Start by clarifying the problem and defining the recurrence relation: the number of ways to reach step n is the sum of ways to reach step n-1 and step n-2. Then, discuss multiple solutions from naive recursion to dynamic programming with O(n) time and O(1) space, highlighting the Fibonacci sequence connection.
Pro tip: Mention that this is essentially the Fibonacci sequence, and you can optimize space to O(1) by only keeping the last two values. Also, discuss how to handle large n with modular arithmetic if needed.
Confirm that n is a positive integer, and that the order of steps matters (e.g., 1+2 and 2+1 are distinct). Also, check if n=0 is considered (usually 1 way).
Let f(n) be the number of ways to reach step n. Then f(n) = f(n-1) + f(n-2), with base cases f(1)=1, f(2)=2.
Explain that a simple recursive solution has exponential time complexity due to overlapping subproblems, which is inefficient for large n.
Show how to use memoization (top-down) or tabulation (bottom-up) to achieve O(n) time and O(n) space. Then optimize to O(1) space by keeping only the last two values.
State time and space complexity, and discuss edge cases like n=0, n=1, and potential integer overflow for large n (suggest modular arithmetic).
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