To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

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Answer 1

Question

Chapter 14, Problem 1P

(a)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(a)

Expert Solution

Explanation of Solution

Given Information: A point of maximum overlap in a set of intervals is a point with the largest number of intervals in the set that overlap it.

Explanation:

Consider that there is no point of maximum overlap in an endpoint of a segment. The maximum overlap occurs in the interior of m segments. Here, the point P is the intersection of those m points.

There must be another point P′ that has the same overlap as P . Therefore, the point P′ is also a point of maximum overlap. Hence, the assumption is not true since there is a point of maximum overlap in an endpoint of maximum overlap.

Hence, it is proved that the there is always a point where maximum overlap has an endpoint of the segment.

(b)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(b)

Expert Solution

Explanation of Solution

Explanation:

Consider a balanced binary tree of endpoints. For inserting the interval, it is necessary to insert the endpoints separately. Consider the endpoints as e . For left endpoint e , increase the value of e by 1 and for right endpoint e , decrease the overlap by 1.

For multiple endpoints with same value, insert the left endpoints with the value before the right endpoints with the value.

Consider that e1,e2,…,en be the sequence of the endpoints and s(i,j) represents the sum. Therefore,

p[e1]+p[ei+1]+…+p[ej]

Where 1≤i≤j≤n .

Here, each node x store the new node that includes the endpoints e1[x],...,er[x] . The sum of the values of all nodes is stored as v[x]=s(1[n],r[x]) and the maximum value id obtained by s(1[n],r[x]) .

For bottom up approach to satisfy the conditions of red black tree following conditions must be hold:

m[x]=max{m[left[x]]v[left[x]]+p[x]v[left[x]]+p[x]+m[right[x]]

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Answer 2

Question

Chapter 14, Problem 1P

(a)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(a)

Expert Solution

Explanation of Solution

Given Information: A point of maximum overlap in a set of intervals is a point with the largest number of intervals in the set that overlap it.

Explanation:

Consider that there is no point of maximum overlap in an endpoint of a segment. The maximum overlap occurs in the interior of m segments. Here, the point P is the intersection of those m points.

There must be another point P′ that has the same overlap as P . Therefore, the point P′ is also a point of maximum overlap. Hence, the assumption is not true since there is a point of maximum overlap in an endpoint of maximum overlap.

Hence, it is proved that the there is always a point where maximum overlap has an endpoint of the segment.

(b)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(b)

Expert Solution

Explanation of Solution

Explanation:

Consider a balanced binary tree of endpoints. For inserting the interval, it is necessary to insert the endpoints separately. Consider the endpoints as e . For left endpoint e , increase the value of e by 1 and for right endpoint e , decrease the overlap by 1.

For multiple endpoints with same value, insert the left endpoints with the value before the right endpoints with the value.

Consider that e1,e2,…,en be the sequence of the endpoints and s(i,j) represents the sum. Therefore,

p[e1]+p[ei+1]+…+p[ej]

Where 1≤i≤j≤n .

Here, each node x store the new node that includes the endpoints e1[x],...,er[x] . The sum of the values of all nodes is stored as v[x]=s(1[n],r[x]) and the maximum value id obtained by s(1[n],r[x]) .

For bottom up approach to satisfy the conditions of red black tree following conditions must be hold:

m[x]=max{m[left[x]]v[left[x]]+p[x]v[left[x]]+p[x]+m[right[x]]

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Answer 3

Question

Chapter 14, Problem 1P

(a)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(a)

Expert Solution

Explanation of Solution

Given Information: A point of maximum overlap in a set of intervals is a point with the largest number of intervals in the set that overlap it.

Explanation:

Consider that there is no point of maximum overlap in an endpoint of a segment. The maximum overlap occurs in the interior of m segments. Here, the point P is the intersection of those m points.

There must be another point P′ that has the same overlap as P . Therefore, the point P′ is also a point of maximum overlap. Hence, the assumption is not true since there is a point of maximum overlap in an endpoint of maximum overlap.

Hence, it is proved that the there is always a point where maximum overlap has an endpoint of the segment.

(b)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(b)

Expert Solution

Explanation of Solution

Explanation:

Consider a balanced binary tree of endpoints. For inserting the interval, it is necessary to insert the endpoints separately. Consider the endpoints as e . For left endpoint e , increase the value of e by 1 and for right endpoint e , decrease the overlap by 1.

For multiple endpoints with same value, insert the left endpoints with the value before the right endpoints with the value.

Consider that e1,e2,…,en be the sequence of the endpoints and s(i,j) represents the sum. Therefore,

p[e1]+p[ei+1]+…+p[ej]

Where 1≤i≤j≤n .

Here, each node x store the new node that includes the endpoints e1[x],...,er[x] . The sum of the values of all nodes is stored as v[x]=s(1[n],r[x]) and the maximum value id obtained by s(1[n],r[x]) .

For bottom up approach to satisfy the conditions of red black tree following conditions must be hold:

m[x]=max{m[left[x]]v[left[x]]+p[x]v[left[x]]+p[x]+m[right[x]]

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Answer 4

Question

Chapter 14, Problem 1P

(a)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(a)

Expert Solution

Explanation of Solution

Given Information: A point of maximum overlap in a set of intervals is a point with the largest number of intervals in the set that overlap it.

Explanation:

Consider that there is no point of maximum overlap in an endpoint of a segment. The maximum overlap occurs in the interior of m segments. Here, the point P is the intersection of those m points.

There must be another point P′ that has the same overlap as P . Therefore, the point P′ is also a point of maximum overlap. Hence, the assumption is not true since there is a point of maximum overlap in an endpoint of maximum overlap.

Hence, it is proved that the there is always a point where maximum overlap has an endpoint of the segment.

(b)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(b)

Expert Solution

Explanation of Solution

Explanation:

Consider a balanced binary tree of endpoints. For inserting the interval, it is necessary to insert the endpoints separately. Consider the endpoints as e . For left endpoint e , increase the value of e by 1 and for right endpoint e , decrease the overlap by 1.

For multiple endpoints with same value, insert the left endpoints with the value before the right endpoints with the value.

Consider that e1,e2,…,en be the sequence of the endpoints and s(i,j) represents the sum. Therefore,

p[e1]+p[ei+1]+…+p[ej]

Where 1≤i≤j≤n .

Here, each node x store the new node that includes the endpoints e1[x],...,er[x] . The sum of the values of all nodes is stored as v[x]=s(1[n],r[x]) and the maximum value id obtained by s(1[n],r[x]) .

For bottom up approach to satisfy the conditions of red black tree following conditions must be hold:

m[x]=max{m[left[x]]v[left[x]]+p[x]v[left[x]]+p[x]+m[right[x]]

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Answer 5

Chapter 14, Problem 1P

(a)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(a)

Expert Solution

Explanation of Solution

Given Information: A point of maximum overlap in a set of intervals is a point with the largest number of intervals in the set that overlap it.

Explanation:

Consider that there is no point of maximum overlap in an endpoint of a segment. The maximum overlap occurs in the interior of m segments. Here, the point P is the intersection of those m points.

There must be another point P′ that has the same overlap as P . Therefore, the point P′ is also a point of maximum overlap. Hence, the assumption is not true since there is a point of maximum overlap in an endpoint of maximum overlap.

Hence, it is proved that the there is always a point where maximum overlap has an endpoint of the segment.

(b)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(b)

Expert Solution

Explanation of Solution

Explanation:

Consider a balanced binary tree of endpoints. For inserting the interval, it is necessary to insert the endpoints separately. Consider the endpoints as e . For left endpoint e , increase the value of e by 1 and for right endpoint e , decrease the overlap by 1.

For multiple endpoints with same value, insert the left endpoints with the value before the right endpoints with the value.

Consider that e1,e2,…,en be the sequence of the endpoints and s(i,j) represents the sum. Therefore,

p[e1]+p[ei+1]+…+p[ej]

Where 1≤i≤j≤n .

Here, each node x store the new node that includes the endpoints e1[x],...,er[x] . The sum of the values of all nodes is stored as v[x]=s(1[n],r[x]) and the maximum value id obtained by s(1[n],r[x]) .

For bottom up approach to satisfy the conditions of red black tree following conditions must be hold:

m[x]=max{m[left[x]]v[left[x]]+p[x]v[left[x]]+p[x]+m[right[x]]

Answer 6

Chapter 14, Problem 1P

(a)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(a)

Expert Solution

Explanation of Solution

Given Information: A point of maximum overlap in a set of intervals is a point with the largest number of intervals in the set that overlap it.

Explanation:

Consider that there is no point of maximum overlap in an endpoint of a segment. The maximum overlap occurs in the interior of m segments. Here, the point P is the intersection of those m points.

There must be another point P′ that has the same overlap as P . Therefore, the point P′ is also a point of maximum overlap. Hence, the assumption is not true since there is a point of maximum overlap in an endpoint of maximum overlap.

Hence, it is proved that the there is always a point where maximum overlap has an endpoint of the segment.

Answer 7

Chapter 14, Problem 1P

(a)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

Answer 8

(a)

Expert Solution

Explanation of Solution

Given Information: A point of maximum overlap in a set of intervals is a point with the largest number of intervals in the set that overlap it.

Explanation:

Consider that there is no point of maximum overlap in an endpoint of a segment. The maximum overlap occurs in the interior of m segments. Here, the point P is the intersection of those m points.

There must be another point P′ that has the same overlap as P . Therefore, the point P′ is also a point of maximum overlap. Hence, the assumption is not true since there is a point of maximum overlap in an endpoint of maximum overlap.

Hence, it is proved that the there is always a point where maximum overlap has an endpoint of the segment.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

(b)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

(b)

Expert Solution

Explanation of Solution

Explanation:

Consider a balanced binary tree of endpoints. For inserting the interval, it is necessary to insert the endpoints separately. Consider the endpoints as e . For left endpoint e , increase the value of e by 1 and for right endpoint e , decrease the overlap by 1.

For multiple endpoints with same value, insert the left endpoints with the value before the right endpoints with the value.

Consider that e1,e2,…,en be the sequence of the endpoints and s(i,j) represents the sum. Therefore,

p[e1]+p[ei+1]+…+p[ej]

Where 1≤i≤j≤n .

Here, each node x store the new node that includes the endpoints e1[x],...,er[x] . The sum of the values of all nodes is stored as v[x]=s(1[n],r[x]) and the maximum value id obtained by s(1[n],r[x]) .

For bottom up approach to satisfy the conditions of red black tree following conditions must be hold:

m[x]=max{m[left[x]]v[left[x]]+p[x]v[left[x]]+p[x]+m[right[x]]

Answer 13

(b)

Program Plan Intro

To show that there will always be a point where maximum overlap is an endpoint of one of the segments.

Answer 14

(b)

Expert Solution

Explanation of Solution

Explanation:

Consider a balanced binary tree of endpoints. For inserting the interval, it is necessary to insert the endpoints separately. Consider the endpoints as e . For left endpoint e , increase the value of e by 1 and for right endpoint e , decrease the overlap by 1.

For multiple endpoints with same value, insert the left endpoints with the value before the right endpoints with the value.

Consider that e1,e2,…,en be the sequence of the endpoints and s(i,j) represents the sum. Therefore,

p[e1]+p[ei+1]+…+p[ej]

Where 1≤i≤j≤n .

Here, each node x store the new node that includes the endpoints e1[x],...,er[x] . The sum of the values of all nodes is stored as v[x]=s(1[n],r[x]) and the maximum value id obtained by s(1[n],r[x]) .