 6.4.1: Is the following code a prefix code? Why or why not? Character m b ...
 6.4.2: Using the code of Exercise 1, decode the string 01101.
 6.4.3: Given the codes Character a e i o u Encoding scheme 00 01 10 110 11...
 6.4.4: Given the codes Character b h q w % Encoding scheme 1000 1001 0 11 ...
 6.4.5: Given the codes Character a p w ( ) Encoding scheme 001 1010 110 11...
 6.4.6: Given the nonprefix codes Character 1 3 5 7 9 Encoding scheme 1 111...
 6.4.7: Write the Huffman codes for a, b, c, and d in the binary tree shown
 6.4.8: Write the Huffman codes for r, s, t, u in the binary tree shown.
 6.4.9: a. Construct the Huffman tree for the following characters and freq...
 6.4.10: a. Construct the Huffman tree for the following characters and freq...
 6.4.11: a. Construct the Huffman tree for the following characters and freq...
 6.4.12: a. Construct the Huffman tree for the following characters and freq...
 6.4.13: Construct the Huffman tree and find the Huffman codes for the follo...
 6.4.14: Construct the Huffman tree and find the Huffman codes for the follo...
 6.4.15: JPEG can achieve various compression levels; the higher the compres...
 6.4.16: Explain why JPEG encoding results in less compression for grayscal...
 6.4.17: Someone does a global substitution on the text file of Exercise 11,...
 6.4.18: Consider the following paragraph. However, in my thoughts I could n...
 6.4.19: Recall the problem posed at the beginning of this chapter. You work...
 6.4.20: In the justification that the Huffman algorithm produces an optimal...
Solutions for Chapter 6.4: Huffman Codes
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 6.4: Huffman Codes
Get Full SolutionsChapter 6.4: Huffman Codes includes 20 full stepbystep solutions. Since 20 problems in chapter 6.4: Huffman Codes have been answered, more than 20066 students have viewed full stepbystep solutions from this chapter. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.