Question

A famous problem, first raised by R. Hamming, is to enumerate, in ascending order with no repetitions, all positive integers with no prime factors other than 2,3 , or 5 . One obvious way to do this is to simply test each integer in turn to see whether it has any factors other than 2,3 , and 5 . But this is very inefficient, since, as the integers get larger, fewer and fewer of them fit the requirement. As an alternative, let us call the required stream of numbers \(S\) and notice the following facts about it. \- S begins with \(1 .\) \- The elements of (scale-stream S 2) are also elements of S. \- The same is true for (scale-stream \(\mathrm{S} 3\) ) and (scale-stream \(5 \mathrm{~S}\) ). \- These are all the elements of \(\mathrm{S}\). Now all we have to do is combine elements from these sources. For this we define a procedure merge that combines two ordered streams into one ordered result stream, eliminating repetitions: (define (merge s1 s2) (cond ((stream-null? s1) s2) ((stream-nul1? s2) s1) (else (let ( s1car (stream-car s1)) (s2car (stream-car s2))) (cond ((< s1car s2car) (cons-stream s1car (merge (stream-cdr s1) s2))) ((> s1car s2car) (cons-stream s2car (merge s1 (stream-cdr s2)))) (else (cons-stream s1car (merge (stream-cdr s1) Then the required stream may be constructed with merge, as follows: (define \(\mathrm{S}\) (cons-stream 1 (merge \(\langle ? ?\rangle\langle ? ?\rangle)\) ) Fill in the missing expressions in the places marked \(\langle ? ?\rangle\) above.

Short answer

Fill with `(scale-stream S 2)` and `(merge (scale-stream S 3) (scale-stream S 5))`. This constructs sequence S correctly.

Step-by-step solution

Understanding the Problem

We are tasked with enumerating all positive integers that have no prime factors other than 2, 3, or 5. This entails generating numbers by multiples of these primes only. This sequence is known as the Hamming numbers or Regular numbers, starting with 1.

Recognize the Components of the Stream

We begin with the smallest Hamming number, which is 1. The subsequent numbers are obtained by multiplying 1 and its successors by 2, 3, and 5, respectively, creating three separate streams that need to be merged: scale-stream S 2, scale-stream S 3, and scale-stream S 5.

Understand the Merge Function

The provided merge function combines two streams into a single sorted stream without repetitions. It processes elements from each stream, adding the smaller element to the result stream, while advancing the stream pointers appropriately.

Formulating the Expression for Stream S

Stream S should be defined using `merge` functions on three scaled streams starting with 1 for both correct scaling and alignment. We need to fill in expressions that reflect these operations: scale-stream S 2, scale-stream S 3, and scale-stream S 5, to reflect all numbers that are multiples of 2, 3, and 5, respectively.

Fill in the Missing Expressions

To construct stream S, initiate it with 1 and combine the scaled streams: `merge (scaled-stream S 2) (merge (scaled-stream S 3) (scaled-stream S 5))`. Specifically, fill as: `(define S (cons-stream 1 (merge (scale-stream S 2) (merge (scale-stream S 3) (scale-stream S 5))))`. This structure generates the correct sequence of Hamming numbers.

Key concepts

Streams and Sequences

Streams and sequences can be a powerful concept when it comes to generating ordered lists of data efficiently without excessive computation. A stream, as we use it here, is a potentially infinite list that is generated incrementally. Unlike traditional lists, which are typically calculated all at once, streams are generated piece by piece as needed. This allows for managing large sequences efficiently. Streams are especially useful in problems like generating Hamming numbers, where the sequence can grow indefinitely. By starting with a known element, such as 1 in this case, successive elements are derived by applying operations, like scaling by a factor, which creates a transform initially specified by the problem. Sequences, on the other hand, are explicitly enumerated series of numbers. In functional programming, these sequences might underpin streams, or be generated through a procedure involving multiple elements like the merge function used here. Understanding streams' lazy evaluation can substantially improve the efficiency when dealing with such potentially infinite lists.

Prime Factors

Prime factors play a crucial role in understanding the generation of specific numeric sequences, like Hamming numbers, from a mathematical perspective. Prime factors of a number are the prime numbers that multiply together to give the original number. When we talk about Hamming numbers, we specifically consider numbers whose prime factors are limited to 2, 3, and 5. This limitation makes determining membership in the set of Hamming numbers easier. Any number that cannot be expressed via multiplication exclusively from these three primes falls outside our desired sequence. In the context of this exercise, the notion of prime factors guides the creation of streams that include numbers like 2, 3, and 5, using operations like scaling and merging. Understanding prime factor decomposition is fundamental not only for generating such sequences but also for ensuring that the resulting stream efficiently represents exactly the numbers required.

Algorithm Efficiency

Algorithm efficiency is central to stream generation, especially when dealing with large or potentially infinite sequences. Instead of testing each integer to ascertain if it fits our criteria (which involves costly primality checks), we develop an efficient algorithm using streams and merge functions. This allows us to create the series of Hamming numbers without redundantly evaluating each number. Using the merge function, we exploit the structure of ordered streams to efficiently combine sequences. By always selecting the smallest available element from the streams, we construct an output stream in ascending order. This method also ensures there are no repetitions, which saves unnecessary computations and storage, optimizing memory usage. Remembering that as numbers get larger, less resultantly fit the criteria underscores why the naive method would be inefficient, highlighting the need for smarter iterative approaches like scaling and merging streams.

Functional Programming

Functional programming emphasizes the use of functions and immutable data structures. It typically avoids mutable state and side effects, focusing instead on building programs through the composition of pure functions. This paradigm is particularly effective for tasks like stream generation, where predictable and repeatable computation is key. In generating Hamming numbers, functional programming approaches help structure the algorithm to achieve conciseness and clarity. The merge function, a recursive process that creates an ordered stream, is a great example of functional programming practice. It leverages the inherently recursive nature of functional languages, which ensure that the resultant stream is properly sorted without manually managing state changes or mutability issues. The separation of concerns and the composability of functions further enhance the robustness of the algorithm. Functional programming enables the reuse of general utility functions, such as scaling or merging, in different parts of the algorithm. This results in a concise, elegant solution that can be easily adapted as required.