miércoles, 19 de febrero de 2014

Chapter #10 Solutions- Allan Weiss - Data Structures and Algorithm Analysis in Java - 3rd Edition

Available Solutions for the following Chapter 10 exercises:

10.1 Show that the greedy algorithm to minimize the mean completion time for
multiprocessor job scheduling works. - Get solution

10.3 A file contains only colons, spaces, newlines, commas, and digits in the following
frequency: colon (100), space (605), newline (100), comma (705), 0 (431),
1 (242), 2 (176), 3 (59), 4 (185), 5 (250), 6 (174), 7 (199), 8 (205), 9 (217).
Construct the Huffman code. . - Get solution

10.4 Part of the encoded file must be a header indicating the Huffman code. Give
a method for constructing the header of size at most O(N) (in addition to the
symbols), where N is the number of symbols.. - Get solution

10.5 Complete the proof that Huffman’s algorithm generates an optimal prefix code.. - Get solution

10.6 Show that if the symbols are sorted by frequency, Huffman’s algorithm can be implemented in linear time. . - Get solution

10.10 Explain how to implement first fit and best fit in O(N logN) time.. - Get solution

10.11 Show the operation of all the bin-packing strategies discussed in Section 10.1.3 on the input 0.42, 0.25, 0.27, 0.07, 0.72, 0.86, 0.09, 0.44, 0.50, 0.68, 0.73, 0.31,0.78, 0.17, 0.79, 0.37, 0.73, 0.23, 0.30. . - Get solution

10.13 Prove Theorem 10.7. . - Get solution

10.14 Prove Theorem 10.8.. - Get solution

10.15 N points are placed in a unit square. Show that the distance between the closest pair is O(N−1/2). . - Get solution

10.16 Argue that for the closest-points algorithm, the average number of points in the strip is O(√ N). (Hint: Use the result of the previous exercise.). - Get solution

10.18 What is the asymptotic running time of quickselect, using a median-of-medianof-three partitioning strategy? . - Get solution

10.19 Show that quickselect with median-of-median-of-seven partitioning is linear.Why is median-of-median-of-seven partitioning not used in the proof? . - Get solution

10.22 Complete the analysis of the sampling algorithm described at the end of Section 10.2.3, and explain how the values of δ and s are chosen. . - Get solution


10.23 Show how the recursive multiplication algorithm computes XY, where X = 1234 and Y = 4321. Include all recursive computations.. - Get solution

10.24 Show how to multiply two complex numbers X = a + bi and Y = c + di using
only three multiplications. . - Get solution

10.25 a. Show that
XLYR + XRYL = (XL + XR)(YL + YR) − XLYL − XRYR
b. This gives an O(N1.59) algorithm to multiply N-bit numbers. Compare this
method to the solution in the text. . - Get solution


10.27 Why is it important that Strassen’s algorithm does not use commutativity in the
multiplication of 2 × 2 matrices? . - Get solution


10.28 Two 70×70 matrices can be multiplied using 143,640 multiplications. Show how
this can be used to improve the bound given by Strassen’s algorithm. . - Get solution

10.29 What is the optimal way to compute A1A2A3A4A5A6, where the dimensions of
the matrices are A1 : 10 × 20, A2 : 20 × 1, A3 : 1 × 40, A4 : 40 × 5, A5 : 5 × 30,
A6: 30 × 15? . - Get solution

10.30 Show that none of the following greedy algorithms for chained matrix multiplication
work. At each step
a. Compute the cheapest multiplication.
b. Compute the most expensive multiplication.
c. Compute the multiplication between the two matrices Mi and Mi+1, such that
the number of columns in Mi is minimized (breaking ties by one of the rules
above). . - Get solution

10.32 Show the optimal binary search tree for the following words, where the frequency
of occurrence is in parentheses: a (0.18), and (0.19), I (0.23), it (0.21), or (0.19). - Get solution


10.35 Write a routine to reconstruct the shortest paths from the algorithm in Section 10.3.4. . - Get solution

10.39 Figure 10.75 shows a routine to flip a coin, assuming that random returns an integer (which is prevalent in many systems). What is the expected performance of the skip list algorithms if the random number generator uses a modulus of the form M = 2B (which is unfortunately prevalent on many systems)? . - Get solution

10.40 a. Use the exponentiation algorithm to prove that 2340 ≡ 1 (mod 341).
b. Show how the randomized primality test works for N = 561 with several choices of A. . - Get solution

10.42 Two point sets are homometric if they yield the same distance set and are not rotations of each other. The following distance set gives two distinct point sets:
{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17 }. Find the two point sets.. - Get solution

10.43 Extend the reconstruction algorithm to find all homometric point sets given a distance set. . - Get solution

10.44 Show the result of α–β pruning of the tree in Figure 10.76. . - Get solution

10.47 The one-dimensional circle packing problem is as follows: You have N circles of radii r1, r2, . . . , rN. These circles are packed in a box such that each circle is tangent to the bottom of the box and are arranged in the original order. The problem is to find the width of the minimum-sized box. Figure 10.77 shows an example with circles of radii 2, 1, 2 respectively. The minimum-sized box has width
4 + 4√ 2. . - Get solution

10.48 Suppose that the edges in an undirected graph G satisfy the triangle inequality:
cu,v + cv,w ≥ cu,w. Show how to compute a traveling salesman tour of cost at most twice optimal. (Hint: Construct a minimum spanning tree.). - Get solution

10.49 You are a tournament director and need to arrange a round robin tournament among N = 2k players. In this tournament, everyone plays exactly one game each day; after N − 1 days, a match has occurred between every pair of players. Give a recursive algorithm to do this. . - Get solution

10.50 a. Prove that in a round robin tournament it is always possible to arrange the players in an order pi1 , pi2 , . . . , piN such that for all 1 ≤ j < N, pij has won the match against pij+1 .
b. Give an O(N logN) algorithm to find one such arrangement. Your algorithm
may serve as a proof for part (a). . - Get solution


10.52 A convex polygon is a polygon with the property that any line segment whose endpoints are on the polygon lies entirely within the polygon. The convex hull




 problem consists of finding the smallest (area) convex polygon that encloses a set of points in the plane. Figure 10.79 shows the convex hull for a set of 40 points. Give an O(N logN) algorithm to find the convex hull. . - Get solution

10.53 Consider the problem of right-justifying a paragraph. The paragraph contains a sequence of words w1, w2, . . . , wN of length a1, a2, . . . , aN, which we wish to break into lines of length L. Words are separated by blanks whose ideal length is b (millimeters), but blanks can stretch or shrink as necessary (but must be >0), so that a line wiwi+1 . . . wj has length exactly L. However, for each blank b we charge |b − b| ugliness points. The exception to this is the last line, for which we charge only if b < b (in other words, we charge only for shrinking), since the last line does not need to be justified. Thus, if bi is the length of the blank between ai and ai+1, then the ugliness of setting any line (but the last) wiwi+1 . . . wj for j > i is
j−1
k=i
|bk − b| = (j − i)|b − b|, where b is the average size of a blank on this line.
This is true of the last line only if b < b, otherwise the last line is not ugly at all.
 
a. Give a dynamic programming algorithm to find the least ugly setting of w1, w2, . . . , wN into lines of length L. (Hint: For i = N, N − 1, . . . , 1, compute the best way to set wi, wi+1, . . . , wN.)
b. Give the time and space complexities for your algorithm (as a function of the number of words, N).
c. Consider the special case where we are using a fixed-width font, and assume the optimal value of b is 1 (space). In this case, no shrinking of blanks is allowed, since the next smallest blank space would be 0. Give a linear-time algorithm to generate the least ugly setting for this case. . - Get solution

10.54 The longest increasing subsequence problem is as follows: Given numbers a1, a2, . . . , aN, find the maximum value of k such that ai1 < ai2 < · · · < aik, and i1 < i2 < · · · < ik. As an example, if the input is 3, 1, 4, 1, 5, 9, 2, 6, 5, the maximum increasing subsequence has length four (1, 4, 5, 9 among others). Give an O(N2) algorithm to solve the longest increasing subsequence problem. . - Get solution

10.55 The longest common subsequence problem is as follows: Given two sequences A = a1, a2, . . . , aM, and B = b1, b2, . . . , bN, find the length, k, of the longest sequence C = c1, c2, . . . , ck such that C is a subsequence (not necessarily continguous) of both A and B. As an example, if A = d,y,n,a,m,i,c
and B = p,r,o,g,r,a,m,m,i,n,g, then the longest common subsequence is a,m,i and has length 3. Give an algorithm to solve the longest common subsequence problem. Your algorithm should run in O(MN) time. . - Get solution


10.57 One form of the knapsack problem is as follows: We are given a set of integers A = a1, a2, . . . , aN and an integer K. Is there a subset of A whose sum is exactly K?
a. Give an algorithm that solves the knapsack problem in O(NK) time.
b. Why does this not show that P = NP? . - Get solution


10.58 You are given a currency system with coins of (decreasing) value c1, c2, . . . , cN cents.
a. Give an algorithm that computes the minimum number of coins required to give K cents in change.
b. Give an algorithm that computes the number of different ways to give K cents in change. . - Get solution

10.59 Consider the problem of placing eight queens on an (eight-by-eight) chess board.
Two queens are said to attack each other if they are on the same row, column, or
(not necessarily main) diagonal.
a. Give a randomized algorithm to place eight nonattacking queens on the board.
b. Give a backtracking algorithm to solve the same problem.
c. Implement both algorithms and compare the running time. . - Get solution





a. Why does this algorithm not work for general graphs?
b. Prove that this algorithm terminates for acyclic graphs.
c. What is the worst-case running time of the algorithm? . - Get solution

10.62 Let A be an N-by-N matrix of zeros and ones. A submatrix S of A is any group of contiguous entries that forms a square.
a. Design an O(N2) algorithm that determines the size of the largest submatrix of ones in A. For instance, in the matrix that follows, the largest submatrix is a
4-by-4 square.
10111000
00010100
00111000
00111010
00111111
01011110
01011110
00011110
b. Repeat part (a) if S is allowed to be a rectangle instead of a square. Largest is measured by area.

10.67 Othello played on a 6-by-6 board is a forced win for black. Prove this by writing
a program. What is the final score if play on both sides is optimal? . - Get solution

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