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Efficient Fibonacci Calculation with Memoization in Standard ML

(* Memoization to store previously computed Fibonacci numbers *) fun make_fib_table () = let (* Create a table to store Fibonacci numbers *) val table = Array.tabulate (100, fn i => if i = 0 orelse i = 1 then i else 0) fun fib n = if n < 0 orelse n >= Array.length table then raise Fail "Index out of bounds" else let (* Check if the value is already computed *) val _ = if Array.sub (table, n) = 0 andalso n > 1 then Array.update (table, n, fib (n - 1) + fib (n - 2)) else () in Array.sub (table, n) end in fib end (* Fibonacci number calculator with memoization *) val fib = (make_fib_table ()) (* Example usage *) val result_10 = fib 10

Explanation:

• Recursive Function with Memoization: The fib function is a classic example of how to use recursion for calculating Fibonacci numbers. Memoization (storing previously computed results) optimizes the recursive solution, demonstrating an essential technique for improving performance in recursive algorithms. This teaches students about the trade-offs between time complexity and space complexity.
• Pattern Matching and Functional Programming: The use of pattern matching and functional constructs like Array.tabulate and Array.update showcases Standard ML’s functional programming features. This helps students become familiar with idiomatic ways to manage state and mutable data in a functional programming paradigm.

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