Binary Search Recursive Approach in DSA Typescript - Time & Space Complexity

Time Complexity: Binary Search Recursive Approach

O(log n)

Understanding Time Complexity

We want to understand how the time taken by recursive binary search changes as the input size grows.

Specifically, how many steps does it take to find a number in a sorted list?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function binarySearch(arr: number[], target: number, left: number, right: number): number {
  if (left > right) return -1;
  const mid = Math.floor((left + right) / 2);
  if (arr[mid] === target) return mid;
  else if (arr[mid] > target) return binarySearch(arr, target, left, mid - 1);
  else return binarySearch(arr, target, mid + 1, right);
}
    

This code searches for a target number in a sorted array by repeatedly dividing the search range in half.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive call that halves the search range each time.
• How many times: At most, the function calls itself until the search range is empty, roughly halving the size each time.

How Execution Grows With Input

Each step cuts the search space in half, so the number of steps grows slowly as the list gets bigger.

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

Pattern observation: Doubling the input size adds only one more step, showing very slow growth.

Final Time Complexity

Time Complexity: O(log n)

This means the number of steps grows very slowly, making the search efficient even for large lists.

Common Mistake

[X] Wrong: "Binary search checks every element one by one, so it is O(n)."

[OK] Correct: Binary search does not check all elements; it cuts the search area in half each time, so it only needs about log n steps.

Interview Connect

Understanding binary search's time complexity shows you can analyze efficient algorithms, a key skill in coding interviews and real projects.

Self-Check

What if we changed the array to be unsorted? How would the time complexity of searching change?

Practice

1. What is the main idea behind the binary search recursive approach?

easy

A. Search each element one by one from start to end

B. Divide the array into halves and search recursively in the correct half

C. Sort the array after each recursive call

D. Use a queue to store elements for searching

Solution

1. Step 1: Understand binary search logic

   Binary search works by dividing the sorted array into two halves and deciding which half to search next based on the middle element.

2. Step 2: Apply recursion in binary search

   The recursive approach calls the same function on the half where the target might be, reducing the search space each time.

3. Final Answer:

   Divide the array into halves and search recursively in the correct half -> Option B

4. Quick Check:

   Binary search = divide and conquer [OK]

Hint: Remember: binary search splits array, then recurses [OK]

Common Mistakes:

• Thinking binary search checks every element
• Sorting array inside recursion
• Using linear search logic

2. Which of the following is the correct TypeScript function signature for a recursive binary search?

easy

A. function binarySearch(arr: number[], target: string): boolean

B. function binarySearch(arr: string[], target: number): number

C. function binarySearch(arr: number[], target: number, left: number, right: number): number

D. function binarySearch(arr: number[], target: number): void

Solution

1. Step 1: Check parameters needed for recursion

   Recursive binary search needs the array, target, and the current left and right indices to search between.

2. Step 2: Verify return type

   The function returns the index of the target if found, so the return type is number.

3. Final Answer:

   function binarySearch(arr: number[], target: number, left: number, right: number): number -> Option C

4. Quick Check:

   Parameters + return type match recursive binary search [OK]

Hint: Recursive binary search needs left and right indices [OK]

Common Mistakes:

• Missing left and right parameters
• Wrong return type (void or boolean)
• Mismatched array and target types

3. What will be the output of the following TypeScript code?

function binarySearch(arr: number[], target: number, left: number, right: number): number {
  if (left > right) return -1;
  const mid = Math.floor((left + right) / 2);
  if (arr[mid] === target) return mid;
  else if (arr[mid] > target) return binarySearch(arr, target, left, mid - 1);
  else return binarySearch(arr, target, mid + 1, right);
}

const nums = [2, 4, 6, 8, 10];
console.log(binarySearch(nums, 6, 0, nums.length - 1));

medium

A. 2

B. 1

C. -1

D. 3

Solution

1. Step 1: Identify middle index and compare

   Initial call: left=0, right=4, mid=2. arr[2] = 6 matches target 6.

2. Step 2: Return index when found

   Since arr[mid] equals target, function returns mid = 2 immediately.

3. Final Answer:

   2 -> Option A

4. Quick Check:

   Target found at index 2 [OK]

Hint: Check middle element first, then recurse [OK]

Common Mistakes:

• Miscomputing mid index
• Returning wrong index
• Confusing array values with indices

4. Find the error in this recursive binary search function:

function binarySearch(arr: number[], target: number, left: number, right: number): number {
  if (left >= right) return -1;
  const mid = Math.floor((left + right) / 2);
  if (arr[mid] === target) return mid;
  else if (arr[mid] > target) return binarySearch(arr, target, left, mid - 1);
  else return binarySearch(arr, target, mid + 1, right);
}

medium

A. The base case condition should be left > right, not left >= right

B. The mid calculation is incorrect

C. The function should return boolean instead of number

D. The recursive calls have wrong parameters

Solution

1. Step 1: Analyze base case condition

   The base case currently returns -1 if left >= right, but it should allow left == right to check that element.

2. Step 2: Correct base case to left > right

   Changing to left > right ensures the search includes the last element before stopping.

3. Final Answer:

   The base case condition should be left > right, not left >= right -> Option A

4. Quick Check:

   Base case must allow left == right [OK]

Hint: Base case stops when left passes right, not equals [OK]

Common Mistakes:

• Stopping search too early with wrong base case
• Miscomputing mid index
• Wrong return type

5. Given a sorted array arr = [1, 3, 5, 7, 9, 11, 13], write a recursive binary search call to find the index of 7. What will be the sequence of mid indices checked?

hard

A. [3, 5]

B. [3, 1]

C. [3, 1, 2]

D. [3]

Solution

1. Step 1: Calculate initial mid index

   left=0, right=6, mid = floor((0+6)/2) = 3. arr[3] = 7 matches target immediately.

2. Step 2: No further recursive calls needed

   Since target found at first mid, no other mid indices are checked.

3. Final Answer:

   [3] -> Option D

4. Quick Check:

   Target found at first mid index 3 [OK]

Hint: First mid often finds target if array is balanced [OK]

Common Mistakes:

• Assuming multiple recursive calls always happen
• Miscomputing mid index
• Confusing indices with values