Question 1

Find a spanning tree of minimum cost for this graph.

Accepted Answer

The minimum weight is 24. One example of a tree is

Question 2

refer to this graph.  
-Using alphabetical ordering, ﬁnd a spanning tree for this graph by using a depth-ﬁrst search.

Accepted Answer

The answer of refer to this graph.  
-Using alphabetical...

Question 3

Suppose you have 5 coins, one of which is counterfeit (either heavier or lighter than the other four). You use a pan balance scale to ﬁnd the bad coin and determine whether it is heavier or lighter. 
(a) Prove that 2 weighings are not enough to guarantee that you ﬁnd the bad coin and determine whether it is heavier or lighter. 
(b) Draw a decision tree for weighing the coins to determine the bad coin (and whether it is heavier or lighter) in the minimum number of weighings.

Accepted Answer

(a) Two weighings yield a 3-ary tree of height 2, which has at most 9 leaves, but 5 coins require a tree with 10 leaves. 
(b) Use the weighing 1 and 2 against 3 and 4 as the root. If the four coins have the same weight, weigh 1 against 5 to determine whether 5 is heavy or light. If 1 and 2 are lighter or heavier than 3 and 4, weigh 1 against 2. If 1 and 2 balance, weigh 3 against 4 to ﬁnd out which of these coins is heavier or lighter; if 1 and 2 do not balance, then immediate information is obtained regarding coins 1 or 2. (The "<" symbol on an edge means that the coins in the left pan weigh less than the coins in the right pan.)

Question 4

Find a spanning tree for the graph K3,4 using a breadth-ﬁrst search. (Assume that the vertices are labeled $u_{1}, u_{2}, u_{3} $ in one set and $v_{1}, v_{2}, v_{3}, v_{4} $ in the other set, and that alphabetical ordering is used in the search, with numerical ordering on the subscripts used to break ties.)

Accepted Answer

Question 5

Use the merge sort to sort the list 3, 8, 12, 4, 1, 5, 9, 6 in increasing order.

Accepted Answer

The answer of Use the merge sort to sort the...

Question 6

ﬁll in the blanks.
-Every full binary tree with 45 vertices has ____ internal vertices.

Accepted Answer

The answer of ﬁll in the blanks.
-Every full binary tree...

Question 7

refer to this graph.  
-Using reverse alphabetical ordering, ﬁnd a spanning tree for the graph by using a breadth-ﬁrst search.

Accepted Answer

The answer of refer to this graph.  
-Using reverse...

Question 8

refer to this graph.  
-Using the ordering C, D, E, F, G, H, I, J, A, B, C, ﬁnd a spanning tree for this graph by using a breadth-ﬁrst search.

Accepted Answer

The answer of refer to this graph.  
-Using the...

Question 9

Write the compound proposition $( \neg p ) \rightarrow ( q \vee ( r \wedge  \neg s ) )$ in inﬁx notation.

Accepted Answer

The answer of Write the compound proposition \(( \neg p...

Question 10

refer to this graph.  
-Using alphabetical ordering, ﬁnd a spanning tree for this graph by using a breadth-ﬁrst search.

Accepted Answer

The answer of refer to this graph.  
-Using alphabetical...

Question 11

There is a tree with degrees 3, 3, 2, 2, 1, 1, 1, 1.

Accepted Answer

A) True 
 B)False

Question 12

ﬁll in the blanks.
-If each edge of $Q _ { 4 }$ has weight 1, then the cost of any spanning tree of minimum cost is ____.

Accepted Answer

The answer of ﬁll in the blanks.
-If each edge of...

Question 13

Use backtracking to ﬁnd a sum of integers in the set {18, 19, 23, 25, 31} that equals 44.

Accepted Answer

The following tree i

Question 14

Use the merge sort to sort the list 4, 8, 6, 1, 5, 7, 3, 2 in increasing order.

Accepted Answer

After splitting the list into eight list

Question 15

The string $p\ r\ q \rightarrow \neg q \triangle p \rightarrow \wedge$ is postﬁx notation for a logic expression; however, there is a misprint. The triangle should be one of these three: $r , V ,  Or  \neg $ Determine which of these three it must be and explain your reasoning.

Accepted Answer

The triangle should

Question 16

ﬁll in the blanks.
-The minimum number of weighings with a pan balance scale needed to guarantee that you ﬁnd the single counterfeit coin and determine whether it is heavier or lighter than the other coins in a group of ﬁve coins is ____.

Accepted Answer

The answer of ﬁll in the blanks.
-The minimum number of...

Question 17

ﬁll in the blanks.
-If T is a full binary tree with 50 leaves, its minimum height is ____.

Accepted Answer

The answer of ﬁll in the blanks.
-If T is a...

Question 18

(a) Set up a binary tree for the following list, in the given order, using alphabetical ordering: SHE, SELLS, SEA, SHELLS, BY, THE, SEASHORE. 
(b) How many comparisons with words in the tree are needed to determine if the word SHARK is in the tree? 
(c) How many comparisons with words in the tree are needed to determine if the word SEAWEED is in the tree? 
(d) How many comparisons with words in the tree are needed to determine if the word SHELLS is in the tree?

Accepted Answer

(a)
@#IMG-DLM&

(b)

Question 19

ﬁll in the blanks.
-There are ____ full 3-ary trees with 6 vertices.

Accepted Answer

The answer of ﬁll in the blanks.
-There are ____ full...

Question 20

Describe the diﬀerence between Prim's algorithm and Kruskal's algorithm for ﬁnding a spanning tree of minimum cost.

Accepted Answer

Using Prim's algorithm, at eac

Exam 11: A: Trees

Find a spanning tree of minimum cost for this graph.

refer to this graph. -Using alphabetical ordering, ﬁnd a spanning tree for this graph by using a depth-ﬁrst search.

Use the merge sort to sort the list 3, 8, 12, 4, 1, 5, 9, 6 in increasing order.

ﬁll in the blanks. -Every full binary tree with 45 vertices has ____ internal vertices.

refer to this graph. -Using reverse alphabetical ordering, ﬁnd a spanning tree for the graph by using a breadth-ﬁrst search.

refer to this graph. -Using the ordering C, D, E, F, G, H, I, J, A, B, C, ﬁnd a spanning tree for this graph by using a breadth-ﬁrst search.

Write the compound proposition (¬p)→(q∨(r∧¬s))( \neg p ) \rightarrow ( q \vee ( r \wedge \neg s ) )(¬p)→(q∨(r∧¬s)) in inﬁx notation.

refer to this graph. -Using alphabetical ordering, ﬁnd a spanning tree for this graph by using a breadth-ﬁrst search.

There is a tree with degrees 3, 3, 2, 2, 1, 1, 1, 1.

ﬁll in the blanks. -If each edge of Q4Q _ { 4 }Q4​ has weight 1, then the cost of any spanning tree of minimum cost is ____.

Use backtracking to ﬁnd a sum of integers in the set {18, 19, 23, 25, 31} that equals 44.

Use the merge sort to sort the list 4, 8, 6, 1, 5, 7, 3, 2 in increasing order.

ﬁll in the blanks. -The minimum number of weighings with a pan balance scale needed to guarantee that you ﬁnd the single counterfeit coin and determine whether it is heavier or lighter than the other coins in a group of ﬁve coins is ____.

ﬁll in the blanks. -If T is a full binary tree with 50 leaves, its minimum height is ____.

ﬁll in the blanks. -There are ____ full 3-ary trees with 6 vertices.

Describe the diﬀerence between Prim's algorithm and Kruskal's algorithm for ﬁnding a spanning tree of minimum cost.

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

Write the compound proposition $( \neg p ) \rightarrow ( q \vee ( r \wedge \neg s ) )$ in inﬁx notation.

ﬁll in the blanks. -If each edge of $Q _ { 4 }$ has weight 1, then the cost of any spanning tree of minimum cost is ____.