Question 6 of 8advanced🔍 AnalyzeNumerical4 marks

Estimate the number of key comparisons required in binary search and linear search if we need to find the details of a person in a sorted database having 2302^{30} (1,073,741,824) records when details of the person being searched lies at the middle position in the database. What do you interpret from your findings?

Correct Answer

Binary Search Calculation: The context states that binary search "divides it in the middle" and "compares the middle element with the key to be searched. If the middle element matches the key, the search is declared successful." Since the details of the person lie at the middle position, the search is successful in the very first comparison. Number of comparisons = 1

Linear Search Calculation: In linear search, elements are checked sequentially from the beginning. To find a record at the middle position in a sorted database, the search must traverse half of the total records. Total records (NN) = 2302^{30}. Number of comparisons = N/2=230/2=229N/2 = 2^{30} / 2 = 2^{29}. Number of comparisons = 536,870,912

Interpretation: Binary search requires only 1 comparison, whereas linear search requires 536,870,912 comparisons. This vast difference highlights that binary search is significantly more efficient than linear search for large sorted databases, reducing the search time and computational effort drastically.

Exercise: EXERCISE | Q: 7 | (Page Page 16)
For More Understanding

Explanation

The textbook context explains that binary search starts by comparing the middle element with the key. If they match, the search terminates immediately. This logic applies directly to the question's scenario where the record is at the middle position, resulting in just 1 comparison. For linear search, standard estimation logic applies where a middle-position item requires N/2 comparisons. The interpretation focuses on the efficiency gap demonstrated by the calculations.

Solution Steps

  1. Step 1: Identify the search position. The target record is at the middle position of the database.

  2. Step 2: Calculate Binary Search comparisons. Since binary search checks the middle element first, and the target is there, comparisons = 1.

  3. Step 3: Calculate Linear Search comparisons. Linear search checks sequentially. For a middle position in 2302^{30} records, comparisons = 230/2=229=536,870,9122^{30} / 2 = 2^{29} = 536,870,912.

  4. Step 4: Compare the results. Binary search (1) is vastly more efficient than Linear search (536,870,912).

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