Floor and Ceil in Sorted Array in DSA Python - Time & Space Complexity

Time Complexity: Floor and Ceil in Sorted Array

O(log n)

Understanding Time Complexity

We want to know how fast we can find the floor or ceil of a number in a sorted array.

How does the time needed change as the array gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

def floor_ceil(arr, x):
    low, high = 0, len(arr) - 1
    floor_val, ceil_val = -1, -1
    while low <= high:
        mid = (low + high) // 2
        if arr[mid] == x:
            return arr[mid], arr[mid]
        elif arr[mid] < x:
            floor_val = arr[mid]
            low = mid + 1
        else:
            ceil_val = arr[mid]
            high = mid - 1
    return floor_val, ceil_val

This code finds the floor and ceil of x in a sorted array using binary search.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop that halves the search space each time.
• How many times: At most, it runs about log base 2 of n times, where n is the array size.

How Execution Grows With Input

Each step cuts the array size roughly in half, so the number of steps grows slowly as the array grows.

Input Size (n)	Approx. Operations
10	About 4 steps
100	About 7 steps
1000	About 10 steps

Pattern observation: Doubling the input size adds only one extra step.

Final Time Complexity

Time Complexity: O(log n)

This means the time to find floor and ceil grows very slowly even if the array becomes very large.

Common Mistake

[X] Wrong: "Since we check elements one by one, the time is O(n)."

[OK] Correct: The code does not check every element; it smartly jumps around by halving the search space each time, so it runs much faster.

Interview Connect

Knowing how to quickly find floor and ceil using binary search shows you understand efficient searching, a key skill in many coding problems.

Self-Check

"What if the array was not sorted? How would the time complexity change?"

Practice

1. What does the floor of a number mean in a sorted array context?

easy

A. The greatest element less than or equal to the number

B. The smallest element greater than the number

C. The average of all elements less than the number

D. The smallest element less than the number

Solution

1. Step 1: Understand floor definition in sorted array

   The floor of a number is the greatest element in the array that is less than or equal to the given number.

2. Step 2: Compare options with definition

   The greatest element less than or equal to the number matches the definition exactly, others describe different concepts.

3. Final Answer:

   The greatest element less than or equal to the number -> Option A

4. Quick Check:

   Floor = greatest ≤ number [OK]

Hint: Floor is always ≤ number, pick greatest such element [OK]

Common Mistakes:

• Confusing floor with ceil
• Choosing smallest element greater than number
• Thinking floor is average of smaller elements

2. Which of the following is the correct Python syntax to find the floor of x in a sorted list arr using binary search?

easy

A. Use bisect_left and check index-1 element

B. Use bisect_right and check index-1 element

C. Use bisect_left and check index element

D. Use bisect_right and check index element

Solution

1. Step 1: Recall bisect_right usage for floor

   bisect_right(arr, x) returns insertion point to the right of existing entries equal to x.

2. Step 2: Find floor index using bisect_right

   The floor is at index bisect_right(arr, x) - 1 if index is valid.

3. Final Answer:

   Use bisect_right and check index-1 element -> Option B

4. Quick Check:

   Floor index = bisect_right - 1 [OK]

Hint: Floor index = bisect_right(arr, x) - 1 [OK]

Common Mistakes:

• Using bisect_left for floor without adjusting index
• Checking index instead of index-1
• Confusing bisect_left and bisect_right roles

3. Given a sorted array arr = [1, 2, 4, 6, 8, 10], write a function to find the floor and ceil of x = 5. What should the function return?

easy

A. [5, 6]

B. [2, 6]

C. [4, 5]

D. [4, 6]

Solution

1. Step 1: Identify floor of 5 in arr

   Floor is greatest element ≤ 5, which is 4.

2. Step 2: Identify ceil of 5 in arr

   Ceil is smallest element ≥ 5, which is 6.

3. Final Answer:

   [4, 6] -> Option D

4. Quick Check:

   Floor = 4, Ceil = 6 [OK]

Hint: Floor ≤ x, Ceil ≥ x; pick closest elements [OK]

Common Mistakes:

• Returning x if not in array
• Confusing floor and ceil values
• Choosing elements not closest to x

4. What is the output of the following code?

import bisect
arr = [1, 3, 5, 7, 9]
x = 6
floor_index = bisect.bisect_right(arr, x) - 1
ceil_index = bisect.bisect_left(arr, x)
print(arr[floor_index], arr[ceil_index])

medium

A. 5 7

B. 5 5

C. 3 7

D. 7 9

Solution

1. Step 1: Calculate floor_index using bisect_right

   bisect_right(arr, 6) returns 3 (position after 5), so floor_index = 3 - 1 = 2, arr[2] = 5.

2. Step 2: Calculate ceil_index using bisect_left

   bisect_left(arr, 6) returns 3 (position of 7), so ceil_index = 3, arr[3] = 7.

3. Final Answer:

   5 7 -> Option A

4. Quick Check:

   Floor = 5, Ceil = 7 [OK]

Hint: Floor uses bisect_right-1, Ceil uses bisect_left [OK]

Common Mistakes:

• Using bisect_left for floor index
• Mixing floor and ceil indices
• Off-by-one errors in indexing

5. The following code tries to find the floor of x in a sorted list arr. What is the error?

import bisect
arr = [2, 4, 6, 8]
x = 1
index = bisect.bisect_right(arr, x) - 1
print(arr[index])

medium

A. IndexError when x is smaller than all elements

B. SyntaxError due to bisect usage

C. Wrong floor value printed

D. No error, code works fine

Solution

1. Step 1: Calculate index for x=1

   bisect_right(arr, 1) returns 0, so index = 0 - 1 = -1.

2. Step 2: Issue with negative index

   arr[-1] accesses the last element 8 due to Python's negative indexing, but since 1 < arr[0], no element ≤1 exists, so 8 is the wrong floor value.

3. Final Answer:

   Wrong floor value printed -> Option C

4. Quick Check:

   index < 0 means no floor [OK]

Hint: Check if index < 0 before accessing floor [OK]

Common Mistakes:

• Not handling case when x < smallest element
• Assuming floor always exists
• Ignoring negative indices in Python lists