Start by clarifying the problem constraints (e.g., array size, value range, duplicates) and then present multiple solutions with trade-offs. Begin with a simple sorting approach, then optimize using a min-heap of size k or Quickselect for average O(n) time. Discuss the choice based on expected input and system constraints.
Pro tip: Mention that Quickselect has O(n) average but O(n^2) worst-case, and you can avoid the worst-case by using a randomized pivot or the Median of Medians algorithm. Also, note that for streaming data or when k is small, a heap is more practical.
Ask about input size, value range, whether duplicates count as separate elements, and if the array can be modified. This shows you consider edge cases and practical limits.
Suggest sorting the array and returning the element at index n-k. This is simple and O(n log n) time, O(1) extra space if in-place, but may be inefficient for large n.
Explain using a min-heap of size k: iterate through the array, push elements, and if heap size exceeds k, pop the smallest. At the end, the heap root is the kth largest. This is O(n log k) time and O(k) space, good for large n and small k.
Describe the Quickselect algorithm: partition the array around a pivot, then recursively search the side that contains the kth largest. Average O(n) time, worst-case O(n^2), but can be mitigated with random pivots.
Discuss when to use each approach: sorting for simplicity, heap for streaming or small k, Quickselect for optimal average time when the array can be modified. Mention that Quickselect modifies the array, which may not be allowed.
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Took me a minute to nail down the key normalization.
Clarify that two strings are in the same shift sequence if the differences between consecutive characters (mod 26) are identical. Then propose a solution that computes a canonical key for each string, such as the tuple of cyclic differences, and groups strings using a hash map. Discuss time and space complexity, and consider edge cases like different lengths and empty strings.
Pro tip: Mention that the difference array can be normalized by taking modulo 26 to handle negative differences, and that using a tuple as a hash key in Python is efficient. Also, note that strings of different lengths cannot be in the same group, so you can skip them early.
Confirm that shifting means adding the same integer k to each character (mod 26), and that strings must be of equal length to be in the same group. Ask if the shift amount can be any integer.
For a string s, compute the sequence of differences d_i = (s[i+1] - s[i]) mod 26 for i from 0 to n-2. This sequence uniquely identifies the shift equivalence class.
Iterate through the list of strings, compute the key for each, and use a dictionary to map each key to a list of strings. Skip strings of different lengths or handle them separately.
Time complexity is O(N * L) where N is number of strings and L is average length. Space is O(N * L) for storing keys. Discuss empty strings, single-character strings, and strings with different lengths.
Walk through a small example to verify correctness. Consider if the key can be further optimized (e.g., using a string representation of differences) and discuss trade-offs.
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Start by clarifying the problem constraints and defining what 'overlapping' means (e.g., intersection area > 0). Then, discuss a sweep-line algorithm that processes events sorted by x-coordinate, maintaining active rectangles and merging them along the y-axis. Finally, analyze the time and space complexity and consider trade-offs with simpler but less efficient approaches.
Pro tip: Mention that the problem is non-trivial because merging rectangles can create new overlaps, so a naive pairwise merge may not yield the minimal set. Emphasize the importance of handling edge cases like touching edges or containment.
Ask about input size, whether rectangles are axis-aligned, if overlapping includes touching edges, and if the output must be minimal or just non-overlapping.
Specify that two rectangles overlap if their projections on both axes overlap. Merging them results in the bounding box of the union.
Sort events (left and right edges) by x-coordinate. Maintain a set of active rectangles and merge intervals along the y-axis at each x-interval.
When active rectangles change, compute the merged y-intervals and emit rectangles for the previous x-interval. Ensure no overlapping output rectangles.
Discuss O(n log n + k) time where k is the number of output rectangles, and O(n) space. Compare with simpler O(n^2) pairwise merge that may not be minimal.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Use the BST property to prune branches: if the current node's value is less than low, skip the left subtree; if greater than high, skip the right subtree. Otherwise, include the node's value and recursively process both subtrees. This yields an efficient O(n) worst-case but often O(log n + k) time where k is the number of nodes in range.
Pro tip: Mention that you can further optimize by stopping recursion when the node's value is out of range, and discuss how this pruning makes the solution efficient for large trees. Also, clarify that the sum can be computed using any traversal (in-order, pre-order) but pruning is key.
Confirm that the tree is a BST, the range is inclusive, and node values are integers. Ask if the tree can be empty or if low > high.
Decide on a recursive DFS approach that leverages BST ordering to prune subtrees that cannot contain values in range.
Write a function that takes a node and returns the sum. If node is null, return 0. If node.val < low, recurse only on right; if node.val > high, recurse only on left; else include node.val and recurse both sides.
Explain that time complexity is O(n) worst-case but typically O(log n + k) where k is nodes in range, and space is O(h) for recursion stack.
Walk through a simple BST and range to verify correctness, including edge cases like empty tree or range outside all values.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.