 3.3.28E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.1E: Give a bigO estimate for the number of operations (where an operat...
 3.3.2E: Give a bigO estimate for the number additions used in this segment...
 3.3.3E: Give a bigO estimate for the number of operations, where an operat...
 3.3.4E: Give a bigO estimate for the number of operations, where an operat...
 3.3.5E: How many comparisons are used by the algorithm given in Exercise 16...
 3.3.6E: a) Use pseudocode to describe the algorithm that puts the first fou...
 3.3.7E: Suppose that an element is known to be among the first four element...
 3.3.9E: Give a bigO estimate for the number of comparisons used by the alg...
 3.3.10E: a) Show that this algorithm determines the number of 1 bits in the ...
 3.3.11E: a) Suppose we have n subsets S1. S2,…, Sn of the set {1, 2, …, n}. ...
 3.3.12E: Consider the following algorithm, which takes as input a sequence o...
 3.3.13E: The conventional algorithm for evaluating a polynomial anxn + an1x...
 3.3.14E: There is a more efficient algorithm (in terms of the number of mult...
 3.3.15E: What is the largest n for which one can solve within one second u p...
 3.3.16E: What is the largest n for which one can solve within a day using an...
 3.3.17E: What is the largest n for which one can solve within a minute using...
 3.3.18E: How much time does an algorithm take to solve a problem of size n i...
 3.3.19E: How much time does an algorithm using 250 operations need if each o...
 3.3.20E: What is the effect in the time required to solve a problem when you...
 3.3.21E: What is the effect in the time required to solve a problem when you...
 3.3.23E: Analyze the averagecase performance of the linear search algorithm...
 3.3.24E: An algorithm is called optimal for the solution of a problem with r...
 3.3.25E: Describe the worstcase time complexity, measured in terms of compa...
 3.3.26E: Describe the worstcase time complexity, measured in terms of compa...
 3.3.27E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.29E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.30E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.31E: Analyze the worstcase time complexity of the algorithm you devised...
 3.3.32E: Determine the worstcase complexity in terms of comparisons of the ...
 3.3.33E: Determine the worstcase complexity in terms of comparisons of the ...
 3.3.34E: How many comparisons does the selection sort (see preamble to Exerc...
 3.3.35E: Find a bigO estimate for the worstcase complexity in terms of num...
 3.3.36E: Show that the greedy algorithm for making change for n cents using ...
 3.3.37E: Find the complexity of a bruteforce algorithm for scheduling the t...
 3.3.38E: Find the complexity of the greedy algorithm for scheduling the most...
 3.3.39E: Describe how the number of comparisons used in the worst case chang...
 3.3.40E: Describe how the number of comparisons used in the worst case chang...
 3.3.41E: An n x n matrix is called upper triangular if aij = 0 whenever i > ...
 3.3.42E: Give a pseudocode description of the algorithm in Exercise 41 for m...
 3.3.43E: An matrix is called upper triangular if whenever .How many multipli...
 3.3.44E: What is the best order to form the product ABC if A, B, and C are m...
 3.3.45E: What is the best order to form the product ABCD if A, B, C. and D a...
 3.3.46E: In this exercise we deal with the problem of string matching.a) Exp...
Solutions for Chapter 3.3: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 3.3
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7th. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 44 problems in chapter 3.3 have been answered, more than 101518 students have viewed full stepbystep solutions from this chapter. Chapter 3.3 includes 44 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.