I: A data structuring technique. Algorithmica, pp. 133-162, 1986.

....in O(log N ) time for a given . Hence, the time complexity of computing R is O(N log N) if preprocessing takes O(N 2 ) time. We can reduce the O(N log N) computing time to O(N) by applying the fractional x y I(left) I(right) I(mid) Q(I) I Figure 4: Hand Probe cascading data structure [CG86] omitted in this version of the paper) We have the following similar results for the family of rectangles and the family of rectilinear convex regions, although the time complexity is increased (we omit the proof in this version of the paper) Lemma 3.4 The touching oracle to conv(P ) can be ....

B. Chazelle and L. Guibas, Fractional Cascading: A Data Structuring Technique. Algorithmica 1: 133-- 162, 1986.

....here, we are given a monotone planar decomposition having N vertices, and a series of K query points. For each query point, we are to return the identifier of the region in which it lies. In main memory, this problem can be solved in optimal time O( N K) log N ) using fractional cascading [7,8]; O(N log N ) is spent on preprocessing and O(K log N ) is needed to perform the queries. We can apply the technique of Lemma 4.1 to the main tree, but the bridge pointers connecting the catalogs make the dag non planar. To get around this, we note that as queries traverse the edges between nodes ....

B. Chazelle and L. J. Guibas, "Fractional Cascading: I. A Data Structuring Technique," Algorithmica 1 (1986), 133--162.

.... Gamma; Delta : So, a four dimensional range tree S can be employed to answer the query Q(p) This gives a sweep plane algorithm of time complexity O(nlog 4 n) two magnitudes of logn slower than claimed. One factor can be removed with the well known fractional cascading technique [5, 19]. For the other one, once again subdivide the cone C=3 by triangulating its quadratic cross section: The two resulting C=6 will have only three boundary planes. Hence, q and S are three dimensional instead of four. 6.2 Scholium: 3 Be C a partition of IR D into k convex cones and D a ....

Bernard Chazelle, Leonidas J. Guibas: "Fractional cascading: I. A data structuring technique", Algorithmica 1, 1986, 133-162.

....L1 neighbor in O( log n) D Gamma1 log log n) amortized time per update. We want to remark here that all our algorithms use classical and well understood data structures, such as range trees, segment trees, and skewer trees. Moreover, we apply the well known technique of fractional cascading ([6, 14]) several times. The rest of this paper is organized as follows. In Section 2, we recall the definition of a range tree. This data structure is used in Section 3 to solve the L1 neighbor problem. In Section 4, we give the data structure for solving the approximate L 2 nearest neighbor ....

....D) O( log n) D ) Now consider the planar case. The algorithm follows a path in the main tree and locates the y coordinate of the query point in the associated structure a binary search tree of some of the nodes on this path. It is well known that layering or fractional cascading (see [6, 15]) can be applied to improve the query time from O( log n) 2 ) to O(log n) i.e. Q(n; 2) O(log n) As a result, the query time for the D dimensional case, where D 2, is improved to O( log n) D Gamma1 ) Hence, we have a data structure for the L1 neighbor problem that has a query time of ....

[Article contains additional citation context not shown here]

B. Chazelle and L. J. Guibas, Fractional cascading I: A data structuring technique, Algorithmica, 1 (1986), pp. 133--162.

....we propose. The only other method for indexing classes appearing in the literature is a scheme called the H tree [25] based on the idea of threading many B trees together to facilitate simultaneous search. This idea is known as fractional cascading in the data structures literature [8]. The H tree scheme, however, offers no good performance guarantees for querying and in addition, the update algorithm is complicated and potentially unbounded. Therefore, we do not include this method in our benchmarking. The solution we proposed in Chapter 5 using rake and contract is too ....

B. Chazelle & L. J. Guibas, "Fractional Cascading: I. A Data Structuring Technique," Algorithmica 1 (1986), 133--162.

....such experimental studies [14,18,30] All involve new datastructures. The approach [18] is based on the H tree datastructure. This data structure threads many B trees together to facilitate simultaneous search. This idea is known as fractional cascading in the data structures literature [5] and is notoriously hard to dynamize. The H tree scheme is heuristic and offers no worstcase performance guarantees for range querying. More importantly, updates are fairly complex and potentially unbounded. The other two approaches are very recent and do share a number of features. The hcC tree ....

B. Chazelle & L. J. Guibas, "Fractional Cascading: I. A Data Structuring Technique," Algorithmica 1 (1986), 133--162.

.... Gamma; Delta : So, a four dimensional range tree S can be employed to answer the query Q(p) This gives a sweep plane algorithm of time complexity O(nlog 4 n) two magnitudes of logn slower than claimed. One factor can be removed with the well known fractional cascading technique [5, 19]. For the other one, once again subdivide the cone C=3 by triangulating its quadratic cross section: The two resulting C=6 will have only three boundary planes. Hence, q and S are three dimensional instead of four. 6.2 Scholium: 3 Be C a partition of IR D into k convex cones and D a ....

Bernard Chazelle, Leonidas J. Guibas: "Fractional cascading: I. A data structuring technique", Algorithmica 1, 1986, 133-162.

....i such that a i = 1, we find the cell of the ffi i grid that contains p (plus the neighboring cells) That is, we perform point location queries in a logarithmic number of grids, but always with the same query point p. In Schwarz and Smid [108] it is shown that the fractional cascading technique [38] can be extended so that all these queries together can be solved in O(log n log log n) time. The main problem is that we have grids with different grid sizes. Therefore, an ordering on the grid cells has to be introduced that is compatible with all these sizes. In [108] such an ordering is ....

....the point having minimum L1 distance to p. This concludes the description of the algorithm for finding the right neighbor of p. For a correctness proof, the reader is referred to [81] It is easy to see that each of the stages can be implemented in O(log 2 n) time. Using fractional cascading [38], this can be improved to O(log n) The range tree can be maintained dynamically in O(log n log log n) amortized time per insertion and deletion. Then, since we need dynamic fractional cascading [94] the query time also becomes O(log n log log n) The generalization to the D dimensional L1 ....

B. Chazelle and L.J. Guibas. Fractional cascading I: A data structuring technique. Algorithmica 1 (1986), pp. 133--162.

....right of the chain. A more efficient scheme is to perform binary search of the y coordinate at the root node and to spend only O(1) time per node as we go down the chain tree, shaving off an O(log n) factor from the query time[58] This scheme is similar to the ones adopted by Chazelle and Guibas[40] in fractional cascading search paradigm and by Willard[124] in his range tree search method. With the linear time algorithm for triangulating a simple polygon due to Chazelle[31] cf. Section 3.6.1) we conclude with the following optimal search structure for planar point location. Theorem 8 ....

....reporting problem, can be solved in O(n log n F) time. Guibas and Seidel[65] showed that merging two convex subdivisions can actually be solved in O(n F) time using topological plane sweep. Most recently Chazelle et al. 37] used hereditary segment trees structure and fractional cascading[40] and solved both segment intersection reporting and counting problems optimally in O(n log n) time and O(n) space. The term F should be included for reporting. The rectangle intersection reporting problem arises in the design of VLSI circuitry, in which each rectangle is used to model a certain ....

B. Chazelle and L. J. Guibas, "Fractional Cascading: I. A Data Structuring Technique," Algorithmica, 1 (1986) 133-186.

....geometry. The naive method, which consists of performing binary search separately in each set, uses space O(n) and has query time O(k log n) More efficient techniques have been developed for specific instances of this repetitive search problem [57,83,101,163,167,168] Chazelle and Guibas [33,34] generalize these solutions and provide a data structuring technique called fractional cascading, which achieves query time O(log n k) using O(n) space. This time bound is optimal, and is faster than the one of the naive method by a logarithmic factor. Fractional cascading can be formalized as ....

....are at most d edges e incident upon v such that x 2 R(e) Clearly, if G has bounded degree, it also has locally bounded degree for any choice of the ranges associated with the edges of G. Let k denote the size of the query tree T . The static fractional cascading technique of Chazelle and Guibas [33,34] achieves O(log n k) query time. Regarding dynamic fractional cascading, Chazelle and Guibas [34] show that insertions and deletions of elements can be supported in O(log n) amortized time, such that the query time is O(log n k log log n) Mehlhorn and Naher s dynamic fractional cascading ....

[Article contains additional citation context not shown here]

B. Chazelle and L.J. Guibas, "Fractional Cascading: I. A Data Structuring Technique," Algorithmica 1 (1986), 133--162.

....p. The entire search from top to bottom in the multi level data structure takes therefore O(log d n) time, resulting in O(log d Gamma1 n) candidate answers. The minimum among these candidates is the final answer to the query. The query time can be improved by applying fractional cascading [8] to the two lower levels of the data structure. This is possible because the bottom level structures are ordered lists (a sequence of intervals) Fractional cascading improves the query time in a 2 level data structure consisting of a segment tree with the one dimensional ordered lists as ....

B. Chazelle and L. Guibas, Fractional cascading I: A data structuring technique, Algorithmica 1 (1986), pp. 133-162.

....efficiently whether k or more points are enclosed by a query rectangle. The naive approach to this problem is to build a range tree [4] on the set S. When a query rectangle R is given, we can answer how many points are inside of R in O(log n) time using the fractional cascading technique of [6]. The preprocessing time and space is O(n log n) Notice that we did not use the parameter k at all. In order to improve the preprocessing time and space and also the query time we use the following observation. Observation 7.1 In order for the query rectangle to contain at least k points, the ....

B.M. Chazelle, L.J. Guibas, "Fractional cascading: I. A data structuring technique", Algorithmica, 1, 133--162, 1986.

....such experimental studies [14,18,30] All involve new data structures. The approach [18] is based on the H tree data structure. This data structure threads many B trees together to facilitate simultaneous search. This idea is known as fractional cascading in the data structures literature [5] and is notoriously hard to make dynamic. The H tree scheme, is heuristic. It offers no worst case performance guarantees for range querying. More importantly, updates are fairly complex and potentially unbounded. The other two approaches are very recent and do share a number of features. The ....

B. Chazelle and L. J. Guibas, "Fractional Cascading: I. A Data Structuring Technique," Algorithmica 1 (1986), 133--162.

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Chazelle B.,Guibas L., Fractional Cascading: a data structuring technique with Geometric Applications, 12 ICALP 1985, pp. 90-100.

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B. Chazelle and L. Guibas, Fractional cascading I: A data structuring technique, Algorithmica 1 (1986), pp. 133-162.

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