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hw3-soln - CSCI 4041 HW#3 Solution Key(Fall 2009 1[30pt...

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CSCI 4041 HW#3 Solution Key (Fall 2009) 1. [30pt] a. [15pt] Successive calls to Tree-Successor constitute a partial traversal of the tree, starting at node x and continuing for next k nodes. The successive calls will involve O(k+h) nodes, which implies a total of O(k+h) distinct edges will be traversed because it is tree. However, each such edge will be traversed at most twice: once going down and once coming up. The complexity of the algorithm is of the order of the number of edge traversals. The partial traversal which is done by k successive calls to Tree-Successor will visit at least k nodes (the k successors of x). Assuming we start at node x. Let y be the k^th successor of x, i.e., the last node obtained in the k successive calls. The traversal may visit some number of additional nodes with keys <= key(x) and keys >= key(y). In the worst case, node x is a leaf at maximum depth h whose successor corresponds to the root. The number of nodes visited with keys <= key(x) is h-2. So, the number of nodes visited is at most k+h-2, and the running time is O(k+h). Similarly, since y is the
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