Let T[n] be an array of n elements. An element x is said to be a majority element in T if the number of i's such that T[i] = x is greater than n/2. Design an algorithm that can decide whether an array T[1..n] includes a majority element and if so find it when the only comparisons allowed between elements are tests of equality. So you cannot assume that an order relationship exists between the elements. What is the time complexity of your algorithm?a. O( n^2 ) b.Nonec.O( n^3 )d.O( n log n )
Question
Let T[n] be an array of n elements. An element x is said to be a majority element in T if the number of i's such that T[i] = x is greater than n/2. Design an algorithm that can decide whether an array T[1..n] includes a majority element and if so find it when the only comparisons allowed between elements are tests of equality. So you cannot assume that an order relationship exists between the elements. What is the time complexity of your algorithm?a. O( n^2 ) b.Nonec.O( n^3 )d.O( n log n )
Solution
The algorithm to find the majority element in an array can be implemented using the Moore’s Voting Algorithm. Here are the steps:
- Initialize an element m and a counter i with m = T[1] and i = 1.
- Process the next element of the array T[j] (j = 2 to n):
- If T[j] == m, then increment the counter i.
- Else decrement the counter i.
- If the counter reaches 0, then set m = T[j] and reset the counter i to 1.
- m will hold the potential majority element. To ensure that m is the majority element, iterate through the array and count its occurrences. If it is greater than n/2, then m is the majority element.
The time complexity of this algorithm is O(n) because each element is processed at most twice. Therefore, the correct answer is not in the options given. The correct answer should be O(n).
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