Algorithms and Data Structures

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N.Wirth. Algorithms and Data Structures. Oberon version 142 PROCEDURE search (x: INTEGER; VAR root: Word); (* ADenS432_List4 *) VAR w1, w2: Word; BEGIN w1 := root; sentinel.key := x; IF w1 = sentinel THEN (*first element*) NEW(root); root.key := x; root.count := 1; root.next := sentinel ELSIF w1.key = x THEN INC(w1.count) ELSE (*search*) REPEAT w2 := w1; w1 := w2.next UNTIL w1.key = x; IF w1 = sentinel THEN (*new entry*) w2 := root; NEW(root); root.key := x; root.count := 1; root.next := w2 ELSE (*found, now reorder*) INC(w1.count); w2.next := w1.next; w1.next := root; root := w1 END END END search The improvement in this search method strongly depends on the degree of clustering in the input data. For a given factor of clustering, the improvement will be more pronounced for large lists. To provide an idea of how much gain can be expected, an empirical measurement was made by applying the above cross-reference program to a short and a relatively long text and then comparing the methods of linear list ordering and of list reorganization. The measured data are condensed into Table 4.2. Unfortunately, the improvement is greatest when a different data organization is needed anyway. We will return to this example in Sect. 4.4. Test 1 Test 2 Number of distinct keys 53 582 Number of occurrences of keys 315 14341 Time for search with ordering 6207 3200622 Time for search with reordering 4529 681584 Improvement factor 1.37 4.70 Table 4.2. Comparsion of List Search Methods.
N.Wirth. Algorithms and Data Structures. Oberon version 143 4.3.3 An Application: Partial Ordering (Topological Sorting) An appropriate example of the use of a flexible, dynamic data structure is the process of topological sorting. This is a sorting process of items over which a partial ordering is defined, i.e., where an ordering is given over some pairs of items but not between all of them. The following are examples of partial orderings: 1. In a dictionary or glossary, words are defined in terms of other words. If a word v is defined in terms of a word w, we denote this by v 〈 w. Topological sorting of the words in a dictionary means arranging them in an order such that there will be no forward references. 2. A task (e.g., an engineering project) is broken up into subtasks. Completion of certain subtasks must usually precede the execution of other subtasks. If a subtask v must precede a subtask w, we write v 〈 w. Topological sorting means their arrangement in an order such that upon initiation of each subtask all its prerequisite subtasks have been completed. 3. In a university curriculum, certain courses must be taken before others since they rely on the material presented in their prerequisites. If a course v is a prerequisite for course w, we write v 〈 w. Topological sorting means arranging the courses in such an order that no course lists a later course as prerequisite. 4. In a program, some procedures may contain calls of other procedures. If a procedure v is called by a procedure w, we write v 〈 w. Topological sorting implies the arrangement of procedure declarations in such a way that there are no forward references. In general, a partial ordering of a set S is a relation between the elements of S. It is denoted by the symbol "〈", verbalized by precedes, and satisfies the following three properties (axioms) for any distinct elements x, y, z of S: 1. if x 〈 y and y 〈 z, then x 〈 z (transitivity) 2. if x 〈 y, then not y 〈 x (asymmetry) 3. not z 〈 z (irreflexivity) For evident reasons, we will assume that the sets S to be topologically sorted by an algorithm are finite. Hence, a partial ordering can be illustrated by drawing a diagram or graph in which the vertices denote the elements of S and the directed edges represent ordering relationships. An example is shown in Fig. 4.13. 2 1 10 6 4 8 9 3 5 7 Fig. 4.13. Partially ordered set The problem of topological sorting is to embed the partial order in a linear order. Graphically, this implies the arrangement of the vertices of the graph in a row, such that all arrows point to the right, as shown in Fig. 4.14. Properties (1) and (2) of partial orderings ensure that the graph contains no loops. This is exactly the prerequisite condition under which such an embedding in a linear order is possible.
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