30 11110 31 11111 2 1 The "Bit-Flips" column shows how many bits need to be flipped at that step. Note that to go from a count of 0 to 1, we only need to flip the last bit. But when we increase the count from 1 to 2, we need to flip the last two bits. (a) Argue why you need O(n2") bit-flips to count from 0 to 2" - 1. (b). How many times is the ith bit flipped when we count from 0 to 2" - 1? Assume that the leftmost bit is i = 0 and the rightmost bit is i = n- 1. (Hint: If you're not sure, compute it by hand for the example above. Can you deduce a pattern?) Show that, in fact, it takes (2") bit flips to go from 0 to 2" – 1. (Hint: Use the answer from part (c)

30 11110 31 11111 2 1 The "Bit-Flips" column shows how many bits need to be flipped at that step. Note that to go from a count of 0 to 1, we only need to flip the last bit. But when we increase the count from 1 to 2, we need to flip the last two bits. (a) Argue why you need O(n2") bit-flips to count from 0 to 2" - 1. (b). How many times is the ith bit flipped when we count from 0 to 2" - 1? Assume that the leftmost bit is i = 0 and the rightmost bit is i = n- 1. (Hint: If you're not sure, compute it by hand for the example above. Can you deduce a pattern?) Show that, in fact, it takes (2") bit flips to go from 0 to 2" – 1. (Hint: Use the answer from part (c)

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Running times of counting. Consider an n-bit counter that counts from 0 to 2" - 1. At each
step, its value is incremented. For example, when n = 5, the counter has the following values:
Step
0
1
⠀
30
31
2
3
4
(a)
(b)
(c)
Value # Bit-Flips
00000
00001
00010
00011
00100
11110
11111
The "Bit-Flips" column shows how many bits need to be flipped at that step. Note that to go from a count
of 0 to 1, we only need to flip the last bit. But when we increase the count from 1 to 2, we need to flip the
last two bits.
1
2
1
3
2
1
(b))
Argue why you need O(n2") bit-flips to count from 0 to 2" - 1.
How many times is the ith bit flipped when we count from 0 to 2" – 1? Assume that the leftmost
bit is i = 0 and the rightmost bit is i = n − 1.
(Hint: If you're not sure, compute it by hand for the example above. Can you deduce a pattern?)
Show that, in fact, it takes (2") bit flips to go from 0 to 2" – 1. (Hint: Use the answer from part

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