Collect Sets Of Ranges
August 25, 2015
Here is a complicated program that mixes the control-break processing with printing to achieve the desired task:
(define (ranges . xs)
(let loop ((xs xs) (in-run #f) (prev #f))
(cond ((null? xs)
(when (and in-run prev)
(display "-")
(display prev))
(newline))
(in-run
(if (= (car xs) (+ 1 prev))
(loop (cdr xs) #t (car xs))
(begin (display "-")
(display prev)
(display ", ")
(display (car xs))
(loop (cdr xs) #f (car xs)))))
((and prev (= (car xs) (+ 1 prev)))
(loop (cdr xs) #t (car xs)))
(else (when prev (display ", "))
(display (car xs))
(loop (cdr xs) #f (car xs))))))
> (ranges 0 1 2 7 21 22 108 109)
0-2, 7, 21-22, 108-109
The in-run variable keeps track of whether or not we are in a run, and the prev variable, initially #f, tracks the previous value in the sequence.
It is easier to separate the tasks of control-break processing and printing. Here is a function that collects groups of integers:
(define (collect . xs)
(if (null? xs) (list)
(let loop ((xs (cdr xs))
(start (car xs))
(prev (car xs))
(zs (list)))
(cond ((null? xs) ; end of input
(if (= start prev)
(reverse (cons prev zs))
(reverse (cons (cons start prev) zs))))
((= (car xs) (+ prev 1)) ; continue run
(loop (cdr xs) start (car xs) zs))
(else ; end run, start new run
(if (= start prev)
(loop (cdr xs) (car xs) (car xs)
(cons prev zs))
(loop (cdr xs) (car xs) (car xs)
(cons (cons start prev) zs))))))))
> (collect 0 1 2 7 21 22 108 109)
((0 . 2) 7 (21 . 22) (108 . 109))
That’s still complicated, but it’s easier to read than ranges, and simpler, too, since there are only three clauses in the cond rather than four. Once you’ve collected the ranges, you can display them or use them in some other way:
(define (display-ranges xs)
(cond ((pair? (car xs))
(display (caar xs))
(display "-")
(display (cdar xs)))
(else (display (car xs))))
(when (pair? (cdr xs))
(display ", ")
(display-ranges (cdr xs))))
> (display-ranges (collect 0 1 2 7 21 22 108 109))
0-2, 7, 21-22, 108-109
If you’re clever, you can pass the break function as a parameter, which makes the collector function much more general:
(define (collect-by break? . xs)
(if (null? xs) (list)
(let loop ((xs (cdr xs))
(start (car xs))
(prev (car xs))
(zs (list)))
(cond ((null? xs) ; end of input
(if (equal? start prev)
(reverse (cons prev zs))
(reverse (cons (cons start prev) zs))))
((break? prev (car xs)) ; continue run
(loop (cdr xs) start (car xs) zs))
(else ; end run, start new run
(if (equal? start prev)
(loop (cdr xs) (car xs) (car xs)
(cons prev zs))
(loop (cdr xs) (car xs) (car xs)
(cons (cons start prev) zs))))))))
Then we can collect and display the ranges like this:
(define (consecutive? a b) (= (+ a 1) b)) > (display-ranges (collect-by consecutive? 0 1 2 7 21 22 108 109) 0-2, 7, 21-22, 108-109
Another approach is to return a list of splits, which is particularly pretty when you use a pattern-match on the input list; xs is the input list, ys is the current run, and xss is the output list in reverse order:
(define (split-between pred? xs)
(let loop ((xs xs) (ys (list)) (xss (list)))
(list-match xs
(() (reverse (cons (reverse ys) xss)))
((x) (reverse (cons (reverse (cons x ys)) xss)))
((x1 x2 . more) (pred? x1 x2) ; x2 starts new group
(loop (cons x2 more) (list) (cons (reverse (cons x1 ys)) xss)))
((x1 x2 . more) (loop (cons x2 more) (cons x1 ys) xss)))))
(define (display-ranges . xs)
(let loop ((xss (split-between (complement consecutive?) xs)))
(cond ((and (pair? (car xss)) (pair? (cdar xss)))
(display (caar xss))
(display "-")
(display (car (reverse (car xss)))))
(else (display (caar xss))))
(when (pair? (cdr xss))
(display ", ")
(loop (cdr xss)))))
> (display-ranges (split-between consecutive? '(0 1 2 7 21 22 108 109))
0-2, 7, 21-22, 108-109
Split-between is generally useful, and might make it into the Standard Prelude someday.
We used list-match and complement from the Standard Prelude. You can run the program at http://ideone.com/DxR3Xp, where you will also find a version of split-between that doesn’t use pattern matching.
Our experience here is instructive. We started with a working program that was a little bit of a mess. First we restructured to separate the control-break from the output, then we generalized the control-break processing to create a new idiom. Good programmers learn to recognize idioms because they simplify the problem and lead to code reuse, which is both economical and likely to reduce bugs.
Programming Praxis 
Keeps track of what is visited, using a Python generator to yield each range found.
In Python.
def collected_ranges(seq): it = iter(seq) last = (next(it),) for e in it: if e - last[-1] == 1: last = (last[0], e) else: yield last last = (e,) yield last def str_range(r): return str(r[0]) if len(r) == 1 else str(r[0]) + "-" + str(r[1]) seq = [0, 1, 2, 7, 21, 22, 108, 109 ] print(", ".join(str_range(r) for r in collected_ranges(seq)))It’s a classic problem that Jackson suggested a simple solution for.
You structure the control flow after the data, i.e. one outer loop for the break points, and one inner loop for a range. The point is to have the state implicitly coded into the control flow, rather than manipulating an explicit state in a single loop. In this case this can be done by “reading ahead” before the loop, and then “reading” at the end of the loop.
To put it simple: Follow the structure of the data. This makes the loops very simple and easy to understand.
def collect(seq): result = [] it = iter(seq) e = next(it, None) while e is not None: fst, snd = e, e e = next(it, None) while e is not None and e == snd + 1: snd = e e = next(it, None) range = (fst, snd) result.append(range) return result print(collect([0, 1, 2, 7, 21, 22, 108, 109])) print(collect([])) print(collect([0, 1, 2, 3, 4, -1, 0, 1, 10])) print(collect([0, 2, 3, 4, -3, -2, -1, 10]))def collectRanges(in: Stream[Int], left: Option[Int] = None, right: Option[Int] = None): Stream[Tuple2[Int, Int]] = { println(in.toList+" "+left+" "+right) if (left == None) if (in.isEmpty) Stream.Empty else collectRanges(in.tail, Some(in.head), Some(in.head)) else if (in.isEmpty) Stream((left.get, right.get)) else if (in.head == right.get + 1) collectRanges(in.tail, left, Some(right.get + 1)) else Stream((left.get, right.get)) #::: collectRanges(in) }Oh yeah, the previous comment is the solution as a Scala Stream
F# solution :
let set = [-1;0; 1; 2; 7; 21; 22; 108; 109; 110; 115 ; 116 ; 117] let f ar = let rec loop arr last range isInRange (acc: string list) = match arr with | a :: tail when range = "" -> loop tail a (a.ToString()) false acc | a :: tail when last = (a - 1) -> loop tail a range true acc | a :: tail -> if(not isInRange) then range :: acc |> loop tail a (a.ToString()) false else (range + "-" + last.ToString()) :: acc |> loop tail a (a.ToString()) false | [] -> if(not isInRange) then range :: acc else (range + "-" + last.ToString()) :: acc loop ar -1 "" false [] |> List.rev let result = f set;; //val result : string list = ["-1-2"; "7"; "21-22"; "108-110"; "115-117"]Here’s another Haskell solution, no fancy output this time, but using a general splitting function with foldl operating on the partially constructed ranges:
import Control.Arrow ((&&&)) split p (x:s) = foldl aux [[x]] s where aux (s@(x:_):acc) y | p x y = (y:s):acc | otherwise = [y]:s:acc ranges = reverse . map (last &&& head) . split ((==).(+1)) main = print $ ranges [0,1,2,7,21,22,108,109] -- [(0,2),(7,7),(21,22),(108,109)]Here’s another Haskell solution, no fancy output this time, but using a general splitting function with foldl operating on the partially constructed ranges:
import Control.Arrow ((&&&)) split p (x:s) = foldl aux [[x]] s where aux (s@(x:_):acc) y | p x y = (y:s):acc | otherwise = [y]:s:acc ranges = reverse . map (last &&& head) . split ((==).(+1)) main = print $ ranges [0,1,2,7,21,22,108,109] -- [(0,2),(7,7),(21,22),(108,109)](No Haskell highlighter – using “text” here)
Just as I would do with a pen and paper, I track the consecutive numbers; their differences is one. Anything other than that marks a breakpoint.
Here is the implementation in JavaScript:
function makeRange(nums){ // store differences betweens the numbers. // This is a way to know consequtive numbers var diffs = []; for (var i = 1; i < nums.length; i++){ diffs.push (nums[i] - nums[i-1]); } // For example, for the numbers : 1,2,3,5,6,9 // The differences are : 1,1,2,1,3 // Numbers that are not 1 mark the break points. // Will now use these break points to construct the ranges var res = []; for (var j = 0, i = 0; j < nums.length; j++){ // for each break point, get the indexes up to, but not including, that point. if (diffs[j] != 1){ var range = nums.slice(i,j+1); res.push(range); i = j + 1; } } return res; }Sample run:
public class Range {
public static String getRange(int a[], int index)
{
if(index == a.length-1)
return “”+a[index];
else{
int i=index;
int start=a[i];
int end=0;
while(a[i+1]==a[i]+1)
i++;
end=a[i];
if(end!=start)
return start+”-“+end+”,”+getRange(a,i+1);
else
return start+”,”+getRange(a,i+1);
}
}
public static void main(String[] args) {
int a[]={1,2,3,4,5,6,8,9,10,-34,-33,36,37,4,8,9,0,1,1,1};
System.out.println(getRange(a,0));
}
}
Another Haskell solution. We allow finding ranges for types other than numbers.
-- True if and only if x is either the predecessor of y or equal to it. isPreq :: (Bounded a, Enum a, Eq a) => a -> a -> Bool x `isPreq` y = x == y || y /= minBound && x == pred y -- Condense a list into a sequence of ranges. The range of a single element -- x is (x,x). ranges :: (Bounded a, Enum a, Eq a) => [a] -> [(a, a)] ranges = foldr step [] where step x [] = [(x,x)] step x ((a,b):cs) | x `isPreq` a = (x,b):cs | otherwise = (x,x):(a,b):cs ------------------------------------------------------------------------------- data Spectrum = Red | Orange | Yellow | Green | Blue | Indigo | Violet deriving (Bounded, Enum, Eq, Show) main :: IO () main = do -- The list of numbers from the problem. let ts = [0, 1, 2, 7, 21, 22, 108, 109] :: [Int] putStrLn $ show ts ++ " =>\n " ++ show (ranges ts) -- The numbers can be repeated and don't have to be increasing. let us = [108, 109, 0, 1, 1, 2, 2, 21, 22, 7] :: [Int] putStrLn $ show us ++ " =>\n " ++ show (ranges us) -- We can generate ranges for types other than numbers. The ordering is -- determined by the declaration of Spectrum. let ss = [Indigo, Violet, Red, Yellow, Green, Blue, Orange] putStrLn $ show ss ++ " =>\n " ++ show (ranges ss)@Globules: nice. Interesting that neither of our fold-based solutions work for infinite lists, whereas my original non-fold-based one does.
So, here is another Haskell solution, using scanl as the looping construct rather than a fold – scanl is like foldl but returns a list of the intermediate results, so we can map over that to extract the actual ranges. To deal with the end of input case, we need to munge the data a little, so we can’t just use concat. Infinite lists are dealt with just fine:
ranges (x:s) = munge $ scanl f ([],(x,x)) s where f (_,(a,b)) c | b+1 == c = ([],(a,c)) | otherwise = ([(a,b)],(c,c)) munge [(x,a)] = x ++ [a] munge ((x,_):s) = x ++ munge s s = 0:1:2:map (+4) s main = (print $ ranges [0,1,2,7,21,22,108,109]) >> (print $ ranges [1]) >> (print $ take 10 $ ranges s)@matthew: Thanks. Yeah, I wasn’t even thinking about infinite lists when I wrote it. (Bad programmer!)
Another annoyance is that in generalizing it to Enum I had to also use Bounded to ensure that pred didn’t throw. But of course now it doesn’t work for Integer. Having maybePred and maybeSucc would be nice.
Aha, yes, I didn’t see that problem. Here’s another iterative solution, this time using unfoldr. It’s rather like Dennis’s nested loop above. Once again, we have to do some jiggerypokery to terminate properly:
import Data.List ranges :: (Eq t, Num t) => [t] -> [(t, t)] ranges [] = [] ranges (a:s) = unfoldr (fmap aux) $ Just(a,a,s) where aux (a,b,[]) = ((a,b),Nothing) aux (a,b,c:s) | b+1 == c = aux (a,c,s) | otherwise = ((a,b),Just(c,c,s)) ss :: [Integer] ss = 0:1:2:map (+4) ss main :: IO() main = (print $ ranges ([0,1,2,7,21,22,108,109] :: [Int])) >> (print $ ranges ([1] :: [Int])) >> (print $ take 10 $ ranges ss)itertools.groupby() from the standard library creates an iterator that breaks an iterable into (key, group) pairs.
Here, a class is used to maintain state–the previous value and the current group number. The class implements the __call__ method to be used as the key func for groupby(). The method applies a predicate to the previous and current items. If the predicate is true, the group number is returned. If the predicate is false, the group number is changed (incremented) first. As a result, each group gets a sequential number assigned.
def collect(seq, predicate): class Grouper: def __init__(self, predicate): self.group_no = 0 self.predicate = predicate self.prev_value = None def __call__(self, new_value): if self.prev_value is not None and not self.predicate(self.prev_value, new_value): self.group_no += 1 self.prev_value = new_value return self.group_no yield from (g for _,g in it.groupby(seq, Grouper(predicate))) [list(g) for g in collect(seq, lambda a,b: a + 1 == b)]This can also be implemented using a generator/coroutine. The generator’s send() method is used as the key func for groupby().
def collect(seq, predicate): def grouper(predicate): group_no = 0 new_value = yield while True: prev_value, new_value = (new_value, (yield group_no)) if not predicate(prev_value, new_value): group_no += 1 gpr = grouper(predicate) next(gpr) yield from (g for _,g in it.groupby(seq, gpr.send)) [list(g) for g in collect(seq, lambda a,b: a + 1 == b)]Definitely the last Haskell solution from me. The pattern here is a sort of map with state, with some special handling of the final state; this can all be nicely captured by the mapAccumL function, as in:
import Data.List ranges [] = [] ranges (x:s) = concat t ++ [a] where (a,t) = mapAccumL f (x,x) s f (m,n) p = if n+1 == p then ((m,p),[]) else ((p,p),[(m,n)]) main = print $ ranges [0,1,2,7,21,22,108,109]Haskell. A job for a right fold with a lazy combining function, creating the shape of the return value first, and filling it up with data from the recursive result later, like any well-behaved tail recursive “modulo cons” program would do.
(the list comprehension is there just to avoid the verbal and verbose “case” statement).
@Will Ness: Very ingenious. We can rewrite Globules original foldr solution (using a case statement here, I like pattern matching) as:
ranges = foldr step [] where step x r = (x,y):s where (_,y):s = case r of [] -> (x,x):r (a,_):_ -> if x+1 == a then r else (x,x):rand get proper behaviour on infinite lists. It does all seem a bit awkward though – I’m not convinced I like any of the functional solutions more than the original nice and simple recursive method.
@matthew: YMMV. I learned to stop worrying, and love the fold. :) It’s easier for me that way: the recursion is already there, `x` is for the value, `r` is for recursive result, and all that’s left for me to do is to decide what goes where. It’s all about the information flow anyway. Here’s (http://stackoverflow.com/a/14403413/849891) where I saw this technique first, in the context of Haskell. In Prolog it’s commonplace, but there we only have the direct recursion (in plain language, that is).
Turnes out there’s also my answer on that SO question, with another solution, a paramophism which might feel more natural.
Also, your “jiggerypokery” `unfoldr` might be related to this (http://stackoverflow.com/q/24519530/849891) “slightly generalizing unfold”, which just might be an apomorphism.
Well, I like a nice fold too, but sometimes things don’t always fit nicely into that pattern. Foldr has nice algebraic properties, but it seems difficult to get right, particularly with lazy lists (and there seems to be a correlation between not working with lazy lists and having a space leak) – maybe that gets easier with practice though, as you suggest. Apomorphism sounds like just the ticket, thanks for that; disjoint unions are an underused datatype.
Incidentally, I came up with this for finding the end points of the ranges, but I’m not sure what to make of it:
So, my original function, which was more or less:
ranges (x:s) = f (x,x) s where f (m,n) [] = [(m,n)] f (m,n) (p:s) | n+1 == p = f (m,p) s | otherwise = (m,n) : f (p,p) sworks nicely with infinite lists and all that, but doesn’t easily fit the foldr recursion scheme because of that extra state parameter – sometimes the recursion is ‘f (m,p) s’ and sometimes it is ‘f (p,p) s’. But we can swap the order of the parameters to f and get:
ranges2 (x:s) = f s (x,x) where f [] (m,n) = [(m,n)] f (p:s) (m,n) | n+1 == p = f s (m,p) | otherwise = (m,n) : f s (p,p)and this does fit the foldr pattern, giving:
where the function on the input list is returning another function that is then applied to initial state. Actually running this with eg. ‘ranges [1..100000000]’ and we get the answer ‘[(1,100000000)]’ in a few seconds, with no memory bloat. Once again, I am very impressed with the Haskell compiler.
On the other hand, all the compiler has really done is inline the definition of foldr, so maybe it’s not that clever.
A solution in Racket Scheme.