Gaston H. Gonnet. Handbook of Algorithms and Data Structures. Addison Wesley, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....string. Search in a trie is fast, since it requires only a single character comparison and pointer traversal for each letter in the query string. In a compact trie, nodes that lead to a single leaf are omitted, as suggested by Sussman [17] Compact tries, and variants such as Patricia tries [9, 14], are fast but space intensive. For this reason, alternatives have been developed, such as the ternary tree [3, 5] where trees rather than arrays are used to represent nodes. However, we have found that ternary trees are neither as fast nor as compact as the structure described below [12] and we ....

G. Gonnet. Handbook of Algorithms and Data structures. Addison-Wesley, Reading, Massachusetts, 1984.

....omit chains of nodes that have only a single descendant and lead to a leaf. This variant of a trie, in which tailing sticks caused by single descendant nodes are eliminated [46] we refer to as a compact trie; we use the array implementation of a compact trie in our experiments. Patricia tries [23, 32], which generalize the idea of collapsing sticks to a single node, are an important variant of a trie that has been widely analysed [21, 26, 38, 47] All single descendant nodes are omitted in a Patricia trie, not just those that lead directly to a leaf node. Singledescendant nodes are eliminated ....

G. Gonnet. Handbook of Algorithms and Data structures. Addison-Wesley, Reading, Massachusetts, 1984.

....of such a study is multifold. First, the model is simple yet prototypical of many sophisticated divide and conquer schemes. Viewing this recurrence from an equally important binary search tree perspective, a large number of extensions and variants (see Devroye [17] and Gonnet and Baeza Yates [29]) can be studied. Second, the inherent phase change of the limit laws from normal to non normal is an interesting phenomenon, which should also occur in many other structures. Third, how sensitive is the limit law of the cost with respect to the toll function For such a simple structure, a ....

....can be implemented in several different ways: two pointers, threaded with or without ag, triply linked (with a pointer to parent) etc. and the nodes can be traversed in di erent orders: inorder, preorder, postorder, breath rst, depth rst, etc. see [2, 10, 11, 12, 21, 22, 30, 49, 62] and [29]. The analysis of the cost of these algorithms then reduces to the calculation of certain parameters on trees like the number of nodes with null (or non null) left (or right) branches, the number of nodes with both non null left and right branches, and the number of nodes that are a left child and ....

G. H. Gonnet and R. Baeza-Yates, Handbook of Algorithms and Data Structures, AddisonWesley, Workingham, 1991.

....of early hashing schemes Early theoretical analysis of hashing schemes was done under the assumption that hash function values are uniformly random and independent. Precise analyses of the average and expected worst case behaviors of the abovementioned schemes have been made, see for example [14, 18]. We mention just a few facts, disregarding asymptotically vanishing terms. Note that some figures depend on implementation details the below hold for the implementations described in Section 5. We first consider the expected number of memory probes needed by the two open addressing schemes ....

Gaston Gonnet. Handbook of Algorithms and Data Structures. Addison-Wesley Publishing Co., London, 1984.

....It is well known how to construct a binary search tree (BST) from n distinct keys to handle the data eciently. Since BST s tend to be balanced (their average height is of order 4:31107 log n [Dev86] operations like the insertion, deletion or look up of keys can be performed at low cost [BYG91]. It is possible to improve the behaviour of BST s by an algorithm due to G.M. Adel son Vel skii and E. M. Landis, the so called AVL trees, which generates binary search trees with a maximal height of 1:4404 log n by means of rotation operations (see [AdVL62] In this note we investigate the ....

....AVL algorithm, to the knowledge of the author it is not known if T h is the same tree as if the AVL algorithm would 7 have been applied. However, this is an important question since the worstcase height of a height balanced tree and the average height of an AVL tree di er by roughly 40 percent [BYG91]. The reason for that heuristic is the belief that there is a speed up of the insertion phase, because no rotation is performed, and also that the total number of rotations is reduced because of automatic rebalancation during the insertion phase, i.e. subtrees out of balance get rebalanced by ....

Baeza-Yates, R., Gonnet, G.H.: Handbook of Algorithms and Data Structures. Addison-Wesley, Reading, Mass., 1991. 8

....the root of T to one of the leaves. Note that T might have internal nodes with only one (left or right) successor. In the same manner it is possible to construct an m ary trie from data which possess an m ary representation, like character strings. For details on the implementation see [12] and [8]. Assuming that the set of keys K is a set of random integers and we use their binary representations to navigate through the trie, we observe that it is much more likely to get a trie which is a balanced tree than a trie which is a linear list. The reason for this fact is that a linear structure ....

G. H. Gonnet and R. Baeza-Yates, Handbook of Algorithms and Data Structures, Addison-Wesley, 1991.

....we could use an unprecise but small summary for the directory. We first investigate two naive summary representations: exact directory and server name. In the exact directory approach, the summary is essentially the cache directory, with each URL represented by its 16 byte MD5 signature [41] [24]. In the server name approach, the summary is the list of the server name component of the URL s in cache. Since on average, the ratio of different URL s to different server names is about 10 to 1 (observed from our traces) the server name approach can cut down the memory by a factor of 10. We ....

....Bits, 32 bits of BitArray Size InBits, and 32 bits of Number of Updates. The header completely specifies the hashing functions used to probe the filter. There are Function Num of hashing functions. The functions are calculated by first taking bits 0 to to to , etc. out of the MD5 signature [41] [24] of the URL, where is Function Bits, and then modular the bits by BitArray Size InBits. If 128 bits are not enough, more bits are generated by computing the MD5 signature of the URL concatenated with itself. The header is followed by a list of 32 bit integers. The most significant bit in an ....

G. Gonnet and R. Baeza-Yates, Handbook of Algorithms and Data Structures. Reading, MA: Addison-Wesley, 1991.

....is created. Then the actual update is performed on the copy, and last a reference to the memory cell holding the copy is returned. In recursive structures such as trees an update leads to shadowing the entire search path. This is also known as path copying [26] In its original form the trie [11, 12] is a data structure where a set of strings from an alphabet containing m characters is stored in an m ary tree and each string corresponds to a unique path. There are several methods for implementing trie structures in the literature [4, 5, 7, 16, 19] One of the drawbacks of these methods is ....

G.H. Gonnet, R.A. Baeza-Yates. Handbook of Algorithms and Data Structures. Addison-Wesley, second edition, 1991.

....even when there is no explicit formula for the solution. We demonstrate this with a brief sketch of a recent analysis of this type [131] Other examples can be found in [270] Example 15.5. Search costs in quadtrees [131] Quadtrees are a well known data structure for multidimensional data storage [168]. Consider a d dimensional data space, and let n points be drawn independently from the uniform distribution in the d dimensional unit cube. We take d fixed and n # #. Suppose that the first n 1 points have already been inserted into the quadtree, and let D n be the search cost (defined as ....

.... methods include [11, 55, 108, 209, 223, 240, 241, 270, 338] Nice general introductions to combinatorial identities, generating functions, and related topics are presented in [81, 351, 377] Further material can be found in [13, 88, 99, 173, 188, 335, 336] A very useful book is the compilation [168]. While it does not discuss methods in too much detail, it lists a wide variety of enumerative results on algorithms and data structures, and gives references where the proofs can be found. Last, but not least in our listing, is Knuth s three volume work [235, 236, 237] While it is devoted ....

G. H. Gonnet and R. Baeza-Yates, Handbook of Algorithms and Data Structures, 2nd ed., Addison-Wesley, 1991.

....of shellsort. We chose shellsort as an example of an application with dynamically changing reference behavior and the potential for false sharing. The sequential shellsort algorithm (or diminishing increment sort) sorts its input by the repetitive ap plication of linear insertion sort [4]. The algorithm is divided into phases, where each phase has an associated distance or increment. During a phase, all elements separated by the increment for that phase are sorted using insertion sort. The increment is reduced after each phase, and must be 1 during the last phase. During the last ....

....phase are sorted using insertion sort. The increment is reduced after each phase, and must be 1 during the last phase. During the last phase, the entire input is sorted using insertion sort. Shellsort does not have a bad worst case, and is particularly efficient when the input is partially sorted [4]. The complexity of shellsort is very difficult to analyze however, since it depends so heavily on the exact sequence of increments. For a sequence of the form 2 k Gamma 1, where 1 k blog Nc, the complexity is known to be O(N 3=2 ) 6] On a multiprocessor, we can implement all of the ....

G. H. Gonnet. Handbook of Algorithms and Data Structures. Addison-Wesley, 1984.

.... (x) 0 X k0 c k x k 1 A 0 X j0 (n; j)x j 1 A and therefore a m;n (m; n) X 0km c k (n; m Gamma k) m; n) X 0km c k m k (m n) k where m k = m(m Gamma 1) Delta Delta Delta (m Gamma k 1) If we now let N = m n and m = ffN , we can use formula (II.46) from [4], to obtain the asymptotic approximation a m;n (m; n) C(ff) Theta 1 N = B(ff(1 Gamma ff) Theta 1 N 2.3.1 The Median Case When m = n, equation (2) can be rewritten as R n;n (z) n; n) X 0in R i;i (z)ffi i (z) n Gamma i; n Gamma i) Gamma R n;n (z)ffi n (z) ....

G.H. Gonnet and R. Baeza-Yates. Handbook of Algorithms and Data Structures. Addison-Wesley, 1991. Second Edition.

.... list, and the search performance moves from O(log n) to O(n) Di erent techniques to have balanced data structures are proposed, such as 2 3 trees, AVL trees and red black trees for the worst case; splay trees for the amortized worst case; and randomized trees and skip lists for the average case [11]. However, all the concept of balancing is based on the implicit assumption of exact searching: we have a search query and want to nd its exact replica in the tree. More sophisticated queries such as range searching are reduced to a couple of exact searches to nd the extremes of the range of ....

G. Gonnet and R. Baeza-Yates. Handbook of Algorithms and Data Structures. Addison-Wesley, 2nd edition, 1991.

....450 s in Irix, 160s in Linux and 133s in Opal. Objects Object creation (which, by itself, does not allocate any backing store) costs 60 s in Mungi. Less than one microsecond of that is for the OT update (on a 4 level B tree, which is sufficient to hold at least 32 million object descriptors [43]) Segment creation in Opal using a recycled inode costs 315 s. Object deletion in Mungi takes 150 s, compared to 900s in Opal. Only the combination of creation, access and deletion could easily be measured in the UNIX systems. The results were about 50 slower in Linux and eight times slower in ....

G. Gonnet and R. Baeza-Yates. Handbook of Algorithms and Data Structures. Addison-Wesley, 2 nd edition, 1990.

....) where #(u) is a strictly positive function of u # (0, 1) with maximum not exceeding 0.07. Thus, random k d trees behave a bit worse than their balanced counterparts, the median k d trees. Surveys of related known probabilistic results are provided by Vitter and Flajolet (1990) and Gonnet and Baeza Yates (1991). Figure 2. Two random k d tree partitions clearly show the elongated rectangles. We propose a minor modification of the insertion procedure, namely, each time a rectangle is split by a newly inserted leaf point, the longest side of its rectangle is cut, that is, the cut is a (k 1) dimensional ....

G. H. Gonnet and R. Baeza-Yates, Handbook of Algorithms and Data Structures, AddisonWesley, Workingham, 1991.

....ref. 11] The WLZW algorithm is based on the idea that sequences of words repeat in text. The repetition arises from the structure of a natural language. This is similar to LZW where compression is based on the assumption that repetitions of sequences of characters occur in text (see refs. 2] [3] and [5] The alphabet of the WLZW compression algorithm consists of words and non words. The adaptation of LZW to word compression raises a problem with the initialization of the compression dictionary. If the classic approach is used then at the beginning of compression the dictionary is ....

G. H. Gonnet, R. Beaza-Yates. Handbook of Algorithms and Data Structures. Addison-Wesley Publishing, 1991

....even though it may be linear in the worst case. O Connor and Snider [81] have related a complexity measure for random strings in cryptology, called maximum order complexity, to the statistical properties of the suffix trees. The notion of suffix tree has been extended to square matrices by Gonnet [48, 49], Giancarlo [43] and Giancarlo and Grossi [46] This data structure can be efficiently deployed in pattern matching algorithms in higher dimensions, an area which is gaining growing interest due to its applications to low level image processing [85] image compression [93] and visual databases ....

....The shaded region is the 5th suffix of the matrix A; b) The 4th prefix of A; c) The shaded submatrix is the 3rd prefix of the 4th suffix of A (the one with bold boundaries) d) A 3 is illustrated as a bold square: the shaded submatrix of A is the 2nd prefix of the 2nd suffix of A 3 . of Gonnet [48, 49], where submatrices were represented as semi infinite spirals called sispirals. The construction in [48, 49] requires O(n 4 ) worst case time and O(n 2 log n) average time (for j Sigmaj = O(1) A more efficient worst case solution, based upon a different linear representation called Lstring , ....

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Gonnet, G. H., and Baeza-Yates, R., Handbook of Algorithms and Data Structures. Addison-Wesley, (1991).

....Moreover, their distribution is even more biased than that of words, being closer to 1.9. We therefore assume that 1 and consider only words, since an analogous proof will hold for separators. On the other hand, more refined versions of Zipf s Law exist, such as the Mandelbrot distribution [Gonnet and Baeza Yates 1991]. This law tries to improve the fit of Zipf s Law for the most frequent values. However, it is mathematically harder to handle and it does not alter the asymptotic results that follow. We analyze the entropy E(d) of such distribution for a vocabulary of v words when d digits are used in the ....

Gonnet, G. H. and Baeza-Yates, R. 1991. Handbook of Algorithms and Data Structures. Addison-Wesley.

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G.H. Gonnet. Handbook of Algorithms and Data Structures. Addison-Wesley, London, 1984.

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Gaston H. Gonnet. Handbook of Algorithms and Data Structures. Addison Wesley, 1990.

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Gonnet G.H. (1983) Handbook of Algorithms and Data Structures. International Computer Science Services.

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G.H. Gonnet, R. Baeza-Yates, Handbook of Algorithms and Data Structures, Addison--Wesley, Workingham, 1991.

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G. H. Gonnet. Handbook of Algorithms and Data Structures. Addison-Wesley, 1984.

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G. H. Gonnet (1991). Handbook of Algorithms and Data Structures (in Pascal and C), AddisonWesley, Reading, MA.

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G. H. Gonnet (1991). Handbook of Algorithms and Data Structures (in Pascal and C), AddisonWesley, Reading, MA.

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Gonnet, G. H. and R. A. Baeza-Yates, Handbook of Algorithms and Data Structures, Second Edition, Addison-Wesley, Reading, MA, 1991.

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