[Verse 1] Calculate the rabbit population, generation by generation F of n equals F of n minus one plus F of n minus two But naive recursion's got a problem, exponential devastation Same subproblems solved repeatedly, that's computational voodoo Tree of calls exploding outward, branches multiply like cancer Every node recalculates what ancestors already knew Memoization holds the answer, cache becomes our sweet enhancer Store results in lookup table, make redundant work taboo [Chorus] Memo-ize, memo-ize, remember what you've done before Hash table keeps the values, don't compute them anymore Top-down with a safety net, or bottom-up explore Fibonacci memoized, efficiency you can't ignore Cache it, stash it, lookup fast Linear time complexity at last [Verse 2] Initialize your memo structure, dictionary or array Check if subproblem's been computed before you start the dance Base cases zero and one return immediately, no delay Higher numbers need the cache-check, give stored values a chance If memo contains the answer, return it in a flash Otherwise compute recursively, then store before you leave Transform exponential nightmare into linear time cash Dynamic programming principles that make computers achieve [Chorus] Memo-ize, memo-ize, remember what you've done before Hash table keeps the values, don't compute them anymore Top-down with a safety net, or bottom-up explore Fibonacci memoized, efficiency you can't ignore Cache it, stash it, lookup fast Linear time complexity at last [Bridge] From two to the power n down to just n operations Overlapping subproblems meet optimal substructure Space-time tradeoff brings computational celebrations Memory investment pays dividends pure [Verse 3] Bottom-up approach builds table from the ground floor rising Fill array positions zero through n in sequence neat Each entry uses previous two, no recursive surprising Iterative solution makes the optimization complete Whether top-down memoization or bottom-up iteration Both eliminate redundancy that plagued the naive way Transform recursive relation into efficient calculation Fibonacci sequence conquered, exponential decay [Chorus] Memo-ize, memo-ize, remember what you've done before Hash table keeps the values, don't compute them anymore Top-down with a safety net, or bottom-up explore Fibonacci memoized, efficiency you can't ignore Cache it, stash it, lookup fast Linear time complexity at last [Outro] Dynamic programming paradigm Memoization sublime Fibonacci optimized for all time
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