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By Gilles Brassard

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5. Prove by mathematical induction that if this tactic is used, then after an arbitrary sequence of merge operations starting from the initial situation, a [0 tree containing k nodes will have a height at most [lg kj. Sec. 9 33 Data Structures The height of the trees can be maintained in an additional array height [I .. N] so that height [i] gives the height of node i in its current tree. Whenever a is the label of a set, height [a] therefore gives the height of the corresponding tree. Initially, height [i] is set to zero for each i.

1. The information attached to a node is shown inside the corresponding box and the arrows show transitions from a node to its successor. Such lists are subject to a number of operations: we might want to insert an additional node, to delete a node, to copy a list, to count the number of elements it contains, and so on. The different computer implementations that are commonly used differ in the quantity of memory required, and in the greater or less ease of carrying out certain operations. Here we content ourselves with mentioning the best-known techniques.

8). This is much better than the first algorithm. However, there exists a third algorithm that gives as great an improvement over the second algorithm as the second does over the first. 9). It will be explained in Chapter 4. function fib3(n) i <-1; j -0; kwhile n > 0 do 0; h - I if n is odd then t v-- jh j e- ih + jk + t i <- ik + t t -- 1h2 h <- 2kh + t k <-k2+ t n -- n div 2 return j Once again, we programmed the three algorithms in Pascal on a CDC CYBER 835 in order to compare their execution times empirically.

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