Insertions and deletions

What is an array?

An array stores values in a sequence of numbered positions. A position is called an index. Indices usually start at 0: the first value is at index 0, the second at index 1. Knowing the index lets you go directly to that position; finding an unknown value may require checking many positions.

Example: In [4, 8, 15], index 1 contains 8. There are three elements, so the valid indices are 0, 1, and 2. A Python list is a dynamic array: its storage can grow when you append values.

Helpful before this chapter

If these ideas are new, begin with:

Programming foundations Time and space complexity

Understand the idea

Packed arrays preserve order by shifting around insertion or deletion. Appending differs from inserting at the beginning because there is no existing suffix to shift.

Work through an example

Insert 9 at index 1 in [4,8,15]: move 15 and 8 right, then write 9. Deleting that position moves the suffix left.

Time and space complexity

Middle insertion/deletion O(n) worst case; dynamic-array append amortized O(1).

Why it works

Insertion shifts right to left so unread values are preserved.

Common mistakes

Wrong shift direction overwrites data.
Removing while scanning can skip the next element.

Chapter lab · Python implementation

Arrays make positions directly accessible. Our lab compacts a sorted list: read examines input; write marks the next output slot. Python lists are resizable arrays of references. Fixed-width address diagrams illustrate slots, rather than promising contiguous Python objects.

This shared lab connects the chapter's topics. Follow the current lesson's worked example first, then trace the complete implementation. Additional algorithms described in topic exercises are variations you can implement independently.

def compact_sorted(values):
    write = 0
    for read in range(len(values)):
        if write == 0 or values[read] != values[write-1]:
            values[write] = values[read]
            write += 1
    return write

def transpose(matrix):
    if not matrix:
        return []
    cols = len(matrix[0])
    if any(len(row) != cols for row in matrix):
        raise ValueError('rectangular matrix required')
    return [[matrix[r][c] for r in range(len(matrix))]
            for c in range(cols)]

Run and test the lab

Copy the implementation and these assertions into a Python 3 file and run it. An assertion failure means the actual result differs from the expectation. Add the current topic's exercise and boundary tests to deepen your understanding.

a=[1,1,2,2,3]; k=compact_sorted(a)
assert a[:k]==[1,2,3]
assert compact_sorted([])==0
assert transpose([[1,2,3],[4,5,6]])==[[1,4],[2,5],[3,6]]
assert transpose([])==[]

CHECK YOUR UNDERSTANDING