Download code from GitHub
In this chapter, we'll revisit a few basic programming techniques, such as recursion and sequences, with Clojure. As we will see, Clojure focuses on the use of higher-order functions to abstract computation, like any other functional programming language. This design can be observed in most, if not all, of the Clojure standard library. In this chapter, we will cover the following topics:
Exploring recursion
Learning about sequences and laziness
Examining zippers
Briefly studying pattern matching
Recursion is one of the central methodologies of computer science. It allows us to elegantly solve problems that have cumbersome non-recursive solutions. Yet, recursive functions are discouraged in quite a few imperative programming languages in favor of non-recursive functions. Clojure does no such thing and completely embraces recursion along with all its pros and cons. In this section, we will explore how to define recursive functions.
In general, a function can be made recursive by simply calling it again from within the body of the function. We can define a simple function to return the first n numbers of the Fibonacci sequence as shown in Example 1.1:
(defn fibo
([n]
(fibo [0N 1N] n))
([xs n]
(if (<= n (count xs))
xs
(let [x' (+ (last xs)
(nth xs (- (count xs) 2)))
xs' (conj xs x')]
(fibo xs' n)))))Example 1.1: A simple recursive function
Note
The Fibonacci sequence is a series of numbers that can be defined as follows:
The first element F0 is 0 and the second element F1 is 1.
The rest of the numbers are the sum of the previous two numbers, that is the nth Fibonacci number Fn = Fn-1 + Fn-2.
In the previously defined fibo function, the last two elements of the list are determined using the nth and last functions, and the sum of these two elements is appended to the list using the conj function. This is done in a recursive manner, and the function terminates when the length of the list, determined by the count function becomes equal to the supplied value n. Also, the values 0N and 1N, which represent BigInteger types, are used instead of the values 0 and 1.This is done because using long or integer values for such a computation could result in an arithmetic overflow error. We can try out this function in the REPL shown as follows:
user> (fibo 10) [0N 1N 1N 2N 3N 5N 8N 13N 21N 34N] user> (last (fibo 100)) 218922995834555169026N
The fibo function returns a vector of the first n Fibonacci numbers as expected. However, for larger values of n, this function will cause a stack overflow:
user> (last (fibo 10000))
StackOverflowError clojure.lang.Numbers.lt (Numbers.java:219)The reason for this error is that there were too many nested function calls. A call to any function requires an additional call stack. With recursion, we reach a point where all of the available stack space in a program is consumed and no more function calls can be performed. A tail call can overcome this limitation by using the existing call stack for a recursive call, which removes the need for allocating a new call stack. This is only possible when the return value of a function is the return value of a recursive call made by the function, in which case an additional call stack is not required to store the state of the function that performs the recursive call. This technique is termed as tail call elimination. In effect, a tail call optimized function consumes a constant amount of stack space.
In fact, the fibo function does indeed make a tail call, as the last expression in the body of the function is a recursive call. Still, it consumes stack space for each recursive call. This is due to the fact that the underlying virtual machine, the JVM, does not perform tail call elimination. In Clojure, tail call elimination has to be done explicitly using a recur form to perform a recursive call. The fibo function we defined earlier can be refined to be tail recursive by using a recur form, as shown in Example 1.2:
(defn fibo-recur
([n]
(fibo-recur [0N 1N] n))
([xs n]
(if (<= n (count xs))
xs
(let [x' (+ (last xs)
(nth xs (- (count xs) 2)))
xs' (conj xs x')]
(recur xs' n)))))Note
Downloading the example code
You can download the example code files from your account at http://www.packtpub.com for all the Packt Publishing books you have purchased. If you purchased this book elsewhere, you can visit http://www.packtpub.com/support and register to have the files e-mailed to you.
Effectively, the fibo-recur function can perform an infinite number of nested recursive calls. We can observe that this function does not blow up the stack for large values of n, shown as follows:
user> (fibo-recur 10) [0N 1N 1N 2N 3N 5N 8N 13N 21N 34N] user> (last (fibo-recur 10000)) 207936...230626N
We should note that a call to fibo-recur can take quite a while to terminate for large values of n. We can measure the time taken for a call to fibo-recur to complete and return a value, using the time macro, as follows:
user> (time (last (fibo-recur 10000)))
"Elapsed time: 1320.050942 msecs"
207936...230626NThe fibo-recur function can also be expressed using the loop and recur forms. This eliminates the need for using a second function arity to pass the [0N 1N] value around, as shown in the fibo-loop function defined in Example 1.3:
(defn fibo-loop [n]
(loop [xs [0N 1N]
n n]
(if (<= n (count xs))
xs
(let [x' (+ (last xs)
(nth xs (- (count xs) 2)))
xs' (conj xs x')]
(recur xs' n)))))Example 1.3: A recursive function defined using loop and recur
Note that the loop
macro requires a vector of bindings (pairs of names and values) to be passed as its first argument. The second argument to the loop form must be an expression that uses the recur form. This nested recur form calls the surrounding expression recursively by passing in the new values for the declared bindings in the loop form. The fibo-loop function returns a value that is equal to that returned by the fibo-recur function, from Example 1.2, shown as follows:
user> (fibo-loop 10) [0N 1N 1N 2N 3N 5N 8N 13N 21N 34N] user> (last (fibo-loop 10000)) 207936...230626N
Another way to handle recursion is by using the trampoline function. The trampoline function takes a function as its first argument, followed by the values of the parameters to be passed to the supplied function. A trampoline form expects the supplied function to return another function, and in such a case, the returned function will be invoked. Thus, a trampoline form manages recursion by obtaining a return value, and invoking the returned value again if it's a function. Thus, the trampoline function avoids using any stack space. Each time the supplied function is invoked, it returns and the result gets stored in the process heap. For example, consider the function in Example 1.4 that calculates the first n numbers of the Fibonacci sequence using a trampoline:
(defn fibo-trampoline [n]
(letfn [(fibo-fn [xs n]
(if (<= n (count xs))
xs
(let [x' (+ (last xs)
(nth xs (- (count xs) 2)))
xs' (conj xs x')]
#(fibo-fn xs' n))))]
(trampoline fibo-fn [0N 1N] n)))Example 1.4: A recursive function defined using trampoline
In the fib-trampoline function, the internal fibo-fn function returns either a sequence, denoted by xs, or a closure that takes no arguments, represented by #(fibo-fn xs' n). This function is equivalent to the fibo-recur function we defined earlier, even in terms of performance, shown as follows:
user> (fibo-trampoline 10) [0N 1N 1N 2N 3N 5N 8N 13N 21N 34N] user> (time (last (fibo-trampoline 10000))) "Elapsed time: 1346.629108 msecs" 207936...230626N
Mutual recursion can also be handled effectively using a trampoline. In mutual recursion, two functions call each other in a recursive manner. For example, consider the function that utilizes two mutually recursive functions in Example 1.5:
(defn sqrt-div2-recur [n]
(letfn [(sqrt [n]
(if (< n 1)
n
(div2 (Math/sqrt n))))
(div2 [n]
(if (< n 1)
n
(sqrt (/ n 2))))]
(sqrt n)))Example 1.5: A simple function that uses mutual recursion
The sqrt-div2-recur function from Example 1.5 defines two mutually recursive functions internally, namely sqrt and div2, that repeatedly square root and halve a given value n until the calculated value is less than 1. The sqrt-div2-recur function declares these two functions using a letfn form and invokes the sqrt function. We can convert this to use a trampoline form as shown in Example 1.6:
(defn sqrt-div2-trampoline [n]
(letfn [(sqrt [n]
(if (< n 1)
n
#(div2 (Math/sqrt n))))
(div2 [n]
(if (< n 1)
n
#(sqrt (/ n 2))))]
(trampoline sqrt n)))Example 1.6: A function that uses mutual recursion using trampoline
In the previous sqrt-div2-trampoline function shown, the functions sqrt and div2 return closures instead of calling a function directly. The trampoline form in the body of the function calls the sqrt function while supplying the value n. Both the sqrt-div2-recur and sqrt-div2-trampoline functions take about the same time to return a value for the given value of n. Hence, using a trampoline form does not have any additional performance overhead, shown as follows:
user> (time (sqrt-div2-recur 10000000000N)) "Elapsed time: 0.327439 msecs" 0.5361105866719398 user> (time (sqrt-div2-trampoline 10000000000N)) "Elapsed time: 0.326081 msecs" 0.5361105866719398
As the preceding examples demonstrate, there are various ways to define recursive functions in Clojure. Recursive functions can be optimized using tail call elimination, by using recur, and mutual recursion, which is done using the trampoline function.
A sequence, shortened as a seq, is essentially an abstraction of a list. This abstraction provides a unified model or interface to interact with a collection of items. In Clojure, all the primitive data structures, namely strings, lists, vectors, maps, and sets can be treated as sequences. In practice, almost everything that involves iteration can be translated into a sequence of computations. A collection is termed as seqable if it implements the abstraction of a sequence. We will learn everything there is to know about sequences in this section.
Sequences can also be lazy. A lazy sequence can be thought of as a possibly infinite series of computed values. The computation of each value is deferred until it is actually needed. We should note that the computation of a recursive function can easily be represented as a lazy sequence. For example, the Fibonacci sequence can be computed by lazily adding the last two elements in the previously computed sequence. This can be implemented as shown in Example 1.7.
(defn fibo-lazy [n]
(->> [0N 1N]
(iterate (fn [[a b]] [b (+ a b)]))
(map first)
(take n)))Example 1.7: A lazy Fibonacci sequence
Note
The threading macro ->> is used to pass the result of a given expression as the last argument to the next expression, in a repetitive manner for all expressions in its body. Similarly, the threading macro -> is used to pass the result of a given expression as the first argument to the subsequent expressions.
The fibo-lazy function from Example 1.7 uses the iterate, map, and take functions to create a lazy sequence. We will study these functions in more detail later in this section. The fibo-lazy function takes a single argument n, which indicates the number of items to be returned by the function. In the fibo-lazy function, the values 0N and 1N are passed as a vector to the iterate function, which produces a lazy sequence. The function used for this iteration creates a new pair of values b and (+ a b) from the initial values a and b.
Next, the map function applies the first function to obtain the first element in each resulting vector. A take form is finally applied to the sequence returned by the map function to retrieve the first n values in the sequence. The fibo-lazy function does not cause any error even when passed relatively large values of n, shown as follows:
user> (fibo-lazy 10) (0N 1N 1N 2N 3N 5N 8N 13N 21N 34N) user> (last (fibo-lazy 10000)) 207936...230626N
Interestingly, the fibo-lazy function in Example 1.7 performs significantly better than the recursive functions from Example 1.2 and Example 1.3, as shown here:
user> (time (last (fibo-lazy 10000)))
"Elapsed time: 18.593018 msecs"
207936...230626NAlso, binding the value returned by the fibo-lazy function to a variable does not really consume any time. This is because this returned value is lazy and not evaluated yet. Also, the type of the return value is clojure.lang.LazySeq, as shown here:
user> (time (def fibo-xs (fibo-lazy 10000))) "Elapsed time: 0.191981 msecs" #'user/fibo-xs user> (type fibo-xs) clojure.lang.LazySeq
We can optimize the fibo-lazy function even further by using memoization, which essentially caches the value returned by a function for a given set of inputs. This can be done using the memoize function, as follows:
(def fibo-mem (memoize fibo-lazy))
The fibo-mem function is a memoized version of the fibo-lazy function. Hence, subsequent calls to the fibo-mem function for the same set of inputs will return values significantly faster, shown as follows:
user> (time (last (fibo-mem 10000))) "Elapsed time: 19.776527 msecs" 207936...230626N user> (time (last (fibo-mem 10000))) "Elapsed time: 2.82709 msecs" 207936...230626N
Note that the memoize function can be applied to any function, and it is not really related to sequences. The function we pass to memoize must be free of side effects, or else any side effects will be invoked only the first time the memoized function is called with a given set of inputs.
Sequences are a truly ubiquitous abstraction in Clojure. The primary motivation behind using sequences is that any domain with sequence-like data in it can be easily modelled using the standard functions that operate on sequences. This infamous quote from the Lisp world reflects on this design:
"It is better to have 100 functions operate on one data abstraction than 10 functions on 10 data structures."
A sequence can be constructed using the cons function. We must provide an element and another sequence as arguments to the cons function. The first function is used to access the first element in a sequence, and similarly the rest function is used to obtain the other elements in the sequence, shown as follows:
user> (def xs (cons 0 '(1 2 3))) #'user/xs user> (first xs) 0 user> (rest xs) (1 2 3)
Note
The first and rest functions in Clojure are equivalent to the car and cdr functions, respectively, from traditional Lisps. The cons function carries on its traditional name.
In Clojure, an empty list is represented by the literal (). An empty list is considered as a truthy value, and does not equate to nil. This rule is true for any empty collection. An empty list does indeed have a type – it's a list. On the other hand, the nil literal signifies the absence of a value, of any type, and is not a truthy value. The second argument that is passed to cons could be empty, in which case the resulting sequence would contain a single element:
user> (cons 0 ()) (0) user> (cons 0 nil) (0) user> (rest (cons 0 nil)) ()
An interesting quirk is that nil can be treated as an empty collection, but the converse is not true. We can use the empty? and nil? functions to test for an empty collection and a nil value, respectively. Note that (empty? nil) returns true, shown as follows:
user> (empty? ()) true user> (empty? nil) true user> (nil? ()) false user> (nil? nil) true
Note
By the truthy value, we mean to say a value that will test positive in a conditional expression such as an if or a when form.
The rest function will return an empty list when supplied an empty list. Thus, the value returned by rest is always truthy. The seq function can be used to obtain a sequence from a given collection. It will return nil for an empty list or collection. Hence, the head, rest and seq functions can be used to iterate over a sequence. The next function can also be used for iteration, and the expression (seq (rest coll)) is equivalent to (next coll), shown as follows:
user> (= (rest ()) nil) false user> (= (seq ()) nil) true user> (= (next ()) nil) true
The sequence function can be used to create a list from a sequence. For example, nil can be converted into an empty list using the expression (sequence nil). In Clojure, the seq? function is used to check whether a value implements the sequence interface, namely clojure.lang.ISeq. Only lists implement this interface, and other data structures such as vectors, sets, and maps have to be converted into a sequence by using the seq function. Hence, seq? will return true only for lists. Note that the list?, vector?, map?, and set? functions can be used to check the concrete type of a given collection. The behavior of the seq? function with lists and vectors can be described as follows:
user> (seq? '(1 2 3)) true user> (seq? [1 2 3]) false user> (seq? (seq [1 2 3])) true
Only lists and vectors provide a guarantee of sequential ordering among elements. In other words, lists and vectors will store their elements in the same order or sequence as they were created. This is in contrast to maps and sets, which can reorder their elements as needed. We can use the sequential? function to check whether a collection provides sequential ordering:
user> (sequential? '(1 2 3)) true user> (sequential? [1 2 3]) true user> (sequential? {:a 1 :b 2}) false user> (sequential? #{:a :b}) false
The associative? function can be used to determine whether a collection or sequence associates a key with a particular value. Note that this function returns true only for maps and vectors:
user> (associative? '(1 2 3)) false user> (associative? [1 2 3]) true user> (associative? {:a 1 :b 2}) true user> (associative? #{:a :b}) false
The behavior of the associative? function is fairly obvious for a map since a map is essentially a collection of key-value pairs. The fact that a vector is also associative is well justified too, as a vector has an implicit key for a given element, namely the index of the element in the vector. For example, the [:a :b] vector has two implicit keys, 0 and 1, for the elements :a and :b respectively. This brings us to an interesting consequence – vectors and maps can be treated as functions that take a single argument, that is a key, and return an associated value, shown as follows:
user> ([:a :b] 1) :b user> ({:a 1 :b 2} :a) 1
Although they are not associative by nature, sets are also functions. Sets return a value contained in them, or nil, depending on the argument passed to them, shown as follows:
user> (#{1 2 3} 1) 1 user> (#{1 2 3} 0) nil
Now that we have familiarized ourselves with the basics of sequences, let's have a look at the many functions that operate over sequences.
There are several ways to create sequences other than using the cons function. We have already encountered the conj function in the earlier examples of this chapter. The conj function takes a collection as its first argument, followed by any number of arguments to add to the collection. We must note that conj behaves differently for lists and vectors. When supplied a list, the conj function adds the other arguments at the head, or start, of the list. In case of a vector, the conj function will insert the other arguments at the tail, or end, of the vector:
user> (conj [1 2 3] 4 5 6) [1 2 3 4 5 6] user> (conj '(1 2 3) 4 5 6) (6 5 4 1 2 3)
The concat function can be used to join or concatenate any number of sequences in the order in which they are supplied, shown as follows:
user> (concat [1 2 3] []) (1 2 3) user> (concat [] [1 2 3]) (1 2 3) user> (concat [1 2 3] [4 5 6] [7 8 9]) (1 2 3 4 5 6 7 8 9)
A given sequence can be reversed using the reverse function, shown as follows:
user> (reverse [1 2 3 4 5 6]) (6 5 4 3 2 1) user> (reverse (reverse [1 2 3 4 5 6])) (1 2 3 4 5 6)
The range function can be used to generate a sequence of values within a given integer range. The most general form of the range function takes three arguments—the first argument is the start of the range, the second argument is the end of the range, and the third argument is the step of the range. The step of the range defaults to 1, and the start of the range defaults to 0, as shown here:
user> (range 5) (0 1 2 3 4) user> (range 0 10 3) (0 3 6 9) user> (range 15 10 -1) (15 14 13 12 11)
We must note that the range function expects the start of the range to be less than the end of the range. If the start of the range is greater than the end of the range and the step of the range is positive, the range function will return an empty list. For example, (range 15 10) will return (). Also, the range function can be called with no arguments, in which case it returns a lazy and infinite sequence starting at 0.
The take and drop functions can be used to take or drop elements in a sequence. Both functions take two arguments, representing the number of elements to take or drop from a sequence, and the sequence itself, as follows:
user> (take 5 (range 10)) (0 1 2 3 4) user> (drop 5 (range 10)) (5 6 7 8 9)
To obtain an item at a particular position in the sequence, we should use the nth function. This function takes a sequence as its first argument, followed by the position of the item to be retrieved from the sequence as the second argument:
user> (nth (range 10) 0) 0 user> (nth (range 10) 9) 9
To repeat a given value, we can use the repeat function. This function takes two arguments and repeats the second argument the number of times indicated by the first argument:
user> (repeat 10 0) (0 0 0 0 0 0 0 0 0 0) user> (repeat 5 :x) (:x :x :x :x :x)
The repeat function will evaluate the expression of the second argument and repeat it. To call a function a number of times, we can use the repeatedly function, as follows:
user> (repeat 5 (rand-int 100)) (75 75 75 75 75) user> (repeatedly 5 #(rand-int 100)) (88 80 17 52 32)
In this example, the repeat form first evaluates the (rand-int 100) form, before repeating it. Hence, a single value will be repeated several times. Note that the rand-int function simply returns a random integer between 0 and the supplied value. On the other hand, the repeatedly function invokes the supplied function a number of times, thus producing a new value every time the rand-int function is called.
A sequence can be repeated an infinite number of times using the cycle function. As you might have guessed, this function returns a lazy sequence to indicate an infinite series of values. The take function can be used to obtain a limited number of values from the resulting infinite sequence, shown as follows:
user> (take 5 (cycle [0])) (0 0 0 0 0) user> (take 5 (cycle (range 3))) (0 1 2 0 1)
The interleave function can be used to combine any number of sequences. This function returns a sequence of the first item in each collection, followed by the second item, and so on. This combination of the supplied sequences is repeated until the shortest sequence is exhausted of values. Hence, we can easily combine a finite sequence with an infinite one to produce another finite sequence using the interleave function:
user> (interleave [0 1 2] [3 4 5 6] [7 8]) (0 3 7 1 4 8) user> (interleave [1 2 3] (cycle [0])) (1 0 2 0 3 0)
Another function that performs a similar operation is the interpose function. The interpose function inserts a given element between the adjacent elements of a given sequence:
user> (interpose 0 [1 2 3])
(1 0 2 0 3)The iterate function can also be used to create an infinite sequence. Note that we have already used the iterate function to create a lazy sequence in Example 1.7. This function takes a function f and an initial value x as its arguments. The value returned by the iterate function will have (f x) as the first element, (f (f x)) as the second element, and so on. We can use the iterate function with any other function that takes a single argument, as follows:
user> (take 5 (iterate inc 5)) (5 6 7 8 9) user> (take 5 (iterate #(+ 2 %) 0)) (0 2 4 6 8)
There are also several functions to convert sequences into different representations or values. One of the most versatile of such functions is the map function. This function maps a given function over a given sequence, that is, it applies the function to each element in the sequence. Also, the value returned by map is implicitly lazy. The function to be applied to each element must be the first argument to map, and the sequence on which the function must be applied is the next argument:
user> (map inc [0 1 2 3]) (1 2 3 4) user> (map #(* 2 %) [0 1 2 3]) (0 2 4 6)
Note that map can accept any number of collections or sequences as its arguments. In this case, the resulting sequence is obtained by passing the first items of the sequences as arguments to the given function, and then passing the second items of the sequences to the given function, and so on until any of the supplied sequences are exhausted. For example, we can sum the corresponding elements of two sequences using the map and + functions, as shown here:
user> (map + [0 1 2 3] [4 5 6])
(4 6 8)The mapv function has the same semantics of map, but returns a vector instead of a sequence, as shown here:
user> (mapv inc [0 1 2 3])
[1 2 3 4]Another variant of the map function is the map-indexed function. This function expects that the supplied function will accept two arguments—one for the index of a given element and another for the actual element in the list:
user> (map-indexed (fn [i x] [i x]) "Hello")
([0 \H] [1 \e] [2 \l] [3 \l] [4 \o])In this example, the function supplied to map-indexed simply returns its arguments as a vector. An interesting point that we can observe from the preceding example is that a string can be treated as a sequence of characters.
The mapcat function is a combination of the map and concat function. This function maps a given function over a sequence, and applies the concat function on the resulting sequence:
user> (require '[clojure.string :as cs]) nil user> (map #(cs/split % #"\d") ["aa1bb" "cc2dd" "ee3ff"]) (["aa" "bb"] ["cc" "dd"] ["ee" "ff"]) user> (mapcat #(cs/split % #"\d") ["aa1bb" "cc2dd" "ee3ff"]) ("aa" "bb" "cc" "dd" "ee" "ff")
In this example, we use the split function from the clojure.string namespace to split a string using a regular expression, shown as #"\d". The split function will return a vector of strings, and hence the mapcat function returns a sequence of strings instead of a sequence of vectors like the map function.
The reduce function is used to combine or reduce a sequence of items into a single value. The reduce function requires a function as its first argument and a sequence as its second argument. The function supplied to reduce must accept two arguments. The supplied function is first applied to the first two elements in the given sequence, and then applied to the previous result and the third element in the sequence, and so on until the sequence is exhausted. The reduce function also has a second arity, which accepts an initial value, and in this case, the supplied function is applied to the initial value and the first element in the sequence as the first step. The reduce function can be considered equivalent to loop-based iteration in imperative programming languages. For example, we can compute the sum of all elements in a sequence using reduce, as follows:
user> (reduce + [1 2 3 4 5]) 15 user> (reduce + []) 0 user> (reduce + 1 []) 1
In this example, when the reduce function is supplied an empty collection, it returns 0, since (+) evaluates to 0. When an initial value of 1 is supplied to the reduce function, it returns 1, since (+ 1) returns 1.
A list comprehension can be created using the for macro. Note that a for form will be translated into an expression that uses the map function. The for macro needs to be supplied a vector of bindings to any number of collections, and an expression in the body. This macro binds the supplied symbol to each element in its corresponding collection and evaluates the body for each element. Note that the for macro also supports a :let clause to assign a value to a variable, and also a :when clause to filter out values:
user> (for [x (range 3 7)] (* x x)) (9 16 25 36) user> (for [x [0 1 2 3 4 5] :let [y (* x 3)] :when (even? y)] y) (0 6 12)
The for macro can also be used over a number of collections, as shown here:
user> (for [x ['a 'b 'c] y [1 2 3]] [x y]) ([a 1] [a 2] [a 3] [b 1] [b 2] [b 3] [c 1] [c 2] [c 3])
The doseq macro has semantics similar to that of for, except for the fact that it always returns a nil value. This macro simply evaluates the body expression for all of the items in the given bindings. This is useful in forcing evaluation of an expression with side effects for all the items in a given collection:
user> (doseq [x (range 3 7)] (* x x)) nil user> (doseq [x (range 3 7)] (println (* x x))) 9 16 25 36 nil
As shown in the preceding example, both the first and second doseq forms return nil. However, the second form prints the value of the expression (* x x), which is a side effect, for all items in the sequence (range 3 7).
The into function can be used to easily convert between types of collections. This function requires two collections to be supplied to it as arguments, and returns the first collection filled with all the items in the second collection. For example, we can convert a sequence of vectors into a map, and vice versa, using the into function, shown here:
user> (into {} [[:a 1] [:c 3] [:b 2]]) {:a 1, :c 3, :b 2} user> (into [] {1 2 3 4}) [[1 2] [3 4]]
We should note that the into function is essentially a composition of the reduce and conj functions. As conj is used to fill the first collection, the value returned by the into function will depend on the type of the first collection. The into function will behave similar to conj with respect to lists and vectors, shown here:
user> (into [1 2 3] '(4 5 6)) [1 2 3 4 5 6] user> (into '(1 2 3) '(4 5 6)) (6 5 4 1 2 3)
A sequence can be partitioned into smaller ones using the partition, partition-all and partition-by functions. Both the partition and partition-all functions take two arguments—one for the number of items n in the partitioned sequences and another for the sequence to be partitioned. However, the partition-all function will also return the items from the sequence, which have not been partitioned as a separate sequence, shown here:
user> (partition 2 (range 11)) ((0 1) (2 3) (4 5) (6 7) (8 9)) user> (partition-all 2 (range 11)) ((0 1) (2 3) (4 5) (6 7) (8 9) (10))
The partition and partition-all functions also accept a step argument, which defaults to the supplied number of items in the partitioned sequences, shown as follows:
user> (partition 3 2 (range 11)) ((0 1 2) (2 3 4) (4 5 6) (6 7 8) (8 9 10)) user> (partition-all 3 2 (range 11)) ((0 1 2) (2 3 4) (4 5 6) (6 7 8) (8 9 10) (10))
The partition function also takes a second sequence as an optional argument, which is used to pad the sequence to be partitioned in case there are items that are not partitioned. This second sequence has to be supplied after the step argument to the partition function. Note that the padding sequence is only used to create a single partition with the items that have not been partitioned, and the rest of the padding sequence is discarded. Also, the padding sequence is only used if there are any items that have not been partitioned. This can be illustrated in the following example:
user> (partition 3 (range 11)) ((0 1 2) (3 4 5) (6 7 8)) user> (partition 3 3 (range 11 12) (range 11)) ((0 1 2) (3 4 5) (6 7 8) (9 10 11)) user> (partition 3 3 (range 11 15) (range 11)) ((0 1 2) (3 4 5) (6 7 8) (9 10 11)) user> (partition 3 4 (range 11 12) (range 11)) ((0 1 2) (4 5 6) (8 9 10))
In this example, we first provide a padding sequence in the second statement as (range 11 12), which only comprises of a single element. In the next statement, we supply a larger padding sequence, as (range 11 15), but only the first item 11 from the padding sequence is actually used. In the last statement, we also supply a padding sequence but it is never used, as the (range 11) sequence is partitioned into sequences of 3 elements each with a step of 4, which will have no remaining items.
The partition-by function requires a higher-order function to be supplied to it as the first argument, and will partition items in the supplied sequence based on the return value of applying the given function to each element in the sequence. The sequence is essentially partitioned by partition-by whenever the given function returns a new value, as shown here:
user> (partition-by #(= 0 %) [-2 -1 0 1 2]) ((-2 -1) (0) (1 2)) user> (partition-by identity [-2 -1 0 1 2]) ((-2) (-1) (0) (1) (2))
In this example, the second statement partitions the given sequence into sequences that each contain a single item as we have used the identity function, which simply returns its argument. For the [-2 -1 0 1 2] sequence, the identity function returns a new value for each item in the sequence and hence the resulting partitioned sequences all have a single element.
The sort function can be used to change the ordering of elements in a sequence. The general form of this function requires a function to compare items and a sequence of items to sort. The supplied function defaults to the compare function, whose behavior changes depending on the actual type of the items being compared:
user> (sort [3 1 2 0]) (0 1 2 3) user> (sort > [3 1 2 0]) (3 2 1 0) user> (sort ["Carol" "Alice" "Bob"]) ("Alice" "Bob" "Carol")
If we intend to apply a particular function to each item in a sequence before performing the comparison in a sort form, we should consider using the sort-by function for a more concise expression. The sort-by function also accepts a function to perform the actual comparison, similar to the sort function. The sort-by function can be demonstrated as follows:
user> (sort #(compare (first %1) (first %2)) [[1 1] [2 2] [3 3]]) ([1 1] [2 2] [3 3]) user> (sort-by first [[1 1] [2 2] [3 3]]) ([1 1] [2 2] [3 3]) user> (sort-by first > [[1 1] [2 2] [3 3]]) ([3 3] [2 2] [1 1])
In this example, the first and second statements both compare items after applying the first function to each item in the given sequence. The last statement passes the > function to the sort-by function, which returns the reverse of the sequence returned by the first two statements.
Sequences can also be filtered, that is transformed by removing some elements from the sequence. There are several standard functions to perform this task. The keep function can be used to remove values from a sequence that produces a nil value for a given function. The keep function requires a function and a sequence to be passed to it. The keep function will apply the given function to each item in the sequence and remove all values that produce nil, as shown here:
user> (keep #(if (odd? %) %) (range 10)) (1 3 5 7 9) user> (keep seq [() [] '(1 2 3) [:a :b] nil]) ((1 2 3) (:a :b))
In this example, the first statement removes all even numbers from the given sequence. In the second statement, the seq function is used to remove all empty collections from the given sequence.
A map or a set can also be passed as the first argument to the keep function since they can be treated as functions, as shown here:
user> (keep {:a 1, :b 2, :c 3} [:a :b :d]) (1 2) user> (keep #{0 1 2 3} #{2 3 4 5}) (3 2)
The filter function can also be used to remove some elements from a given sequence. The filter function expects a predicate function to be passed to it along with the sequence to be filtered. The items for which the predicate function does not return a truthy value are removed from the result. The filterv function is identical to the filter function, except for the fact that it returns a vector instead of a list:
user> (filter even? (range 10)) (0 2 4 6 8) user> (filterv even? (range 10)) [0 2 4 6 8]
Both the filter and keep functions have similar semantics. However, the primary distinction is that the filter function returns a subset of the original elements, whereas keep returns a sequence of non nil values that are returned by the function supplied to it, as shown in the following example:
user> (keep #(if (odd? %) %) (range 10)) (1 3 5 7 9) user> (filter odd? (range 10)) (1 3 5 7 9)
Note that in this example, if we passed the odd? function to the keep form, it would return a list of true and false values, as these values are returned by the odd? function.
Also, a for macro with a :when clause is translated into an expression that uses the filter function, and hence a for form can also be used to remove elements from a sequence:
user> (for [x (range 10) :when (odd? x)] x)
(1 3 5 7 9)A vector can be sliced using the subvec function. By sliced, we mean to say that a smaller vector is selected from the original vector depending on the values passed to the subvec function. The subvec function takes a vector as its first argument, followed by the index indicating the start of the sliced vector, and finally another optional index that indicates the end of the sliced vector, as shown here:
user> (subvec [0 1 2 3 4 5] 3) [3 4 5] user> (subvec [0 1 2 3 4 5] 3 5) [3 4]
Maps can be filtered by their keys using the select-keys function. This function requires a map as the first argument and a vector of keys as a second argument to be passed to it. The vector of keys passed to this function indicates the key-value pairs to be included in the resulting map, as shown here:
user> (select-keys {:a 1 :b 2} [:a]) {:a 1} user> (select-keys {:a 1 :b 2 :c 3} [:a :c]) {:c 3, :a 1}
Another way to select key-value pairs from a map is to use the find function, as shown here:
user> (find {:a 1 :b 2} :a)
[:a 1]
take-while and drop-while are analogous to the take and drop functions, and require a predicate to be passed to them, instead of the number of elements to take or drop. The take-while function takes elements as long as the predicate function returns a truthy value, and similarly the drop-while function will drop elements for the same condition:
user> (take-while neg? [-2 -1 0 1 2]) (-2 -1) user> (drop-while neg? [-2 -1 0 1 2]) (0 1 2)
lazy-seq and lazy-cat are the most elementary constructs to create lazy sequences. The value returned by these functions will always have the type clojure.lang.LazySeq. The lazy-seq function is used to wrap a lazily computed expression in a cons form. This means that the rest of the sequence created by the cons form is lazily computed. For example, the lazy-seq function can be used to construct a lazy sequence representing the Fibonacci sequence as shown in Example 1.8:
(defn fibo-cons [a b] (cons a (lazy-seq (fibo-cons b (+ a b)))))
Example 1.8: A lazy sequence created using lazy-seq
The fibo-cons function requires two initial values, a and b, to be passed to it as the initial values, and returns a lazy sequence comprising the first value a and a lazily computed expression that uses the next two values in the sequence, that is, b and (+ a b). In this case, the cons form will return a lazy sequence, which can be handled using the take and last functions, as shown here:
user> (def fibo (fibo-cons 0N 1N)) #'user/fibo user> (take 2 fibo) (0N 1N) user> (take 11 fibo) (0N 1N 1N 2N 3N 5N 8N 13N 21N 34N 55N) user> (last (take 10000 fibo)) 207936...230626N
Note that the fibo-cons function from Example 1.8 recursively calls itself without an explicit recur form, and yet it does not consume any stack space. This is because the values present in a lazy sequence are not stored in a call stack, and all the values are allocated on the process heap.
Another way to define a lazy Fibonacci sequence is by using the lazy-cat function. This function essentially concatenates all the sequences it is supplied in a lazy fashion. For example, consider the definition of the Fibonacci sequence in Example 1.9:
(def fibo-seq (lazy-cat [0N 1N] (map + fibo-seq (rest fibo-seq))))
Example 1.9: A lazy sequence created using lazy-cat
The fibo-seq variable from Example 1.9 essentially calculates the Fibonacci sequence using a lazy composition of the map, rest, and + functions. Also, a sequence is required as the initial value, instead of a function as we saw in the definition of fibo-cons from Example 1.8. We can use the nth function to obtain a number from this sequence as follows:
user> (first fibo-seq) 0N user> (nth fibo-seq 1) 1N user> (nth fibo-seq 10) 55N user> (nth fibo-seq 9999) 207936...230626N
As shown previously, fibo-cons and fibo-seq are concise and idiomatic representations of the infinite series of numbers in the Fibonacci sequence. Both of these definitions return identical values and do not cause an error due to stack consumption.
An interesting fact is that most of the standard functions that return sequences, such as map and filter, are inherently lazy. Any expression that is built using these functions is lazy, and hence never evaluated until needed. For example, consider the following expression that uses the map function:
user> (def xs (map println (range 3))) #'user/xs user> xs 0 1 2 (nil nil nil)
In this example, the println function is not called when we define the xs variable. However, once we try to print it in the REPL, the sequence is evaluated and the numbers are printed out by calling the println function. Note that xs evaluates to (nil nil nil) as the println function always returns nil.
Sometimes, it is necessary to eagerly evaluate a lazy sequence. The doall and dorun functions are used for this exact purpose. The doall function essentially forces evaluation of a lazy sequence along with any side effects of the evaluation. The value returned by doall is a list of all the elements in the given lazy sequence. For example, let's wrap the map expression from the previous example in a doall form, shown as follows:
user> (def xs (doall (map println (range 3)))) 0 1 2 #'user/xs user> xs (nil nil nil)
Now, the numbers are printed out as soon as xs is defined, as we force evaluation using the doall function. The dorun function has similar semantics as the doall function, but it always returns nil. Hence, we can use the dorun function instead of doall when we are only interested in the side effects of evaluating the lazy sequence, and not the actual values in it. Another way to call a function with some side effects over all values in a collection is by using the run! function, which must be passed a function to call and a collection. The run! function always returns nil, just like the dorun form.
Now that we are well versed with sequences, let's briefly examine zippers. Zippers are essentially data structures that help in traversing and manipulating trees. In Clojure, any collection that contains nested collections is termed as a tree. A zipper can be thought of as a structure that contains location information about a tree. Zippers are not an extension of trees, but rather can be used to traverse and realize a tree.
Note
The following namespaces must be included in your namespace declaration for the upcoming examples:
(ns my-namespace
(:require [clojure.zip :as z]
[clojure.xml :as xml]))The following examples can be found in src/m_clj/c1/zippers.clj of the book's source code.
We can define a simple tree using vector literals, as shown here:
(def tree [:a [1 2 3] :b :c])
The vector tree is a tree, comprised of the nodes :a, [1 2 3], :b, and :c. We can use the vector-zip function to create a zipper from the vector tree as follows:
(def root (z/vector-zip tree))
The variable root defined previously is a zipper and contains location information for traversing the given tree. Note that the vector-zip function is simply a combination of the standard seq function and the seq-zip function from the clojure.zip namespace. Hence, for trees that are represented as sequences, we should use the seq-zip function instead. Also, all other functions in the clojure.zip namespace expect their first argument to be a zipper.
To traverse the zipper, we must use the clojure.zip/next function, which returns the next node in the zipper. We can easily iterate over all the nodes in the zipper using a composition of the iterate and clojure.zip/next functions, as shown here:
user> (def tree-nodes (iterate z/next root)) #'user/tree-nodes user> (nth tree-nodes 0) [[:a [1 2 3] :b :c] nil] user> (nth tree-nodes 1) [:a {:l [], :pnodes ... }] user> (nth tree-nodes 2) [[1 2 3] {:l [:a], :pnodes ... }] user> (nth tree-nodes 3) [1 {:l [], :pnodes ... }]
As shown previously, the first node of the zipper represents the original tree itself. Also, the zipper will contain some extra information, other than the value contained in the current node, which is useful in navigating across the given tree. In fact, the return value of the next function is also a zipper. Once we have completely traversed the given tree, a zipper pointing to the root of the tree will be returned by the next function. Note that some information in a zipper has been truncated from the preceding REPL output for the sake of readability.
To navigate to the adjacent nodes in a given zipper, we can use the down, up, left, and right functions. All of these functions return a zipper, as shown here:
user> (-> root z/down) [:a {:l [], :pnodes ... }] user> (-> root z/down z/right) [[1 2 3] {:l [:a], :pnodes ... }] user> (-> root z/down z/right z/up) [[:a [1 2 3] :b :c] nil] user> (-> root z/down z/right z/right) [:b {:l [:a [1 2 3]], :pnodes ... }] user> (-> root z/down z/right z/left) [:a {:l [], :pnodes ... }]
The down, up, left, and right functions change the location of the root zipper in the [:a [1 2 3] :b :c] tree, as shown in the following illustration:
The preceding diagram shows a zipper at three different locations in the given tree. Initially, the location of the zipper is at the root of the tree, which is the entire vector. The down function moves the location to the first child node in the tree. The left and right functions move the location of the zipper to other nodes at the same level or depth in the tree. The up function moves the zipper to the parent of the node pointed to by the zipper's current location.
To obtain the node representing the current location of a zipper in a tree, we must use the node function, as follows:
user> (-> root z/down z/right z/right z/node) :b user> (-> root z/down z/right z/left z/node) :a
To navigate to the extreme left or right of a tree, we can use the leftmost and rightmost functions, respectively, as shown here:
user> (-> root z/down z/rightmost z/node) :c user> (-> root z/down z/rightmost z/leftmost z/node) :a
The lefts and rights functions return the nodes that are present to the left and right, respectively, of a given zipper, as follows:
user> (-> root z/down z/rights) ([1 2 3] :b :c) user> (-> root z/down z/lefts) nil
As the :a node is the leftmost element in the tree, the rights function will return all of the other nodes in the tree when passed a zipper that has :a as the current location. Similarly, the lefts function for the zipper at the :a node will return an empty value, that is nil.
The root function can be used to obtain the root of a given zipper. It will return the original tree used to construct the zipper, as shown here:
user> (-> root z/down z/right z/root) [:a [1 2 3] :b :c] user> (-> root z/down z/right r/left z/root) [:a [1 2 3] :b :c]
The path function can be used to obtain the path from the root element of a tree to the current location of a given zipper, as shown here:
user> (def e (-> root z/down z/right z/down)) #'user/e user> (z/node e) 1 user> (z/path e) [[:a [1 2 3] :b :c] [1 2 3]]
In the preceding example, the path of the 1 node in tree is represented by a vector containing the entire tree and the subtree [1 2 3]. This means that to get to the 1 node, we must pass through the root and the subtree [1 2 3].
Now that we have covered the basics of navigating across trees, let's see how we can modify the original tree. The insert-child function can be used to insert a given element into a tree as follows:
user> (-> root (z/insert-child :d) z/root) [:d :a [1 2 3] :b :c] user> (-> root z/down z/right (z/insert-child 0) z/root) [:a [0 1 2 3] :b :c]
We can also remove a node from the zipper using the remove function. Also, the replace function can be used to replace a given node in a zipper:
user> (-> root z/down z/remove z/root) [[1 2 3] :b :c] user> (-> root z/down (z/replace :d) z/root) [:d [1 2 3] :b :c]
One of the most noteworthy examples of tree-like data is XML. Since zippers are great at handling trees, they also allow us to easily traverse and modify XML content. Note that Clojure already provides the xml-seq function to convert XML data into a sequence. However, treating an XML document as a sequence has many strange implications.
One of the main disadvantages of using xml-seq is that there is no easy way to get to the root of the document from a node if we are iterating over a sequence. Also, xml-seq only helps us iterate over the XML content; it doesn't deal with modifying it. These limitations can be overcome using zippers, as we will see in the upcoming example.
For example, consider the following XML document:
<countries>
<country name="England">
<city>Birmingham</city>
<city>Leeds</city>
<city capital="true">London</city>
</country>
<country name="Germany">
<city capital="true">Berlin</city>
<city>Frankfurt</city>
<city>Munich</city>
</country>
<country name="France">
<city>Cannes</city>
<city>Lyon</city>
<city capital="true">Paris</city>
</country>
</countries>The document shown above contains countries and cities represented as XML nodes. Each country has a number of cities, and a single city as its capital. Some information, such as the name of the country and a flag indicating whether a city is a capital, is encoded in the XML attributes of the nodes.
Note
The following example expects the XML content shown previously to be present in the resources/data/sample.xml file, relative to the root of your Leiningen project.
Let's define a function to find out all the capital cities in the document, as shown in Example 1.10:
(defn is-capital-city? [n]
(and (= (:tag n) :city)
(= "true" (:capital (:attrs n)))))
(defn find-capitals [file-path]
(let [xml-root (z/xml-zip (xml/parse file-path))
xml-seq (iterate z/next (z/next xml-root))]
(->> xml-seq
(take-while #(not= (z/root xml-root) (z/node %)))
(map z/node)
(filter is-capital-city?)
(mapcat :content))))Example 1.10: Querying XML with zippers
Firstly, we must note that the parse function from the clojure.xml namespace reads an XML document and returns a map representing the document. Each node in this map is another map with the :tag, :attrs, and :content keys associated with the XML node's tag name, attributes, and content respectively.
In Example 1.10, we first define a simple function, is-capital-city?, to determine whether a given XML node has the city tag, represented as :city. The is-capital-city? function also checks whether the XML node contains the capital attribute, represented as :capital. If the value of the capital attribute of a given node is the "true" string, then the is-capital-city? function returns true.
The find-capitals function performs most of the heavy lifting in this example. This function first parses XML documents present at the supplied path file-path, and then converts it into a zipper using the xml-zip function. We then iterate over the zipper using the next function until we arrive back at the root node, which is checked by the take-while function. We then map the node function over the resulting sequence of zippers using the map function, and apply the filter function to find the capital cities among all the nodes. Finally, we use the mapcat function to obtain the XML content of the filtered nodes and flatten the resulting sequence of vectors into a single list.
When supplied a file containing the XML content we described earlier, the find-capitals function returns the names of all capital cities in the document:
user> (find-capitals "resources/data/sample.xml")
("London" "Berlin" "Paris")As demonstrated previously, zippers are apt for dealing with trees and hierarchical data such as XML. More generally, sequences are a great abstraction for collections and several forms of data, and Clojure provides us with a huge toolkit for dealing with sequences. There are several more functions that handle sequences in the Clojure language, and you are encouraged to explore them on your own.
In this section, we will examine pattern matching in Clojure. Typically, functions that use conditional logic can be defined using the if, when, or cond forms. Pattern matching allows us to define such functions by declaring patterns of the literal values of their parameters. While this idea may appear quite rudimentary, it is a very useful and powerful one, as we shall see in the upcoming examples. Pattern matching is also a foundational programming construct in other functional programming languages.
In Clojure, there is no pattern matching support for functions and forms in the core language. However, it is a common notion among Lisp programmers that we can easily modify or extend the language using macros. Clojure takes this approach as well, and thus pattern matching is made possible using the match and defun macros. These macros are implemented in the core.match (https://github.com/clojure/core.match) and
defun (https://github.com/killme2008/defun) community libraries. Both of these libraries are also supported on ClojureScript.
Note
The following library dependencies are required for the upcoming examples:
[org.clojure/core.match "0.2.2" :exclusions [org.clojure/tools.analyzer.jvm]] [defun "0.2.0-RC"]
Also, the following namespaces must be included in your namespace declaration:
(ns my-namespace
(:require [clojure.core.match :as m]
[defun :as f]))The following examples can be found in src/m_clj/c1/match.clj of the book's source code.
Let's consider a simple example that we can model using pattern matching. The XOR logic function returns a true value only when its arguments are exclusive of each other, that is, when they have differing values. In other words, the XOR function will return false when both of its arguments have the same values. We can easily define such a function using the match macro, as shown in Example 1.11:
(defn xor [x y]
(m/match [x y]
[true true] false
[false true] true
[true false] true
[false false] false))Example 1.11: Pattern matching using the match macro
The xor function from Example 1.11 simply matches its arguments, x and y, against a given set of patterns, such as [true true] and [true false]. If both the arguments are true or false, then the function returns false, or else it returns true. It's a concise definition that relies on the values of the supplied arguments, rather than the use of conditional forms such as if and when. The xor function can be defined alternatively, and even more concisely, by the defun macro, as shown in Example 1.12:
(f/defun xor ([true true] false) ([false true] true) ([true false] true) ([false false] false))
Example 1.12: Pattern match using the defun macro
The definition of the xor function that uses the defun macro simply declares the actual values as its arguments. The expression to be returned is thus determined by the values of its inputs. Note that the defun macro rewrites the definition of the xor function to use the match macro. Hence, all patterns supported by the match macro can also be used with the defun macro. Both the definitions of the xor function, from Example 1.11 and Example 1.12, work as expected, as shown here:
user> (xor true true) false user> (xor true false) true user> (xor false true) true user> (xor false false) false
The xor function will throw an exception if we try to pass values that have not been declared as a pattern:
user> (xor 0 0)
IllegalArgumentException No matching clause: [0 0] user/xor ...We can define a simple function to compute the nth number of the Fibonacci sequence using the defun macro, as shown in Example 1.13:
(f/defun fibo
([0] 0N)
([1] 1N)
([n] (+ (fibo (- n 1))
(fibo (- n 2)))))Note the use of the variable n in the function's pattern rules. This signifies that any value other than 0 and 1 will match with the pattern definition that uses n. The fibo function defined in Example 1.13 does indeed calculate the nth Fibonacci sequence, as shown here:
user> (fibo 0) 0N user> (fibo 1) 1N user> (fibo 10) 55N
However, the definition of fibo, shown in Example 1.13, cannot be optimized by tail call elimination. This is due to the fact that the definition of fibo is tree recursive. By this, we mean to say that the expression (+ (fibo ...) (fibo ...)) requires two recursive calls in order to be evaluated completely. In fact, if we replace the recursive calls to the fibo function with recur expressions, the resulting function won't compile. It is fairly simple to convert tree recursion into linear recursion, as shown in Example 1.14:
(f/defun fibo-recur ([a b 0] a) ([a b n] (recur b (+ a b) (dec n))) ([n] (recur 0N 1N n)))
Example 1.14: A tail recursive function with pattern matching
It is fairly obvious from the definition of the fibo-recur function, from Example 1.14, that it is indeed tail recursive. This function does not consume any stack space, and can be safely called with large values of n, as shown here:
user> (fibo-recur 0) 0N user> (fibo-recur 1) 1N user> (fibo-recur 10) 55N user> (fibo-recur 9999) 207936...230626N
As the preceding examples show us, pattern matching is a powerful tool in functional programming. Functions that are defined using pattern matching are not only correct and expressive, but can also achieve good performance. In this respect, the core.match and defun libraries are indispensible tools in the Clojure ecosystem.
In this chapter, we introduced a few programming constructs that can be used in the Clojure language. We've explored recursion using the recur, loop, and trampoline forms. We've also studied the basics of sequences and laziness, while describing the various functions in the Clojure language that are used in creating, transforming, and filtering sequences. Next, we had a look at zippers, and how they can be used to idiomatically handle trees and hierarchical data such as XML. Finally, we briefly explored pattern matching using the core.match and defun libraries.
In the next chapter, we will explore concurrency and parallelism. We will study the various data structures and functions that allow us to leverage these concepts in Clojure in ample detail.
