D. C. Kozen, The Design and Analysis of Algorithms, Springer, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of L, in our case the Fiedler value. This can be computed in polynomial time as discussed in [25] Then we can compute the corresponding eigenvector x symbolically by solving (L zI)x = 0 mod p(z) where p(z) is the characteristic polynomial of L. Gaussian elimination over a field is in P [21], so if p(z) is irreducible we obtain a solution x where each component x i is given by a polynomial in z with bounded integer coe#cients. We note that letting z be any eigenvalue will force x to be a true eigenvector. If p(z) is reducible, we try the above. If we fail to solve the equation, we ....

D. C. Kozen, The Design and Analysis of Algorithms, Springer-Verlag, New York, 1992.

....set with this set. In an amortized setting a classical data structure of Tarjan supports a sequence of m finds and at most n unions on a universe of n elements in O(n mff(m n; n; log n) time where ff(m;n; l) minfk j Ak (b c) lg and A i (j) is Ackermann s function as described in [6]. We refine the analysis of this data structure and show that in fact the cost of each find is proportional to the size of the corresponding set. Specifically, we show that one can pay for a sequence of union and find operations by charging a constant to each participating element and O(ff(m; n; ....

....a classical data structure supports a sequence of m finds and at most n unions on a universe of n elements in O(n mff(m n; n; log n) time where ff(m; n; l) minfk j A k (b c) lg and Smid s data structure supports insert in constant time. A i (j) is Ackermann s function as defined in [6]. This data structure uses a tree to represent each set and its efficiency is due to two simple heuristics. The first heuristic called union by rank makes the root of the set of smaller rank a child of the root of the set of larger rank to carry out a union . The second heuristic called path ....

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D. Kozen. The design and analysis of algorithms, 1992.

....) To find P mistake (r) which is the probability that any process is not infected by any other process, we need to range over all members. For any two processes p and q, P mistake (p, r) and P mistake (q, r) are not independent. We will bound P mistake (r) using the Inclusion Exclusion Principle [11]: Pr(# i A i ) i Pr(A i ) Using this principle, we can bound P mistake (r) as follows: f)P mistake (p, r) n P (k r = n f) 3) From this we can calculate how many rounds are needed to achieve a certain quality of detection. The calculation is best done not recursively, such as ....

Dexter Kozen. The Design and Analysis of Algorithms. Springer Verlag, 1991.

....(a; r) 1 Gamma P (k r = n) However, for any two members a and b, P incomp (a; r) and P incomp (b; r) are not independent. P incomp (r) the probability that any member is not infected by sequence number information from any other member can be bound using the Inclusion Exclusion Principle [16]: P r( i A i ) iP r(A i ) From this bound we calculate how many micro steps are needed to achieve stability detection. The calculation is done iteratively starting at the first micro step. B. Analysis when the size of gossip subset is three When the size of gossip subset is greater than ....

D. Kozen. The Design and Analysis of Algorithms. Springer Verlag, 1991.

....our algorithms with random skip lists. Regarding the analysis of the performance of our algorithms, we will exploit the large amount of known results about random BSTs, after we will have explicitly noticed and proved that a RBST is always a random BST. In contrast, Aragon and Seidel (see also [21, 16]) made use of Mulmuley games to analyze the performance of randomized treaps. The paper is organized as follows. In Sections 2 and 3, the insertion and deletion algorithms are described and their main properties stated. We present the analysis of the performance of the basic operations in Section ....

D. C. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, 1992.

....the dynamic version of the problem where edges can be inserted and deleted. 2: Array initialization Section III.8.1 of [15] contains a description of how a bitvector can be intitialized in worst case constant time. 3: Sink Exercise 23.1 6 in [6] 4: Scorpion Exercise 3 of homework 2 in [13], from which also the following paragraph is taken. This is a counterexample to an earlier version of the famous Anderaa Rosenberg conjecture, which stated that any nontrivial graph property that is invariant under graph isomorphism requires n 2 ) probes of the adjacency matrix to test. ....

....of this idea has been described by Overmars in [16] for the design of dynamic data structures. 2: Binomial Heaps A proof that binomial heaps support merging in amortized constant time was given in [10] 3: Dijkstra s Algorithm Folklore. 4: Prim s Algorithm Exercise 2 of homework 4 in [13]. Prim s algorithm for computing a minimum spanning tree is, e.g. described in Section 24.2 of [6] 5: The Stable Marriage Problem Exercise 1 of homework 6 in [13] Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n 2 ) solution in [8] 1 Algorithms ....

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Dexter C. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, 1992.

....them in one step. The task of deciding whether the input is in the language or not, is equivalent to finding whether there exists a directed path in this graph between the starting instantaneous description and the final one. This can be achieved in time proportional to the size of the graph [10, 30]. Hence, we obtain the following inclusion: DSPACE[s(n) # # c 0 DT IME[2 cs(n) In particular, we have the following inclusion which is conjectured to be strict. L # P. 16 2.2.2 The Random Access Machine The Random Access Machine (RAM for short) is a computational model that is much ....

....compute the product of two n n matrices in log n time using n 3 processors because all the entries of the product matrix are independent of each other, and hence can be computed in parallel. Computing A n can be achieved in O(log 2 n) using n 4 processors by repeatedly squaring the matrix [10, 30]. The matrix based repeated squaring algorithm for Shortest Path Problem can be solved in # log n steps for any # 0 with poly(n) processors on a PRAM that allows unbounded fan in min operations (but only bounded fan in additions) because multiplying k matrices (for any fixed constant k) can ....

Dexter L. Kozen. The Design and Analysis of Algorithms. Springer, Berlin, 1992.

....e = u; v) of G with an edge (f i ; f j ) in E . The edge (f i ; f j ) in G is called the dual edge of (u; v) in G. Let E 0 E be a subset of edges in G. It is well known that (V; E 0 ) is a spanning tree of G if and only if (V ; E Gamma E 0 ) is a spanning tree in G [22]. Thus the edges in (E Gamma T ) form a spanning tree in G which we denote T y . An example is shown in figure 3(a) If T has bounded height p N then every edge in (E Gamma T ) and therefore the corresponding edge in (E Gamma T ) 5 determines a cycle in T with at most 2 p ....

D. Kozen. The Design and Analysis of Algorithms. Springer, Berlin, 1992.

....0 and allow for 1 access per cycle. Moreover, all of them can be accessed in parallel. Trying to assign the static objects of a program to these banks in such a way that no conflicts arise during program execution 2 is essentially the same problem. Since the first problem is known to be NP[13], the second one must also be NP. Moreover, the second problem is just a special case of the more general problem addressed in this document, where banks can have different waitstates and allow for parallel access. This prohibits the possibility of finding an optimal solution for non trivial ....

D. C. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, USA, 1992.

.... ( Ng, 1984; Ng and Tarski, 1977] semantics and logics of programs ( Kozen, 1981b; Pratt, 1988] automata and formal language theory ( Kuich, 1987; Kuich and Salomaa, 1986] and the design and analysis of algorithms ( Aho et al. 1975; Tarjan, 1981; Mehlhorn, 1984; Iwano and Steiglitz, 1990; Kozen, 1991b] As discussed in Section 13, Kleene algebra plays a prominent role in dynamic algebra as an algebraic model of program behavior. Beginning with the monograph of [Conway, 1971] many authors have contributed over the years to the development of the algebraic theory; see [Backhouse, 1975; ....

....binary relations on a set. In fact it is the latter interpretation that makes Kleene algebra a suitable choice for modeling programs in dynamic algebras. Other more unusual interpretations are the min; algebra used in shortest path algorithms (see [Aho et al. 1975; Tarjan, 1981; Mehlhorn, 1984; Kozen, 1991b] and KAs of convex polyhedra used in computational geometry as described in [Iwano and Steiglitz, 1990] Axiomatization of the equational theory of the regular sets is a central question going back to the original paper of [Kleene, 1956] A completeness theorem for relational algebras was ....

Dexter Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, 1991.

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D. C. Kozen, The Design and Analysis of Algorithms, Springer, 1991.

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Kozen, D. C. (1992). The Design and Analysis of Algorithms. Springer-Verlag.

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Dexter C. Kozen. The Design and Analysis of Algorithms. Springer Verlag, 1992.

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D. Kozen. The design and analysis of algorithms. Springer-Verlag, 1992.

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D. C. Kozen. Design and Analysis of Algorithms. Springer-Verlag, 1992.

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Dexter Kozen. The Design and Analysis of Algorithms. Springer Verlag, 1991.

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D. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, 1991.

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D. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, 1991.

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D. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, Nev York, 1991.

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D. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, 1991.

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Kozen, D.C., "The Design and Analysis of Algorithms", Springer-Verlag, New York, 1992

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Kozen, D.C.: The Design and Analysis of Algorithms, Springer-Verlag, New York, 1992

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D. C. Kozen. Design and Analysis of Algorithms. Springer-Verlag, 1992.

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D. Kozen, The Design and Analysis of Algorithms, Springer-Verlag, 1991.

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D. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, 1991.

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