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What is the time complexity of the top-down memoization approach for the Jump Game problem in the worst case, and why is the common belief that it is O(n) incorrect?

medium🪤 Complexity Trap Q5 of Q15

Greedy Algorithms - Jump Game (Can Reach End?)

What is the time complexity of the top-down memoization approach for the Jump Game problem in the worst case, and why is the common belief that it is O(n) incorrect?

AO(n) because each position is visited once

BO(n^2) because each position may recursively explore up to n next positions

CO(2^n) due to exponential recursion without memoization

DO(n log n) due to binary search optimization

Step-by-Step Solution

Solution:

Step 1: Analyze recursion calls
Each position calls recursive checks for all reachable next positions, up to n in worst case.
Step 2: Memoization reduces repeated calls but nested loops remain
Memoization avoids repeated work but nested for-loop over reachable positions causes O(n^2) complexity.
Final Answer:
Option B -> Option B
Quick Check:
Nested recursion with memo -> O(n^2), not O(n) [OK]

Quick Trick: Nested recursion loops cause O(n^2) time [OK]

Common Mistakes:

MISTAKES

Assuming memoization makes it O(n)
Ignoring nested loop over reachable positions

Trap Explanation:

PITFALL

Candidates think memoization means linear time, but nested loops cause quadratic time.

Interviewer Note:

CONTEXT

Tests understanding of recursion + memoization complexity traps.

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