Algorithms and Data Structures

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N.Wirth. Algorithms and Data Structures. Oberon version 198 occupies k words and that each pointer occupies one word of storage. What is the gain in storage when using a binary tree versus an n-ary tree? 4.15. Assume that a tree is built upon the following definition of a recursive data structure (see Exercise 4.1). Formulate a procedure to find an element with a given key x and to perform an operation P on this element. RECTYPE Tree = RECORD x: INTEGER; left, right: Tree END 4.16. In a certain file system a directory of all files is organized as an ordered binary tree. Each node denotes a file and specifies the file name and, among other things the date of its last access, encoded as an integer. Write a program that traverses the tree and deletes all files whose last access was before a certain date. 4.17. In a tree structure the frequency of access of each element is measured empirically by attributing to each node an access count. At certain intervals of time, the tree organization is updated by traversing the tree and generating a new tree by using the program of straight search and insertion, and inserting the keys in the order of decreasing frequency count. Write a program that performs this reorganization. Is the average path length of this tree equal to, worse, or much worse than that of an optimal tree? 4.18. The method of analyzing the tree insertion algorithm described in Sect. 4.5 can also be used to compute the expected numbers C n of comparisons and M n of moves (exchanges) which are performed by Quicksort, sorting n elements of an array, assuming that all n! permutations of the n keys 1, 2, ... n are equally likely. Find the analogy and determine C n and M n . 4.19. Draw the balanced tree with 12 nodes which has the maximum height of all 12-node balanced trees. In which sequence do the nodes have to be inserted so that the AVL-insertion procedure generates this tree? 4.20. Find a sequence of n insertion keys so that the procedure for insertion in an AVL-tree performs each of the four rebalancing acts (LL, LR, RR, RL) at least once. What is the minimal length n for such a sequence? 4.21. Find a balanced tree with keys 1 ... n and a permutation of these keys so that, when applied to the deletion procedure for AVL-trees, it performs each of the four rebalancing routines at least once. What is the sequence with minimal length n? 4.22. What is the average path length of the Fibonacci-tree T n ? 4.23. Write a program that generates a nearly optimal tree according to the algorithm based on the selection of a centroid as root. 4.24. Assume that the keys 1, 2, 3, ... are inserted into an empty B-tree of order 2. Which keys cause page splits to occur? Which keys cause the height of the tree to increase? If the keys are deleted in the same order, which keys cause pages to be merged (and disposed) and which keys cause the height to decrease? Answer the question for (a) a deletion scheme using balancing, and (b) a scheme whithout balancing (upon underflow, a single item only is fetched from a neighboring page). 4.25. Write a program for the search, insertion, and deletion of keys in a binary B-tree. Use the node type and the insertion scheme shown above for the binary B-tree. 4.26. Find a sequence of insertion keys which, starting from the empty symmetric binary B-tree, causes
N.Wirth. Algorithms and Data Structures. Oberon version 199 the shown procedure to perform all four rebalancing acts (LL, LR, RR, RL) at least once. What is the shortest such sequence? 4.27. Write a procedure for the deletion of elements in a symmetric binary B-tree. Then find a tree and a short sequence of deletions causing all four rebalancing situations to occur at least once. 4.28. Formulate a data structure and procedures for the insertion and deletion of an element in a priority search tree. The procedures must maintain the specified invariants. Compare their performance with that of the radix priority search tree. 4.29. Design a module with the following procedures operating on radix priority search trees: — insert a point with coordinates x, y. — enumerate all points within a specified rectangle. — find the point with the least x-coordinate in a specified rectangle. — find the point with the largest y-coordinate within a specified rectangle. — enumerate all points lying within two (intersecting) rectangles. References [4.1] G.M. Adelson-Velskii and E.M. Landis. Doklady Akademia Nauk SSSR, 146, (1962), 263-66; English translation in Soviet Math, 3, 1259-63. [4.2] R. Bayer and E.M. McCreight. Organization and Maintenance of Large Ordered Indexes. Acta Informatica, 1, No. 3 (1972), 173-89. [4.3] R. Bayer and E.M. McCreight. Binary B-trees for Virtual memory. Proc. 1971 ACM SIGFIDET Workshop, San Diego, Nov. 1971, pp. 219-35. [4.4] R. Bayer and E.M. McCreight. Symmetric Binary B-trees: Data Structure and Maintenance Algorithms. Acta Informatica, 1, No. 4 (1972), 290-306. [4.5] T.C. Hu and A.C. Tucker. SIAM J. Applied Math, 21, No. 4 (1971) 514-32. [4.6] D. E. Knuth. Optimum Binary Search Trees. Acta Informatica, 1, No. 1 (1971), 14-25. [4.7] W.A. Walker and C.C. Gotlieb. A Top-down Algorithm for Constructing Nearly Optimal Lexicographic Trees, in: Graph Theory and Computing (New York: Academic Press, 1972), pp. 303-23. [4.8] D. Comer. The ubiquitous B-tree. ACM Comp. Surveys, 11, 2 (June 1979), 121-137. [4.9] J. Vuillemin. A unifying look at data structures. Comm. ACM, 23, 4 (April 1980), 229-239. [4.10] E.M. McCreight. Priority search trees. SIAM J. of Comp. (May 1985)
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Algorithms and Data Structures

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