Floor and Ceil in Sorted Array in DSA Javascript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the floor of a number in a sorted array?

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

beginner

What is the ceil of a number in a sorted array?

The ceil of a number is the smallest element in the array that is greater than or equal to that number.

intermediate

Why is binary search useful for finding floor and ceil in a sorted array?

Binary search helps find floor and ceil efficiently by repeatedly dividing the search space in half, reducing time complexity to O(log n).

intermediate

If the target number is smaller than all elements in the array, what is the floor and ceil?

Floor does not exist (no element ≤ target), ceil is the smallest element in the array.

intermediate

If the target number is larger than all elements in the array, what is the floor and ceil?

Floor is the largest element in the array, ceil does not exist (no element ≥ target).

What does the floor of 7 mean in the sorted array [1, 3, 5, 8, 10]?

A The number 7 itself

B The smallest number ≥ 7

C The largest number ≤ 7

D The smallest number in the array

Floor is the greatest element less than or equal to the target, so here it is 5.

What is the ceil of 6 in the sorted array [2, 4, 6, 8, 10]?

A 4

B 10

C 8

D 6

Ceil is the smallest element greater than or equal to 6, which is 6 itself.

If the target is 0 and the array is [5, 10, 15], what is the floor?

A 0

B No floor exists

C 5

D 15

No element is less than or equal to 0, so floor does not exist.

Which algorithm is best to find floor and ceil in a sorted array efficiently?

A Binary search

B Linear search

C Bubble sort

D Insertion sort

Binary search reduces the search space quickly, making it efficient for this task.

If the target is 20 and the array is [1, 5, 10, 15], what is the ceil?

A No ceil exists

B 20

C 1

D 15

No element is greater than or equal to 20, so ceil does not exist.

Explain how to find the floor and ceil of a number in a sorted array using binary search.

Think about how to move left or right in the array to find closest smaller or larger elements.

Describe what happens when the target number is smaller than all elements or larger than all elements in the array regarding floor and ceil.

Consider the array boundaries and what floor and ceil mean.

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 or equal to the number

C. The average of all elements less than the number

D. The smallest element strictly 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 with options

   The greatest element less than or equal to the number matches this definition exactly, while others describe ceil or incorrect 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 must be strictly less than number
• Mixing floor with average or median

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

easy

A. Use Math.floor(x) directly on array

B. Use a for loop from 0 to arr.length and return first element greater than x

C. Use arr.indexOf(x) to find floor

D. Use a while loop with start = 0, end = arr.length - 1, update mid and check arr[mid] <= x

Solution

1. Step 1: Understand binary search for floor

   Binary search uses start and end pointers, calculates mid, and checks if arr[mid] <= x to move start or end accordingly.

2. Step 2: Evaluate options

   Use a while loop with start = 0, end = arr.length - 1, update mid and check arr[mid] <= x correctly describes binary search logic for floor. Options A, B, and C are incorrect or irrelevant.

3. Final Answer:

   Use a while loop with start=0, end=arr.length-1, update mid and check arr[mid] ≤ x -> Option D

4. Quick Check:

   Binary search with arr[mid] ≤ x finds floor [OK]

Hint: Binary search with arr[mid] ≤ x finds floor efficiently [OK]

Common Mistakes:

• Using linear search instead of binary search
• Using indexOf which only finds exact matches
• Misusing Math.floor which works on numbers, not arrays

3. What is the output of the following JavaScript code?

const arr = [1, 3, 5, 7, 9];
function findFloor(arr, x) {
  let start = 0, end = arr.length - 1, floor = -1;
  while (start <= end) {
    let mid = Math.floor((start + end) / 2);
    if (arr[mid] <= x) {
      floor = arr[mid];
      start = mid + 1;
    } else {
      end = mid - 1;
    }
  }
  return floor;
}
console.log(findFloor(arr, 6));

medium

A. 6

B. 5

C. 7

D. -1

Solution

1. Step 1: Trace binary search for floor of 6

   Initial start=0, end=4, floor=-1. mid=2 (arr[2]=5) ≤ 6, floor=5, start=3. mid=3 (arr[3]=7) > 6, end=2. Loop ends.

2. Step 2: Return floor value

   Floor found is 5, which is the greatest element ≤ 6.

3. Final Answer:

   5 -> Option B

4. Quick Check:

   Floor of 6 in [1,3,5,7,9] is 5 [OK]

Hint: Floor is last arr[mid] ≤ x during binary search [OK]

Common Mistakes:

• Returning mid index instead of value
• Confusing floor with ceil
• Not updating floor variable correctly

4. Identify the bug in this JavaScript function to find the ceil of a number x in a sorted array arr:

function findCeil(arr, x) {
  let start = 0, end = arr.length - 1, ceil = -1;
  while (start <= end) {
    let mid = Math.floor((start + end) / 2);
    if (arr[mid] >= x) {
      ceil = arr[mid];
      end = mid + 1;
    } else {
      start = mid + 1;
    }
  }
  return ceil;
}

medium

A. The line end = mid + 1; should be end = mid - 1;

B. The line start = mid + 1; should be start = mid - 1;

C. The initial value of ceil should be 0 instead of -1

D. The while loop condition should be start < end instead of start <= end

Solution

1. Step 1: Analyze binary search update for ceil

   When arr[mid] ≥ x, we update ceil and must move end to mid - 1 to search left side for smaller ceil.

2. Step 2: Identify incorrect update

   The code incorrectly sets end = mid + 1, which moves end right, causing infinite loop or wrong result.

3. Final Answer:

   The line end = mid + 1; should be end = mid - 1; -> Option A

4. Quick Check:

   For ceil, move end left with end = mid - 1 [OK]

Hint: For ceil, move end left; for floor, move start right [OK]

Common Mistakes:

• Moving end pointer in wrong direction
• Confusing ceil logic with floor
• Incorrect initial ceil value

5. Given a sorted array arr = [2, 4, 6, 8, 10], write a function to find both floor and ceil of a number x. What will be the floor and ceil of x = 5?

hard

A. Floor = 5, Ceil = 5

B. Floor = 6, Ceil = 4

C. Floor = 4, Ceil = 6

D. Floor = 2, Ceil = 10

Solution

1. Step 1: Understand floor and ceil definitions

   Floor is greatest element ≤ x, ceil is smallest element ≥ x.

2. Step 2: Find floor and ceil for x=5 in arr

   Elements ≤ 5 are 2 and 4; greatest is 4 (floor). Elements ≥ 5 are 6, 8, 10; smallest is 6 (ceil).

3. Final Answer:

   Floor = 4, Ceil = 6 -> Option C

4. Quick Check:

   Floor ≤ 5 = 4, Ceil ≥ 5 = 6 [OK]

Hint: Floor ≤ x max, Ceil ≥ x min in sorted array [OK]

Common Mistakes:

• Swapping floor and ceil values
• Assuming floor or ceil equals x if not present
• Choosing wrong boundary elements