Ada Solution Manual

This ? le contains the exercises, hints, and solutions for Chapter 1 of the book Introduction to the Design and digest of Algorithms, 2nd edition, by A. Levitin. The problems that might be ch allenging for at to the lowest degree some students be marked by those that might be di? delirium for a majority of students are marked by . Exercises 1. 1 1. Do some research on al-Khorezmi ( also al-Khwarizmi), the man from whose name the word algorithmic programic program is derived. In particular, you should learn what the origins of the words algorithm and algebra keep in common. 2. pr star 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. economise down thrust directions for going from your school to your home with the clearcutness required by an algorithm. b. Write down a recipe for cooking your favorite dish with the precision required by an algorithm. 4. Desig n an algorithm for swapping dickens 3 figure of speech non-zero whole rime n, m. Besides using arithmetic operations, your algorithm should non use both(prenominal) temporary variable. 5. Design an algorithm for computing gcd(m, n) using Euclids algorithm. 6. kick upst aerate the equality gcd(m, n) = gcd(n, m forward-looking n) for every pair of positive integers m and n. 7. What does Euclids algorithm do for a pair of numbers in which the ? rst number is teentsyer than the certify one? What is the largest number of times this rat happen during the algorithms execution on such an remark? 8. What is the smallest and the largest number of divisions possible in the algorithm for determining a height number? 9. a. Euclids algorithm, as presented in Euclids treatise, uses subtractions rather than integer divisions. Write a pseudocode for this version of Euclids algorithm. b.Euclids game (see Bog) starts with devil unequal positive numbers on the dialog box. Two players mo ve in turn. On each move, a player has to write on the board a positive number equal to the difference of two numbers already on the board this number must be new, i. e. , contrastive from all the numbers already on the board. The player who cannot move loses the game. Should you occupy to move ? rst or second in this game? 10. The extended Euclids algorithm determines not only the greatest common cistron d of two positive integers m and n but also integers (not of necessity positive) x and y, such that mx + ny = d. a. Look up a explanation of the extended Euclids algorithm (see, e. g. , KnuI, p. 13) and implement it in the language of your choice. b. turn 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 doorsteps T present are n consoles in a hallway, numbered sequentially from 1 to n. Initially all the locker doors are closed. You suffer n passes by the lockers, each time starti ng 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 clean it if it is open, you close it. For example, later 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 ridiculous ones 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. afterward the last pass, which locker doors are open and which are closed?How much of them are open? 2 Hints to Selected Exercises 1. 1 1. It is probably faster to do this by searching the Web, but your library should be able to help, too. 2. wiz can ? nd arguments supporting each view. There is a well-established precept pertinent to the matter, though scienti? c details or mathematical expressions of them are not patentable. (Why do you think this is t he case? ) But should this preclude granting patents for all algorithms? 3. You may assume that you are writing your algorithms for a human rather than a machine.Still, make sure that your descriptions do not contain obvious ambiguities. Knuth (KnuI, p. 6) provides an evoke comparison between cooking recipes and algorithms. 6. Prove that if d divides both m and n (i. e. , m = sd and n = td for some positive integers s and t), past 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. Perform one looping 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 snub the total number of distinct numbers that can be create verbally on the board, starting with an initial pair m, n where m n ? 1. You should exploit a connection 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 go out have no solutions. 11. Tracing the algorithm by hand for, say, n = 10 and studying its out dress should help answering both questions. 3 Solutions to Exercises 1. 1. Al-Khwarizmi (9th century C. E. ) was a great Arabic scholar, most 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 legal 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 employ as computer programs (e. g. , Cha00). 3. n/a 4.ALGORITHM Exchange valueswithoutT(a,b) // flip-flop the two values without using temporary variable // scuttlebutttwo n umbers 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 install that if d divides two integers u and v, it also divides both u + v and u ? v. By de? nition of division, at that place embody 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 ternary 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 chemical 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 formerly 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 bit i 0) && (numbersj-1 index)) numbersj = numbersj-1 j = j 1 numbersj = index 3. Align the name 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 stai n to the right and resume the comparisons. 4. a. If we represent each of the rivers banks and each of the two islands by vertices and the bridges by edges, we will get the following graphical record 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 room (i. e. , a sequence of adjacent vertices) in this multigraph that traverses all the edges exactly once and returns to a starting vertex. Such routes are called Eulerian circuits if a channel traverses all the edges exactly once but does not return to its starting vertex, it is called an Eulerian path. b.Euler turn up 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 (m ulti)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 passport 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 take. 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 s everal 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 train stops, and a trip that requires the smallest number of train changes. Note that the ? rst criterion requires the information intimately the expected traveling time between move 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 sets 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 flier (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, gap, N) if col N then print board else col