From a18e3cb83b1ca1e5a7d5a5e531c08d092d7b3b2c Mon Sep 17 00:00:00 2001 From: Lusina <12752833+BrianLusina@users.noreply.github.com> Date: Tue, 10 Feb 2026 10:25:00 +0300 Subject: [PATCH 1/4] feat(algorithms, top-k-elements): smallest range covering elements from k lists using a heap --- algorithms/top_k_elements/__init__.py | 0 .../smallest_range_covering_k_lists/README.md | 142 ++++++++++++++++++ .../__init__.py | 46 ++++++ ...vering_elements_from_k_lists_example_1.png | Bin 0 -> 31675 bytes ...vering_elements_from_k_lists_example_2.png | Bin 0 -> 33374 bytes ...vering_elements_from_k_lists_example_3.png | Bin 0 -> 32375 bytes ..._elements_from_k_lists_heap_solution_1.png | Bin 0 -> 17103 bytes ...elements_from_k_lists_heap_solution_10.png | Bin 0 -> 40508 bytes ...elements_from_k_lists_heap_solution_11.png | Bin 0 -> 29142 bytes ...elements_from_k_lists_heap_solution_12.png | Bin 0 -> 28769 bytes ...elements_from_k_lists_heap_solution_13.png | Bin 0 -> 40829 bytes 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algorithms/top_k_elements/smallest_range_covering_k_lists/test_smallest_range_covering_elements_from_k_lists.py diff --git a/algorithms/top_k_elements/__init__.py b/algorithms/top_k_elements/__init__.py new file mode 100644 index 00000000..e69de29b diff --git a/algorithms/top_k_elements/smallest_range_covering_k_lists/README.md b/algorithms/top_k_elements/smallest_range_covering_k_lists/README.md new file mode 100644 index 00000000..7b3e65eb --- /dev/null +++ b/algorithms/top_k_elements/smallest_range_covering_k_lists/README.md @@ -0,0 +1,142 @@ +# Smallest Range Covering Elements from K Lists + +You are given k sorted lists of integers, nums, where each list in nums is in non-decreasing order. Your task is to find +the smallest range that contains at least one element from each of the k lists. + +A range [a,b] is considered smaller than another range [c,d] if `b − a < d − c`, or `a List[int]: + # min heap that will store (value, list index, element index) for the smallest elements + min_heap: List[Tuple[int, int, int]] = [] + heapq.heapify(min_heap) + # Initialize the maximum value among the current elements in the heap + max_value = float("-inf") + # Initialize range boundaries with placeholders + range_start = 0 + range_end = float("inf") + + number_of_lists = len(nums) + + # insert the first elements from each list into the min-heap and update max_value to be the maximum value seen so far + for list_idx in range(number_of_lists): + # (value, list index, index within the list) + value = nums[list_idx][0] + heapq.heappush(min_heap, (value, list_idx, 0)) + # Track the max value among the current elements + max_value = max(max_value, value) + + # Continue processing until we can't proceed further + while len(min_heap) == number_of_lists: + # Pop the smallest element from the heap (min_val from one of the lists) + min_value, row, col = heapq.heappop(min_heap) + + # Update the smallest range if the current range is smaller + if max_value - min_value < range_end - range_start: + range_start = min_value + range_end = max_value + + # If possible, add the next element from the same row to the heap + if col + 1 < len(nums[row]): + next_value = nums[row][col + 1] + # Push the next element from the same list + heapq.heappush(min_heap, (next_value, row, col + 1)) + # Update max_val if needed + max_value = max(max_value, next_value) + + # If any list runs out of elements, we can't form a complete range anymore + + # Return the smallest range found + return [range_start, range_end] diff --git a/algorithms/top_k_elements/smallest_range_covering_k_lists/images/examples/smallest_range_covering_elements_from_k_lists_example_1.png 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E0eQoQ;s5{u literal 0 HcmV?d00001 diff --git a/algorithms/top_k_elements/smallest_range_covering_k_lists/test_smallest_range_covering_elements_from_k_lists.py b/algorithms/top_k_elements/smallest_range_covering_k_lists/test_smallest_range_covering_elements_from_k_lists.py new file mode 100644 index 00000000..ff36eae3 --- /dev/null +++ b/algorithms/top_k_elements/smallest_range_covering_k_lists/test_smallest_range_covering_elements_from_k_lists.py @@ -0,0 +1,28 @@ +import unittest +from typing import List +from parameterized import parameterized +from algorithms.top_k_elements.smallest_range_covering_k_lists import smallest_range + +SMALLEST_RANGE_COVERING_ELEMENTS_FROM_K_LISTS_TESTS = [ + ([[4, 10, 15, 24, 26], [0, 9, 12, 20], [5, 18, 22, 30]], [20, 24]), + ([[1, 2, 3], [1, 2, 3], [1, 2, 3]], [1, 1]), + ([[1, 5, 8], [4, 12], [7, 8, 10]], [4, 7]), + ([[2, 6, 10], [1, 5, 9], [4, 8, 12]], [4, 6]), + ([[1, 3, 7], [2, 4, 8], [5, 6, 9]], [3, 5]), + ([[1, 5], [3, 7], [4, 6]], [3, 5]), + ([[1, 5], [4, 6], [6, 8], [11, 15]], [5, 11]), + ([[1, 5]], [1, 1]), + ([[1, 9], [3, 8], [4, 4]], [1, 4]), + ([[1, 2], [3, 4], [8, 8]], [2, 8]), +] + + +class SmallestRangeCoveringElementsFromKListsTestCase(unittest.TestCase): + @parameterized.expand(SMALLEST_RANGE_COVERING_ELEMENTS_FROM_K_LISTS_TESTS) + def test_smallest_range(self, nums: List[List[int]], expected: List[int]): + actual = smallest_range(nums) + self.assertEqual(expected, actual) + + +if __name__ == "__main__": + unittest.main() From e5caaa00d484f4dc0d0acc607802e06382091ea8 Mon Sep 17 00:00:00 2001 From: Lusina <12752833+BrianLusina@users.noreply.github.com> Date: Tue, 10 Feb 2026 10:51:43 +0300 Subject: [PATCH 2/4] feat(algorithms, top-k-elements): smallest range covering alements from k lists, two pointer and brute force --- .../smallest_range_covering_k_lists/README.md | 153 ++++++++++++++++++ .../__init__.py | 78 ++++++++- ...ts_from_k_lists_two_pointer_solution_1.png | Bin 0 -> 57017 bytes ...ts_from_k_lists_two_pointer_solution_2.png | Bin 0 -> 62242 bytes ...ts_from_k_lists_two_pointer_solution_3.png | Bin 0 -> 35681 bytes ...ts_from_k_lists_two_pointer_solution_4.png | Bin 0 -> 33298 bytes ...ts_from_k_lists_two_pointer_solution_5.png | Bin 0 -> 53160 bytes ...ts_from_k_lists_two_pointer_solution_6.png | Bin 0 -> 43871 bytes ...ts_from_k_lists_two_pointer_solution_7.png | Bin 0 -> 38799 bytes ...ts_from_k_lists_two_pointer_solution_8.png | Bin 0 -> 32693 bytes ...ts_from_k_lists_two_pointer_solution_9.png | Bin 0 -> 52124 bytes ...st_range_covering_elements_from_k_lists.py | 20 ++- 12 files changed, 249 insertions(+), 2 deletions(-) create mode 100644 algorithms/top_k_elements/smallest_range_covering_k_lists/images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_1.png create mode 100644 algorithms/top_k_elements/smallest_range_covering_k_lists/images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_2.png create mode 100644 algorithms/top_k_elements/smallest_range_covering_k_lists/images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_3.png create mode 100644 algorithms/top_k_elements/smallest_range_covering_k_lists/images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_4.png create mode 100644 algorithms/top_k_elements/smallest_range_covering_k_lists/images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_5.png create mode 100644 algorithms/top_k_elements/smallest_range_covering_k_lists/images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_6.png create mode 100644 algorithms/top_k_elements/smallest_range_covering_k_lists/images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_7.png create mode 100644 algorithms/top_k_elements/smallest_range_covering_k_lists/images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_8.png create mode 100644 algorithms/top_k_elements/smallest_range_covering_k_lists/images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_9.png diff --git a/algorithms/top_k_elements/smallest_range_covering_k_lists/README.md b/algorithms/top_k_elements/smallest_range_covering_k_lists/README.md index 7b3e65eb..9037342d 100644 --- a/algorithms/top_k_elements/smallest_range_covering_k_lists/README.md +++ b/algorithms/top_k_elements/smallest_range_covering_k_lists/README.md @@ -53,6 +53,69 @@ Output: [1,1] ### Optimal Brute Force +We need to find the smallest range that contains at least one number from each of the k sorted lists. At first glance, +a simple brute force solution comes to mind, i.e., checking every combination of elements from the lists to find the +smallest range. However, that would involve too many comparisons and will lead to TLE. Instead, we can refine this +process into something more manageable. + +At any moment, we need to select one number from each list. So, to find the smallest range, we need to minimize the +difference between the largest and smallest numbers chosen at each step. The important point here is that, at any time, +our range is defined by the smallest number chosen and the largest number chosen. + +So we need to select the smallest number among the current numbers picked from each list and move forward by choosing the +next number from the same list that gave us this smallest number. This makes sense because moving forward in any other +list would only increase the range, which we want to avoid. We repeat this process of updating the smallest number and +checking if the new range is smaller than our previously found range. If it is, we update the range. + +We continue this until we reach the end of one of the lists because, at that point, it’s no longer possible to select a +number from each list. + +Algorithm steps: + +- Initialize k to the number of lists in nums and create an array indices to keep track of the current index of each + list, initializing all to 0. +- Initialize an array range to store the smallest range, starting with {0, INT_MAX}. +- Enter an infinite loop: + - Initialize curMin to INT_MAX, curMax to INT_MIN, and minListIndex to 0. + - Iterate over each list to find the current minimum and maximum values: + - For each list i, retrieve the current element using indices[i]. + - Update curMin if the current element is less than curMin, and set minListIndex to i. + - Update curMax if the current element is greater than curMax. + - After checking all lists, if the difference curMax - curMin is smaller than the current range (range[1] - range[0]), + update range to {curMin, curMax}. + - Move to the next element in the list that had the minimum value by incrementing indices[minListIndex]. + - If the updated index equals the size of nums[minListIndex], break the loop (all elements have been processed). +- Return the smallest range stored in range. + +> Note: Due to Python's relatively slower execution speed, the optimal brute-force solution will lead to a Time Limit +> Exceeded (TLE) error when using Python3. However, this same solution will perform adequately in other programming +> languages. + +#### Complexity Analysis + +Let `n` be the total number of elements across all lists and k be the number of lists. + +##### Time Complexity + +In each iteration of the while (true) loop, we traverse all k lists to find the current minimum and maximum. This +takes O(k) time. + +The loop continues until at least one of the lists is fully traversed. In the worst case, every element from every list +is visited, and the total number of elements across all lists is n. Therefore, the loop runs O(n) times. + +Overall, the time complexity becomes O(n⋅k). + +##### Space Complexity + +The space complexity is dominated by the indices and range arrays, both of which have size proportional to k, the number +of lists. + +The indices array stores the current index of each list, so it takes O(k) space. + +The range array also stores two integers, so it takes O(1) space. + +Hence, the overall space complexity is O(k). + ### Heap (Priority Queue) The core idea of this solution is to find the smallest range that includes at least one number from each of the k sorted @@ -140,3 +203,93 @@ the output range (two integers) is negligible and does not contribute to the ove ### Two-Pointer +Since we need a range that includes one number from each of the k lists, we can think of this as a subarray problem. +However, the numbers are spread across multiple lists. To simplify, we can combine all the lists into a single sorted +list of numbers. When merging, we also keep track of which list each number came from, since the problem requires at +least one number from each original list in the final range. + +Once we have the merged list, the problem becomes finding the smallest range (or subarray) in this list that contains at +least one element from each of the original k lists. This is a common scenario for a sliding window or two-pointer +approach: we want to expand and shrink the window (subarray) dynamically to find the minimum range that meets the criteria. + +The right pointer will expand the window by moving forward in the merged list, and the left pointer will shrink the +window once we know the window contains at least one element from each list. + +As the right pointer moves through the merged list, we need to ensure that the current subarray includes at least one +number from each list. So we keep track of how many lists are "covered" by the current subarray (i.e., how many of the +k lists have at least one number in the current window). + +Once all lists are covered, the window between the left and right pointers represents a valid range. We then check if +this range is the smallest we've found so far. + +After finding a valid range, we need to shrink the window (move the left pointer forward) to see if we can make the range +even smaller while still keeping one number from each list in the subarray. As we move the left pointer forward, we check +if we lose coverage from any list. If we do, we stop shrinking and start expanding the window again by moving the right +pointer. + +We will continue this until we can no longer expand the window (i.e., the right pointer reaches the end of the merged list). +By this point, we have explored all possible ranges, and the smallest valid range is our final answer. + +![Solution 2.1](./images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_1.png) +![Solution 2.2](./images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_2.png) +![Solution 2.3](./images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_3.png) +![Solution 2.4](./images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_4.png) +![Solution 2.5](./images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_5.png) +![Solution 2.6](./images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_6.png) +![Solution 2.7](./images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_7.png) +![Solution 2.8](./images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_8.png) +![Solution 2.9](./images/solutions/smallest_range_covering_elements_from_k_lists_two_pointer_solution_9.png) + +Algorithm steps: + +- Initialize an empty array `merged` to store pairs of numbers and their respective list indices. +- Merge all lists into `merged`: + - For each list in `nums`, iterate through its numbers and add each number along with its list index to merged. +- Sort the `merged` array to facilitate the two-pointer technique. +- Initialize a frequency map `freq` to keep track of how many times each list is represented in the current window. +- Set the `left` pointer to 0, `count` to 0, and initialize `rangeStart` to 0 and `rangeEnd` to INT_MAX. +- Use a `right` pointer to iterate through the merged array: + - Increment the count for the list index in `freq` for `merged[right]`. + - If the count for this list index becomes 1, increment `count` (indicating a new list is represented). +- When all lists are represented (i.e., `count == nums.size()`): + - Calculate the current range as `curRange = merged[right].first - merged[left].first`. + - If `curRange` is smaller than the previously found range (`rangeEnd - rangeStart`): + - Update rangeStart and rangeEnd to the current numbers. + - Decrement the frequency count for the leftmost number (i.e., `merged[left]`). + - If this list index's frequency becomes 0, decrement `count` (indicating that a list is no longer represented). + - Move the `left` pointer to the right to attempt shrinking the window. +- After completing the iteration, return the smallest range as a array containing `rangeStart` and `rangeEnd`. + +#### Complexity Analysis + +Let `n` be the total number of elements across all lists and `k` be the number of lists. + +##### Time Complexity + +The first nested loop iterates over `k` lists, and for each list, it iterates through its elements. In the worst case, +this requires `O(n)` time since we are processing all elements once. + +After merging, we sort the merged array which contains n elements. Sorting has a time complexity of `O(nlog(n))`. + +The two-pointer approach iterates through the merged list once (with the right pointer) and may also move the left +pointer forward multiple times. In total, each pointer will traverse the merged list at most `n` times. + +Combining these steps, the overall time complexity is: `O(nlog(n))` + +##### Space Complexity + +We create a merged array to hold n elements, which requires `O(n)` space. + +We use an unordered map (`freq`) that can potentially store `k` elements (one for each list). Thus, this requires `O(k)` +space. + +Some extra space is used when we sort an array. The space complexity of the sorting algorithm (S) depends on the +programming language. + +- In Python, the sort method sorts a list using the Timsort algorithm which is a combination of Merge Sort and Insertion + Sort and has a space complexity of `O(n)`. +- In C++, the sort() function is implemented as a hybrid of Quick Sort, Heap Sort, and Insertion Sort, with a worst-case + space complexity of `O(log(n))`. +- In Java, Arrays.sort() is implemented using a variant of the Quick Sort algorithm which has a space complexity of `O(log(n))`. + +Combining these, the overall space complexity is: `O(n)` diff --git a/algorithms/top_k_elements/smallest_range_covering_k_lists/__init__.py b/algorithms/top_k_elements/smallest_range_covering_k_lists/__init__.py index 07562b9b..e804f5b7 100644 --- a/algorithms/top_k_elements/smallest_range_covering_k_lists/__init__.py +++ b/algorithms/top_k_elements/smallest_range_covering_k_lists/__init__.py @@ -1,4 +1,5 @@ -from typing import List, Tuple +from collections import defaultdict +from typing import List, Tuple, DefaultDict import heapq @@ -44,3 +45,78 @@ def smallest_range(nums: List[List[int]]) -> List[int]: # Return the smallest range found return [range_start, range_end] + + +def smallest_range_two_pointer(nums: List[List[int]]) -> List[int]: + merged: List[Tuple[int, int]] = [] + + # merge all lists with their list index + for list_idx, num_list in enumerate(nums): + for num in num_list: + merged.append((num, list_idx)) + + # sor the merged list + merged.sort() + + # Two pointers to track the smallest range + freq: DefaultDict[int, int] = defaultdict(int) + left, count = 0, 0 + range_start, range_end = 0, float("inf") + + for right in range(len(merged)): + val = merged[right][1] + freq[val] += 1 + if freq[val] == 1: + count += 1 + + # when all lists are represented, try to shrink the window + while count == len(nums): + current_range = merged[right][0] - merged[left][0] + if current_range < range_end - range_start: + range_start = merged[left][0] + range_end = merged[right][0] + + freq[merged[left][1]] -= 1 + if freq[merged[left][1]] == 0: + count -= 1 + + left += 1 + + return [range_start, range_end] + + +def smallest_range_brute_force(nums: List[List[int]]) -> List[int]: + k = len(nums) + # Stores the current index of each list + indices = [0] * k + # To track the smallest range + range_list = [0, float("inf")] + + while True: + cur_min, cur_max = float("inf"), float("-inf") + min_list_index = 0 + + # Find the current minimum and maximum values across the lists + for i in range(k): + current_element = nums[i][indices[i]] + + # Update the current minimum + if current_element < cur_min: + cur_min = current_element + min_list_index = i + + # Update the current maximum + if current_element > cur_max: + cur_max = current_element + + # Update the range if a smaller one is found + if cur_max - cur_min < range_list[1] - range_list[0]: + range_list[0] = cur_min + range_list[1] = cur_max + + # Move to the 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b/algorithms/top_k_elements/smallest_range_covering_k_lists/test_smallest_range_covering_elements_from_k_lists.py index ff36eae3..03e723fc 100644 --- a/algorithms/top_k_elements/smallest_range_covering_k_lists/test_smallest_range_covering_elements_from_k_lists.py +++ b/algorithms/top_k_elements/smallest_range_covering_k_lists/test_smallest_range_covering_elements_from_k_lists.py @@ -1,7 +1,11 @@ import unittest from typing import List from parameterized import parameterized -from algorithms.top_k_elements.smallest_range_covering_k_lists import smallest_range +from algorithms.top_k_elements.smallest_range_covering_k_lists import ( + smallest_range, + smallest_range_two_pointer, + smallest_range_brute_force, +) SMALLEST_RANGE_COVERING_ELEMENTS_FROM_K_LISTS_TESTS = [ ([[4, 10, 15, 24, 26], [0, 9, 12, 20], [5, 18, 22, 30]], [20, 24]), @@ -23,6 +27,20 @@ def test_smallest_range(self, nums: List[List[int]], expected: List[int]): actual = smallest_range(nums) self.assertEqual(expected, actual) + @parameterized.expand(SMALLEST_RANGE_COVERING_ELEMENTS_FROM_K_LISTS_TESTS) + def test_smallest_range_two_pointers( + self, nums: List[List[int]], expected: List[int] + ): + actual = smallest_range_two_pointer(nums) + self.assertEqual(expected, actual) + + @parameterized.expand(SMALLEST_RANGE_COVERING_ELEMENTS_FROM_K_LISTS_TESTS) + def test_smallest_range_brute_force( + self, nums: List[List[int]], expected: List[int] + ): + actual = smallest_range_brute_force(nums) + self.assertEqual(expected, actual) + if __name__ == "__main__": unittest.main() From c8905e77c236048c340d130c80de159658a9e652 Mon Sep 17 00:00:00 2001 From: github-actions <${GITHUB_ACTOR}@users.noreply.github.com> Date: Tue, 10 Feb 2026 07:52:07 +0000 Subject: [PATCH 3/4] updating DIRECTORY.md --- DIRECTORY.md | 3 +++ 1 file changed, 3 insertions(+) diff --git a/DIRECTORY.md b/DIRECTORY.md index 701993c6..8f16308c 100644 --- a/DIRECTORY.md +++ b/DIRECTORY.md @@ -312,6 +312,9 @@ * [Test Find All Subsets](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/subsets/find_all_subsets/test_find_all_subsets.py) * Taxi Numbers * [Taxi Numbers](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/taxi_numbers/taxi_numbers.py) + * Top K Elements + * Smallest Range Covering K Lists + * [Test Smallest Range Covering Elements From K Lists](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/top_k_elements/smallest_range_covering_k_lists/test_smallest_range_covering_elements_from_k_lists.py) * Two Pointers * Array 3 Pointers * [Test Array 3 Pointers](https://github.com/BrianLusina/PythonSnips/blob/master/algorithms/two_pointers/array_3_pointers/test_array_3_pointers.py) From d8e1ec66f061d86277fe3715819c69763f5cfdad Mon Sep 17 00:00:00 2001 From: Lusina <12752833+BrianLusina@users.noreply.github.com> Date: Tue, 10 Feb 2026 10:59:21 +0300 Subject: [PATCH 4/4] Update algorithms/top_k_elements/smallest_range_covering_k_lists/__init__.py Co-authored-by: coderabbitai[bot] <136622811+coderabbitai[bot]@users.noreply.github.com> --- .../top_k_elements/smallest_range_covering_k_lists/__init__.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/algorithms/top_k_elements/smallest_range_covering_k_lists/__init__.py b/algorithms/top_k_elements/smallest_range_covering_k_lists/__init__.py index e804f5b7..b3b7804c 100644 --- a/algorithms/top_k_elements/smallest_range_covering_k_lists/__init__.py +++ b/algorithms/top_k_elements/smallest_range_covering_k_lists/__init__.py @@ -55,7 +55,7 @@ def smallest_range_two_pointer(nums: List[List[int]]) -> List[int]: for num in num_list: merged.append((num, list_idx)) - # sor the merged list + # sort the merged list merged.sort() # Two pointers to track the smallest range

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