Algorithm AnalysisStart with 102 coins on a table, 98 showing heads and 4 showing tails. There are two ways tochange the coins:• flip over any ten coins, or• place n+ 1 additional coins, all showing tails, on the table where n is the number of headscurrently showing on the table.For example, you might begin by flipping nine heads and one tail, yielding 90 heads and 12 tails,then add 91 tails, yielding 90 heads and 103 tails.(a) Model this situation as a state machine, carefully defining the set of states, the start state, andthe possible state transitions.(b) Optionally, explain how to reach a state with exactly one tail showing.(c) Consider the following derived variables:C ::= the number of coins on the table H ::= the number of heads on the tableT::= the number of tails on the table C, ::= parity (C)H2 ::= parity(H)T2 ::= parity(T)Here the parity : Z → {0,1} function is defined as parity(n) = 0 when n is even and 1otherwise.Which of these variables is strictly decreasing. Optionally, prove your conclusion.(d) Someone claims that this algorithm/process should be able to reach a state where there isexactly one head. Prove using the Invariant Principle that they are wrong. Optionally, verifythat the predicate you use is indeed a preserved invariant.(e) Unrelated to the above problem, describe the asymptotic behavior of f(n) ::= n+lnn+(lnn)-by relating it using the Big-Theta relation to one fo the following: n,n' logn,n, 1,2", 2" In".

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Algorithm Analysis Start with 102 coins on a table, 98 showing heads and 4 showing tails. There are two ways to change the coins: • flip over any ten coins, or • place n+ 1 additional coins, all showing tails, on the table where n is the number of heads currently showing on the table. For example, you might begin by flipping nine heads and one tail, yielding 90 heads and 12 tails, then add 91 tails, yielding 90 heads and 103 tails. (a) Model this situation as a state machine, carefully defining the set of states, the start state, and the possible state transitions. (b) Optionally, explain how to reach a state with exactly one tail showing. (c) Consider the following derived variables:

Algorithm Analysis Start with 102 coins on a table, 98 showing heads and 4 showing tails. There are two ways to change the coins: • flip over any ten coins, or • place n+ 1 additional coins, all showing tails, on the table where n is the number of heads currently showing on the table. For example, you might begin by flipping nine heads and one tail, yielding 90 heads and 12 tails, then add 91 tails, yielding 90 heads and 103 tails. (a) Model this situation as a state machine, carefully defining the set of states, the start state, and the possible state transitions. (b) Optionally, explain how to reach a state with exactly one tail showing. (c) Consider the following derived variables:

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

Algorithm Analysis
Start with 102 coins on a table, 98 showing heads and 4 showing tails. There are two ways to
change the coins:
• flip over any ten coins, or
• place n+ 1 additional coins, all showing tails, on the table where n is the number of heads
currently showing on the table.
For example, you might begin by flipping nine heads and one tail, yielding 90 heads and 12 tails,
then add 91 tails, yielding 90 heads and 103 tails.
(a) Model this situation as a state machine, carefully defining the set of states, the start state, and
the possible state transitions.
(b) Optionally, explain how to reach a state with exactly one tail showing.
(c) Consider the following derived variables:
C ::= the number of coins on the table H ::= the number of heads on the table
T::= the number of tails on the table C, ::= parity (C)
H2 ::= parity(H)
T2 ::= parity(T)
Here the parity : Z → {0,1} function is defined as parity(n) = 0 when n is even and 1
otherwise.
Which of these variables is strictly decreasing. Optionally, prove your conclusion.
(d) Someone claims that this algorithm/process should be able to reach a state where there is
exactly one head. Prove using the Invariant Principle that they are wrong. Optionally, verify
that the predicate you use is indeed a preserved invariant.
(e) Unrelated to the above problem, describe the asymptotic behavior of f(n) ::= n+lnn+(lnn)-
by relating it using the Big-Theta relation to one fo the following: n,n' logn,n, 1,2", 2" In".

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