What is the tightest upper bound on the worst case runtime of the following algorithm from the choices below? a=n WHILE (a & (а - 1)) a = a-1 END-WHILE O 0(1) O O( log(n)) O O(n) O O(n²)
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- The algorithm of Euclid computes the greatest common divisor (GCD) of two integer numbers a and b. The following pseudo-code is the original version of this algorithm. Algorithm Euclid(a,b)Require: a, b ≥ 0Ensure: a = GCD(a, b) while b ̸= 0 do t ← b b ← a mod b a ← tend whilereturn a We want to estimate its worst case running time using the big-Oh notation. Assuming the Euclidian division of two integers of n bits is executed in O(n^2), give a big-Oh notation of the worst-case running time of the algorithm Euclid.The algorithm of Euclid computes the greatest common divisor (GCD) of two integer numbers a and b. The following pseudo-code is the original version of this algorithm. Algorithm Euclid(a,b)Require: a, b ≥ 0Ensure: a = GCD(a, b) while b ̸= 0 do t ← b b ← a mod b a ← tend whilereturn a We want to estimate its worst case running time using the big-Oh notation. Answer the following questions: a. Let x be a integer stored on n bits. How many bits will you need to store x/2? b. We note that if a ≥ b, then a mod b < a/2. Assume the values of the input integers a and b are encoded on n bits. How many bits will be used to store the values of a and b at the next iteration of the While loop? c. Deduce from this observation, the maximal number iterations of the While loop the algorithm will do.Determine the big-O worst-case runtime for each algorithm. for (k = 0; k < N; k++) for (i = 0; i < N; i++) { for (j = 0; j < N; j++) { if (Arr[i] < Arr[j] && Arr[i]<Arr[k]) { ++count } else { ++count } } }
- When the order of growth of the running time of an algorithm is N log N, the doubling test will lead to the hypothesis that the running time is ~ a N for a constant a. Isn’tthat a problem?The algorithm of Euclid computes the greatest common divisor (GCD) of two integer numbers a and b. The following pseudo-code is the original version of this algorithm. Algorithm 1 Euclid1(a,b)Require: a, b > 0Ensure: a = GCD(a, b) while a ̸= b do if a > b then a ← a − b else b ← b − a end if end whilereturn a We want to prove the correctness of this algorithm using the loop invariant technique. a. In a first attempt to prove the correctness of this algorithm, we propose the following loop invariant property: “GCD(a, b) is a factor of a and b”. Explain why this property willnot help you as is. b. Use your previous observation to propose another loop invariant property that willhelp you to prove the correctness of the version of Euclid’s algorithm presented above. c. Show that your loop invariant property is true before executing the While loop forthe first time (i.e., initialization property)Conceptual Question: How do you find the Big Oh, Big Omega, and Big Theta of an Algorithm? What do each of them mean? Can you provide an example for solving for each with some algorithm? Like... (Java) for (i = 1; i < n; i++) { j = i; while ((j > 0) && (s[j] < s[j-1])) { temp = s[j]; s[j] = s[j-1]; s[j-1] = temp; } j--; } And what about if Algorithm is recurssive like... public int runBinarySearchRecursively(int[] sortedArray, int key, int low, int high) { int middle = (low + high) / 2; if (high < low) { return -1; } if (key == sortedArray[middle]) { return middle; } else if (key < sortedArray[middle]) { return runBinarySearchRecursively(sortedArray, key, low, middle - 1); } else { return runBinarySearchRecursively(sortedArray, key, middle + 1, high); } }
- Consider the problem of counting, in a given text, the number of substrings that start with an A and end with a B. For example, there are four such substrings in CABAAXBYA.a. Design a brute-force algorithm for this problem and determine its efficiency class.b. Design a more efficient algorithm for this problem with complexity O (n)The algorithm consists of the following steps.1. Input: A weighted graph G = (V, E, w)2. Output: A maximal weighted matching M of G3. M ← Ø4. E ← E5. V ← V6. while E = Ø7. select at random any v ∈ V8. let e ∈ E be the heaviest edge incident to v9. M ← M ∪ e10. V ← V {v}11. E ← E \ {e and all adjacent edges to e}make two ways of implementing this algorithm in Python. don't copy bartleby old answer its wrong need correct ans otherwise you will get downvoteThe algorithm of Euclid computes the greatest common divisor (GCD) of two integer numbers a and b. The following pseudo-code is the original version of this algorithm. Algorithm Euclid(a,b)Require: a, b ≥ 0Ensure: a = GCD(a, b) while b ̸= 0 do t ← b b ← a mod b a ← tend whilereturn a What is the maximum number of iterations of the While loop the algorithm will do?
- Give an O(n^2)-time algorithm to nd the longest monotonically increasing subsequence of a sequence of n numbers.Illustrate your algorithm on the sequence:8, 3, 7, 5, 9, 3, 4, 1, 9, 2, 6.Theoretical Complexity Analysis Using the asymtotic notations we explored, kindly, derive the complexity of the worst-case time of this algorithm: def stringMatch(subStr,strLine): if subStr in strLine: return strLine.find(subStr) return 0Select the asymptotic worst-case time complexity of the following algorithm:AlgorithmInput: a1, a2, ..., an, a sequence of numbersn, the length of the sequencex, a numberOutput: ??For i = 1 to n-1For j = i+1 to nFor k = 1 to nIf ((ai)^2 + (aj)^2 = (ak)^2) Return( "True" )End-forEnd-forEnd-forReturn( "False" )a. ?(1)b. ?(?)c. ?(?2)d. ?(?3)