Floor and Ceil in Sorted Array in DSA Go - 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.

func floorCeil(arr []int, x int) (int, int) {
    low, high := 0, len(arr)-1
    floor, ceil := -1, -1
    for low <= high {
        mid := low + (high-low)/2
        if arr[mid] == x {
            return arr[mid], arr[mid]
        } else if arr[mid] < x {
            floor = arr[mid]
            low = mid + 1
        } else {
            ceil = arr[mid]
            high = mid - 1
        }
    }
    return floor, ceil
}

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 loop that halves the search space each time (binary search).
• How many times: The loop runs about log base 2 of n times, where n is the array size.

How Execution Grows With Input

Each step cuts the array 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 more step.

Final Time Complexity

Time Complexity: O(log n)

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

Common Mistake

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

[OK] Correct: The code uses binary search, which skips many elements each step, so it does not check all elements.

Interview Connect

Knowing how to use binary search to quickly find floor and ceil is a useful skill that shows you understand efficient searching.

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?

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 the floor definition

   The floor of a number in a sorted array is the greatest element 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 this definition exactly, while 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
• Thinking floor is strictly less than number
• Mixing floor with average or median

2. Which Go code snippet correctly declares a function to find the floor in a sorted array?

easy

A. func floor(arr int[], x int) int { /* code */ }

B. func floor(arr []int, x int) int { /* code */ }

C. function floor(arr []int, x int) int { /* code */ }

D. func floor(arr []int, x int) void { /* code */ }

Solution

1. Step 1: Check Go function syntax

   In Go, functions are declared with func, parameter types follow names, and return type is after parameters.

2. Step 2: Validate each option

   func floor(arr []int, x int) int { /* code */ } uses correct Go syntax: func floor(arr []int, x int) int. func floor(arr int[], x int) int { /* code */ } uses incorrect array syntax, C uses wrong keyword, D has wrong return type.

3. Final Answer:

   func floor(arr []int, x int) int { /* code */ } -> Option B

4. Quick Check:

   Go func syntax = func floor(arr []int, x int) int { /* code */ } [OK]

Hint: Go arrays use []type, func keyword, return type after params [OK]

Common Mistakes:

• Using 'function' keyword instead of 'func'
• Incorrect array type syntax
• Wrong return type declaration

3. What is the output of the following Go code snippet?

arr := []int{1, 3, 5, 7, 9}
x := 6
floor := -1
for _, val := range arr {
    if val <= x {
        floor = val
    } else {
        break
    }
}
fmt.Println(floor)

medium

A. 5

B. 6

C. 7

D. -1

Solution

1. Step 1: Trace the loop over array elements

   The loop checks each element: 1 ≤ 6 (floor=1), 3 ≤ 6 (floor=3), 5 ≤ 6 (floor=5), 7 ≤ 6? No, break loop.

2. Step 2: Determine final floor value

   Last floor assigned is 5, so printed output is 5.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   Floor of 6 in arr = 5 [OK]

Hint: Floor is last element ≤ x before break [OK]

Common Mistakes:

• Assuming floor equals x if not present
• Not breaking loop early
• Confusing floor with ceil

4. Identify the bug in this Go code to find the ceil of x in a sorted array arr:

func ceil(arr []int, x int) int {
    ceil := -1
    for _, val := range arr {
        if val >= x {
            ceil = val
        }
    }
    return ceil
}

medium

A. It uses wrong comparison operator

B. It initializes ceil with -1 incorrectly

C. It returns ceil before loop ends

D. It does not break after finding first valid ceil

Solution

1. Step 1: Analyze loop behavior for ceil

   The loop updates ceil whenever val ≥ x, but does not stop after first valid ceil, so it ends with the last such value, which may not be the smallest ceil.

2. Step 2: Identify correct fix

   To find the smallest ceil, the loop should break immediately after first val ≥ x is found.

3. Final Answer:

   It does not break after finding first valid ceil -> Option D

4. Quick Check:

   Ceil needs first ≥ x, break loop [OK]

Hint: Break loop on first val ≥ x for correct ceil [OK]

Common Mistakes:

• Not breaking loop early
• Confusing ceil with floor logic
• Incorrect initialization of ceil

5. Given a sorted array arr := []int{2, 4, 6, 8, 10} and a number x = 5, write a Go function that returns both floor and ceil of x. What is the correct output of floor and ceil?

hard

A. Floor: -1, Ceil: -1

B. Floor: 6, Ceil: 4

C. Floor: 4, Ceil: 6

D. Floor: 5, Ceil: 5

Solution

1. Step 1: Find floor of 5 in arr

   Floor is greatest element ≤ 5. Elements ≤ 5 are 2 and 4, greatest is 4.

2. Step 2: Find ceil of 5 in arr

   Ceil is smallest element ≥ 5. Elements ≥ 5 are 6, 8, 10, smallest is 6.

3. Final Answer:

   Floor: 4, Ceil: 6 -> Option C

4. Quick Check:

   Floor ≤ x ≤ Ceil = 4 ≤ 5 ≤ 6 [OK]

Hint: Floor ≤ x ≤ Ceil in sorted array [OK]

Common Mistakes:

• Swapping floor and ceil values
• Assuming floor or ceil equals x if not present
• Returning -1 when valid floor or ceil exists