Question

Memoization (also called tabulation) is a technique that enables a procedure to record, in a local table, values that have previously been computed. This technique can make a vast difference in the performance of a program. A memoized procedure maintains a table in which values of previous calls are stored using as keys the arguments that produced the values. When the memoized procedure is asked to compute a value, it first checks the table to see if the value is already there and, if so, just returns that value. Otherwise, it computes the new value in the ordinary way and stores this in the table. As an example of memoization, recall from section \(1.2 .2\) the exponential process for computing Fibonacci numbers: \(\begin{aligned}(\text { define }(f i b ~ n)& \\\\(\text { cond }((=\mathrm{n})&0) \\ &((=\mathrm{n} 1)&1) \\ &(e l s e(+(f i b(-\mathrm{n} 1))\\\ & &(f i b(-\mathrm{n}&2))))) \end{aligned}\) The memoized version of the same procedure is \(\begin{aligned}&\text { (define memo-fib } \\\&\text { (memoize (lambda ( } \mathrm{n} \text { ) } \\\&\begin{array}{rll}\text { (cond } & ((=\mathrm{n} & 0) & 0) \\ & ((=\mathrm{n} & 1) & 1)\end{array} \\\& & (\text { else (+ (memo-fib (-n 1)) } \\\& & \end{aligned}\) where the memoizer is defined as (define (memoize f) (let ((table (make-table))) (lambda (x) (let ((previously-computed-result (lookup x table))) (or previously-computed-result (let ((result (f x))) (insert! x result table) result)))))) Draw an environment diagram to analyze the computation of (memo-fib 3 ). Explain why memo-fib computes the \(n\)th Fibonacci number in a number of steps proportional to \(n\). Would the scheme still work if we had simply defined memo-fib to be (memoize fib)?

Short answer

`memo-fib` operates linearly, checking previously computed values in a table. Redefining would cause inefficiency due to independent lambda environments.

Step-by-step solution

Setup Environment

Begin by initializing the environment for the execution of the `memo-fib` function. This includes creating a memory table to store computed results and binding the `memo-fib` name to the memoized Fibonacci function.

memo-fib Initialization

The `memo-fib` function is defined using a lambda expression that takes an argument \( n \). The computation is wrapped in a memoizer that handles caching results.

Evaluate (memo-fib 3)

When `(memo-fib 3)` is called, the memoizer first checks the table for a pre-computed result for \( n = 3 \). Since the table is initially empty, the computation proceeds.

Compute memo-fib(2)

The computation requires evaluating `(memo-fib 2)`. Again, the table is checked and found empty for \( n = 2 \), so it proceeds to compute this value.

Base Cases Computation

To find `(memo-fib 2)`, both base cases `(memo-fib 1)` and `(memo-fib 0)` need evaluation. These are base cases: `(memo-fib 1)` returns 1 and `(memo-fib 0)` returns 0, as per the Fibonacci sequence.

Insert Result into Table

The result of `(memo-fib 2)` is calculated as 1 by adding results from `(memo-fib 1)` and `(memo-fib 0)`. It is then stored in the table to avoid recalculating in future calls.

Compute memo-fib(3) Completion

With the result of `(memo-fib 2)` stored, computing `(memo-fib 3)` involves adding the results of `(memo-fib 2)` and `(memo-fib 1)` (which is already known). The result is 2, storing it in the table for future reference.

Memoization Efficiency

Analysis of `memo-fib` shows that each new computation checks the table first, performing actual computation only for not-yet-computed values. This makes the process linear, with steps proportional to \( n \), rather than exponential.

Redefining memo-fib Analysis

If `memo-fib` were defined as `(memoize fib)`, it would indeed work, though it would still recompute values for each call because the table within `memoize` would not share values across separate instances of the lambda function.

Key concepts

Fibonacci Sequence

The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding ones, often starting with 0 and 1. This pattern is easy to understand but becomes exponentially intensive to compute as the series progresses using naive recursion.
This is because each function call generates two more calls until base cases are reached. Imagine computing \( F(n) \):

• For each \( F(n) \), you compute \( F(n-1) \) and \( F(n-2) \).
• Each \( F(n-1) \) requires \( F(n-2) \) and \( F(n-3) \), and so on.
• This branching results in a tree structure where calculations overlap extensively.

To gain efficiency, memoization preserves previously computed values, thereby transforming this exponential task into a linear one. As demonstrated, using memoization with the Fibonacci sequence, you reduce redundant computations, storing results to reuse later in the process.

Functional Programming

Functional Programming is a paradigm focused on writing functions that avoid changing state and mutable data. It's all about composing pure functions and using higher-order functions such as `map`, `filter`, and `reduce`.
Key Characteristics:

• Immutable State: No variables that can be changed.
• Pure Functions: Given the same input, they always return the same output without side effects.
• Higher-Order Functions: Functions are first-class citizens. They can be passed as arguments or returned from other functions.

In our exercise, the use of `lambda` within the `memo-fib` function is a testament to the power of functional programming. It encapsulates the Fibonacci calculation function without relying on external state changes, and it pairs well with memoization to efficiently manage state and computation.

Algorithm Optimization

Algorithm Optimization involves making your code more efficient, reducing resource consumption, or improving execution speed. Memoization is a standout technique for optimizing recursive algorithms like the Fibonacci sequence.

Benefits of memoization:

• Reduces Time Complexity: By storing results of expensive function calls and returning the cached result when the same inputs recur, the time complexity for the Fibonacci sequence calculation becomes linear, \( O(n) \), from exponential, \( O(2^n) \).
• Increases Efficiency: By avoiding unnecessary recalculations, the overall efficiency of the algorithm increases substantially.

To effectively optimize a recursive algorithm, focus on:

• Identifying overlapping subproblems that can benefit from cached results.
• Implementing a storage strategy (a dictionary or table) to hold previously calculated outputs.

Algorithm optimization, using techniques like memoization, is crucial for making systems more robust, scalable, and responsive, especially for tasks involving intensive computations.