The log y part is the hint they're basically handing you.
Start by clarifying the problem constraints (e.g., integer vs. floating-point, negative exponents, overflow) and then explain the binary exponentiation (exponentiation by squaring) algorithm. Walk through the iterative or recursive implementation, emphasizing how it reduces the number of multiplications to O(log y) by halving the exponent at each step.
Pro tip: Mention handling of edge cases like y=0, negative exponents, and integer overflow, and discuss whether to use recursion or iteration based on stack depth and performance. Also, note that for modular exponentiation, you can apply the modulo at each multiplication to prevent overflow.
Ask about input types (integer, float), range of y, negative exponents, and whether the result should be modulo something. Confirm expected behavior for y=0 and y<0.
Describe how to use the binary representation of the exponent: square the base and halve the exponent, multiplying the result when the current bit is 1. This yields O(log y) multiplications.
Write pseudocode or actual code for the chosen approach. For iterative: initialize result=1, while y>0: if y odd, result*=x; x*=x; y//=2. For recursive: if y==0 return 1; half=power(x, y//2); return half*half*(x if y odd else 1).
State that time complexity is O(log y) due to halving the exponent each iteration/recursion, and space complexity is O(1) for iterative or O(log y) for recursive due to call stack.
Mention handling negative exponents by computing the positive power and taking reciprocal, and modular exponentiation for large numbers. Also, note potential integer overflow and how to mitigate (e.g., using long long or modulo).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clarifying the problem constraints (e.g., input size, duplicates, sorted output) and then propose an efficient solution using a min-heap of size k, which runs in O(n log k) time and O(k) space. Discuss alternative approaches like sorting or quickselect, and analyze their trade-offs to demonstrate depth.
Pro tip: Mention that for very large n, a min-heap of size k is optimal when k is small, but if k is close to n, sorting or quickselect might be better; also note that quickselect has O(n) average time but O(n^2) worst-case, so randomization or median-of-medians can mitigate that.
Ask about input size, whether duplicates count separately, if the output needs to be sorted, and if the input can be modified. This shows attention to detail and avoids incorrect assumptions.
Suggest sorting the list and taking the last k elements, which is simple but O(n log n) time. Acknowledge its inefficiency for large n.
Explain that maintaining a min-heap of size k while iterating through the list yields O(n log k) time and O(k) space. For each element, if it's larger than the heap's root, replace the root and heapify.
Mention quickselect (average O(n)) and its worst-case O(n^2), and possibly a max-heap if k is large. Compare trade-offs based on constraints.
Summarize time and space complexity for each approach, and discuss edge cases like k > n, k = 0, negative numbers, and duplicates.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.