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Volumn 31, Issue 6, 2002, Pages 1687-1722

Phase change of limit laws in the quicksort recurrence under varying toll functions

Author keywords

Analysis of algorithms; Binary search trees; Contraction method; Limit distribution; Method of moments; Quicksort

Indexed keywords

ALGORITHMS; FUNCTIONS; MATHEMATICAL MODELS; METHOD OF MOMENTS;

EID: 0036766889     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S009753970138390X     Document Type: Article
Times cited : (59)

References (76)
  • 1
    • 0001324627 scopus 로고
    • Asymptotic fringe distributions for general families of random trees
    • D. Aldous, Asymptotic fringe distributions for general families of random trees, Ann. Appl. Probab., 1 (1991), pp. 228-266.
    • (1991) Ann. Appl. Probab. , vol.1 , pp. 228-266
    • Aldous, D.1
  • 2
    • 0025503457 scopus 로고
    • A note on the expected behaviour of binary tree traversals
    • A. Andersson, A note on the expected behaviour of binary tree traversals, Comput. J., 33 (1990), pp. 471-472.
    • (1990) Comput. J. , vol.33 , pp. 471-472
    • Andersson, A.1
  • 3
    • 0032221886 scopus 로고    scopus 로고
    • Normal approximations of the number of records in geometrically distributed random variables
    • Z.-D. Bai, H.-K. Hwang, and W.-Q. Liang, Normal approximations of the number of records in geometrically distributed random variables, Random Structures Algorithms, 13 (1998), pp. 319-334.
    • (1998) Random Structures Algorithms , vol.13 , pp. 319-334
    • Bai, Z.-D.1    Hwang, H.-K.2    Liang, W.-Q.3
  • 7
    • 0000485156 scopus 로고
    • Some asymptotic theory for the bootstrap
    • P. J. Bickel and P. A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist., 9 (1981), pp. 1196-1217.
    • (1981) Ann. Statist. , vol.9 , pp. 1196-1217
    • Bickel, P.J.1    Freedman, P.A.2
  • 10
    • 0022110696 scopus 로고
    • The expected performance of traversal algorithms in binary trees
    • K. Brinck, The expected performance of traversal algorithms in binary trees. Comput. J., 28 (1985), pp. 426-432.
    • (1985) Comput. J. , vol.28 , pp. 426-432
    • Brinck, K.1
  • 11
    • 0019563785 scopus 로고
    • Analysis of algorithms on threaded trees
    • K. Brinck and N. Y. Foo, Analysis of algorithms on threaded trees, Comput. J., 24 (1981), pp. 148-155.
    • (1981) Comput. J. , vol.24 , pp. 148-155
    • Brinck, K.1    Foo, N.Y.2
  • 12
    • 0016538471 scopus 로고
    • Nonrecursive traversals of trees
    • W. A. Burkhard, Nonrecursive traversals of trees, Comput. J., 18 (1975), pp. 227-230.
    • (1975) Comput. J. , vol.18 , pp. 227-230
    • Burkhard, W.A.1
  • 13
    • 0012683566 scopus 로고
    • Corrigendum: Nonrecursive traversals of trees
    • W. A. Burkhard, Corrigendum: Nonrecursive traversals of trees, Comput. J., 20 (1977), p. 352.
    • (1977) Comput. J. , vol.20 , pp. 352
    • Burkhard, W.A.1
  • 14
    • 0035539863 scopus 로고    scopus 로고
    • Phase changes in random m-ary search trees and generalized quicksort
    • H.-H. Chern and H.-K. Hwang, Phase changes in random m-ary search trees and generalized quicksort, Random Structures Algorithms, 19 (2001), pp. 316-358.
    • (2001) Random Structures Algorithms , vol.19 , pp. 316-358
    • Chern, H.-H.1    Hwang, H.-K.2
  • 15
    • 0036660151 scopus 로고    scopus 로고
    • An asymptotic theory for Cauchy-Euler differential equations with applications to the analysis of algorithms
    • to appear
    • H.-H. Chern, H.-K. Hwang, and T.-H. Tsai, An asymptotic theory for Cauchy-Euler differential equations with applications to the analysis of algorithms, J. Algorithms, to appear.
    • J. Algorithms
    • Chern, H.-H.1    Hwang, H.-K.2    Tsai, T.-H.3
  • 18
    • 84990652727 scopus 로고
    • Limit laws for local counters in random binary search trees
    • L. Devroye, Limit laws for local counters in random binary search trees, Random Structures Algorithms, 2 (1991), pp. 303-315.
    • (1991) Random Structures Algorithms , vol.2 , pp. 303-315
    • Devroye, L.1
  • 19
    • 0032107999 scopus 로고    scopus 로고
    • Universal limit laws for depths in random trees
    • L. Devroye, Universal limit laws for depths in random trees, SIAM J. Comput., 28 (1998), pp. 409-432.
    • (1998) SIAM J. Comput. , vol.28 , pp. 409-432
    • Devroye, L.1
  • 20
    • 0037783432 scopus 로고    scopus 로고
    • Limit laws for sums of functions of subtrees of random binary search trees
    • to appear
    • L. Devroye, Limit laws for sums of functions of subtrees of random binary search trees, SIAM J. Comput., to appear.
    • SIAM J. Comput.
    • Devroye, L.1
  • 21
    • 0001851198 scopus 로고
    • Application of the method of moments in probability and statistics
    • (San Antonio, TX, 1987), AMS, Providence, RI
    • P. Diaconis, Application of the method of moments in probability and statistics, in Moments in Mathematics (San Antonio, TX, 1987), AMS, Providence, RI, 1987, pp. 125-142.
    • (1987) Moments in Mathematics , pp. 125-142
    • Diaconis, P.1
  • 22
    • 0033440049 scopus 로고    scopus 로고
    • Total path length for random recursive trees
    • R. P. Dobrow and J. A. Fill, Total path length for random recursive trees. Combin. Probab. Comput., 8 (1999), pp. 317-333.
    • (1999) Combin. Probab. Comput. , vol.8 , pp. 317-333
    • Dobrow, R.P.1    Fill, J.A.2
  • 23
    • 0015771171 scopus 로고
    • Simple algorithms for traversing a tree without an auxiliary stack
    • B. Dwyer, Simple algorithms for traversing a tree without an auxiliary stack, Inform. Process. Lett., 2 (1974), pp. 143-145.
    • (1974) Inform. Process. Lett. , vol.2 , pp. 143-145
    • Dwyer, B.1
  • 24
    • 0012640855 scopus 로고
    • A note on traversal algorithms for triply linked trees
    • T. I. Fenner and G. Loizou, A note on traversal algorithms for triply linked trees, BIT, 21 (1981), pp. 153-156.
    • (1981) BIT , vol.21 , pp. 153-156
    • Fenner, T.I.1    Loizou, G.2
  • 25
    • 0030304083 scopus 로고    scopus 로고
    • On the distribution of binary search trees under the random permutation model
    • J. A. Fill, On the distribution of binary search trees under the random permutation model, Random Structures Algorithms, 8 (1996), pp. 1-25.
    • (1996) Random Structures Algorithms , vol.8 , pp. 1-25
    • Fill, J.A.1
  • 27
    • 0030364144 scopus 로고    scopus 로고
    • On the distribution for the duration of a randomized leader election algorithm
    • J. A. Fill, H. Mahmoud, and W. Szpankowski, On the distribution for the duration of a randomized leader election algorithm, Ann. Appl. Probabl., 6 (1996), pp. 1260-1283.
    • (1996) Ann. Appl. Probabl. , vol.6 , pp. 1260-1283
    • Fill, J.A.1    Mahmoud, H.2    Szpankowski, W.3
  • 29
    • 0000640693 scopus 로고
    • Singularity analysis of generating functions
    • P. Flajolet and A. Odlyzko, Singularity analysis of generating functions, SIAM J. Discrete Math., 3 (1990), pp. 216-240.
    • (1990) SIAM J. Discrete Math. , vol.3 , pp. 216-240
    • Flajolet, P.1    Odlyzko, A.2
  • 30
    • 19544388116 scopus 로고    scopus 로고
    • On the analysis of linear probing hashing
    • P. Flajolet, P. Poblete, and A. Viola, On the analysis of linear probing hashing. Algorithmica, 22 (1998), pp. 490-515.
    • (1998) Algorithmica , vol.22 , pp. 490-515
    • Flajolet, P.1    Poblete, P.2    Viola, A.3
  • 32
    • 0023045001 scopus 로고
    • Eliminating the flag in threaded binary search trees
    • D. Gordon, Eliminating the flag in threaded binary search trees, Inform. Process. Lett., 23 (1986), pp. 209-214.
    • (1986) Inform. Process. Lett. , vol.23 , pp. 209-214
    • Gordon, D.1
  • 33
    • 0037163972 scopus 로고    scopus 로고
    • Sorting algorithms for broadcast communications: Mathematical analysis
    • to appear
    • P. Grabner and H. Prodinger, Sorting algorithms for broadcast communications: Mathematical analysis, Theoret. Comput. Sci., to appear.
    • Theoret. Comput. Sci.
    • Grabner, P.1    Prodinger, H.2
  • 34
    • 0011927813 scopus 로고    scopus 로고
    • Fifty years of the Wiener index
    • I. Gutman, S. Klavzar, and B. Mohar, eds.
    • I. Gutman, S. Klavzar, and B. Mohar, eds., Fifty years of the Wiener index. Match, 35 (1997), pp. 1-259.
    • (1997) Match , vol.35 , pp. 1-259
  • 35
    • 0000975791 scopus 로고
    • Combinatorial analysis of quicksort algorithm
    • P. Hennequin, Combinatorial analysis of quicksort algorithm, RAIRO Inform. Théor. Appl., 23 (1989), pp. 317-333.
    • (1989) RAIRO Inform. Théor. Appl. , vol.23 , pp. 317-333
    • Hennequin, P.1
  • 36
    • 0004069039 scopus 로고
    • Analyse en moyenne d'algorithme, tri rapide et arbres de recherche
    • Ph.D. thesis, Ecole Polytechnique, Palaiseau, France
    • P. Hennequin, Analyse en moyenne d'algorithme, tri rapide et arbres de recherche, Ph.D. thesis, Ecole Polytechnique, Palaiseau, France, 1991.
    • (1991)
    • Hennequin, P.1
  • 37
    • 0002217386 scopus 로고
    • Quicksort
    • C. A. R. Hoare, Quicksort, Comput. J., 5 (1962), pp. 10-15.
    • (1962) Comput. J. , vol.5 , pp. 10-15
    • Hoare, C.A.R.1
  • 39
    • 0033141151 scopus 로고    scopus 로고
    • Asymptotics of poisson approximation to random discrete distributions: An analytic approach
    • H.-K. Hwang, Asymptotics of Poisson approximation to random discrete distributions: An analytic approach, Adv. in Appl. Probabl., 31 (1999), pp. 448-491.
    • (1999) Adv. in Appl. Probabl. , vol.31 , pp. 448-491
    • Hwang, H.-K.1
  • 42
    • 0012704406 scopus 로고
    • A contribution to the analysis of in situ permutation
    • P. Kirschenhofer, H. Prodinger, and R. F. Tichy, A contribution to the analysis of in situ permutation, Glas. Mat. Ser. III, 22 (1987), pp. 269-278.
    • (1987) Glas. Mat. Ser. III , vol.22 , pp. 269-278
    • Kirschenhofer, P.1    Prodinger, H.2    Tichy, R.F.3
  • 45
    • 0014735325 scopus 로고
    • An algorithm for in-situ permutation
    • I. D. G. MacLeod, An algorithm for in-situ permutation, Austral. Comput. J., 2 (1970), pp. 16-19.
    • (1970) Austral. Comput. J. , vol.2 , pp. 16-19
    • MacLeod, I.D.G.1
  • 46
    • 0000580577 scopus 로고
    • An ideal metric and the rate of convergence to a self similar process
    • M. Maejima and S. T. Rachey, An ideal metric and the rate of convergence to a self similar process, Ann. Probab., 15 (1987), pp. 708-727.
    • (1987) Ann. Probab. , vol.15 , pp. 708-727
    • Maejima, M.1    Rachey, S.T.2
  • 47
    • 0001889115 scopus 로고
    • Limiting distributions for path lengths in recursive trees
    • H. M. Mahmoud, Limiting distributions for path lengths in recursive trees, Probab. Engrg. Inform. Sci., 5 (1991), pp. 53-59.
    • (1991) Probab. Engrg. Inform. Sci. , vol.5 , pp. 53-59
    • Mahmoud, H.M.1
  • 50
    • 0000065736 scopus 로고
    • Analysis of quickselect: An algorithm for order statistics
    • H. M. Mahmoud, R. Modarres, and R. T. Smythe, Analysis of quickselect: An algorithm for order statistics, RAIRO Inform. Théor. Appl., 29 (1995), pp. 255-276.
    • (1995) RAIRO Inform. Théor. Appl. , vol.29 , pp. 255-276
    • Mahmoud, H.M.1    Modarres, R.2    Smythe, R.T.3
  • 51
    • 0018753097 scopus 로고
    • Traversing binary trees simply and cheaply
    • J. M. Morris, Traversing binary trees simply and cheaply, Inform. Process. Lett., 9 (1979), pp. 197-200.
    • (1979) Inform. Process. Lett. , vol.9 , pp. 197-200
    • Morris, J.M.1
  • 52
    • 0003623464 scopus 로고    scopus 로고
    • Ph.D. Dissertation, Institut für Mathematische Stochastik, Universität Freiburg, Freiburg, Germany
    • R. Neininger, Limit Laws for Random Recursive Structures Algorithms, Ph.D. Dissertation, Institut für Mathematische Stochastik, Universität Freiburg, Freiburg, Germany, 1999: also available online from www.stochastik.uni-freiburg.de/homepages/neiniger/.
    • (1999) Limit Laws for Random Recursive Structures Algorithms
    • Neininger, R.1
  • 53
    • 0035539864 scopus 로고    scopus 로고
    • On a multivariate contraction method for random recursive structures with applications to quicksort
    • R. Neininger, On a multivariate contraction method for random recursive structures with applications to quicksort, Random Structures Algorithms, 19 (2001), pp. 498-524.
    • (2001) Random Structures Algorithms , vol.19 , pp. 498-524
    • Neininger, R.1
  • 54
    • 0036566412 scopus 로고    scopus 로고
    • On binary search tree recursions with monomials as toll functions
    • R. Neininger, On binary search tree recursions with monomials as toll functions, J. Comput. Appl. Math., 142 (2002), pp. 185-196.
    • (2002) J. Comput. Appl. Math. , vol.142 , pp. 185-196
    • Neininger, R.1
  • 56
    • 0004155195 scopus 로고    scopus 로고
    • A general contraction theorem and asymptotic normality in combinatorial structures
    • preprint, Institut für Mathematische Stochastik, Universität Freiburg, Freiburg, Germany
    • R. Neininger and L. Rüschendorf, A General Contraction Theorem and Asymptotic Normality in Combinatorial Structures, preprint, Institut für Mathematische Stochastik, Universität Freiburg, Freiburg, Germany, 2001; also available online from http://www.stochastik.uni-freiburg.de/homepages/neininger.
    • (2001)
    • Neininger, R.1    Rüschendorf, L.2
  • 57
    • 0036567083 scopus 로고    scopus 로고
    • Binary search tree recursions with harmonic toll functions
    • to appear
    • A. Panholzer and H. Prodinger, Binary search tree recursions with harmonic toll functions, J. Comput. Appl. Math., to appear.
    • J. Comput. Appl. Math.
    • Panholzer, A.1    Prodinger, H.2
  • 58
    • 0033267675 scopus 로고    scopus 로고
    • Normal convergence problem? Two moments and a recurrence may be the clues
    • B. Pittel, Normal convergence problem? Two moments and a recurrence may be the clues, Ann. Appl. Probab., 9 (1999), pp. 1260-1302.
    • (1999) Ann. Appl. Probab. , vol.9 , pp. 1260-1302
    • Pittel, B.1
  • 59
    • 0000789371 scopus 로고
    • How to select a loser
    • H. Prodinger, How to select a loser, Discrete Math., 120 (1993), pp. 149-159.
    • (1993) Discrete Math. , vol.120 , pp. 149-159
    • Prodinger, H.1
  • 61
    • 21144466500 scopus 로고
    • A new ideal metric with applications to multivariate stable limit theorems
    • S. T. Rachev and L. Rüschendorf, A new ideal metric with applications to multivariate stable limit theorems, Probab. Theory Related Fields, 94 (1992), pp. 163-187.
    • (1992) Probab. Theory Related Fields , vol.94 , pp. 163-187
    • Rachev, S.T.1    Rüschendorf, L.2
  • 62
    • 0001131997 scopus 로고
    • Probability metrics and recursive algorithms
    • S. T. Rachev and L. Rüschendorf, Probability metrics and recursive algorithms, Adv. in Appl. Probab., 27 (1995), pp. 770-799.
    • (1995) Adv. in Appl. Probab. , vol.27 , pp. 770-799
    • Rachev, S.T.1    Rüschendorf, L.2
  • 64
    • 0002497513 scopus 로고
    • A limiting distribution for quicksort
    • M. Régnier, A limiting distribution for quicksort, RAIRO Inform. Théor. Appl., 23 (1989), pp. 335-343.
    • (1989) RAIRO Inform. Théor. Appl. , vol.23 , pp. 335-343
    • Régnier, M.1
  • 65
    • 0015599078 scopus 로고
    • An improved algorithm for traversing binary trees without auxiliary stack
    • J. M. Robson, An improved algorithm for traversing binary trees without auxiliary stack, Inform. Process. Lett., 2 (1973), pp. 12-14.
    • (1973) Inform. Process. Lett. , vol.2 , pp. 12-14
    • Robson, J.M.1
  • 66
    • 0002731988 scopus 로고
    • A limit theorem for "Quicksort"
    • U. Rösler, A limit theorem for "Quicksort," RAIRO Inform. Théor. Appl., 25 (1991), pp. 85-100.
    • (1991) RAIRO Inform. Théor. Appl. , vol.25 , pp. 85-100
    • Rösler, U.1
  • 67
    • 38249008257 scopus 로고
    • A fixed point theorem for distributions
    • U. Rösler, A fixed point theorem for distributions, Stochastic Process. Appl., 42 (1992), pp. 195-214.
    • (1992) Stochastic Process. Appl. , vol.42 , pp. 195-214
    • Rösler, U.1
  • 68
    • 9644259154 scopus 로고    scopus 로고
    • The analysis of stochastic divide and conquer algorithms
    • U. Rösler, The analysis of stochastic divide and conquer algorithms, Algorithmica, 29 (2001), pp. 238-261.
    • (2001) Algorithmica , vol.29 , pp. 238-261
    • Rösler, U.1
  • 69
    • 0002015483 scopus 로고    scopus 로고
    • The contraction method for recursive algorithms
    • U. Rösler and L. Rüschendorf, The contraction method for recursive algorithms, Algorithmica, 29 (2001), pp. 3-33.
    • (2001) Algorithmica , vol.29 , pp. 3-33
    • Rösler, U.1    Rüschendorf, L.2
  • 70
    • 0039031027 scopus 로고    scopus 로고
    • Limiting distribution of the costs of partial match retrievals in multidimensional tries
    • W. Schachinger, Limiting distribution of the costs of partial match retrievals in multidimensional tries, Random Structures Algorithms, 17 (2000), pp. 428-459.
    • (2000) Random Structures Algorithms , vol.17 , pp. 428-459
    • Schachinger, W.1
  • 71
    • 0004244040 scopus 로고
    • Gurland Publishing, New York
    • R. Sedgewick, Quicksort, Gurland Publishing, New York, 1980.
    • (1980) Quicksort
    • Sedgewick, R.1
  • 75
    • 0034180003 scopus 로고    scopus 로고
    • Introspective sorting and selection revisited
    • J. D. Valois, Introspective sorting and selection revisited, Software-Practice and Experience, 30 (2000), pp. 617-638.
    • (2000) Software-Practice and Experience , vol.30 , pp. 617-638
    • Valois, J.D.1
  • 76
    • 0000143519 scopus 로고
    • Ideal metrics in the problem of approximating distributions of sums of independent random variables
    • V. M. Zolotarev, Ideal metrics in the problem of approximating distributions of sums of independent random variables, Theory Probab. Appl., 22 (1977), pp. 433-449.
    • (1977) Theory Probab. Appl. , vol.22 , pp. 433-449
    • Zolotarev, V.M.1


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