Solving Mathematical Problems with Recursion in Clojure

7.7.2 Solving Mathematical Problems§

In this section, we delve into solving mathematical problems using recursion in Clojure. As experienced Java developers, you are likely familiar with iterative approaches to problem-solving. However, Clojure’s functional paradigm encourages us to think recursively, allowing us to express solutions more elegantly and concisely. We’ll explore how to calculate permutations, combinations, and solve the classic Towers of Hanoi problem using recursion.

Understanding Recursion in Clojure§

Recursion is a fundamental concept in functional programming, where a function calls itself to solve smaller instances of the same problem. In Clojure, recursion is often preferred over iteration due to its alignment with functional programming principles, such as immutability and pure functions.

Tail Recursion and the recur Keyword§

Clojure optimizes recursive calls using tail recursion, which prevents stack overflow by reusing the current function’s stack frame. The recur keyword is used to perform tail-recursive calls efficiently.

(defn factorial [n]
  (loop [acc 1, n n]
    (if (zero? n)
      acc
      (recur (* acc n) (dec n)))))

In this example, factorial calculates the factorial of a number using a loop and recur, ensuring the recursion is tail-optimized.

Solving Permutations§

Permutations involve rearranging elements of a set in all possible orders. Recursively generating permutations can be elegantly expressed in Clojure.

Recursive Permutation Function§

Let’s define a function to generate permutations of a list:

(defn permutations [coll]
  (if (empty? coll)
    '(())
    (for [x coll
          perm (permutations (remove #{x} coll))]
      (cons x perm))))

Explanation:

• Base Case: If the collection is empty, return a list containing an empty list.
• Recursive Case: For each element x in the collection, remove x and recursively find permutations of the remaining elements. Prepend x to each permutation.

Comparing with Java§

In Java, generating permutations typically involves more boilerplate code, often using loops and mutable data structures. Clojure’s recursive approach is more concise and expressive.

// Java example for generating permutations
public static List<List<Integer>> permutations(List<Integer> list) {
    if (list.isEmpty()) {
        List<List<Integer>> result = new ArrayList<>();
        result.add(new ArrayList<>());
        return result;
    }
    List<List<Integer>> result = new ArrayList<>();
    for (int i = 0; i < list.size(); i++) {
        List<Integer> remaining = new ArrayList<>(list);
        Integer element = remaining.remove(i);
        for (List<Integer> perm : permutations(remaining)) {
            perm.add(0, element);
            result.add(perm);
        }
    }
    return result;
}

Solving Combinations§

Combinations involve selecting elements from a set without regard to order. Recursive solutions can efficiently generate combinations.

Recursive Combination Function§

Here’s a function to generate combinations of a given size:

(defn combinations [coll k]
  (cond
    (zero? k) '(())
    (empty? coll) '()
    :else (let [[x & xs] coll]
            (concat
              (map #(cons x %) (combinations xs (dec k)))
              (combinations xs k)))))

Explanation:

• Base Cases: Return a list containing an empty list if k is zero. Return an empty list if the collection is empty.
• Recursive Case: Use the first element x and recursively find combinations of size k-1 from the rest of the collection. Also, find combinations of size k without x.

Comparing with Java§

Java solutions for combinations often involve nested loops and manual management of indices, making Clojure’s recursive approach more intuitive.

// Java example for generating combinations
public static List<List<Integer>> combinations(List<Integer> list, int k) {
    List<List<Integer>> result = new ArrayList<>();
    if (k == 0) {
        result.add(new ArrayList<>());
        return result;
    }
    if (list.isEmpty()) {
        return result;
    }
    Integer first = list.get(0);
    List<Integer> rest = list.subList(1, list.size());
    for (List<Integer> comb : combinations(rest, k - 1)) {
        List<Integer> newComb = new ArrayList<>(comb);
        newComb.add(0, first);
        result.add(newComb);
    }
    result.addAll(combinations(rest, k));
    return result;
}

Solving the Towers of Hanoi§

The Towers of Hanoi is a classic problem that demonstrates the power of recursion. The goal is to move a stack of disks from one peg to another, following specific rules.

Recursive Solution for Towers of Hanoi§

Here’s a Clojure function to solve the Towers of Hanoi problem:

(defn towers-of-hanoi [n source target auxiliary]
  (when (pos? n)
    (towers-of-hanoi (dec n) source auxiliary target)
    (println "Move disk from" source "to" target)
    (towers-of-hanoi (dec n) auxiliary target source)))

Explanation:

• Base Case: If n is zero, do nothing.
• Recursive Case: Move n-1 disks from the source to the auxiliary peg, move the nth disk to the target peg, and finally move the n-1 disks from the auxiliary to the target peg.

Comparing with Java§

Java solutions for the Towers of Hanoi often involve more verbose code due to explicit stack management, whereas Clojure’s recursion provides a cleaner approach.

// Java example for solving Towers of Hanoi
public static void towersOfHanoi(int n, char source, char target, char auxiliary) {
    if (n > 0) {
        towersOfHanoi(n - 1, source, auxiliary, target);
        System.out.println("Move disk from " + source + " to " + target);
        towersOfHanoi(n - 1, auxiliary, target, source);
    }
}

Try It Yourself§

To deepen your understanding, try modifying the provided Clojure functions:

• Permutations: Modify the permutations function to handle duplicate elements.
• Combinations: Extend the combinations function to return combinations with repetition.
• Towers of Hanoi: Add a counter to track the number of moves made.

Visualizing Recursion with Diagrams§

Let’s visualize the recursive process of solving the Towers of Hanoi using a flowchart:

Diagram Explanation: This flowchart illustrates the recursive steps involved in solving the Towers of Hanoi problem. The process involves moving n-1 disks, transferring the nth disk, and then moving the n-1 disks again.

Key Takeaways§

• Recursion in Clojure is a powerful tool for solving mathematical problems, offering concise and expressive solutions.
• Tail recursion with the recur keyword optimizes recursive calls, preventing stack overflow.
• Permutations and combinations can be elegantly expressed using recursive functions, contrasting with Java’s iterative approaches.
• Towers of Hanoi demonstrates the elegance of recursion, simplifying complex problem-solving.

Exercises§

1. Permutations with Duplicates: Modify the permutations function to handle lists with duplicate elements.
2. Combinations with Repetition: Extend the combinations function to allow repeated elements in combinations.
3. Towers of Hanoi with Move Counter: Add a counter to the towers-of-hanoi function to track the number of moves.

By exploring these exercises, you’ll gain a deeper understanding of recursion and its applications in solving mathematical problems with Clojure.

Quiz: Mastering Recursion in Clojure§

By mastering recursion in Clojure, you’ll be well-equipped to tackle a wide range of mathematical problems with elegance and efficiency.