Hello! I need your help with the question attached (3rd Question). Please show the work done. Thank you! Consider the following algorithmic procedure:Input :Non-negative integers x, yOutput :Non-negative integer ar:= x;s:= y;a := 0;while s + 0 doif 3 | s thenr:=r+r+r;s:= s/3;endelse if 3| (s - 1) thena := a+r;r:=r+r+r;s:= (s – 1)/3;endelsea := a+r+r;r:=r+r+r;s:= (s – 2)/3;endendreturn aAlgorithm 1: MultiplyWe can model Algorithm 1 as a state machine whose states are triples of non-negative integers(r, s,a). The initial state is (x, y, 0). The transitions are given by the rule & that for s > 0:(3r, s/3,a)8(r, s, a) = {(3r, (s – 1)/3,a+r)if 3 | sif 3 | (s – 1)(3r, (s – 2)/3,a+ 2r) otherwise.1. List the sequence of steps that appears in the execution of the algorithm for inputs x = 5and y = 10. In other words, walk us through the algorithm starting from the initial state andending with the final state.2. Verify the partial correctness of Algorithm 1.(a) Define a predicate P on the states that you believe is a preserved invariant. (Suggestion:Think about why the loop condition is s # 0 and look at your sequence of steps fromPart 1.)(b) Prove that P is a preserved invariant.(c) Apply the Invariant Principle to prove that if s = 0, then a = xy.3. Verify that Algorithm 1 terminates.(a) Define a strictly decreasing derived variable on the states.(b) Prove that the algorithm terminates after at most 1+ log, y executions of the body of thedo statement.

3. (a) There are three derived variables r, s, and a. Out of all the three variables r and a are…

Answered: Consider the following algorithmic…

Consider the following algorithmic procedure: Input :Non-negative integers x, y Output :Non-negative integer r:= x; s:= y; a := 0; while s #0 do if 3 |s then

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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