I need to Develop a function called bounds which takes a nested list of numbers as its only argument (ie: a tree). Bounds should return the largest &smallest value in the tree. Eg:
(bounds '(1 (-2 17 (4) -8 (-6 13) (-8 17))))
Using clojure and not using the flatten function but using recursion to visit each node
(defn maxv [tree]
(cond
(number? tree) tree
(tree? tree)
(let [newmax (maxv (first tree)) ]
;;let newmax be the first in the tree
(if (< newmax (maxv (first (rest tree))))
;; if the next in the tree is smaller
(= (newmax (maxv (first (rest tree)))))
;;change newmax to this
(recur (maxv (rest tree)))))
;; recur through the rest
this is what i have i think i am being too java ish
flatten is very efficient, so there is no benefit in implementing it yourself.
If your nested list is not too big, you could apply (juxt min max) directly to the flattened list:
(defn bounds [coll]
(apply (juxt min max) (flatten coll)))
For large input collections I would recommend using reduce instead of apply:
(defn bounds [coll]
(reduce (fn [[minv maxv :as res] v]
(if res [(min minv v) (max maxv v)] [v v]))
nil
(flatten coll)))
If you really need pure recursive implementation, here is an example for you:
(defn bounds [coll]
(loop [coll coll [minv maxv :as res] nil]
(if-let [[h & ts] (seq coll)]
(if (sequential? h)
(recur (concat h ts) res)
(recur ts (if res [(min minv h) (max maxv h)] [h h])))
res)))
All three implementation will yield you a tuple containing min and max values:
(bounds '(1 (-2 17 (4) -8 (-6 13) (-8 17)))) ; => [-8 17]
Update: note on sum-tree implementation from comments
It's a very bad idea to use multiple recursion in your code, because it'll blow your stack very quickly:
(sum-tree (range 7000)) ; => java.lang.StackOverflowError
Try using tail recursion with recur, or higher-order functions instead:
(defn sum-tree [tree]
(if (number? tree)
tree
(reduce + (map sum-tree tree))))
Related
problem formulation
Informally speaking, I want to write a function which, taking as input a function that generates binary factorizations and an element (usually neutral), creates an arbitrary length factorization generator. To be more specific, let us first define the function nfoldr in Clojure.
(defn nfoldr [f e]
(fn rec [n]
(fn [s]
(if (zero? n)
(if (empty? s) e)
(if (seq s)
(if-some [x ((rec (dec n)) (rest s))]
(f (list (first s) x))))))))
Here nil is used with the meaning "undefined output, input not in function's domain". Additionally, let us view the inverse relation of a function f as a set-valued function defining inv(f)(y) = {x | f(x) = y}.
I want to define a function nunfoldr such that inv(nfoldr(f , e)(n)) = nunfoldr(inv(f) , e)(n) when for every element y inv(f)(y) is finite, for each binary function f, element e and natural number n.
Moreover, I want the factorizations to be generated as lazily as possible, in a 2-dimensional sense of laziness. My goal is that, when getting some part of a factorization for the first time, there does not happen (much) computation needed for next parts or next factorizations. Similarly, when getting one factorization for the first time, there does not happen computation needed for next ones, whereas all the previous ones get in effect fully realized.
In an alternative formulation we can use the following longer version of nfoldr, which is equivalent to the shorter one when e is a neutral element.
(defn nfoldr [f e]
(fn [n]
(fn [s]
(if (zero? n)
(if (empty? s) e)
((fn rec [n]
(fn [s]
(if (= 1 n)
(if (and (seq s) (empty? (rest s))) (first s))
(if (seq s)
(if-some [x ((rec (dec n)) (rest s))]
(f (list (first s) x)))))))
n)))))
a special case
This problem is a generalization of the problem of generating partitions described in that question. Let us see how the old problem can be reduced to the current one. We have for every natural number n:
npt(n) = inv(nconcat(n)) = inv(nfoldr(concat2 , ())(n)) = nunfoldr(inv(concat2) , ())(n) = nunfoldr(pt2 , ())(n)
where:
npt(n) generates n-ary partitions
nconcat(n) computes n-ary concatenation
concat2 computes binary concatenation
pt2 generates binary partitions
So the following definitions give a solution to that problem.
(defn generate [step start]
(fn [x] (take-while some? (iterate step (start x)))))
(defn pt2-step [[x y]]
(if (seq y) (list (concat x (list (first y))) (rest y))))
(def pt2-start (partial list ()))
(def pt2 (generate pt2-step pt2-start))
(def npt (nunfoldr pt2 ()))
I will summarize my story of solving this problem, using the old one to create example runs, and conclude with some observations and proposals for extension.
solution 0
At first, I refined/generalized the approach I took for solving the old problem. Here I write my own versions of concat and map mainly for a better presentation and, in the case of concat, for some added laziness. Of course we can use Clojure's versions or mapcat instead.
(defn fproduct [f]
(fn [s]
(lazy-seq
(if (and (seq f) (seq s))
(cons
((first f) (first s))
((fproduct (rest f)) (rest s)))))))
(defn concat' [s]
(lazy-seq
(if (seq s)
(if-let [x (seq (first s))]
(cons (first x) (concat' (cons (rest x) (rest s))))
(concat' (rest s))))))
(defn map' [f]
(fn rec [s]
(lazy-seq
(if (seq s)
(cons (f (first s)) (rec (rest s)))))))
(defn nunfoldr [f e]
(fn rec [n]
(fn [x]
(if (zero? n)
(if (= e x) (list ()) ())
((comp
concat'
(map' (comp
(partial apply map)
(fproduct (list
(partial partial cons)
(rec (dec n))))))
f)
x)))))
In an attempt to get inner laziness we could replace (partial partial cons) with something like (comp (partial partial concat) list). Although this way we get inner LazySeqs, we do not gain any effective laziness because, before consing, most of the computation required for fully realizing the rest part takes place, something that seems unavoidable within this general approach. Based on the longer version of nfoldr, we can also define the following faster version.
(defn nunfoldr [f e]
(fn [n]
(fn [x]
(if (zero? n)
(if (= e x) (list ()) ())
(((fn rec [n]
(fn [x] (println \< x \>)
(if (= 1 n)
(list (list x))
((comp
concat'
(map' (comp
(partial apply map)
(fproduct (list
(partial partial cons)
(rec (dec n))))))
f)
x))))
n)
x)))))
Here I added a println call inside the main recursive function to get some visualization of eagerness. So let us demonstrate the outer laziness and inner eagerness.
user=> (first ((npt 5) (range 3)))
< (0 1 2) >
< (0 1 2) >
< (0 1 2) >
< (0 1 2) >
< (0 1 2) >
(() () () () (0 1 2))
user=> (ffirst ((npt 5) (range 3)))
< (0 1 2) >
< (0 1 2) >
< (0 1 2) >
< (0 1 2) >
< (0 1 2) >
()
solution 1
Then I thought of a more promising approach, using the function:
(defn transpose [s]
(lazy-seq
(if (every? seq s)
(cons
(map first s)
(transpose (map rest s))))))
To get the new solution we replace the previous argument in the map' call with:
(comp
(partial map (partial apply cons))
transpose
(fproduct (list
repeat
(rec (dec n)))))
Trying to get inner laziness we could replace (partial apply cons) with #(cons (first %) (lazy-seq (second %))) but this is not enough. The problem lies in the (every? seq s) test inside transpose, where checking a lazy sequence of factorizations for emptiness (as a stopping condition) results in realizing it.
solution 2
A first way to tackle the previous problem that came to my mind was to use some additional knowledge about the number of n-ary factorizations of an element. This way we can repeat a certain number of times and use only this sequence for the stopping condition of transpose. So we will replace the test inside transpose with (seq (first s)), add an input count to nunfoldr and replace the argument in the map' call with:
(comp
(partial map #(cons (first %) (lazy-seq (second %))))
transpose
(fproduct (list
(partial apply repeat)
(rec (dec n))))
(fn [[x y]] (list (list ((count (dec n)) y) x) y)))
Let us turn to the problem of partitions and define:
(defn npt-count [n]
(comp
(partial apply *)
#(map % (range 1 n))
(partial comp inc)
(partial partial /)
count))
(def npt (nunfoldr pt2 () npt-count))
Now we can demonstrate outer and inner laziness.
user=> (first ((npt 5) (range 3)))
< (0 1 2) >
(< (0 1 2) >
() < (0 1 2) >
() < (0 1 2) >
() < (0 1 2) >
() (0 1 2))
user=> (ffirst ((npt 5) (range 3)))
< (0 1 2) >
()
However, the dependence on additional knowledge and the extra computational cost make this solution unacceptable.
solution 3
Finally, I thought that in some crucial places I should use a kind of lazy sequences "with a non-lazy end", in order to be able to check for emptiness without realizing. An empty such sequence is just a non-lazy empty list and overall they behave somewhat like the lazy-conss of the early days of Clojure. Using the definitions given below we can reach an acceptable solution, which works under the assumption that always at least one of the concat'ed sequences (when there is one) is non-empty, something that holds in particular when every element has at least one binary factorization and we are using the longer version of nunfoldr.
(def lazy? (partial instance? clojure.lang.IPending))
(defn empty-eager? [x] (and (not (lazy? x)) (empty? x)))
(defn transpose [s]
(lazy-seq
(if-not (some empty-eager? s)
(cons
(map first s)
(transpose (map rest s))))))
(defn concat' [s]
(if-not (empty-eager? s)
(lazy-seq
(if-let [x (seq (first s))]
(cons (first x) (concat' (cons (rest x) (rest s))))
(concat' (rest s))))
()))
(defn map' [f]
(fn rec [s]
(if-not (empty-eager? s)
(lazy-seq (cons (f (first s)) (rec (rest s))))
())))
Note that in this approach the input function f should produce lazy sequences of the new kind and the resulting n-ary factorizer will also produce such sequences. To take care of the new input requirement, for the problem of partitions we define:
(defn pt2 [s]
(lazy-seq
(let [start (list () s)]
(cons
start
((fn rec [[x y]]
(if (seq y)
(lazy-seq
(let [step (list (concat x (list (first y))) (rest y))]
(cons step (rec step))))
()))
start)))))
Once again, let us demonstrate outer and inner laziness.
user=> (first ((npt 5) (range 3)))
< (0 1 2) >
< (0 1 2) >
(< (0 1 2) >
() < (0 1 2) >
() < (0 1 2) >
() () (0 1 2))
user=> (ffirst ((npt 5) (range 3)))
< (0 1 2) >
< (0 1 2) >
()
To make the input and output use standard lazy sequences (sacrificing a bit of laziness), we can add:
(defn lazy-end->eager-end [s]
(if (seq s)
(lazy-seq (cons (first s) (lazy-end->eager-end (rest s))))
()))
(defn eager-end->lazy-end [s]
(lazy-seq
(if-not (empty-eager? s)
(cons (first s) (eager-end->lazy-end (rest s))))))
(def nunfoldr
(comp
(partial comp (partial comp eager-end->lazy-end))
(partial apply nunfoldr)
(fproduct (list
(partial comp lazy-end->eager-end)
identity))
list))
observations and extensions
While creating solution 3, I observed that the old mechanism for lazy sequences in Clojure might not be necessarily inferior to the current one. With the transition, we gained the ability to create lazy sequences without any substantial computation taking place but lost the ability to check for emptiness without doing the computation needed to get one more element. Because both of these abilities can be important in some cases, it would be nice if a new mechanism was introduced, which would combine the advantages of the previous ones. Such a mechanism could use again an outer LazySeq thunk, which when forced would return an empty list or a Cons or another LazySeq or a new LazyCons thunk. This new thunk when forced would return a Cons or perhaps another LazyCons. Now empty? would force only LazySeq thunks while first and rest would also force LazyCons. In this setting map could look like this:
(defn map [f s]
(lazy-seq
(if (empty? s) ()
(lazy-cons
(cons (f (first s)) (map f (rest s)))))))
I have also noticed that the approach taken from solution 1 onwards lends itself to further generalization. If inside the argument in the map' call in the longer nunfoldr we replace cons with concat, transpose with some implementation of Cartesian product and repeat with another recursive call, we can then create versions that "split at different places". For example, using the following as the argument we can define a nunfoldm function that "splits in the middle" and corresponds to an easy-to-imagine nfoldm. Note that all "splitting strategies" are equivalent when f is associative.
(comp
(partial map (partial apply concat))
cproduct
(fproduct (let [n-half (quot n 2)]
(list (rec n-half) (rec (- n n-half))))))
Another natural modification would allow for infinite factorizations. To achieve this, if f generated infinite factorizations, nunfoldr(f , e)(n) should generate the factorizations in an order of type ω, so that each one of them could be produced in finite time.
Other possible extensions include dropping the n parameter, creating relational folds (in correspondence with the relational unfolds we consider here) and generically handling algebraic data structures other than sequences as input/output. This book, which I have just discovered, seems to contain valuable relevant information, given in a categorical/relational language.
Finally, to be able to do this kind of programming more conveniently, we could transfer it into a point-free, algebraic setting. This would require constructing considerable "extra machinery", in fact almost making a new language. This paper demonstrates such a language.
I have function
(defn goneSeq [inseq uptil]
(loop [counter 0 newSeq [] orginSeq inseq]
(if (== counter uptil)
newSeq
(recur (inc counter) (conj newSeq (first orginSeq)) (rest orginSeq)))))
(defn insert [sorted-seq n]
(loop [currentSeq sorted-seq counter 0]
(cond (empty? currentSeq) (concat sorted-seq (vector n))
(<= n (first currentSeq)) (concat (goneSeq sorted-seq counter) (vector n) currentSeq)
:else (recur (rest currentSeq) (inc counter)))))
that takes in a sorted-sequence and insert the number n at its appropriate position for example: (insert [1 3 4] 2) returns [1 2 3 4].
Now I want to use this function with reduce to sort a given sequence so something like:
(reduce (insert seq n) givenSeq)
What is thr correct way to achieve this?
If the function works for inserting a single value, then this would work:
(reduce insert [] givenSeq)
for example:
user> (reduce insert [] [0 1 2 30.5 0.88 2.2])
(0 0.88 1 2 2.2 30.5)
Also, it should be noted that sort and sort-by are built in and are better than most hand-rolled solutions.
May I suggest some simpler ways to do insert:
A slowish lazy way is
(defn insert [s x]
(let [[fore aft] (split-with #(> x %) s)]
(concat fore (cons x aft))))
A faster eager way is
(defn insert [coll x]
(loop [fore [], coll coll]
(if (and (seq coll) (> x (first coll)))
(recur (conj fore x) (rest coll))
(concat fore (cons x coll)))))
By the way, you had better put your defns in bottom-up order, if possible. Use declare if there is mutual recursion. You had me thinking your solution did not compile.
I have the following functions that check for odd parity in sequence
(defn countOf[a-seq elem]
(loop [number 0 currentSeq a-seq]
(cond (empty? currentSeq) number
(= (first currentSeq) elem) (recur (inc number) (rest currentSeq))
:else (recur number (rest currentSeq))
)
)
)
(defn filteredSeq[a-seq elemToRemove]
(remove (set (vector (first a-seq))) a-seq)
)
(defn parity [a-seq]
(loop [resultset [] currentSeq a-seq]
(cond (empty? currentSeq) (set resultset)
(odd? (countOf currentSeq (first currentSeq))) (recur (concat resultset (vector(first currentSeq))) (filteredSeq currentSeq (first currentSeq)))
:else (recur resultset (filteredSeq currentSeq (first currentSeq)))
)
)
)
for example (parity [1 1 1 2 2 3]) -> (1 3) that is it picks odd number of elements from a sequence.
Is there a better way to achieve this?
How can this be done with reduce function of clojure
First, I decided to make more idiomatic versions of your code, so I could really see what it was doing:
;; idiomatic naming
;; no need to rewrite count and filter for this code
;; putting item and collection in idiomatic argument order
(defn count-of [elem a-seq]
(count (filter #(= elem %) a-seq)))
;; idiomatic naming
;; putting item and collection in idiomatic argument order
;; actually used the elem-to-remove argument
(defn filtered-seq [elem-to-remove a-seq]
(remove #(= elem-to-remove %) a-seq))
;; idiomatic naming
;; if you want a set, use a set from the beginning
;; destructuring rather than repeated usage of first
;; use rest to recur when the first item is guaranteed to be dropped
(defn idiomatic-parity [a-seq]
(loop [result-set #{}
[elem & others :as current-seq] a-seq]
(cond (empty? current-seq)
result-set
(odd? (count-of elem current-seq))
(recur (conj result-set elem) (filtered-seq elem others))
:else
(recur result-set (filtered-seq elem others)))))
Next, as requested, a version that uses reduce to accumulate the result:
;; mapcat allows us to return 0 or more results for each input
(defn reducing-parity [a-seq]
(set
(mapcat
(fn [[k v]]
(when (odd? v) [k]))
(reduce (fn [result item]
(update-in result [item] (fnil inc 0)))
{}
a-seq))))
But, reading over this, I notice that the reduce is just frequencies, a built in clojure function. And my mapcat was really just a hand-rolled keep, another built in.
(defn most-idiomatic-parity [a-seq]
(set
(keep
(fn [[k v]]
(when (odd? v) k))
(frequencies a-seq))))
In Clojure we can refine our code, and as we recognize places where our logic replicates the built in functionality, we can simplify the code and make it more clear. Also, there is a good chance the built in is better optimized than our own work-alikes.
Is there a better way to achieve this?
(defn parity [coll]
(->> coll
frequencies
(filter (fn [[_ v]] (odd? v)))
(map first)
set))
For example,
(parity [1 1 1 2 1 2 1 3])
;#{1 3}
How can this be done with reduce function of clojure.
We can use reduce to rewrite frequencies:
(defn frequencies [coll]
(reduce
(fn [acc x] (assoc acc x (inc (get acc x 0))))
{}
coll))
... and again to implement parity in terms of it:
(defn parity [coll]
(let [freqs (frequencies coll)]
(reduce (fn [s [k v]] (if (odd? v) (conj s k) s)) #{} freqs)))
I'm working on 4clojure problems and a similar issue keeps coming up. I'll write a solution that works for all but one of the test cases. It's usually the one that is checking for lazy evaluation. The solution below works for all but the last test case. I've tried all kinds of solutions and can't seem to get it to stop evaluating until integer overflow. I read the chapter on lazy sequences in Joy of Clojure, but I'm having a hard time implementing them. Is there a rule of thumb I'm forgetting, like don't use loop or something like that?
; This version is non working at the moment, will try to edit a version that works
(defn i-between [p k coll]
(loop [v [] coll coll]
(let [i (first coll) coll (rest coll) n (first coll)]
(cond (and i n)
(let [ret (if (p i n) (cons k (cons i v)) (cons i v))]
(recur ret coll))
i
(cons i v )
:else v))))
Problem 132
Ultimate solution for those curious:
(fn i-between [p k coll]
(letfn [(looper [coll]
(if (empty? coll) coll
(let [[h s & xs] coll
c (cond (and h s (p h s))
(list h k )
(and h s)
(list h )
:else (list h))]
(lazy-cat c (looper (rest coll))))
))] (looper coll)))
When I think about lazy sequences, what usually works is thinking about incremental cons'ing
That is, each recursion step only adds a single element to the list, and of course you never use loop.
So what you have is something like this:
(cons (generate first) (recur rest))
When wrapped on lazy-seq, only the needed elements from the sequence are realized, for instance.
(take 5 (some-lazy-fn))
Would only do 5 recursion calls to realize the needed elements.
A tentative, far from perfect solution to the 4clojure problem, that demonstrates the idea:
(fn intercalate
[pred value col]
(letfn [(looper [s head]
(lazy-seq
(if-let [sec (first s)]
(if (pred head sec)
(cons head (cons value (looper (rest s) sec)))
(cons head (looper (rest s) sec)))
(if head [head] []))))]
(looper (rest col) (first col))))
There, the local recursive function is looper, for each element tests if the predicate is true, in that case realizes two elements(adds the interleaved one), otherwise realize just one.
Also, you can avoid recursion using higher order functions
(fn [p v xs]
(mapcat
#(if (p %1 %2) [%1 v] [%1])
xs
(lazy-cat (rest xs) (take 1 xs))))
But as #noisesmith said in the comment, you're just calling a function that calls lazy-seq.
(defn insert [s k]
(let [spl (split-with #(< % k) s)]
(concat (first spl) (list k) (last spl))))
(defn insert-sort [s]
(reduce (fn [s k] (insert s k)) '() s))
(insert-sort (reverse (range 5000)))
throws a stack over flow error. What am I doing wrong here?
Same issue as with Recursive function causing a stack overflow. Concat builds up a bunch of nested lazy sequences like (concat (concat (concat ...))) without doing any actual work, and then when you force the first element all the concats must get resolved at once, blowing the stack.
Your reduce creates new list each time.
My implementation:
(defn- insert [el seq]
(if (empty? seq) (cons el seq)
(if (< el (first seq)) (cons el seq)
(cons (first seq) (insert el (rest seq))))))
(defn insertion-sort
([seq sorted]
(if (empty? seq) sorted
(recur (rest seq) (insert (first seq) sorted))))
([seq]
(insertion-sort seq nil)))
As the main answer suggests, the list concat is the offender. Calling "doall", with that list as input... will result in an ISeq :
;;insertion sort helper
(defn insert [s k]
;;find the insert point
(let [spl (split-with #(< % k) s)
ret (concat (first spl) (list k) (last spl))]
(doall ret)))
;;insertion sort
(defn insert-sort [s]
(reduce (fn [s k] (insert s k)) '() s))
But wait... Is the sequence still lazy ?
The following hack of the above code, interestingly, indicates that the sequence is indeed still lazy !
;;insertion sort helper
(defn insert [s k]
;;find the insert point
(let [spl (split-with #(< % k) s)
ret (concat (first spl) (list k) (last spl))
ret2 (doall ret)
_ (println "final " (.getClass ret2))]
ret2))
;;insertion sort
(defn insert-sort [s]
(reduce (fn [s k] (insert s k)) '() s))
So, if the list is still lazy, then why does the use of doall fix anything ?
The "doall" function is not gauranteed to return a "non lazy" list, but rather, it gaurantees that the list which it DOES return will have been evaluated by a full walk through through.
Thus, the essence of the problem is the multiple function calls, the laziness is certainly related to this aspect of the code in your original question, but it is not the "primary" source of the overflow.