solu7 - This le contains the exercises, hints, and...

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This fle contains the exercises, hints, and solutions For Chapter 7 oF the book ”Introduction to the Design and Analysis oF Algorithms,” 2nd edition, by A. Levitin. The problems that might be challenging For at least some students are marked by ± ; those that might be difficult For a majority oF students are marked by ² . Exercises 7.1 1. Is it possible to exchange numeric values oF two variables, say, u and v , without using any extra storage? 2. Will the comparison counting algorithm work correctly For arrays with equal values? 3. Assuming that the set oF possible list values is { a, b, c, d }, sort the Following list in alphabetical order by the distribution counting algorithm: b, c, d, c, b, a, a, b. 4. Is the distribution counting algorithm stable? 5. Design a one-line algorithm For sorting any array oF size n whose values are n distinct integers From 1 to n. 6. ± The ancestry problem asks to determine whether a vertex u is an ancestor oF vertex v in a given binary (or, more generally, rooted ordered) tree oF n vertices. Design a O ( n ) input enhancement algorithm that provides sufficient inFormation to solve this problem For any pair oF the tree’s vertices in constant time. 7. ² The Following technique, known as virtual initialization , provides a time-efficient way to initialize just some elements oF a given array A [0 ..n 1] so that For each oF its elements, we can say in constant time whether it has been initialized and, iF it has been, with which value. This is done by utilizing a variable counter For the number oF initialized elements in A and two auxiliary arrays oF the same size, say B [0 ..n 1] and C [0 ..n 1] , defned as Follows. B [0] ,...,B [ counter 1] contain the indices oF the el- ements oF A that were initialized: B [0] contains the index oF the element initialized frst, B [1] contains the index oF the element initialized second, and so on. ±urthermore, iF A [ i ] was the k th element (0 k counter 1) to be initialized, C [ i ] contains k. a. Sketch the state oF arrays A [0 .. 7] ,B [0 .. 7] , and C [0 .. 7] aFter the three assignments A [3] x ; A [7] z ; A [1] y. b. In general, how can we check with this scheme whether A [ i ] has been initialized and, iF it has been, with which value? 1
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8. a. Write a program for multiplying two sparse matrices, a p -by- q matrix A and a q -by- r matrix B. b. Write a program for multiplying two sparse polynomials p ( x ) and q ( x ) of degrees m and n , respectively. 9. Write a program that plays the game of tic-tac-toe with the human user by storing all possible positions on the game’s 3-by-3 board along with the best move for each of them. 2
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Hints to Exercises 7.1 1. Yes, it is possible. How? 2. Check the algorithm’s pseudocode to see what it does upon encountering equal values. 3. Trace the algorithm on the input given (see Figure 7.2 for an example).
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solu7 - This le contains the exercises, hints, and...

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