Ada Solution Manual

This ? le contains the exercises, hints, and solutions for Chapter 1 of the book Introduction to the Design and Analysis of Algorithms, 2nd edition, by A. Levitin. The problems that might be ambitious for at least some students atomic material body 18 marked by those that might be di? cult for a majority of students are marked by . Exercises 1. 1 1. Do some re inquisition on al-Khorezmi (also al-Khwarizmi), the man from whose name the word algorithmic programic rule is derived. In particular, you should learn what the origins of the words algorithm and algebra have in common. 2. given up that the of? cial purpose of the U.S. patent system is the promotion of the useful arts, do you think algorithms are patentable in this country? Should they be? 3. a. Write down driving directions for deviation from your school to your home with the precision required by an algorithm. b. Write down a recipe for cooking your favorite dish with the precision required by an algorithm. 4. Design a n algorithm for swapping two 3 digit non-zero integers n, m. Besides using arithmetic operations, your algorithm should not use some(prenominal) temporary variable. 5. Design an algorithm for computing gcd(m, n) using Euclids algorithm. 6.Prove the equality gcd(m, n) = gcd(n, m fashionable n) for every reduplicate of positive integers m and n. 7. What does Euclids algorithm do for a pair of numbers in which the ? rst number is sm all in aller than the second star? What is the largest number of times this can happen during the algorithms execution on such an input? 8. What is the smallest and the largest number of divisions possible in the algorithm for determining a prime number? 9. a. Euclids algorithm, as presented in Euclids treatise, uses subtractions kind of than integer divisions. Write a pseudocode for this version of Euclids algorithm. b.Euclids game (see Bog) break ups with two unequal positive numbers on the board. Two players move in turn. On each move, a player ha s to write on the board a positive number equal to the difference of two numbers already on the board this number must be revolutionary, i. e. , different from all the numbers already on the board. The player who cannot move loses the game. Should you choose to move ? rst or second in this game? 10. The extended Euclids algorithm determines not only the greatest common divisor d of two positive integers m and n entirely also integers (not necessarily positive) x and y, such that mx + ny = d. a. Look up a description of the extended Euclids algorithm (see, e. g. , KnuI, p. 13) and fulfil it in the language of your choice. b. Modify your program for ? nding integer solutions to the Diophantine equation ax + by = c with any set of integer coef? cients a, b, and c. 11. Locker doors There are n storage lockers in a hallway, numbered sequentially from 1 to n. Initially all the locker doors are closed. You nark n passes by the lockers, each time first with locker 1. On the ith pass, i = 1, 2, . . . n, you toggle the door of every ith locker if the door is closed, you rough it if it is sensory(a), you close it. For example, after the ? rst pass every door is open on the second pass you only toggle the even-numbered lockers (2, 4, . . . ) so that after the second pass the even doors are closed and the odd whizzs are open the third time through, you close the door of locker 3 (opened from the ? rst pass), open the door of locker 6 (closed from the second pass), and so on. After the last pass, which locker doors are open and which are closed?How many of them are open? 2 Hints to Selected Exercises 1. 1 1. It is probably faster to do this by searching the Web, that your library should be able to help, too. 2. One can ? nd arguments supporting either view. There is a well-established principle pertinent to the matter, though scienti? c facts or mathematical expressions of them are not patentable. (Why do you think this is the case? ) But should this preclude gran ting patents for all algorithms? 3. You may assume that you are writing your algorithms for a benevolent rather than a machine.Still, make sure that your descriptions do not contain obvious ambiguities. Knuth (KnuI, p. 6) provides an interesting comparison between cooking recipes and algorithms. 6. Prove that if d divides some(prenominal) m and n (i. e. , m = sd and n = td for some positive integers s and t), then it also divides both n and r = m mod n and vice versa. Use the formula m = qn + r (0 ? r n) and the fact that if d divides two integers u and v, it also divides u + v and u ? v. (Why? ) 7. do one iteration of the algorithm for two arbitrarily chosen integers m n. 9. a.Use the equality gcd(m, n) = gcd(m ? n, n) for m ? n 0. b. The key is to ? gure out the total number of distinct numbers that can be written on the board, starting with an initial pair m, n where m n ? 1. You should exploit a continuative of this question to the question of part (a). Considering small examples, especially those with n = 1 and n = 2, should help, too. 10. Of course, for some coef? cients, the equation will have no solutions. 11. Tracing the algorithm by hand for, say, n = 10 and studying its outcome should help answering both questions. 3 Solutions to Exercises 1. 1. Al-Khwarizmi (9th century C. E. ) was a great Arabic scholar, around famous for his algebra textbook. In fact, the word algebra is derived from the Arabic title of this book while the word algorithm is derived from a translation of Al-Khwarizmis last name (see, e. g. , KnuI, pp. 1-2, Knu96, pp. 88-92, 114). 2. This level-headed issue has yet to be settled. The current legal state of a? airs distinguishes mathematical algorithms, which are not patentable, from other algorithms, which may be patentable if use as computer programs (e. g. , Cha00). 3. n/a 4.ALGORITHM Exchange valueswithoutT(a,b) //exchange the two values without using temporary variable //Inputtwo numbers a,b. //Outputexchange values of a,b a=a+b b=a-b a=a-b ALGORITHM Euclid (m,n) // Computes gcd(m. n) by Euclids algorithm // Input Two nonnegative, not-both-zero integers m and n // Output Greatest common divisor of m and n while n ? 0 do r 5. 6. Let us ? rst point that if d divides two integers u and v, it also divides both u + v and u ? v. By de? nition of division, there exist integers s and t such that u = sd and v = td. Therefore u v = sd td = (s t)d, i. . , d divides both u + v and u ? v. 4 Also note that if d divides u, it also divides any integer multiple ku of u. Indeed, since d divides u, u = sd. Hence ku = k(sd) = (ks)d, i. e. , d divides ku. Now we can prove the assertion in question. For any pair of positive integers m and n, if d divides both m and n, it also divides both n and r = m mod n = m ? qn. Similarly, if d divides both n and r = m mod n = m ? qn, it also divides both m = r + qn and n. Thus, the two pairs (m, n) and (n, r) have the same ? nite nonempty set of common divisors, including the largest element in the set, i. . , gcd(m, n) = gcd(n, r). 7. For any input pair m, n such that 0 ? m n, Euclids algorithm simply swaps the numbers on the ? rst iteration gcd(m, n) = gcd(n, m) because m mod n = m if m n. Such a swap can happen only once since gcd(m, n) = gcd(n, m mod n) implies that the ? rst number of the new pair (n) will be greater than its second number (m mod n) after every iteration of the algorithm. 8. Algorithm While i 0) && (numbersj-1 index)) numbersj = numbersj-1 j = j 1 numbersj = index 3. Align the pattern with the beginning of the text.Compare the corresponding characters of the pattern and the text left-to right until either all the pattern characters are matched (then stopthe search is successful) or the algorithm runs out of the texts characters (then stopthe search is unsuccessful) or a mismatching pair of characters is encountered. In the latter case, shift the pattern one position to the right and resume the comparisons. 4. a. If we re present each of the rivers banks and each of the two islands by vertices and the bridges by edges, we will get the following chart 0 a a b c b c d d (This is, in fact, a multigraph, not a graph, because it has more than one edge between the same pair of vertices. But this doesnt matter for the issue at hand. ) The question is whether there exists a path (i. e. , a rank of adjacent vertices) in this multigraph that traverses all the edges exactly once and returns to a starting bloom. Such paths are called Eulerian spells if a path traverses all the edges exactly once but does not return to its starting vertex, it is called an Eulerian path. b.Euler proved that an Eulerian circuit exists in a connected (multi)graph if and only if all its vertices have even degrees, where the degree of a vertex is de? ned as the number of edges for which it is an endpoint. Also, an Eulerian path exists in a connected (multi)graph if and only if it has exactly two vertices of odd degrees such a path must start at one of those two vertices and end at the other. Hence, for the multigraph of the puzzle, there exists neither an Eulerian circuit nor an Eulerian path because all its four vertices have odd degrees. If we are to be satis? d with an Eulerian path, two of the multigraphs vertices must be made even. This can be accomplished by adding one new bridge connecting the same places as the existing bridges. For example, a new bridge between the two islands would make possible, among others, the walk a ? b ? c ? a ? b ? d ? c ? b ? d a a b c b c d d If we want a walk that returns to its starting point, all the vertices in the 21 corresponding multigraph must be even. Since a new bridge/edge changes the parity of two vertices, at least two new bridges/edges will be necessitate. For example, here is one such enhancement a a c b c d d This would make possible a ? b ? c ? a ? b ? d ? c ? b ? d ? a, among several other such walks. 5. A Hamiltonian circuit is marked on the graph below 6. a. At least three reasonablecriteria come to mind the fastest trip, a trip with the smallest number of inculcate stops, and a trip that requires the smallest number of train changes. Note that the ? rst criterion requires the information about the expected traveling time between stations and the time needed for train changes whereas the other two criteria do not require such information. . A natural approach is to mimic subway plans by representing stations by vertices of a graph, with two vertices connected by an edge if there is a train line between the corresponding stations. If the time spent on changing a train is to be taken into account (e. g. , because the station in question is on more than one line), the station should be represented by more then one vertex. 22 7. procedure Queens(unused, board, col, N) if col N then print board else col