Discrete Mathematics With Applicat...

5th Edition

EPP + 1 other

ISBN: 9781337694193

Textbook Problem

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Show that given an array a [ b o t ] , a [ b o t + 1 ] , … , a [ t o p ] of length k. if mid = ⌊ ( b o t + t o p ) / 2 ⌋ then

a. the subarray a [ m i d + 1 ] , a [ m i d + 2 ] , … , a [ t o p ] has length ⌊ k / 2 ⌋ .
b. the subarray a [ b o t ] , a [ b o t + 1 ] , … , a [ m i d ] has length ⌊ k / 2 ⌋ .

To determine

(a)

Show that given an array a[bot],a[bot+1],...,a[top] of length k, if mid=⌊(bot+top)/2⌋ then the subarray a[mid],a[mid+1],...,a[top] has length ⌊k/2⌋.

Explanation

Given information:

an array a[bot],a[bot+1],...,a[top] of length k.

Proof:

The array a[bot],a[bot+1],...,a[top] has length k and thus a[bot],a[bot+1],...,a[top] contains k elements.

By the number of elements in a list theorem, we also know that the number of integers from bot to top top is top − bot + 1 and thus a[bot],a[bot+1],...,a[top] contains top + bot − 1 elements.

top+bot−1=k

We then assign mid to the lower of the two middle indices of the array.

mid=⌊bot+top2⌋

By the number of the elements in a list theorem, we also know that the number of integers from mid + 1 to top top is top − ( mid + 1) − 1 = top − mid and thus a[mid+1],a[mid+2],...,a[top] contains top − mid elements.

Length array = top − mid

=top−⌊bot+top2⌋

=top−⌊k+12⌋ top+bot−1=k

=k−⌊k+12⌋ top is the kth element in the array

FIRST CASE: k even

If k is even, then k

To determine

(b)

Show that given an array a[bot],a[bot+1],...,a[top] of length k, if mid=⌊(bot+top)/2⌋ then subarray a[bot],a[bot+1],...,a[mid] has length ⌈k/2⌉.

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Chapter 11 Solutions

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Sect-11.1 P-1TY Sect-11.1 P-2TY Sect-11.1 P-3TY Sect-11.1 P-4TY Sect-11.1 P-5TY Sect-11.1 P-6TY Sect-11.1 P-1ES Sect-11.1 P-2ES Sect-11.1 P-3ES Sect-11.1 P-4ES

Sect-11.1 P-5ES Sect-11.1 P-6ES Sect-11.1 P-7ES Sect-11.1 P-8ES Sect-11.1 P-9ES Sect-11.1 P-10ES Sect-11.1 P-11ES Sect-11.1 P-12ES Sect-11.1 P-13ES Sect-11.1 P-14ES Sect-11.1 P-15ES Sect-11.1 P-16ES Sect-11.1 P-17ES Sect-11.1 P-18ES Sect-11.1 P-19ES Sect-11.1 P-20ES Sect-11.1 P-21ES Sect-11.1 P-22ES Sect-11.1 P-23ES Sect-11.1 P-24ES Sect-11.1 P-25ES Sect-11.1 P-26ES Sect-11.1 P-27ES Sect-11.1 P-28ES Sect-11.2 P-1TY Sect-11.2 P-2TY Sect-11.2 P-3TY Sect-11.2 P-4TY Sect-11.2 P-5TY Sect-11.2 P-6TY Sect-11.2 P-1ES Sect-11.2 P-2ES Sect-11.2 P-3ES Sect-11.2 P-4ES Sect-11.2 P-5ES Sect-11.2 P-6ES Sect-11.2 P-7ES Sect-11.2 P-8ES Sect-11.2 P-9ES Sect-11.2 P-10ES Sect-11.2 P-11ES Sect-11.2 P-12ES Sect-11.2 P-13ES Sect-11.2 P-14ES Sect-11.2 P-15ES Sect-11.2 P-16ES Sect-11.2 P-17ES Sect-11.2 P-18ES Sect-11.2 P-19ES Sect-11.2 P-20ES Sect-11.2 P-21ES Sect-11.2 P-22ES Sect-11.2 P-23ES Sect-11.2 P-24ES Sect-11.2 P-25ES Sect-11.2 P-26ES Sect-11.2 P-27ES Sect-11.2 P-28ES Sect-11.2 P-29ES Sect-11.2 P-30ES Sect-11.2 P-31ES Sect-11.2 P-32ES Sect-11.2 P-33ES Sect-11.2 P-34ES Sect-11.2 P-35ES Sect-11.2 P-36ES Sect-11.2 P-37ES Sect-11.2 P-38ES Sect-11.2 P-39ES Sect-11.2 P-40ES Sect-11.2 P-41ES Sect-11.2 P-42ES Sect-11.2 P-43ES Sect-11.2 P-44ES Sect-11.2 P-45ES Sect-11.2 P-46ES Sect-11.2 P-47ES Sect-11.2 P-48ES Sect-11.2 P-49ES Sect-11.2 P-50ES Sect-11.2 P-51ES Sect-11.3 P-1TY Sect-11.3 P-2TY Sect-11.3 P-3TY Sect-11.3 P-1ES Sect-11.3 P-2ES Sect-11.3 P-3ES Sect-11.3 P-4ES Sect-11.3 P-5ES Sect-11.3 P-6ES Sect-11.3 P-7ES Sect-11.3 P-8ES Sect-11.3 P-9ES Sect-11.3 P-10ES Sect-11.3 P-11ES Sect-11.3 P-12ES Sect-11.3 P-13ES Sect-11.3 P-14ES Sect-11.3 P-15ES Sect-11.3 P-16ES Sect-11.3 P-17ES Sect-11.3 P-18ES Sect-11.3 P-19ES Sect-11.3 P-20ES Sect-11.3 P-21ES Sect-11.3 P-22ES Sect-11.3 P-23ES Sect-11.3 P-24ES Sect-11.3 P-25ES Sect-11.3 P-26ES Sect-11.3 P-27ES Sect-11.3 P-28ES Sect-11.3 P-29ES Sect-11.3 P-30ES Sect-11.3 P-31ES Sect-11.3 P-32ES Sect-11.3 P-33ES Sect-11.3 P-34ES Sect-11.3 P-35ES Sect-11.3 P-36ES Sect-11.3 P-37ES Sect-11.3 P-38ES Sect-11.3 P-39ES Sect-11.3 P-40ES Sect-11.3 P-41ES Sect-11.3 P-42ES Sect-11.3 P-43ES Sect-11.4 P-1TY Sect-11.4 P-2TY Sect-11.4 P-3TY Sect-11.4 P-4TY Sect-11.4 P-5TY Sect-11.4 P-1ES Sect-11.4 P-2ES Sect-11.4 P-3ES Sect-11.4 P-4ES Sect-11.4 P-5ES Sect-11.4 P-6ES Sect-11.4 P-7ES Sect-11.4 P-8ES Sect-11.4 P-9ES Sect-11.4 P-10ES Sect-11.4 P-11ES Sect-11.4 P-12ES Sect-11.4 P-13ES Sect-11.4 P-14ES Sect-11.4 P-15ES Sect-11.4 P-16ES Sect-11.4 P-17ES Sect-11.4 P-18ES Sect-11.4 P-19ES Sect-11.4 P-20ES Sect-11.4 P-21ES Sect-11.4 P-22ES Sect-11.4 P-23ES Sect-11.4 P-24ES Sect-11.4 P-25ES Sect-11.4 P-26ES Sect-11.4 P-27ES Sect-11.4 P-28ES Sect-11.4 P-29ES Sect-11.4 P-30ES Sect-11.4 P-31ES Sect-11.4 P-32ES Sect-11.4 P-33ES Sect-11.4 P-34ES Sect-11.4 P-35ES Sect-11.4 P-36ES Sect-11.4 P-37ES Sect-11.4 P-38ES Sect-11.4 P-39ES Sect-11.4 P-40ES Sect-11.4 P-41ES Sect-11.4 P-42ES Sect-11.4 P-43ES Sect-11.4 P-44ES Sect-11.4 P-45ES Sect-11.4 P-46ES Sect-11.4 P-47ES Sect-11.4 P-48ES Sect-11.4 P-49ES Sect-11.4 P-50ES Sect-11.4 P-51ES Sect-11.5 P-1TY Sect-11.5 P-2TY Sect-11.5 P-3TY Sect-11.5 P-4TY Sect-11.5 P-5TY Sect-11.5 P-1ES Sect-11.5 P-2ES Sect-11.5 P-3ES Sect-11.5 P-4ES Sect-11.5 P-5ES Sect-11.5 P-6ES Sect-11.5 P-7ES Sect-11.5 P-8ES Sect-11.5 P-9ES Sect-11.5 P-10ES Sect-11.5 P-11ES Sect-11.5 P-12ES Sect-11.5 P-13ES Sect-11.5 P-14ES Sect-11.5 P-15ES Sect-11.5 P-16ES Sect-11.5 P-17ES Sect-11.5 P-18ES Sect-11.5 P-19ES Sect-11.5 P-20ES Sect-11.5 P-21ES Sect-11.5 P-22ES Sect-11.5 P-23ES Sect-11.5 P-24ES Sect-11.5 P-25ES Sect-11.5 P-26ES

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