7. By the tree method, prove that the following four ways of saying that the presi- dent is queasy are logically equivalent. Note: It's enough to test validity of a cycle of four arguments. (4) 3x (PxA Qx) Vx Vy [(Px Py) ⇒x=y] (b) Yx (Px-Qx) a 3x Vy (Py+y=x) (c) 3x [Px ^ Qx a Vy (Py-y=x)] (4) 3x [Qx^ Vy (Py+y=x)]

7. By the tree method, prove that the following four ways of saying that the presi- dent is queasy are logically equivalent. Note: It's enough to test validity of a cycle of four arguments. (4) 3x (PxA Qx) Vx Vy [(Px Py) ⇒x=y] (b) Yx (Px-Qx) a 3x Vy (Py+y=x) (c) 3x [Px ^ Qx a Vy (Py-y=x)] (4) 3x [Qx^ Vy (Py+y=x)]

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Part I For each of the following two recurrences, construct a recursion tree and use it to solve the recurrence. Your solution should include a chart similar to examples in lecture and studio with columns for the depth, number of nodes, work per node, and work per level at each level of the tree; a summation giving the total work of the recurrence; and ● a closed-form asymptotic solution to the recurrence. You may assume for simplicity that the input size n is a power of b, the denominator of the size in the recursive term, and that T(1) = d for some constant d. You may hand-draw or typset the chart, but please typeset the summation and its closed-form solution. 1. T(n) = 3T(n/2) + n² 2. T(n) = 4T(n/2) + n²
(c) Draw the recursion tree for the recurrence T(n) = 3T (|n/V3|) + cn, where c is a constant, and determine a good asymptotic upper bound on its solution. Verify your bound by the substitution method.
Suppose (X[mX]+Y [mY ]) > 0. What can you infer about each of the four subproblems (namely, (X1, Y1), (X1, Y2), (X2, Y1), and (X2, Y2))? Clearly identify the ones that do not have to solved anymore, and those that still have to be solved. Briefly justify.
4.Consider F = {BC -> D, B -> E, CE -> D, E -> CA, BF -> G} and R(A,B,C,D,E,F,G) a)Find all many keys for R
Problem 2. Using the procedure TREE-SUCCESSOR and TREE-MINIMUM, write a function F(x), where x is a node in a binary search tree, to produce the output that INORDER- TREE-WALK function would produce. Determine the upper bound running time complexity of F(x) and prove its correctness.
5. Prove that the following langauges are not regular (a) L (b) L = {a"b* : n > k} (c) L = {a"b™c* :n+m> k > 0} (d) L = {w e {a, b}* : 2n.(w) = 3n,(w)} {ww: w E {a, b}*}
True or False (If your answer to the question is "False", explain why, and provide correction when possible). (a) Let h(n) be the heuristics for the node n, h(m) be the heuristics for the node m, d(m,n) be the actual minimal cost from node m to n in a graph. A* satisfies the monotone restriction iff d(m,n) <= |h(n)-h(m)|. (b) If an A* heuristics is admissible then it satisfies the monotone restriction. (c) Best-first search guarantees optimality in its returned solution. (d) Least-cost-first search guarantees optimality in its returned solution. (e) If all edges are with unit cost, then Breadth-first search guarantees optimality in its returned solution.
It was claimed that:(a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N. Show that this is actually the case.
(c) Consider a sort of items, according to their keys, that inserts all the items one at a time into an initially empty regular binary search tree and then applies an in-order traversal to complete the sort. Assume that all items have distinct keys. Using big-Theta notation.what is the worst-case complexity of the sort? What is the average-case complexity of the sort? Now answer the same two questions if an AVL tree is used instead of a regular binary search tree.
Solving recurrences Solve the following recurrence relations and give a Θbound for eachof them. Do not use the Master Theorem. You must use the recursion tree method. For each recurrence,state the branching factor, the height of the tree, the size of the subproblems at depth k, and the number of subproblems at depth k. You can draw the recursion tree to help find the solution, but do not include the tree in your submission as it will not be graded.(a) T (n) = 4T (n/3)+n(b) T (n) = 3T (n/3)+n(c) T (n) = 4T (n/4)+√n(d) T (n) = 16T (n/4)+n2(e) T (n) = 3T (n/4)+1
(b) Solve the recurrence relation : T (n) =T O(n). 3 using the recursion tree method.
True or False (If your answer to the question is "False", explain why, and provide correction when possible). (a) Let h(n) be the heuristics for the node n, h(m) be the heuristics for the node m, d(m,n) be the actual minimal cost from node m to n in a graph. A* satisfies the monotone restriction iff d(m,n)