Find Minimum in Rotated Sorted Array in DSA Go - Time & Space Complexity

Time Complexity: Find Minimum in Rotated Sorted Array

O(log n)

Understanding Time Complexity

We want to know how fast we can find the smallest number in a rotated 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 findMin(nums []int) int {
    left, right := 0, len(nums)-1
    for left < right {
        mid := left + (right-left)/2
        if nums[mid] > nums[right] {
            left = mid + 1
        } else {
            right = mid
        }
    }
    return nums[left]
}

This code finds the minimum value by using a binary search approach on a rotated sorted array.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The loop that halves the search range each time.
• How many times: The loop runs until the search range is reduced to one element, about log base 2 of n times.

How Execution Grows With Input

Each time the array size doubles, the number of steps increases by one.

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

Pattern observation: The steps grow slowly as input size grows, increasing by one each time the input doubles.

Final Time Complexity

Time Complexity: O(log n)

This means the time to find the minimum grows slowly, adding only a few extra steps even if the array gets much bigger.

Common Mistake

[X] Wrong: "The loop runs through every element, so it is O(n)."

[OK] Correct: The code cuts the search space in half each time, so it does not check every element one by one.

Interview Connect

Understanding this time complexity shows you can use efficient search methods on special arrays, a skill often valued in problem solving.

Self-Check

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

Practice

1. What is the main idea behind finding the minimum element in a rotated sorted array?

easy

A. Use a hash map to store elements and find minimum

B. Sort the array again to find the minimum

C. Traverse the array linearly to find the smallest element

D. Use binary search to find the point where order breaks

Solution

1. Step 1: Understand the rotated sorted array property

   A rotated sorted array is sorted but rotated at some pivot point, causing a break in order.

2. Step 2: Use binary search to find the break point

   Binary search helps find the point where the order breaks, which is where the minimum element lies.

3. Final Answer:

   Use binary search to find the point where order breaks -> Option D

4. Quick Check:

   Binary search finds minimum efficiently [OK]

Hint: Look for the pivot where order breaks using binary search [OK]

Common Mistakes:

• Trying to sort the array again wastes time
• Using linear search ignores the sorted property
• Using extra space unnecessarily

2. Which Go code snippet correctly initializes variables for binary search to find minimum in a rotated sorted array nums?

easy

A. start, end := -1, len(nums)-1

B. start, end := 0, len(nums)-1

C. start, end := 1, len(nums)

D. start, end := 0, len(nums)

Solution

1. Step 1: Understand array indexing in Go

   Arrays are zero-indexed, so start should be 0 and end should be last index (length - 1).

2. Step 2: Check each option for correct initialization

   start, end := 0, len(nums)-1 correctly sets start=0 and end=len(nums)-1, which is valid for binary search.

3. Final Answer:

   start, end := 0, len(nums)-1 -> Option B

4. Quick Check:

   Correct index range initialization [OK]

Hint: Start at 0 and end at length minus one for array indices [OK]

Common Mistakes:

• Using length as end index causes out-of-range errors
• Starting at 1 misses the first element
• Negative start index is invalid

3. What will be the output of the following Go code snippet?

nums := []int{4,5,6,7,0,1,2}
start, end := 0, len(nums)-1
for start < end {
  mid := (start + end) / 2
  if nums[mid] > nums[end] {
    start = mid + 1
  } else {
    end = mid
  }
}
fmt.Println(nums[start])

medium

A. 0

B. 4

C. 7

D. 2

Solution

1. Step 1: Trace the binary search loop

   Initial start=0, end=6. Mid=(0+6)/2=3, nums[3]=7 > nums[6]=2, so start=4.

2. Step 2: Continue loop until start meets end

   Now start=4, end=6. Mid=(4+6)/2=5, nums[5]=1 <= nums[6]=2, so end=5. Next mid=(4+5)/2=4, nums[4]=0 <= nums[5]=1, so end=4. Loop ends when start=4, end=4.

3. Final Answer:

   0 -> Option A

4. Quick Check:

   Minimum element found at index 4 is 0 [OK]

Hint: Compare mid with end to decide search side [OK]

Common Mistakes:

• Confusing mid calculation causing infinite loop
• Incorrectly updating start or end pointers
• Printing nums[end] instead of nums[start]

4. Identify the bug in this Go code to find minimum in rotated sorted array nums:

start, end := 0, len(nums)-1
for start <= end {
  mid := (start + end) / 2
  if nums[mid] > nums[end] {
    start = mid
  } else {
    end = mid - 1
  }
}
fmt.Println(nums[start])

medium

A. All of the above

B. Loop condition should be start < end, not <=

C. end should be set to mid, not mid - 1

D. start should be set to mid + 1, not mid

Solution

1. Step 1: Analyze loop condition and pointer updates

   The loop uses start <= end which can cause infinite loop when start == end. Also, start = mid does not move start forward properly, and end = mid - 1 skips the mid element incorrectly.

2. Step 2: Correct the loop and pointer updates

   Loop should be start < end. When nums[mid] > nums[end], start = mid + 1 to move forward. Else, end = mid to include mid in next search.

3. Final Answer:

   All of the above -> Option A

4. Quick Check:

   Proper loop and pointer updates avoid infinite loops [OK]

Hint: Use start < end and adjust pointers by +1 or no change [OK]

Common Mistakes:

• Using <= causes infinite loops
• Not moving start pointer forward correctly
• Skipping mid element by subtracting 1 from end

5. Given a rotated sorted array nums that may contain duplicates, which modification is needed in the binary search approach to find the minimum element?

hard

A. Ignore duplicates and apply normal binary search

B. If nums[mid] == nums[end], increase start by 1 to skip duplicate

C. If nums[mid] == nums[end], reduce end by 1 to skip duplicate

D. Sort the array first to handle duplicates

Solution

1. Step 1: Understand duplicates affect binary search

   Duplicates can cause nums[mid] == nums[end], making it unclear which side to search.

2. Step 2: Modify binary search to handle duplicates

   When nums[mid] == nums[end], safely reduce end by 1 to skip duplicate without missing minimum.

3. Final Answer:

   If nums[mid] == nums[end], reduce end by 1 to skip duplicate -> Option C

4. Quick Check:

   Handle duplicates by shrinking search space carefully [OK]

Hint: When equal, reduce end to skip duplicates safely [OK]

Common Mistakes:

• Ignoring duplicates causes wrong results
• Increasing start instead of reducing end misses minimum
• Sorting wastes time and is unnecessary