QUESTION 1 Prove by induction that the solution to the following recurrence captures the running time...
QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE r=1(no induction required, just use the definition of the determinants) Theorem 4.3. The determinant of an n × n matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1 Sr S n, we have ar-1 ar-1 ar-1 ar+1 ar+1 ar+1 an an rt whenever k is a scalar and u, v, and each a are row vectors in F". Proof. The proof...
Q) prove correctness the recurrence relation for case n = 2^x using a proof bt induction. T(n) if n <= 1 then ....... 0 if n > . 1 . then ............1+4T(n/2) hint : when n = 2^x each of recursive calls in a given instnace of repetitiveRecursion in on the subproblem of the smae size the equation n = j-i +1 may be helpful in expressiong the problem size in terms of parameters i and j the closed-form expression...
Need answers for 1-5 Consider the following recurrence relation: H(n) = {0 if n lessthanorequalto 0 1 if n = 1 or n = 2 H(n - 1) + H (n - 2)-H(n - 3) if n > 2. (a) Compute H(n) for n = 1, 2, ...., 10. (b) Using the pattern from part (a), guess what H(100) is. 2. Consider the recurrence relation defined in Example 3.3 (FROM TEXT BOOK, also discussed in class and shown in slides)...
The following algorithm (Rosen pg. 363) is a recursive version of linear search, which has access to a global list of distinct integers a_1, a_2,..., a_n. procedure search(i, j, x : i,j, x integers, 1 < i < j < n) if a_i = x then return i else if i = j then 4. return 0 else return search(i + 1, j, x) Prove that this algorithm correctly solves the searching problem when called with parameters i = 1...
Part I: Induction (90 pt.) (90 pt., 15 pt. each) Prove each of the following statements using induction, strong induction, or structural induction. For each statement, answer the following questions. a. (3 pt.) Complete the basis step of the proof. b. (3 pt.) What is the inductive hypothesis? c. (3 pt.) What do you need to show in the inductive step of the proof? d. (6 pt.) Complete the inductive step of the proof. 5. Let bo, bu, b2,... be...
a) Prove that running time T(n)=n3+30n+1 is O(n3) [1 mark] b) Prove that running time T(n)=(n+30)(n+5) is O(n2) [1 mark] c) Count the number of primitive operation of algorithm unique1 on page 174 of textbook, give a big-Oh of this algorithm and prove it. [2 mark] d) Order the following function by asymptotic growth rate [2 mark] a. 4nlogn+2n b. 210 c. 3n+100logn d. n2+10n e. n3 f. nlogn
DISCRETE MATHEMATIC For question 1, Use mathematical induction to prove the statements are correct for n ∈ Z+(set of positive integers). 1. Prove that for n ≥ 1 1 + 8 + 15 + ... + (7n - 6) = [n(7n - 5)]/2 For question 2, Use a direct proof, proof by contraposition or proof by contradiction. 2. Let m, n ≥ 0 be integers. Prove that if m + n ≥ 59 then (m ≥ 30 or n ≥...
6. Consider the recurrence relation T(n) = 2T(n-1) + 5 for integers n 1 and T(O) = 0. Find a closed-form solution Using induction, prove your solution correct for all integers n 20.
4. Suppose T (n) satisfies the recurrence equations T(n) = 2 * T( n/2 ) + 6 * n, n 2 We want to prove that T (n)-o(n * log(n)) T(1) = 3 (log (n) is log2 (n) here and throughout ). a. compute values in this table for T (n) and n*log (n) (see problem #7) T(n) | C | n * log(n) 2 4 6 b. based on the values in (a) find suitable "order constants" C and...
For each of the following problems write a recurrence relation describing the running time of each of the following algorithms and determine the asymptotic complexity of the function defined by the recurrence relation. Justify your solution using substitution and carefully computing lower and upper bounds for the sums. Simplify and express your answer as Θ(n k ) or Θ(n k (log n)) wherever possible. If the algorithm takes exponential time, then just give exponential lower bounds. 5. func5 (A,n) /*...