Problem Description

You are provided with an array of integers, nums, which is sorted and contains no duplicates. Your task is to transform this array into a more compact representation using a list of ranges. A range [a, b] includes all integers that lie between a and b, inclusive. For example, the range [2, 5] represents the numbers 2, 3, 4, 5.

You need to come up with the smallest list of such ranges that together cover exactly the same set of numbers as in the original array nums. For each range in that list:

• If the range contains a single number, say a, it should be represented by just that number as a string, e.g., "a".
• If the range contains more than one number, a through b, it should be represented as the string "a->b".

The output should be a list of these string representations, arranged in ascending order, showcasing the continuous and discrete ranges found in nums.

Intuition

To generate the list of ranges efficiently, we can employ a two-pointer approach, which is a common technique used in array traversals. Here's the thought process that leads to the solution:

• Since the array is sorted and contains unique integers, any consecutive elements that differ by 1 can be considered part of the same range.

• We can keep track of the start of the current range with pointer i and use another pointer j to explore the extent of the range.

• Starting with the left pointer i fixed at the beginning of the array, we move the right pointer j forward as long as consecutive elements are part of the same range (i.e., nums[j + 1] is exactly 1 greater than nums[j]).

• Once we find that nums[j + 1] is not a direct successor of nums[j], we've identified a complete range. This range may be just a single number if i and j are at the same position, or it could be a sequence of numbers if j has moved from its starting position.

• For each range, we use a helper function f(i, j) to format the range's representation as a string, based on the start and end indices i, j of the range in the array.

• Once the range is added to the result list, we reset the left pointer i to be one position ahead of j, since j + 1 is where our last range ended, and continue the search for the next range.

• We repeat this process until the end of the array is reached.

This method allows us to break down the problem into smaller parts, making the task of finding and representing ranges in the array both straightforward and efficient.

Solution Approach

To implement the solution, we apply the two-pointer technique as previously discussed. The code structure follows a clear pattern that helps us to identify and construct the ranges:

1. Initialization: We create an empty list ans to store the range strings, and initiate our pointers i and j to 0. The pointer i serves as the start of a new range, and j is used to find the end of this range.

2. Iterating through the array: We proceed with a while loop that runs as long as i is less than the length of the array.

3. Exploration within the range: Inside the while loop:

   • We assign j to the same value as i to start exploring the range.
   • While j + 1 is within the bounds of the array and the element at j + 1 is one greater than the element at j, we increment j. This step essentially "gathers" all the consecutive numbers into one range.

4. Range formation: Once j stops moving, we reached the end of a range. We then call our helper function f(i, j) which takes the indices i and j and returns the string representation:

   • If i == j, this means the range consists of a single number, so we simply convert nums[i] to a string.
   • Otherwise, we format the string to be "nums[i]->nums[j]" to represent a range spanning multiple numbers.

5. Storing the range and moving on: After calling the helper function f(i, j), we add the resulting string to our ans list, indicating that this range is complete.

6. Advance the i pointer: We then set i to j + 1 to begin a new range and repeat the process.

7. Return the result: Once the entire array has been traversed, all ranges have been found, formatted, and added to ans. Finally, we return this list as our output.

This algorithm ensures that we only pass through the array once, therefore, the time complexity is O(n) where n is the length of the array, making the solution time-efficient. Space complexity is O(1) since we don't use any additional data structures that grow with the size of the input; the space used is just for the output list and the pointers.

The solution is a neat application of the two-pointer pattern, which is valuable for problems involving consecutive elements or pairs within a single array.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach. Consider the following sorted integer array with no duplicates:

nums = [0, 1, 2, 4, 5, 7]

We want to convert this array into a compact list of ranges and represent each range as a string. Here is how the provided method works:

1. Initialization:

   • The output list ans is initialized to [].
   • Set pointer i = 0 and pointer j = 0.

2. Iterating through the array (i < len(nums)):

   • i starts at 0. The first number is nums[i] = 0.

3. Exploration within the range:

   • We set j to the same value as i, so j = 0.
   • The condition j + 1 < len(nums) and nums[j + 1] == nums[j] + 1 allows j to move forward. Since nums[j+1] = 1 is one greater than nums[j] = 0, we increment j to 1.
   • Continue moving j to 2 since nums[j+1] = 2 is one greater than nums[j] = 1.
   • j cannot move past 2 because nums[j+1] = 4 is not one greater than nums[j] = 2.

4. Range formation:

   • With j stopped at 2, we call the helper function f(i, j) to get the range string.
   • Since i = 0 and j = 2, the range is 0, 1, 2, so ans receives the string "0->2".

5. Storing the range and moving on:

   • The string "0->2" is added to the ans list.
   • i is then set to j + 1, which is 3.

(Repeat steps 2-4 for the next set of ranges)

6. The same process repeats with the new value of i:

   • Set i = j + 1 = 3 and so nums[i] = 4. Since the next number 5 is a direct successor, j is incremented to 4.
   • The range nums[i] to nums[j] is 4->5, so "4->5" is added to ans.
   • Increment i to j + 1 = 5 where nums[i] = 7. Since 7 is not followed by an immediate successor, this is a single-value range.

7. Final range:

   • The final number 7 forms a range by itself.
   • ans receives the string "7".

8. Return the result:

   • The process concludes since i has reached the end of the array.
   • The final output list is ans = ["0->2", "4->5", "7"].

By this approach, the original array [0, 1, 2, 4, 5, 7] is succinctly represented as ["0->2", "4->5", "7"], where each string in the list represents a compact range from the array.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(n), where n is the number of elements in the input list nums. This is because the code iterates through the list only once, with a while loop that progresses from one range-starting element to the next. Within this loop, there is another while loop that finds the end of the current range, effectively advancing the outer loop's index j. Even though there is this nested loop, each element is considered only once for the range formation, which guarantees linear time complexity.

The space complexity of the given code is also O(n). In the worst case scenario, when there are no consecutive sequences, the ans list would contain the same number of elements as nums, with each element converted into a string. There is no additional space used that scales with the input size, except for the ans list which stores the resulting ranges.

Learn more about how to find time and space complexity quickly using problem constraints.