D. E. Willard. New trie data structures which support very fast search operations. JCSS, 28(3):379-94, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....thus runs in O(n log n) time and O(n log n) space. Using van Emde Boas trees [16] for the entry and exit lists reduces the running time to O(n log n log log n) but as each slice s entry and exit list requires O(n) space, the total space increases to O(n ) Using Willard s q fast tries [17] yields time O(n log 1:5 n) and space O(n log n) In general, however, if we set the arity of T to some d and use a dictionary that on N items admits predecessor and successor queries, insertions, and deletions in time D(N ) then Q(N) O(D(N) log d N d) D(N) log d N to nd the rst ....

D. E. Willard. New trie data structures which support very fast search operations. JCSS, 28(3):379-94, 1984.

....Recently, there has been a growin interest in solving problems in computational geometry on a grid, i.e. points (or line segments with endpoints) in U d = 0. u 1] d. See for example, Karlsson [7] Karlsson and Munro [8] Keil and Kirkpatrick [10] Mfiller [15] Overmars [16] and Willard [20,21]. In some sense, computational geometry on a grid shows the complexity of a problem after sorting. But apart from being of great theoretical interest, it is our belief that structures on grids are potentially practical. Indeed, analysis of algorithms with a restricted domain is of interest for ....

D.E. Willard, New Trie Data Structures Which Support Very Fast Search Operations, J. Cornput. $yst. $ci. 28 (1984), 379-394

.... [Has87] Related research Apart from the sorted table binary search solution, previous linear space solutions to the static dictionary problem implementable on AC RAMs were constructed by Tarjan and Yao [TY79] achieving constant query time if w = c log n for some constant c, and by Willard [Wil84] and Karlsson [Kar84] achieving query time O( w) Previous lower bounds for the static dictionary problem have been obtained by Fich and Miltersen [FM95,Mil96] The former paper gives lower bounds for the case where the computational instructions allowed are addition, subtraction and ....

D. E. Willard. New trie data structures which support very fast search operations. Journal of Computer and Systems Sciences, 28:379--394, 1984.

....smaller indexed 2 3 trees, improving their performance from O(log n) to O(log n) amortized time. Structural decomposition differs from ideas used to solve decomposable search problems [BS80] and from other ad hoc instances of data structure decomposition (e.g. Die91, DR91, DS87, Tsa84, Wil83, Wil84] in that these previous results use only one level of decomposition, whereas bootstrapping is recursive. The rest of this thesis applies data structural bootstrapping to build efficient catenable heap ordered deques and confluently persistent deques. Chapter 2 discusses catenable heap ordered ....

D. E. Willard. New trie data structures which support very fast search operations. Journal of Computer and System Sciences, 28(3):379--94, 1984.

.... Hastad s switching lemma [15] Related research Apart from the sorted table, previous linear space solutions to the static dictionary problem implementable on AC RAMs were constructed by Tarjan and Yao [22] achieving constant query time if w = c log n for some constant c, and by Willard [25] and Karlsson [16] achieving query time O( Previous lower bounds for the static dictionary problem have been obtained by Fich and Miltersen [12, 20] The former paper gives lower bounds for the case where the computational instructions allowed are addition, subtraction and multiplication (the ....

D. E. Willard. New trie data structures which support very fast search operations. Journal of Computer and Systems Sciences, 28:379--394, 1984.

....[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 space O(jdom Dj) and an O( p log N) time bound is achieved. Many applications of dictionaries require insertions and deletions but cannot afford O(N) storage. For these applications we improve the time bound from O( p log N) to O(log log N ) Our solution is based on v. Emde Boas ....

D. Willard: "New Trie Data Structures which Support Very Fast Search Operations", Journal of Computer and System Sciences 28, 395-419, 1984 9

....can be solved using only space polynomial in the dictionary size (denoted by n) on a RAM with (log jU j) bit memory words. Here we are restricting the space used in terms of the dictionary size in order to exclude the trivial bit vector solution and a family of its generalizations, due to Willard [37], that process dictionary operations in constant time and use space depending on the size of the universe. The Dictionary Problem has been extensively studied and has a rich variety of solutions. Balanced search trees solve the problem in O(log n) time per operation and use O(n) space ....

....Single Cons operations can be implemented in O( p log F ) time and O(1) amortized space or, alternately, in O(1) time and O(F ffl ) space, where F denotes the total number of Cons operations performed and ffl is any positive constant. This implementation is based on Willard s data structure [37] for maintaining a dictionary in a small universe. Universal hashing [8] and dynamic perfect hashing [13] offer alternate implementations that require O(1) randomized amortized time and O(1) amortized space per Cons operation. We develop a data structure that performs a cascade of f Cons ....

D.E.Willard. New trie data structures which support very fast search operations. J. Comp. Sys. Sci., 28, 1984, 379-394.

....2 3 trees so that their leaves store smaller indexed 2 3 trees, improving their performance from O(log n) to O(log 3 n) amortized time. Structural decomposition differs from ideas used to solve decomposable search problems [5] and from other ad hoc instances of data structure decomposition (e.g. [14, 15, 16, 40, 42, 43]) in that these previous results use only one level of decomposition, whereas bootstrapping is recursive. Non destructive (or, in the parlance of Section 1.1, confluently persistent) deques have many uses in high level programming languages such as LISP, ML, and Scheme [35] in which ....

D. E. Willard. New trie data structures which support very fast search operations. Journal of Computer and System Sciences, 28(3):379--94, 1984.

....smaller indexed 2 3 trees, improving their performance from O(log n) to O(log n) amortized time. Structural decomposition differs from ideas used to solve decomposable search problems [BS80] and from other ad hoc instances of data structure decomposition (e.g. Die91, DR91, DS87, Tsa84, Wil83, Wil84] in that these previous results use only one level of decomposition, whereas bootstrapping is recursive. 1.4 Summary The rest of this thesis applies data structural bootstrapping to build efficient catenable heap ordered deques and confluently persistent deques. Chapter 2 discusses catenable ....

D. E. Willard. New trie data structures which support very fast search operations. Journal of Computer and System Sciences, 28(3):379--94, 1984.

....tradeoffs between the space randomization and update times can be achieved. Theorem 1. Given N distinct integers in the universe 1 Delta Delta Delta U , 1. The q fast trie uses O(N ) space and performs insertions, deletions, Succ and Pred operations in O( p log U ) time deterministically [18]. 2. The Van Emde Boas data structure uses O(U ) space and performs insertions, deletions, Succ and Pred operations in O(log log U ) time deterministically. With randomization, space becomes O(N ) with no change to other bounds [16] The stabbing problem we consider can be equivalently formalized ....

....Consider now the path of length H leading each of those nodes. Each node on that path is also not deleted and uses at most O(F (U ) space (to store the array of pointers to its children) Thus, at most N c Theta 2 Theta H Theta O(F (U ) O(H N c F (U ) space is used for T in total. As in [18], we can safely assume that one can allocate or deallocate the memory space (i.e. an uninitialized array of size F (U ) of any internal node in T in O(1) time. That completes the description of T ; it remains to show how queries and updates are implemented using this pruned trie. The Stab(p) ....

D. E. Willard. New trie data structures which support very fast search operations. Journal of Computer and System Science, 28:379--394, 1984.

....The data structure introduced by Tarjan and Yao [14] allows element location in O(1) expected time and O(log n jU j) in the worst case, where jU j is the size of the universe. This structure does not support range and neighbour queries efficiently. The P fast trie and the Q fast trie by Willard [17] support element retrieval and nearest neighbour search in O( p log jU j) time in the worst case using O i n p log jU j2 p log jU j j and O(n) space respectively. A range search requires O( p log jU j s) time, where s is the size of the output. The Y fast trie by Willard [16] is a ....

D. E. Willard. New trie data structures which support very fast search operations. Journal of Computer and System Sciences, 28:379--394, 1984.

....space If we allow the cost of member queries to be O( p log n) the space complexity can be reduced to O(n 2 fflw ) or if U denotes the size of the universe to O(n U ffl ) We use the data structure above, with two modifications. First, we apply an idea similar to that of Willard [17] and Karlsson [6] to reduce the number of nodes in the trie by keeping (traditional) 2 3 trees of height Theta( p log n) at the bottom of the trie. All 2 3 trees have the same height and updates in the trie corresponds to splitting and merging of 2 3 trees. Most keys will be stored in the 2 3 ....

D. E. Willard. New trie data structures which support very fast search operations. Journal of Computer and System Sciences, 28:379--394,

....With the problem size chosen as N = P i S i , Yellin and Jutla presented an abstract algorithm that requires O(N 2 = log N) dictionary operations. Two implementations were given. One used dynamic perfect hashing [1] giving randomized expected time O(N 2 = log N ) The other used q tries [4], giving a worst case running time of O(N 2 = p log N ) They stated that they knew of no previous works that show a sub quadratic complexity . 1 On leave from School of Computing and Information Technology, Griffith University, Queensland, Australia 4111 Preprint submitted to Information ....

D. E. Willard, New trie data structures which support very fast search operations. J. Comp. Sys. Sci. 28 (1984) 379--394.

.... Hastad s switching lemma [12] Related research Apart from the sorted table, previous linear space solutions to the static dictionary problem implementable on AC 0 RAMs were constructed by Tarjan and Yao [19] achieving constant query time if w = c log n for some constant c, and by Willard [21] and Karlsson [13] achieving query time O( p w) Previous lower bounds for the static dictionary problem have been obtained by Fich and Miltersen [9, 17] The former paper gives lower bounds for the case where the computational instructions allowed are addition, subtraction and multiplication ....

D. E. Willard. New trie data structures which support very fast search operations. Journal of Computer and Systems Sciences, 28:379--394, 1984.

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