H. Kaplan and R. E. Tarjan. Purely functional representations of catenable sorted lists. In Proc. 28th Annual ACM Symposium on the Theory of Computing, pages 202--211, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....4, d = 0) 0.21 0.28 0.31 1.22 1.77 4.17 8.01 arithmetic (a = 28, b = 4, d = 32) 0.16 0.24 0.28 0.82 1.21 3.83 7.31 Figure 8. Random indexing and updating (one update followed by 100 look ups repeated 1000 times) adapt an implementation of ordered lists. Instances based on finger search trees [12, 7], for example, support cons in Q(1) time and allow to access or update the i th element in Q(log i) time. 8 Conclusion and future work We have presented a purely functional implementation of onesided flexible arrays based on weight balanced multiway trees. The data structure is simple to ....

Haim Kaplan and Robert E. Tarjan. Purely functional representations of catenable sorted lists. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pages 202--211, Philadelphia, Pennsylvania, May 1996.

....property using on a collection of 2 3 trees. Skip Lists by Pugh [18] also support finger searching. More recently, Brodal [5] has investigated finger search trees designed to improve insertion and deletion time. Of special note are the purely functional catenable sorted lists of Kaplan and Tarjan [12]. Their design not only has the finger search property, but it also requires very little space overhead. We will contrast our design with theirs later. Challenges and results. Supporting finger search in balanced search trees can be challenging. The main di#culty is in shifting the finger fast ....

.... 02 184) which also includes a discussion on how the hands can be used to improve performance in database applications by utilize the pre fetching capability available in many modern computer architectures. Finally, we note that the purely functional catenable sorted lists of Kaplan and Tarjan [12] also support finger searches in worst case O(log d) time and with a logarithmic space overhead. We provide a brief comparison between their design and ours in Appendix C. ....

H. Kaplan and R. E. Tarjan. Purely functional representations of catenable sorted lists. In Proc. 28th Annual ACM Symposium on the Theory of Computing, pages 202--211, 1996.

....property using on a collection of 2 3 trees. Skip Lists by Pugh [18] also support finger searching. More recently, Brodal [5] has investigated finger search trees designed to improve insertion and deletion time. Of special note are the purely functional catenable sorted lists of Kaplan and Tarjan [12]. Their design not only has the finger search property, but it also requires very little space overhead. We will contrast our design with theirs later. Challenges and results. Supporting finger search in balanced search trees can be challenging. The main difficulty is in shifting the finger fast ....

.... 02 184) which also includes a discussion on how the hands can be used to improve performance in database applications by utilize the pre fetching capability available in many modern computer architectures. Finally, we note that the purely functional catenable sorted lists of Kaplan and Tarjan [12] also support finger searches in worst case O(logd) time and with a logarithmic space overhead. We provide a brief comparison between their design and ours in Appendix C. ....

H. Kaplan and R. E. Tarjan. Purely functional repre- sentations of catenable sorted lists. In Proc. Sth An- nual ACM Symposium on the Theory of Computing, pages 202-211, 1996.

....take O(log n) time. The building blocks of this data structure are simpler implementations of heaps with inferior time bounds which are of independent interest. All these data structures are built upon redundant binary counters, which proved useful in developing several recent data structures [7, 5, 11]. Redundant binary counters are described in Section 3. The most simplest data structure which we present is the fat heap. A fat heap is an interesting and simple generaliza tion of the classical binomial queue [6] that uses redundant binary counters rather than a regular one. Fat heaps support ....

....REDUNDANT COUNTERS Two redundant counters form the heart of our heap structure. These counters are based on the redundant binary representation of Knuth and Clancy [7] extended to support increments and decrements of arbitrary digits. Similar counters are used by Brodal [5] and Kaplan and Tarjan [11]. In this section we describe simple pointer based implementation of a redundant b ary counter (for b 2) that supports incrementing and decrementing an arbitrary digit in O(1) time. A b ary redundant representation (b ary RR) of a nonnegative integer x is a sequence of digits dn ; d0 , ....

H. Kaplan and R. E. Tarjan. Purely functional representations of catenable sorted lists. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 202-211. ACM Press, 1996.

....our structure generalizes to allow catenation, which no one knows how to implement efficiently using incremental recopying. Also, our structure can be extended to support access, insertion, and deletion d positions away from the end of a list in O(log d) amortized time, by applying the ideas in [9]. 4 Catenable Deques In this section we show how to extend our ideas to support catenation. Specif ically, we describe a data structure for catenable deques that achieves an 0(1) amortized time bound for PUSH, POP, INJECT, EJECT, and CATENATE. Our structure is based upon an analogous structure ....

....Another research direction is to design a confluently persistent representation of sorted lists such that accesses or updates d positions from an end take O(log d) time, and catenation takes O(1) time. The best structure so far developed for this problem has a doubly logarithmic catenation time [9]; it is strictly functional, and the time bounds are worst case. ....

H. Kaplan and R. E. Tarjan. Purely functional representations of catenable sorted lists. In Proceedings of the 2Sth Annual ACM Symposium on Theory of Computing, pages 202-211. ACM Press, 1996.

....achieve worst case bounds. In essence the representation is based on a segmented number system where the digits are drawn from the set f 1 2 ; 1; 2g. AVL trees offer a slightly greater degree of freedom since also trees of successive heights may be linked, ie 1 2 and 1 may be added to 2. H. Kaplan and R.E. Tarjan (1996) informally describe three purely functional implementations of 2 3 finger search trees. All three solutions are based on the Numerical Representations as Higher Order Nested Datatypes 47 double spine view and the first two are superficially similar to the data structure of Sec. 8. The first ....

Kaplan, Haim, & Tarjan, Robert E. 1996 (May). Purely functional representations of catenable sorted lists. Pages 202--211 of: Proceedings of the twenty-eighth annual ACM symposium on the theory of computing.

....data structures is implemented as a pair of lists, and the general organization of the data structure with a system of colors is analogous to a redundant numerical representation where the colors red, yellow and green can be seen, respectively as the digits 2,0, and 1. This analogy is described in [KT96]. The data structures developed with these techniques provide most of the basic operations on sequences with the same asymptotic behavior as those provided by its imperative counterparts. Efficient functional implementations of priority queues have also been designed with some of these techniques ....

....to finger search trees (see next topic) Afterwards, we plan to study the relation between some of the data structures described here and the use of fingers in search trees [GMPR77, Kos81] The relation of the recursive slowdown deques of section 5. 2 with finger search trees have been sketched in [KT96]. Also, the randomaccess list implementation as a forest of leaf tree mentioned before can be seen equivalent to the presence of a finger in a leaf tree. Consider, for example, a leftist left perfect leaf tree. If we start at the leftmost leaf and proceed upwards trough the left spine until the ....

Haim Kaplan and Robert E. Tarjan. Purely Functional Representations of Catenable Sorted Lists. In ACM Symposium on Theory of Computing, pages 202--211, May 1996.

....confluently persistent [85] Such data structures have been denoted purely functional data structures. Some recently developed purely functional data structures are: queues and deques [84] random access lists [83] catenable lists [67] priority queues [22] and catenable finger search trees [68]. A survey on the design of functional data structures can be found in the thesis of Okasaki [85] It remains an interesting open problem if there exists a construction which can remove the amortization from the node splitting technique of Driscoll et al. 44] for making data structures fully ....

Haim Kaplan and Robert Endre Tarjan. Purely functional representations of catenable sorted lists. In Proc. 28th Ann. ACM Symp. on Theory of Computing (STOC), pages 202--211, 1996.

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H. Kaplan and R. E. Tarjan. Purely functional representations of catenable sorted lists. In Proc. 28th Annual ACM Symposium on Theory of Computing, pages 202-211. ACM Press, 1996.

....in which numbers have more than one representation and a single digit change is all that is needed to add one. Clancy and Knuth [9] used this idea in an implementation of finger search trees. Descriptions of such redundant representations as well as other applications can be found in [2, 9, 28]. The Clancy Knuth method represents numbers in base two but using three digits, 0,1, and 2. A redundant binary representation (RBR) of a non negative number x is a sequence of digits d n , d n Gamma1 , d 0 with d i 2 f0; 1; 2g and x = P n i=0 d i 2 . Such a representation is in ....

....uses only one subdeque instead of two, thus leading to a linear recursive structure. A final open problem is to devise a purely functional implementation of finger search trees (random access lists) with constant time catenation. Our best solution to this problem has O(log log n) catenation time [28]. Acknowledgements We thank Adam Buchsbaum, David Wagner, Ian Munro, and Chris Okasaki for their vital contributions to this paper. Adam Buchsbaum engaged in extensive and fruitful discussions concerning our ideas. David Wagner suggested the idea of the color invariant as an alternative to the ....

H. Kaplan and R. E. Tarjan. Purely functional representations of catenable sorted lists. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 202--211. ACM Press, 1996.

No context found.

H. Kaplan and R. E. Tarjan. Purely functional representations of catenable sorted lists. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 202-211. ACM Press, 1996.

....our structure generalizes to allow catenation, which no one knows how to implement efficiently using incremental recopying. Also, our structure can be extended to support access, insertion, and deletion d positions away from the end of a list in O(log d) amortized time, by applying the ideas in [9]. 4 Catenable Deques In this section we show how to extend our ideas to support catenation. Specifically, we describe a data structure for catenable deques that achieves an O(1) amortized time bound for push, pop, inject, eject, and catenate. Our structure is based upon an analogous structure of ....

....Another research direction is to design a confluently persistent representation of sorted lists such that accesses or updates d positions from an end take O(log d) time, and catenation takes O(1) time. The best structure so far developed for this problem has a doubly logarithmic catenation time [9]; it is strictly functional, and the time bounds are worst case. ....

H. Kaplan and R. E. Tarjan. Purely functional representations of catenable sorted lists. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 202--211. ACM Press, 1996.

No context found.

H. Kaplan and R. E. Tarjan. Purely functional representations of catenable sorted lists. In Proc. 28th Annual ACM Symposium on the Theory of Computing, pages 202--211, 1996.

No context found.

H. Kaplan and R. Tarjan. Purely functional representations of catenable sorted lists. In Proc. of the 28th Annual 113 ACM Symposium on the Theory of Computing, pages 202--211, May 1996.

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