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mathematics
statistics

Questions and Answers of Statistics

Use mathematical induction to prove that breadth-first search visits vertices in order of their level in the resulting spanning tree.
Use pseudocode to describe a variation of breadth-first search that assigns the integer m to the mth vertex visited in the search.
Show that if G is a directed graph and T is a spanning tree constructed using depth-first search, then every edge not in the spanning tree is a forward edge connecting an ancestor to a descendant, a
Find the distance between each pair of spanning trees shown in Figures 3(c) and 4 of the graph G shown in Figure 2.
Suppose that T1 and T2 are spanning trees of a simple graph G. Moreover, suppose that e1 is an edge in T1 that is not in T2. Show that there is an edge e2 in T2 that is not in T1 such that T1 remains
For each of the directed graphs in Exercises 18-23 of Section 10.5 either find a rooted spanning tree of the graph or determine that no such tree exists.
Give an algorithm to build a rooted spanning tree for connected directed graphs in which each vertex has the same in-degree and out-degree.
Use Exercise 60 to construct an algorithm for determining whether a directed graph contains a circuit. In exercise Show that if G is a directed graph and T is a spanning tree constructed using
Find a spanning tree for each of these graphs. a) K5 b) K4,4 c) K1,6 d) Q3 e) C5 f) W5
Draw all the spanning trees of the given simple graphs.
The roads represented by this graph are all unpaved. The lengths of the roads between pairs of towns are represented by edge weights. Which roads should be paved so that there is a path of paved
Devise an algorithm similar to Prim's algorithm for constructing a maximum spanning tree of a connected weighted graph.
Find a maximum spanning tree for the weighted graph in Exercise 2.
Find a maximum spanning tree for the weighted graph in Exercise 4.
Devise an algorithm for finding the second shortest spanning tree in a connected weighted graph.
Show that there is a unique minimum spanning tree in a connected weighted graph if the weights of the edges are all different.
Find a spanning tree with minimal total weight containing the edges {e, i} and {g, k} in the weighted graph in Figure 3.
Express the algorithm devised in Exercise 22 in pseudocode. In exercise Describe an algorithm for finding a spanning tree with minimal weight containing a specified set of edges in a connected
Use Sollin's algorithm to produce a minimum spanning tree for the weighted graph shown ina) Figure 1.b) Figure 3.
Prove that Sollin's algorithm produces a minimum spanning tree in a connected undirected weighted graph.
Showthat if there are r trees in the forest at some intermediate step of Sollin's algorithm, then at least [r/2] edges are added by the next iteration of the algorithm.
Prim's algorithm to find a minimum spanning tree for the given weighted graph.
Show that Sollin's algorithm requires at most log n iterations to produce a minimum spanning tree from a connected undirected weighted graph with n vertices.
Show that if G is a weighted graph with distinct edge weights, then for every simple circuit of G, the edge of maximum weight in this circuit does not belong to any minimum spanning tree of G.
Prove that the reverse-delete algorithm always produces a minimum spanning tree when given as input a weighted graph with distinct edge weights.
Use Kruskal's algorithm to design the communications network described at the beginning of the section.
Use Kruskal's algorithm to find a minimum spanning tree for the weighted graph in Exercise 3.
Find a connected weighted simple graph with the fewest edges possible that has more than one minimum spanning tree.
a) Explain how to use preorder, inorder, and postorder traversals to find the prefix, infix, and postfix forms of an arithmetic expression. b) Draw the ordered rooted tree that represents ((x − 3)
a) Describe the Huffman coding algorithm for constructing an optimal code for a set of symbols, given the frequency of these symbols. b) Use Huffman coding to find an optimal code for these symbols
Drawthe game tree for nim if the starting position consists of two piles with one and four stones, respectively. Who wins the game if both players follow an optimal strategy?
a) Define a rooted tree and the root of such a tree. b) Define the parent of a vertex and a child of a vertex in a rooted tree. c) What are an internal vertex, a leaf, and a subtree in a rooted
a) What is a binary search tree? b) Describe an algorithm for constructing a binary search tree. c) Form a binary search tree for the words vireo, warbler, egret, grosbeak, nuthatch, and kingfisher.
Show that a simple graph is a tree if and only if it contains no simple circuits and the addition of an edge connecting two nonadjacent vertices produces a new graph that has exactly one simple
Give an upper bound and a lower bound for the number of leaves in a B-tree of degree k with height h.
Draw Bk for k = 0, 1, 2, 3, 4.
Find the height of Bk. Prove that your answer is correct.
What is the degree of the root of Bk? Prove that your answer is correct.
Draw an Sk-tree for k = 0, 1, 2, 3, 4.
Suppose that T is an Sk-tree with handle v. Show that T can be obtained from disjoint trees T0, T1, . . . , Tk−1, with roots r0, r1, . . . , rk−1, respectively, where v is not in any of these
List the vertices of the ordered rooted trees in Figures 3 and 9 of Section 11.3 in level order.
Devise an algorithm for constructing a rooted tree from the universal addresses of its leaves.
Which of these graphs are cacti?a)b) c)
Show that a cactus is formed if we add a circuit containing new edges beginning and ending at a vertex of a tree.
Show that every tree with at least one edge must have at least two pendant vertices.
Find a degree-constrained spanning tree of the given graph where each vertex has degree less than or equal to 3, or show that such a spanning tree does not exist.a.b.
A tree with n vertices is called graceful if its vertices can be labeled with the integers 1, 2, . . . , n such that the absolute values of the difference of the labels of adjacent vertices are all
How many nonisomorphic caterpillars are there with six vertices?
Suppose that in a long bit string the frequency of occurrence of a 0 bit is 0.9 and the frequency of a 1 bit is 0.1 and bits occur independently. a) Construct a Huffman code for the four blocks of
Suppose that e is an edge in a weighted graph that is incident to a vertex v such that the weight of e does not exceed the weight of any other edge incident to v. Show that there exists a minimum
Show that if no two edges in a weighted graph have the same weight, then the edge with least weight incident to a vertex v is included in every minimum spanning tree.
Show that a subgraph T = (V, F) of the graph G = (V, E) is an arborescence of G rooted at r if and only if T contains r, T has no simple circuits, and for every vertex v ∈ V other than r, deg −
In this exercise we will develop an algorithm to find the strong components of a directed graph G = (V, E). Recall that a vertex w ∈ V is reachable from a vertex v ∈ V if there is a directed path
What is the sum of the degrees of the vertices of a tree with n vertices?
Show that every tree is a planar graph.
Show that every forest can be colored using two colors.
Prove the absorption law x + xy = x using the other laws in Table 5.
Show that F(x, y, z) = xy + xz + yz has the value 1 if and only if at least two of the variables x, y, and z have the value 1.
Deal with the Boolean algebra {0, 1} with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example 8. 14. Verify the law of the
Show that these identities hold.
Prove or disprove these equalities. a) x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z b) x + (y ⊕ z) = (x + y) ⊕ (x + z) c) x ⊕ (y + z) = (x ⊕ y) + (x ⊕ z)
Suppose that F is a Boolean function represented by a Boolean expression in the variables x1, . . . , xn. Show that
b) Translate the equation in part (a) into a propositional equivalence by changing each 0 into an F, each 1 into a T, each Boolean sum into a disjunction, each Boolean product into a conjunction,
How many different Boolean functions F(, , ) are there such that F(x, y, z) = F( x, y, z) for all values of the Boolean variables x, y, and z?
Show that you obtain De Morgan's laws for propositions (in Table 6 in Section 1.3) when you transform De Morgan's laws for Boolean algebra in Table 6 into logical equivalences.
Show that in a Boolean algebra, the idempotent laws x ∨ x = x and x ∧ x = x hold for every element x.
Show that in a Boolean algebra, the complement of the element 0 is the element 1 and vice versa.
Show that in a Boolean algebra, if x ∨ y = 0, then x = 0 and y = 0, and that if x ∧ y = 1, then x = 1 and y = 1.
Show that a complemented, distributive lattice is a Boolean algebra.
Use a table to express the values of each of these Boolean functions. a) F(x, y, z) = y b) F(x, y, z) = x + yz c) F(x, y, z) = xy + () d) F(x, y, z) = x(yz + y z)
Use a 3-cube Q3 to represent each of the Boolean functions in Exercise 5 by displaying a black circle at each vertex that corresponds to a 3-tuple where this function has the value 1.
Find a Boolean product of the Boolean variables x, y, and z, or their complements, that has the value 1 if and only ifa) x = y = 0, z = 1.b) x = 0, y = 1, z = 0.c) x = 0, y = z = 1.d) x = y = z = 0.
Express each of the Boolean functions in Exercise 12 using the operators + and .a) x + y + z
Show that a) = x ↓ x. b) xy = (x ↓ x) ↓ (y ↓ y). c) x + y = (x ↓ y) ↓ (x ↓ y).
Express each of the Boolean functions in Exercise 3 using the operator |.
Show that the set of operators {+, ·} is not functionally complete.
Find the sum-of-products expansions of these Boolean functions.a) F(x, y, z) = x + y + zb) F(x, y, z) = (x + z)yc) F(x, y, z) = xd) F(x, y, z) = x y
Find the sum-of-products expansion of the Boolean function F(w, x, y, z) that has the value 1 if and only if an odd number of w, x, y, and z have the value 1.
Show that the Boolean sum y1 + y2 +· · ·+ yn, where yi = xi or yi = xi, has the value 0 for exactly one combination of the values of the variables, namely, when xi = 0 if yi = xi and xi = 1 if yi
In Exercises 1-2 find the output of the given circuit.1.2.
Construct a circuit for a full subtractor using AND gates, OR gates, and inverters. A full subtractor has two bits and a borrowas input, and produces as output a difference bit and a borrow.
Construct a circuit that compares the two-bit integers (x1x0)2 and (y1y0)2, returning an output of 1 when the first of these numbers is larger and an output of 0 otherwise.
Use NAND gates to construct circuits with these outputs. a) b) x + y c) xy d) x ⊕ y
Construct a half adder using NAND gates.
Construct a multiplexer using AND gates, OR gates, and inverters that has as input the four bits x0, x1, x2, and x3 and the two control bits c0 and c1. Set up the circuit so that xi is the output,
Design a circuit that implements majority voting for five individuals.
Show how the sum of two five-bit integers can be found using full and half adders.
a) Draw a K-map for a function in two variables and put a 1 in the cell representing y. b) What are the minterms represented by cells adjacent to this cell?
Draw the 4-cube Q4 and label each vertex with the minterm in the Boolean variables w, x, y, and z associated with the bit string represented by this vertex. For each literal in these variables,
a) Draw a K-map for a function in four variables. Put a 1 in the cell that represents wxyz. b) Which minterms are represented by cells adjacent to this cell?
Find the cells in a K-map for Boolean functions with five variables that correspond to each of these products. a) x1x2x3x4 b) x1x3x5 c) x2x4 d) x3x4 e) x3 f) x5
a) How many cells does a K-map in six variables have? b) How many cells are adjacent to a given cell in a K-map in six variables?
Which rows and which columns of a 4 × 16 map for Boolean functions in six variables using the Gray codes 1111, 1110, 1010, 1011, 1001, 1000, 0000, 0001, 0011, 0010, 0110, 0111, 0101, 0100, 1100,
Suppose that there are five members on a committee, but that Smith and Jones always vote the opposite of Marcus. Design a circuit that implements majority voting of the committee using this
Use the Quine-McCluskey method to simplify the sumof- products expansions in Exercise 12. a) xyz + x yz b) xyz + xyz + xyz + xyz c) xyz + xyz + xy z + xyz + x yz d) xyz + xyz + xy z + xyz + xyz + x y
Use the Quine-McCluskey method to simplify the sumof- products expansions in Exercise 14. a) xyz + x yz b) xyz + xyz + xyz + xyz c) xyz + xyz + xy z + xyz + x yz d) xyz + xyz + xy z + xyz + xyz + x y
Use the method from Exercise 26 to simplify the productof- sums expansion (x + y + z)(x + y + z)(x + y + z) (x + y + z)(x + y + z).
Build a circuit using OR gates, AND gates, and inverters that produces an output of 1 if a decimal digit, encoded using a binary coded decimal expansion, is divisible by 3, and an output of 0
Draw the K-maps of these sum-of-products expansions in two variables. a) x b) xy + c) xy + x + y +
Find a minimal sum-of-products expansion, given the K-map shown with don't care conditions indicated with ds.

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