D. H. Greene and D. E. Knuth. Mathematics for the analysis of algorithms. Birkhauser, Boston, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....function. Similarly, by induction, # # n # (# 2) Thus A # # # # # # # 1###n # # # # ##1 n # . This completes the proof. Remark. The case # = 2 has been encountered in di#erent contexts and di#erent elementary methods of proof have been used; see [44, 63, 69, 70]. Besides these elementary proofs, we can also give an analytic proof along the line of Mellin transform (by deriving a uniform O estimate of n#1 n # e nu ) and singularity analysis. Actually, we can show that if a n wn # , w C 0 , # 1, then we have A n #(#)w n # . This is ....

..... # (n 1) n 1) # = 1.05836 . 2 = 1.28986 . 26 Only the formul for d ##26216were previously obtained by Pfeifer [72] and Roos [76] All other results are new. For relevant materials on permutations, see Erdos and Turan [31] Wilf [92] Greene and Knuth [44], Arratia and Tavare [8] Knopfmacher and Warlimont [63] and Arratia et al. 7] and on record values, Nevzorov [68] and Borovkov and Pfeifer [18] Random mappings. Structurally, random mappings are set of cycles of rooted labeled trees. In terms of the notation of 3.1, the generating function ....

Greene, D. H. and Knuth, D. E. (1990) Mathematics for the analysis of algorithms. third edn. Birkhauser, Boston/Basel/Berlin. 43

.... [23] Fill [12] Hu man coding (see Glassey and Karp [18] Chang and Thomas [7] binomial group testing (see Hwang et al. 30] O Geran et al. 43] dynamic programming (see Fredman and Knuth [16] dichotomous search problems (see Wong [56] Morris [41] Carlitz [6] Gal [17] Greene and Knuth [20]) design of electrical circuits (see Pelling and Rogers [46, 45] divide and conquer problems (see Chen et al. 8] etc. Hammersley and Grimmett [22] proved that if g(n) is nonincreasing then f(n) 1 k n g(k) f(n) n) where (1) f(1) and (n) bn=2c) dn=2e) g(n) n 2) 2) ....

....point of view, this means that the divide and conquer in each step can be more exible if the merge cost is linear. Some concrete problems involving essentially f(n) are listed below. Optimal search in a sorted array: see Wong [56] Morris [41] Carlitz [6] Gal [17] Greene and Knuth [20]; Best case cost of quicksort and worst case cost of mergesort: see Flajolet and Golin [13] Golin and Sedgewick [19] Chen et al. 8] The function F (n) satis es (with g(n) n) F (n) F (bn=2c) F (dn=2e) bn=2c ) minf2 ) n 2 1 k n (k) where (n) denotes the ....

[Article contains additional citation context not shown here]

D. H. Greene and D. E. Knuth, \Mathematics for the Analysis of Algorithms," Third Edition, Birkhauser, Boston, 1990.

....n = mg valid for the widest possible range of m, but to show that for m lying in the interval n Sigma O(oe n ) very precise asymptotic formulae can be obtained. These formulae are in close connection with our results in [17] Although local limit theorems receive a constant research interest [2, 3, 7, 14, 13, 24], our approach and results, especially Theorem 1, seem rarely discussed in a systematic manner. Recall that a lattice random variable X is said to be of maximal span h if X takes only values of the form b hk, k 2 Z, for some constants b and h 0; and there does not exist b and h h such ....

.... (w) we have the relation ]P (w; z) Am [z ]C (z) where A k : z 1 When m i n, namely, there exist two constants 0 1 2 1 such that 1 n m 2 n, the first term A k is easily treated by singularity analysis (cf. 10] and the second term by the saddle point method (cf. [7, 14, 20, 21]) From there a local limit theorem as above can be obtained. 13 5 Examples Let us consider some typical examples. More examples can be found in [17, 19] and the references cited there. Example 1. Connected components in random mappings. By random mapping (cf. 21] we mean a random ....

D. H. Greene and D. E. Knuth, Mathematics for the analysis of algorithms, Birkhauser, Boston/Basel/Berlin, third edition, 1990.

.... [23] Fill [12] Hu#man coding (see Glassey and Karp [18] Chang and Thomas [7] binomial group testing (see Hwang et al. 30] O Geran et al. 43] dynamic programming (see Fredman and Knuth [16] dichotomous search problems (see Wong [56] Morris [41] Carlitz [6] Gal [17] Greene and Knuth [20]) design of electrical circuits (see Pelling and Rogers [46, 45] divide and conquer problems (see Chen et al. 8] etc. Hammersley and Grimmett [22] proved that if g(n) is nonincreasing then f(n) 1#k#n g(k) #(n) #(#n 2#) #(#n 2#) g(n) n 2) 2) g(n) n ....

....point of view, this means that the divide and conquer in each step can be more flexible if the merge cost is linear. Some concrete problems involving essentially f(n) are listed below. Optimal search in a sorted array: see Wong [56] Morris [41] Carlitz [6] Gal [17] Greene and Knuth [20]; Best case cost of quicksort and worst case cost of mergesort: see Flajolet and Golin [13] Golin and Sedgewick [19] Chen et al. 8] The function F (n) satisfies (with g(n) n) F (n) F (#n 2#) F (#n 2#) F (2 ) min 2 = F (2 #log ) n 1#k n #(k) ....

[Article contains additional citation context not shown here]

D. H. Greene and D. E. Knuth, "Mathematics for the Analysis of Algorithms," Third Edition, Birkhauser, Boston, 1990.

....for n = m valid for the widest possible range of m, but to show that for m lying in the interval n O(# n ) very precise asymptotic formulae can be obtained. These formulae are in close connection with our results in [17] Although local limit theorems receive a constant research interest [2, 3, 7, 14, 13, 24], our approach and results, especially Theorem 1, seem rarely discussed in a systematic manner. Recall that a lattice random variable X is said to be of maximal span h if X takes only values of the form b hk, k Z, for some constants b and h 0; and there does not exist b # and h # h such ....

....the relation ]P (w, z) Am [z ]C (z) where A k : z 1 . When m n, namely, there exist two constants 0 # 1 # 2 1 such that # 1 n # 2 n, the first term A k is easily treated by singularity analysis (cf. 10] and the second term by the saddle point method (cf. [7, 14, 20, 21]) From there a local limit theorem as above can be obtained. 13 5 Examples Let us consider some typical examples. More examples can be found in [17, 19] and the references cited there. Example 1. Connected components in random mappings. By random mapping (cf. 21] we mean a random ....

D. H. Greene and D. E. Knuth, Mathematics for the analysis of algorithms, Birkhauser, Boston/Basel/Berlin, third edition, 1990.

....by induction, fi fi[z fi fi M ( 2) Thus jA n j = fi fi fi fi fi fi fi 1 n fi fi fi fi 1 This completes the proof. Remark. The case fl = 2 has been encountered in different contexts and different elementary methods of proof have been used; see [44, 63, 69, 70]. Besides these elementary proofs, we can also give an analytic proof along the line of Mellin transform (by deriving a uniform O estimate of n1 n Gammanu ) and singularity analysis. Actually, we can show that if a n wn , w 2 C n f0g, fl 1, then we have A n we i(fl)w This is ....

....fi fi = 1:05836 : Gamma 2 = 1:28986 : 26 Only the formulae for d K and d L for Omega Gamma 26220 were previously obtained by Pfeifer [72] and Roos [76] All other results are new. For relevant materials on permutations, see Erdos and Tur an [31] Wilf [92] Greene and Knuth [44], Arratia and Tavar e [8] Knopfmacher and Warlimont [63] and Arratia et al. 7] and on record values, Nevzorov [68] and Borovkov and Pfeifer [18] Random mappings. Structurally, random mappings are set of cycles of rooted labeled trees. In terms of the notation of x 3.1, the generating function ....

Greene, D. H. and Knuth, D. E. (1990) Mathematics for the analysis of algorithms. third edn. Birkhauser, Boston/Basel/Berlin. 43

....analogue of the problem considered in [KORSW] of determing the number of partitions of an n element set with distinct block sizes. In a partition of a set we require that blocks be pairwise disjoint. Similar problems within the context of cycles of permutations were solved by Greene and Knuth [GK] and concerning irreducible factors of polynomials over a nite eld by the present authors in [KW] 2. Unordered Coverings with distinct block sizes Let W (n) denote the numbers of covers of N with the added restriction that the sizes of the blocks be distinct and set W (0) 1. Then we can write ....

D H Greene and D E Knuth, Mathematics for the Analysis of Algorithms, 2nd ed., Birkhauser Boston, 1982.

....any k Query is answered in O(1) worst case time. For paths with more than k edges, we exploit the following combinatorial property: if a subset H of vertices is picked at random from a graph G, then a sufficiently long path will intersect H with high probability. This property appears already in [13], and later on it has been used may times in designing efficient algorithms for transitive closure and shortest paths (see e.g. 4, 14, 16, 27, 28] The following theorem is from [27] Theorem 2 Let H V be a set of vertices chosen uniformly at random. Then the probability that a given simple ....

D. H. Greene and D.E. Knuth. Mathematics for the analysis of algorithms. Birkhauser, 1982.

....our techniques is that both weight increases and weight decreases can be supported with exactly the same code. Surprisingly, the algorithm is rather simple and thus amenable to ecient implementations. Techniques. Di erently from other approaches, which were more based on properties of long paths [2, 4, 11, 13], tree data structures [13] and dynamic reevaluation of products of real valued matrices [3, 4] our method hinges on the notion of uniform paths: informally, we say that a path is uniform if all of its subpaths are shortest paths. Note that as a special case a shortest path is uniform, and thus ....

D. H. Greene and D.E. Knuth. Mathematics for the analysis of algorithms. Birkhauser, 1982.

No context found.

D. H. Greene and D. E. Knuth. Mathematics for the analysis of algorithms. Birkhauser, Boston, 1981.

No context found.

D. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, Birkhauser, Boston 1990. 21

No context found.

D. H. Greene and D. E. Knuth. Mathematics for the analysis of algorithms. Progress in computer science. Birkhuser, 2d edition, 1982.

No context found.

Greene, D. H., Knuth, D. E.: Mathematics for the Analysis of Algorithms, Birkhauser, 1981.

No context found.

D. H. Greene and D. E. Knuth. Mathematics for the Analysis of Algorithms, Birkhauser Verlag, 1990

No context found.

D. H. Greene and D.E. Knuth. Mathematics for the analysis of algorithms. Birkhauser, 1982.

No context found.

D. H. Greene and D. E. Knuth. Mathematics for the Analysis of Algorithms. Birkhauser, Basel, 1982.

No context found.

D Greene and D Knuth. Mathematics for the Analysis of Algorithms. Birkhauser, 1990.

No context found.

D.H. Greene and D.E. Knuth. Mathematics for the analysis of algorithms, Birkhauser, 1990.

No context found.

D.H. Greene and D.E. Knuth. Mathematics for the analysis of algorithms, Birkhauser, 1990.

No context found.

Greene, D. H., and Knuth, D. E. Mathematics for the analysis of algorithms. Birkhauser, Boston, 1981.

No context found.

D. H. Greene and D. E. Knuth, "Mathematics for the analysis of algorithms", Birkhauser, Boston, 1981.

No context found.

D. H. Greene and D. E. Knuth, Mathematics for the analysis of algorithms, Birkhauser (Boston, 1981).

No context found.

D. H. Greene and D. E. Knuth, Mathematics for the analysis of algorithms, Birkhauser (Boston, 1981).

No context found.

D. H. Greene and D. E. Knuth, "Mathematics for the analysis of algorithms", Birkhauser, Boston, 1981.

No context found.

Greene, D. H., and Knuth, D. E. Mathematics for the analysis of algorithms. Birkhauser, Boston, 1981.

First 50 documents  Next 50

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