Algorithms and Data Structures

More documents

Recommendations

Info

N.Wirth. Algorithms and Data Structures. Oberon version 98 from both ends inward; compare its performance with that of the procedure given in this text. 2.11. Note that in a (two-way) natural merge we do not blindly select the least value among the available keys. Instead, upon encountering the end of a run, the tail of the other run is simply copied onto the output sequence. For example, merging of 2, 4, 5, 1, 2, ... 3, 6, 8, 9, 7, ... results in the sequence 2, 3, 4, 5, 6, 8, 9, 1, 2, ... instead of 2, 3, 4, 5, 1, 2, 6, 8, 9, ... which seems to be better ordered. What is the reason for this strategy? 2.12. A sorting method similar to the Polyphase is the so-called Cascade merge sort [2.1 and 2.9]. It uses a different merge pattern. Given, for instance, six sequences T0 ... T5, the cascade merge, also starting with a "perfect distribution" of runs on T0 ... T4, performs a five-way merge from T0 ... T4 onto T5 until T4 is empty, then (without involving T5) a four-way merge onto T4, then a three-way merge onto T3, a two-way merge onto T2, and finally a copy operation from T0 onto T1. The next pass operates in the same way starting with a five-way merge to T0, and so on. Although this scheme seems to be inferior to Polyphase because at times it chooses to leave some sequences idle, and because it involves simple copy operations, it surprisingly is superior to Polyphase for (very) large files and for six or more sequences. Write a well structured program for the Cascade merge principle. References [2.1] B. K. Betz and Carter. Proc. ACM National Conf. 14, (1959), Paper 14. [2.2] R.W. Floyd. Treesort (Algorithms 113 and 243). Comm. ACM, 5, No. 8, (1962), 434, and Comm. ACM, 7, No. 12 (1964), 701. [2.3] R.L. Gilstad. Polyphase Merge Sorting - An Advanced Technique. Proc. AFIPS Eastern Jt. Comp. Conf., 18, (1960), 143-48. [2.4] C.A.R. Hoare. Proof of a Program: FIND. Comm. ACM, 13, No. 1, (1970), 39-45. [2.5] C.A.R. Hoare. Proof of a Recursive Program: Quicksort. Comp. J., 14, No. 4 (1971), 391-95. [2.6] C.A.R. Hoare. Quicksort. Comp.J., 5. No.1 (1962), 10-15. [2.7] D.E. Knuth. The Art of Computer Programming. Vol. 3. Reading, Mass.: Addison- Wesley, 1973. [2.8] H. Lorin. A Guided Bibliography to Sorting. IBM Syst. J., 10, No. 3 (1971), 244-254. [2.9] D.L. Shell. A Highspeed Sorting Procedure. Comm. ACM, 2, No. 7 (1959), 30-32. [2.10] R.C. Singleton. An Efficient Algorithm for Sorting with Minimal Storage (Algorithm 347). Comm. ACM, 12, No. 3 (1969), 185. [2.11] M.H. Van Emden. Increasing the Efficiency of Quicksort (Algorithm 402). Comm. ACM, 13, No. 9 (1970), 563-66, 693. [2.12] J.W.J. Williams. Heapsort (Algorithm 232) Comm. ACM, 7, No. 6 (1964), 347-48.
N.Wirth. Algorithms and Data Structures. Oberon version 99 3 Recursive Algorithms 3.1 Introduction An object is said to be recursive, if it partially consists or is defined in terms of itself. Recursion is encountered not only in mathematics, but also in daily life. Who has never seen an advertising picture which contains itself? Fig. 3.1. A picture with a recursion Recursion is a particularly powerful technique in mathematical definitions. A few familiar examples are those of natural numbers, tree structures, and of certain functions: 1. Natural numbers: (a) 0 is a natural number. (b) the successor of a natural number is a natural number. 2. Tree structures: (a) ∅ is a tree (called the empty tree). (b) If t 1 and t 2 are trees, then the structure consisting of a node with two descendants t 1 and t 2 is also a (binary) tree. 3. The factorial function f(n): f(0) = 1 f(n) = n × f(n - 1) for n > 0 The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions. Recursive algorithms, however, are primarily appropriate when the problem to be solved, or the function to be computed, or the data structure to be processed are already defined in recursive terms. In general, a recursive program P can be expressed as a composition P of a sequence of statements S (not containing P) and P itself. P ≡ P[S, P] The necessary and sufficient tool for expressing programs recursively is the procedure or subroutine, for it allows a statement to be given a name by which this statement may be invoked. If a procedure P contains an explicit reference to itself, then it is said to be directly recursive; if P contains a reference to another procedure Q, which contains a (direct or indirect) reference to P, then P is said to be indirectly recursive. The use of recursion may therefore not be immediately apparent from the program text.
Page 1 and 2:
Algorithms and Data Structures © N
Page 3 and 4:
N.Wirth. Algorithms and Data Struct
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48: N.Wirth. Algorithms and Data Struct
Page 97: N.Wirth. Algorithms and Data Struct
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211:
show all

Algorithms and Data Structures

Create successful ePaper yourself

Delete template?

Save as template?