B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Notice that the number of second level data structures that we have to query is only O(log n) It follows that the total query time to report the points inside an axis parallel query window is O(log 2 n t k) where k is the number of reported points. Using a technique called fractional cascading [44, 45] this can be reduced to O(logn t k) It seems at first sight that the data structure uses a lot of storage, because there is a linear number of second level data structures one per node in the first level tree. Fortunately, many of the second level data structures contain only a small number ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmlea, 1:163-191, 1986.

....to Overmars [14] the datastructure can be implemented such that finding those k object boundery segments intersecting a (bounding) box takes time O(k log 2 n) The 2.5: intersection 2. 6: outside application of dynamic fractional cascading improves the query time to O(k log n log log n) [4, 5, 12]. Insertion in the dynamic datastructure can be done in time O(logn log log n) The boundaries of n objects of constant complexity are inserted, so O(nlognloglogn) time in total is needed. The datastructure requires O(nlogn) storage. We store with each object ei the intersection points of the ....

....is O(n) so at most O(n) queries will be performed. As mentioned above according to Overmars [14] windowing in a set of line segments segments takes time O(k log 2 n) using a dynamic datastructure. The application of dynamic fractional cascading improves the query time to O(k log n log log n) [4, 5, 12]. At most O(n) isolated parts are reported in total, lemma 2.5. Insertion in the dynamic datastructure can be done in time O(log n log log n) The dynamic datastructure requires O(n log n) storage. Thus, Li) can be computed from , L ) by deletion from . L ) of the marked (fragments of) ....

[Article contains additional citation context not shown here]

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163-191, 1986.

....6 An optimal data structure for fixed radius ray shooting The query time and the space needed by the data structure of Lemma 8 is sub optimal by a log nfactor. We can obtain an optimal data structure by avoiding the duplication in the storage of lower envelopes, and by using fractional cascading [3, 4]. Consider a region r. It has at most three doors dl, d2, and d3. At most two of the three possible pairs of doors permit an x monotone path, so there are at most two auxiliary data structures associated with r. Consider one of these structures, say, for dl and d2. Either dl and d2 are both doors ....

....structure for searching in A, All auxiliary data structures are based on lists of breakpoints, sorted by x coordinate. All the lists being searched during a query are searched with the same search key, namely the x coordinate of the query arc center. We can therefore apply fractional cascading [3, 4]. This adds additional keys to the lists, as well as cross pointers between the list stored in a node and the one stored in the parent node. As this is a standard application of fractional cascading on a tree, we do not discuss the details, and observe only that the space requirement increases by ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163-191, 1986.

....a graph and we want to search for an element in each dictionary in a path in this graph. A number of fundamental two dimensional geometric searching problems arising in the implementation of geographic information systems can be solved with data structures based on this iterative search approach [7]. These problems include: line intersection queries on a polygon P , to report the edges of P intersected by a query line . ray shooting queries on a polygon P , to report the first edge of P intersected by a query ray; point location on a planar subdivision, to report the region ....

....O(log n k) the answer verification time is O(log n k) 5. 5 Applications Our authenticated fractional cascading scheme can be used to design authenticated data structures for various fundamental two dimensional geometric search problems, where iterative search is implicitly performed (see [7]) In all of these problems, the underlying catalog graph has degree bounded by a small constant. In the following, n denotes the problem size. Theorem 12 There is an authenticated data structure for answering line intersection queries on a polygon that can be constructed in O(n log n) time and ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986.

....G 0035, and and by the Office of Naval Research and the Advanced Research Projects Agency under contract N00014 91 J 4052, ARPA order 8225. I Introduction Fractional cascading is a preprocessing technique that allows efficient searching of the sanhe key in a collection of catalogs (sorted lists) [3, 4]. More formally, there is a catalog associated with each node of a graph G. Given a search argument and a search path in G, the goal is to find the smallest entry in each of the catalogs of the nodes on the search path. The search path can be either explicit, in which it is specified before ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163 191, 1986.

....space time trade o#. This technique was adapted to external storage by [SR95, ASV99] In the same paper, Chazelle introduced the concept of a hive graph. This structure was used to improve the search cost of many previous data structures. It was later generalized by Chazelle and Guibas [CG86a, CG86b] to the technique of fractional cascading. Other problems solved in [Cha86] using filtering search included point enclosure, segment intersection, and k nearest neighbors. 2.1.6 Orthogonal Geometric Range Search Most of the techniques for orthogonal range search are based on range trees. The ....

B. Chazelle and L. J. Guibas. Fractional cascading: II.Applications. Algorithmica, 1:163--191, 1986.

....6 An optimal data structure for fixed radius ray shooting The query time and the space needed by the data structure of Lemma 8 is sub optimal by a log n factor. We can obtain an optimal data structure by avoiding the duplication in the storage of lower envelopes, and by using fractional cascading [3, 4]. Consider a region r. It has at most three doors d 1 , d 2 , and d 3 . At most two of the three possible pairs of doors permit an x monotone path, so there are at most two auxiliary data structures associated with r. Consider one of these structures, say, for d 1 and d 2 . Either d 1 and d 2 are ....

....structure for searching in . All auxiliary data structures are based on lists of breakpoints, sorted by x coordinate. All the lists being searched during a query are searched with the same search key, namely the x coordinate of the query arc center. We can therefore apply fractional cascading [3, 4]. This adds additional keys to the lists, as well as cross pointers between the list stored in a node and the one stored in the parent node. As this is a standard application of fractional cascading on a tree, we do not discuss the details, and observe only that the space requirement increases by ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986.

.... into a constant number of axis monotone curves, the data structure can be implemented so that finding those A object boundary segments intersecting a (bounding) box takes time O(A log 2 n) 14] The application of dynamic fractional cascading improves the query time to O(A log n log log n) [4, 5, 12]. Insertion in the dynamic data structure can be done in time O(log n log log n) The boundaries of n objects of constant complexity are inserted, so O(n log n log log n) time in total is needed. The data structure requires O(n log n) storage. We store with each object e i the intersection ....

....A(L) is O(n) so at most O(n) queries will be performed. As mentioned above according to Overmars [14] windowing in a set of line segments takes time O(A log 2 n) using a dynamic data structure. The application of dynamic fractional cascading improves the query time to O(A log n log log n) [4, 5, 12]. By Lemma 2.5 at most O(n) isolated parts are reported in total. Insertion in the dynamic data structure can be done in time O(log n log log n) The dynamic data structure requires O(n log n) storage. Thus, U(L i ) can be computed from U(L i Gamma1 ) by deletion from U(L i Gamma1 ) of the marked ....

[Article contains additional citation context not shown here]

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986.

....can both be modeled through the same algorithmic pattern, the preprocessing algorithms are considerably di#erent, and thus have separate implementations. 6. 5 Optimal Chain Method The chain method has been refined to optimality by Edelsbrunner, Guibas, and Stolfi [27] using fractional cascading [18, 19]. We recall that, in the optimal chain method, the search structure requires O(n) space, can be constructed in O(n) time, and allows queries in O(log n) time. We can implement also the optimal chain method within the binary space partition search algorithmic pattern described in the previous ....

B. Chazelle and L. J. Guibas. Fractional cascading II. Applications. Algorithmica, 1(3):163--191, 1986.

....a point location query. We can maintain a semi dynamic set of t interior disjoint vertical trapezoids and answer point location queries in O#log t# time per query and O#log t# amortized time per insertion, using a data structure of size O#t log t# based on a segment tree with fractional cascading [10, 11, 39]. This approach adds O#p log t# to the overall running time of our hidden surface removal algorithm; the total insertion time O#t log t# is dominated by other terms. Although this approach is slower than pixel marking, it can be used when the set of pixels is presented online instead of being xed ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. ############ 1:163-191, 1986.

....log n ) Repeating the procedure (with independently chosen random bits) m log n times increases the probability of nding the mincut to 1 exp m . Example 9. 2 (Fractional Cascading) The problem of searching for a key in many ordered lists arises very frequently in computational geometry (see [5] for applications) Chazelle and Guibas [4] introduced fractional cascading as a general technique for solving this problem. Their work uni ed some earlier work in this area and gave a general strategy for improving upon the naive method of doing independent searches for the same key in separate ....

B. Chazelle and L. Guibas. Fractional cascading: Ii. applications. Algorithmica, 1:163 - 191, 1986.

....a point location query. We can maintain a semi dynamic set of t interior disjoint vertical trapezoids and answer point location queries in O(log t) time per query and O(log t) amortized time per insertion, using a data structure of size O(t log t) based on a segment tree with fractional cascading [10, 11, 39]. This approach adds O(p log t) to the overall running time of our hidden surface removal algorithm; the total insertion time O(t log t) is dominated by other terms. Although this approach is slower than pixel marking, it can be used when the set of pixels is presented online instead of being xed ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica 1:163-191, 1986.

....where K is the number of intersections for the reporting problem and K = 1 for the intersection counting problem. We describe their method in section 2. They also state that the space can be reduced to linear by streaming [7] and the time to O(n log n K) by a dynamic form of fractional cascading [4, 5], which they admit is Supported in part by an NSERC Research Grant 1 complicated. This paper presents an alternative way to reduce space that yields a much simpler approach to reducing the time. The red blue intersection problem was first considered while researchers were searching for general ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986.

.... into a constant number of axis monotone curves, the data structure can be implemented so that nding those A object boundary segments intersecting a (bounding) box takes time O(A log 2 n) 14] The application of dynamic fractional cascading improves the query time to O(A log n log log n) [4, 5, 12]. Insertion in the dynamic data structure can be done in time O(log n log log n) The boundaries of n objects of constant complexity are inserted, so O(n log n log log n) time in total is needed. The data structure requires O(n log n) storage. We store with each object e i the intersection ....

....A(L) is O(n) so at most O(n) queries will be performed. As mentioned above according to Overmars [14] windowing in a set of line segments takes time O(A log 2 n) using a dynamic data structure. The application of dynamic fractional cascading improves the query time to O(A log n log log n) [4, 5, 12]. By Lemma 2.5 at most O(n) isolated parts are reported in total. Insertion in the dynamic data structure can be done in time O(log n log log n) The dynamic data structure requires O(n log n) storage. Thus, U(L i ) can be computed from U(L i 1 ) by deletion from U(L i 1 ) of the marked ....

[Article contains additional citation context not shown here]

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163-191, 1986.

....map M that is not monotone, fictitious regularization edges are added to M and point location in M is reduced to point location in the resulting refinement M 0 of M . Hence, the chain method is ordinary for general maps. In the bridged chain method [24] the technique of fractional cascading [16, 17] is applied to the sets of y coordinates of the separators. Fractional cascading establishes bridges between the separator of a node and the separators of its children such that there are O(1) vertices between any two consecutive bridges. Hence, except for the separator of the root, Step 1 can ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986. 31

....the task is therefore to determine whether p is above (resp. below) the lower (resp. upper) envelope of I(v) This amounts to locating the envelope arcs below and above p and then deciding whether p is indeed contained in I(v) By using the fractional cascading technique of Chazelle and Guibas [5, 6] we are able to answer each inside query in Sigma in O(log n) time. This gives us the following: Theorem 3 Given a sequence Sigma of n disk insert updates and point inside queries, one can answer all the inside queries in Sigma in O(n log n) time. Combining this with the discussion from the ....

B. Chazelle and L.J. Guibas, Fractional cascading: II. Applications, Algorithmica, 1(1986), 163--191.

.... query for position i by determining if p is inside I(w) for each w that is a left fringe node of i (answering true if and only if it is inside them all) In the full version of this paper we use this observation, together with the fractional cascading technique of Chazelle and Guibas [5, 6], to answer each inside query in Sigma in O(log n) time. This gives us the following: Theorem 3 Given a sequence Sigma of n disk insert updates and point inside queries, one can answer all the inside queries in Sigma in O(n log n) time. Combining this with the discussion from the previous ....

B. Chazelle and L.J. Guibas, Fractional cascading: II. Applications, Algorithmica, 1:163--191, 1986.

....and and by the Office of Naval Research and the Advanced Research Projects Agency under contract N00014 91 J 4052, ARPA order 8225. 1 Introduction Fractional cascading is a preprocessing technique that allows efficient searching of the same key in a collection of catalogs (sorted lists) [3, 4]. More formally, there is a catalog associated with each node of a graph G. Given a search argument y and a search path in G, the goal is to find the smallest entry y in each of the catalogs of the nodes on the search path. The search path can be either explicit, in which it is specified before ....

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986.

No context found.

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986.

No context found.

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986.

No context found.

B. Chazelle and L. J. Guibas, Fractional Cascading: II. Applications, Algorithmica, 1, 1986, pp.163--191.

No context found.

B. Chazelle and L.J. Guibas. Fractional Cascading: II. Applications, Algorithmica, 1, pp.163-191, 1986.

No context found.

B. Chazelle and L. J. Guibas, Fractional Cascading: II. Applications, Algorithmica, 1, 1986, pp.163-191.

No context found.

B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163--191, 1986.

No context found.

B. Chazelle and L. J. Guibas. Fractional cascading: II. applications. Algorithmica, 1:163--191, 1986.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC