. Describe an algorithm that inserts an integer x in the appropriate position into the list a1,a2, . . . , an of integers that are in increasing order

1. Break Down the Problem

To insert an integer $x$ into a sorted list of integers $a_1, a_2, \ldots, a_n$, we need to:

1. Find the correct position for $x$ in the list such that the list remains sorted.
2. Shift the elements of the list as necessary to insert $x$.

2. Relevant Concepts

The algorithm can utilize a binary search method to efficiently find the correct insertion index in a sorted list. This is because a binary search has a time complexity of $O(\log n)$, making it more efficient than a linear search for larger lists.

3. Analysis and Detail

Here’s the algorithm outlined step-by-step:

1. Initialize variables:

   • Let $left = 0$
   • Let $right = n$ (size of list)

2. Binary Search to find the position:

   • While $left < right$:
     • Calculate $mid = \left( left + right \right) // 2$ (integer division)
     • If $a[mid] < x$:
       • Set $left = mid + 1$
     • Else:
       • Set $right = mid$

3. Insertion:

   • Once the position is found, $left$ will be the index where $x$ should be inserted.
   • Shift elements from index $left$ to $n$ one position to the right to make space for $x$.
   • Insert $x$ at index $left$.

4. Verify and Summarize

The binary search effectively reduces the number of comparisons needed to find the correct position for $x$, ensuring the list remains sorted after the insertion. The shifting of elements ensures that no data is lost, and the time complexity for this operation is $O(n)$ for the shifting part after $O(\log n)$ for the search.

Final Answer

The algorithm to insert an integer $x$ into a sorted list $a_1, a_2, \ldots, a_n$ is as follows:

1. Initialize $left = 0$ and $right = n$.
2. Perform a binary search until $left < right$:
   • Calculate $mid = (left + right) // 2$.
   • If $a[mid] < x$, set $left = mid + 1$; otherwise, set $right = mid$.
3. Shift elements $a[left]$ to $a[n-1]$ to the right.
4. Insert $x$ at index $left$.

This algorithm maintains the sorted order of the list after insertion.