To show that an array is Monge if and only if for all i = 1 , 2 , .... , m − 1 and j = 1 , 2 , ..... , n − 1 it has A [ i , j ] + A [ i + 1 , j + 1 ] ≤ A [ i , j + 1 ] + A [ i + 1 , j 0 ] .

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Answer 1

Question

Chapter 4, Problem 6P

(a)

Program Plan Intro

To show that an array is Monge if and only if for all i=1,2,....,m−1 and j=1,2,.....,n−1 it has A[i,j]+A[i+1,j+1]≤A[i,j+1]+A[i+1,j0].

(b)

Program Plan Intro

To modify one element in order that it becomes aMonge array.

(c)

Program Plan Intro

To prove that array is Monge if f(1)≤f(2)≤........≤f(m) for any m*n .

(d)

Program Plan Intro

To explain the computation of the leftmost minimum in the odd-numbered rows of A in O(m+n) time.

(e)

Program Plan Intro

To give the recurrence relation that computes leftmost minimum in O(m+nlgm) time.

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A unimodal array is an array that has a sequence of monotonically increasing integers followed by a sequence of monotonically decreasing integers, assuming all elements in the array are unique. Example: А- (4, 5, 8, 9, 10, 11, 7, 3, 2, 1}: A is a unimodal array because there is an increasing sequence followed by a decreasing sequence and the maximum element is 11. B= {11, 9, 8, 7, 5, 4, 3, 2, 1}: B is not a unimodal array because there is no increasing sequence It is simply a decreasing sequence and the maximum element is 11. C= {1, 2, 3, 4, 5, 7, 8, 9, 11}: C is not a unimodal array because There is an increasing sequence, but there is no decreasing sequence and the maximum element is 11. Design an efficient algorithm with the lowest possible complexity to state whether a given array is unimodal or not, and explain why your algorithm is efficient. Analyze the complexity of your algorithm.

20. Given a 4-element array with distinct elements, say {x1, x2, x3, x4}, it is known that x1 is not the minimum. How to use 2 comparisons to find an element that is neither the maximum nor the minimum? Solution:

Multiplying Factors For a pair of integers (x,y) and an integer k, the multiplying factor is defined as minimum positive integer m such that LCM(x,y) * m is divisible by k. Here LCM(x,y) represents the smallest positive integer divisible by both x and y, their least common multiple. Given an array arr of n integers and an integer k, find the sum of multiplying factors over all pairs of integers (arr[i], arr[j]) and k0 <= i,j < n. Example Suppose n=3, arr =[4,5,6] and k=12. Pair Multiplying Factor Remarks (4, 3 LCM(4,4)**3=4**3 =12 4 ) LCM(5,5)**12=5** 12=60 (5, 12 LCM(6,6)**2=6**2 5) (6, 2 12 6 )

Answer 2

Question

Chapter 4, Problem 6P

(a)

Program Plan Intro

To show that an array is Monge if and only if for all i=1,2,....,m−1 and j=1,2,.....,n−1 it has A[i,j]+A[i+1,j+1]≤A[i,j+1]+A[i+1,j0].

(b)

Program Plan Intro

To modify one element in order that it becomes aMonge array.

(c)

Program Plan Intro

To prove that array is Monge if f(1)≤f(2)≤........≤f(m) for any m*n .

(d)

Program Plan Intro

To explain the computation of the leftmost minimum in the odd-numbered rows of A in O(m+n) time.

(e)

Program Plan Intro

To give the recurrence relation that computes leftmost minimum in O(m+nlgm) time.

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Answer 3

Question

Chapter 4, Problem 6P

(a)

Program Plan Intro

To show that an array is Monge if and only if for all i=1,2,....,m−1 and j=1,2,.....,n−1 it has A[i,j]+A[i+1,j+1]≤A[i,j+1]+A[i+1,j0].

(b)

Program Plan Intro

To modify one element in order that it becomes aMonge array.

(c)

Program Plan Intro

To prove that array is Monge if f(1)≤f(2)≤........≤f(m) for any m*n .

(d)

Program Plan Intro

To explain the computation of the leftmost minimum in the odd-numbered rows of A in O(m+n) time.

(e)

Program Plan Intro

To give the recurrence relation that computes leftmost minimum in O(m+nlgm) time.

Expert Solution & Answer

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Answer 4

Question

Chapter 4, Problem 6P

(a)

Program Plan Intro

To show that an array is Monge if and only if for all i=1,2,....,m−1 and j=1,2,.....,n−1 it has A[i,j]+A[i+1,j+1]≤A[i,j+1]+A[i+1,j0].

(b)

Program Plan Intro

To modify one element in order that it becomes aMonge array.

(c)

Program Plan Intro

To prove that array is Monge if f(1)≤f(2)≤........≤f(m) for any m*n .

(d)

Program Plan Intro

To explain the computation of the leftmost minimum in the odd-numbered rows of A in O(m+n) time.

(e)

Program Plan Intro

To give the recurrence relation that computes leftmost minimum in O(m+nlgm) time.

Expert Solution & Answer

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Answer 5

Chapter 4, Problem 6P

(a)

Program Plan Intro

To show that an array is Monge if and only if for all i=1,2,....,m−1 and j=1,2,.....,n−1 it has A[i,j]+A[i+1,j+1]≤A[i,j+1]+A[i+1,j0].

(b)

Program Plan Intro

To modify one element in order that it becomes aMonge array.

(c)

Program Plan Intro

To prove that array is Monge if f(1)≤f(2)≤........≤f(m) for any m*n .

(d)

Program Plan Intro

To explain the computation of the leftmost minimum in the odd-numbered rows of A in O(m+n) time.

(e)

Program Plan Intro

To give the recurrence relation that computes leftmost minimum in O(m+nlgm) time.

Answer 6

Chapter 4, Problem 6P

(a)

Program Plan Intro

To show that an array is Monge if and only if for all i=1,2,....,m−1 and j=1,2,.....,n−1 it has A[i,j]+A[i+1,j+1]≤A[i,j+1]+A[i+1,j0].

Answer 7

Chapter 4, Problem 6P

(a)

Program Plan Intro

To show that an array is Monge if and only if for all i=1,2,....,m−1 and j=1,2,.....,n−1 it has A[i,j]+A[i+1,j+1]≤A[i,j+1]+A[i+1,j0].

Answer 8

(b)

Program Plan Intro

To modify one element in order that it becomes aMonge array.

Answer 9

(b)

Program Plan Intro

To modify one element in order that it becomes aMonge array.

Answer 10

(c)

Program Plan Intro

To prove that array is Monge if f(1)≤f(2)≤........≤f(m) for any m*n .

Answer 11

(c)

Program Plan Intro

To prove that array is Monge if f(1)≤f(2)≤........≤f(m) for any m*n .

Answer 12

(d)

Program Plan Intro

To explain the computation of the leftmost minimum in the odd-numbered rows of A in O(m+n) time.

Answer 13

(d)

Program Plan Intro

To explain the computation of the leftmost minimum in the odd-numbered rows of A in O(m+n) time.

Answer 14

(e)

Program Plan Intro

To give the recurrence relation that computes leftmost minimum in O(m+nlgm) time.

Answer 15

(e)

Program Plan Intro

To give the recurrence relation that computes leftmost minimum in O(m+nlgm) time.

Answer 16

Expert Solution & Answer

Answer 17

Expert Solution & Answer

Answer 18

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Answer 19

Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.2 - Prob. 1E Ch. 4.2 - Prob. 2E Ch. 4.2 - Prob. 3E Ch. 4.2 - Prob. 4E Ch. 4.2 - Prob. 5E

To show that an array is Monge if and only if for all i = 1 , 2 , .... , m − 1 and j = 1 , 2 , ..... , n − 1 it has A [ i , j ] + A [ i + 1 , j + 1 ] ≤ A [ i , j + 1 ] + A [ i + 1 , j 0 ] .

Concept explainers

To show that an array is Monge if and only if for all i=1,2,....,m−1 and j=1,2,.....,n−1 it has A[i,j]+A[i+1,j+1]≤A[i,j+1]+A[i+1,j0].

To modify one element in order that it becomes aMonge array.

To prove that array is Monge if f(1)≤f(2)≤........≤f(m) for any m*n .

To explain the computation of the leftmost minimum in the odd-numbered rows of A in O(m+n) time.

To give the recurrence relation that computes leftmost minimum in O(m+nlgm) time.

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Chapter 4 Solutions