R. Tarjan. Amortized Computational Complexity. SIAM Journal of Algebraic Discrete Methods, 6:306--318, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....applications, the potential of an empty structure is zero and the potential is always non negative. It follows that, for any sequence of operations starting with an empty structure, the total actual cost of the operations is bounded above by the sum of their amortized costs. See the survey paper [14] for a more complete discussion of amortized analysis. 3 Noncatenable Deques In this section we describe an implementation of persistent noncatenable deques with a constant amortized time bound per operation. The structure is based on the analogous Kaplan Tarjan structure [8] but is much ....

....buffers are of constant size, whereas in [13] and [7] they are noncatenable deques. Second, the skeleton of the present structure is a binary tree, instead of a tree extension of a redundant digital numbering system as in [7] The amortized analysis uses the standard potential function method of [14] rather than the more complicated debit mechanism used in [13] For catenable steques (EJECT iS not allowed) we have a simpler structure that has a stack as its skeleton rather than a binary tree. It is based on the same recursive decomposition of lists as in [8, Section 4] Our new structure ....

R. E. Tarjan. Amortized computational complexity. SIAM J. Algebraic Discrete Methods, 6(2):306-318, 1985.

....charge this cost to the insertions and deletions by using a potential function that is O(log n) times the number of nodes whose swap times are in the future. In this extended abstract we omit the details of the analysis, which uses the standard potential function paradigm of amortized analysis [15]. The result is an amortized bound of O(1) per find rain and O(log 2 n) per insert and delete. Each update can increase the 4 potential by O(log2n) advancing the current time decreases the potential by an amount that pays for the advance. A variant of this data structure has the items stored ....

R. E. Tarjan. Amortized computational complexity. SIAM J. on Algebraic and Discrete Methods, 6(2):306- 318, 1985.

....2.2 Amortized analysis Often we are not so much interested in the running time of a single operation, but rather in the overall running time of a sequence of operations. To make this precise, we consider amortized time bounds. This concept is explained in detail in an article by Tarjan [Tar85] and in the textbook by Mehlhorn [Meh84a] Assume for example that we have a data structure supporting the operations I, D and Q. For a sequence of operations consisting of i, d and q operations of the respective type. Let n be an upper bound for the input size parameter of this data structure, ....

R. E. Tarjan, Amortized computational complexity, SIAM J. Algebraic Discrete Methods 6 (1985), no. 2, 306--318.

....link cut operations and various queries (such as finding the lowest common ancestor of two nodes) in logarithmic time. As shown in [18] they can be modified to support ordered trees and expand contract operations. In the description of time bounds we use standard concepts of amortized complexity [34]. In the rest of this section, the SPQR tree presented in [11, 12, 13, 14] is described. Let G be a biconnected graph. A split pair of G is either a pair of adjacent vertices or a separation pair. In the former case the split pair is said trivial, in the latter non trivial. A split component of ....

R. E. Tarjan. Amortized computational complexity. SIAM J. Algcb,'aic Disc,ctc Methods, 6(2):306 318, 1985.

....intriguing issue, while dealing with incremental algorithms, is the evaluation of time performances. The time cost of a dynamic algorithm may be expressed in terms of worst case cost (i.e. an upper bound to the time spent for a single operation performed on the data structure) or amortized cost [11], which provides the average time spent per operation over the worst possible squc of operations. A possible way to evaluate such a cost is to bound the worst case total 2 PRELIMINARIES cost of any sequence of operations and then dividing it by the length of the sequence. Note that the amortized ....

....claim we will show that the total time spent for performing a sequence of m edge insertions is O(mn) thus improving on the O(m 2) bound obtained by performing m topological orderings of the graph. Equivalently the cost of an insertion amortized over a sequence of O(m) insertions is only O(n) see [11] for examples of amortized analysis) The space requirements are favourable: the algorithm uses O(n m) space for the representation of the graph and O(n) space to store the topological order; O(n) additional space is required during the execution of the update operation. Given a graph G let ord ....

R. E. Tarjan. Amortized computational complexity. S'IAM J. Alg. Disc. Meth., 6:306 318, 1985.

....9) and the comparisons to update s u (Step 7) in Delete vertex each remaining unit of work corresponds to the destruction of a binomial tree edge, i.e. a previous comparison. It therefore suffices to bound the number of comparisons. We do this using the accounting method of amortized analysis [6, 16]. Define 1 credit to be the work required for 1 comparison. We will maintain the following invariant after each operation: 6 Credit Invariant (i) Every heap element has C = O(1) credits. ii) Every root of rank k has an additional k 4 credits. iii) Every nonroot of rank k with a marked ....

R.E. Tarjan, Amortized computational complexity, SIAM J. on Algebraic and Discrete Methods 6, 2, 1985, pp.306--318.

....heap ordered queues in the sensitivity analysis of minimum spanning trees, shortest path trees, and minimum cost network flow on planar graphs. Larmore and Hirschberg [LH85] and Cole and Siegal [CS84] independently show how to implement heap ordered queues in O(1) amortized time per operation [Tar85] Gajewska and Tarjan [GT86] modify their techniques to produce heap ordered deques with O(1) time per operation; they give both amortized and worst case solutions. Applications of heap ordered deques include computing all pairs shortest path information [Fre91] and external farthest neighbors ....

R. E. Tarjan. Amortized computational complexity. SIAM Journal on Algebraic Discrete Methods, 6(2):306--18, 1985.

....9) and the comparisons to update s u (Step 7) in Delete vertex each remaining unit of work corresponds to the destruction of a binomial tree edge, i.e. a previous comparison. It therefore suces to bound the number of comparisons. We do this using the accounting method of amortized analysis [6, 17]. De ne 1 credit to be the work required for 1 comparison. We will maintain the following invariant after each operation: Credit Invariant (i) Every heap element has C = O(1) credits. ii) Every root of rank k has an additional k 4 credits. iii) Every nonroot of rank k with a marked parent ....

Tarjan, R.E.: Amortized computational complexity. SIAM J. on Algebraic and Discrete Methods 6 (1985) 306-318

....weight and w s new parent gains weight; two paths of nodes need to be updated: the one upward from node x and the one upward from w s new parent. All the nodes on the two paths should revise their weights to re ect the changes. To facilitate the amortized analysis, we use an accounting method [22], where we charge C units of cost to a level node w that changes its parent. Since we only change the weights of one of the N bottom level elements on level 0, and in the worst case the element will change its parent, we charge C 0 to each dynamic weight update operation. The credits ....

R. E. Tarjan. Amortized Computational Complexity, SIAM Journal on Algebraic and Discrete Methods, 6(2): 306-318, 1985.

....two heap trees with backbones of length s and t takes time T (s, t) s t i 1. In our application of finding pairs we are more interested in bounding the total time required to do a sequence of melds rather than bounding the time of each individual meld. We therefore turn to amortized analysis [137]. On a forest F of heap trees we define the potential function #(F ) to be the sum of the lengths of the backbones of the heap trees in the forest. Melding two heap trees with backbones of length s and t, as illustrated in Figure 4.5, changes the potential of the forest with ## = i 1 (s t) The ....

R. E. Tarjan. Amortized computational complexity. SIAM Journal on Algebraic and Discrete Methods, 6:306--318, 1985.

....two heap trees with backbones of length s and t takes time T (s, t) s t i 1. In our application of finding pairs we are more interested in bounding the total time required to do a sequence of melds rather than bounding the time of each individual meld. We therefore turn to amortized analysis [179]. On a forest F of heap trees we define the potential function #(F ) to be the sum of the lengths of the backbones of the heap trees in the forest. Melding two heap trees with backbones of length s and t, as illustrated in Figure 7.5, changes the potential of the forest with ## = i 1 (s t) The ....

R. E. Tarjan. Amortized computational complexity. SIAM Journal on Algebraic and Discrete Methods, 6:306--318, 1985.

....weight and w s new parent gains weight; two paths of nodes need to be updated: the one upward from node x and the one upward from w s new parent. All the nodes on the two paths should revise their weights to reflect the changes. To facilitate the amortized analysis, we use an accounting method [22], where we charge C units of cost to a level node w that changes its parent. Since we only change the weights of one 14 4 MODIFICATION TO ACHIEVE O(LOG of the N bottom level elements on level 0, and in the worst case the element will change its parent, we charge C 0 to each dynamic weight ....

R. E. Tarjan. Amortized Computational Complexity, SIAM Journal on Algebraic and Discrete Methods, 6(2): 306--318, 1985.

....charge this cost to the insertions and deletions by using a potential function that is O(log n) times the number of nodes whose swap times are in the future. In this extended abstract we omit the details of the analysis, which uses the standard potential function paradigm of amortized analysis [15]. The result is an amortized bound of O(1) per nd min and O(log 2 n) per insert and delete. Each update can increase the potential by O(log 2 n) advancing the current time decreases the potential by an amount that pays for the advance. A variant of this data structure has the items stored ....

R. E. Tarjan. Amortized computational complexity. SIAM J. on Algebraic and Discrete Methods, 6(2):306-318, 1985.

....A useful cost measure is therefore given by T:x = number of unfoldings of (iii) and (iv) required for the evaluation of a 6 x. Given cost measure T , we want to derive a logarithmic bound for the amortized cost A of a 6 , given by A:x = T:x : a 6 x) x; where is a potential function [6]. For the sake of brevity it it left implicit that T:x and A:x depend on a. 2 Setting out for an inductive derivation, we rst calculate a recurrence relation for A. To this end, we take of the form: h i = 0 :ht; a; ui = t :t:u :u: Note that a does not occur in the right hand side ....

Tarjan R.E. Amortized computational complexity. SIAM Journal on Algebraic and Discrete Methods 6 (1985) 306-318.

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R.E. Tarjan. Amortized computational complexity. SIAM Journal on Algenraic and Discrete Methods, 6:306--318, 1985.

No context found.

R.E. Tarjan. Amortized computational complexity. SIAM Journal on Algenraic and Discrete Methods, 6:306--318, 1985.

No context found.

R.E. Tarjan. Amortized computational complexity. SIAM Journal on Algebraic and Discrete Methods, 6:306-- 318, 1985.

....simple amortization argument using a potential function equal to the absolute value of the difference in stack sizes shows that this gives a lineartime simulation of a deque by a constant number of stacks: k deque operations starting from an empty deque are simulated by O(k) stack operations. See [44] for a discussion of amortization and potential functions. This simple idea is the essence of Stoss s tape simulation. The idea of representing a queue by two stacks in this way appears in [5, 20, 22] this representation of a deque appears in [19, 21, 23, 39] The second idea is to use ....

....to record the results. This avoids the need to maintain the stack of stacks structures in our representations, and also allows the buffers to be shorter. Okasaki called the resulting general method implicit recursive slowdown . He argues that the standard techniques of amortized analysis [44] do not suffice in this case because of the need to deal with persistence. His idea is in fact much more general than recursive slow down, however, and the standard techniques [44] do indeed suffice for an analysis. Working with Okasaki, we have devised even simpler versions of our structures that ....

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R. E. Tarjan. Amortized computational complexity. SIAM J. Algebraic Discrete Methods, 6(2):306-- 318, 1985.

....applications, the potential of an empty structure is zero and the potential is always non negative. It follows that, for any sequence of operations starting with an empty structure, the total actual cost of the operations is bounded above by the sum of their amortized costs. See the survey paper [17] for a more complete discussion of amortized analysis. 3. Noncatenable Deques. In this section we describe an implementation of persistent noncatenable deques with a constant amortized time bound per operation. The structure is based on the analogous Kaplan Tarjan structure [11, 10] but is much ....

....are of constant size, whereas in [16] and [8] they are noncatenable deques. Second, the skeleton of the present structure is a binary tree, instead of a tree extension of a redundant digital numbering system as in [8] Also, our amortized analysis uses the standard potential function method of [17] rather than the more complicated debit mechanism used in [16] Another related structure is that of [10, Section 5] which represents purely functional, real time deques as pairs of triples rather than 5 tuples, but otherwise is similar to (but simpler than) the structure of [8, Section 9] It is ....

R. E. Tarjan, Amortized computational complexity, SIAM J. Algebraic Discrete Methods, 6 (1985), pp. 306-318.

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R. Tarjan. Amortized Computational Complexity. SIAM Journal of Algebraic Discrete Methods, 6:306--318, 1985.

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Tarjan, R. (1985). Amortized computational complexity. SIAM J. Algebr. Discrete Methods 6: 306Y318.

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R. E. Tarjan. Amortized computational complexity. SIAM J. Appl. Discrete Math, 6:306--318, 1985.

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Tarjan R.E. Amortized computational complexity. SIAM Journal on Algebraic and Discrete Methods 6 (1985) 306--318.

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R. Tarjan. Amortized Computational Complexity. SIAM Journal of Algebraic Discrete Methods, 6:306--318, 1985.

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R.E. Tarjan. Amortized computational complexity. SIAM Journal on Discrete Mathematics, 6(2), 1985.

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