нУЁh&=∙;hы !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWX>Closed intervals of totally ordered types, e.g. time intervals(c) Lackmann PhymetricGPL-3olaf.klinke@phymetric.deexperimental Safe-Inferred 89:абцдег:ю;Yclosed-intervalsThe result of 6ь, either the empty sequence, a singleton or two subsequences of roughly the same sizeclosed-intervalsSearch tree of intervals.closed-intervals/class of Intervals whose bounds can be adjustedclosed-intervals!Time types supporting differencesclosed-intervalsZuences support ь efficiently only in the case when the sequence has the property that for any split xs = ys <> zs< into non-empty parts the convex hull of each part is the and ы of the leftmost and rightmost element, respectively. This property is guaranteed by (п but does not hold in the case where the sequence contains nested intervals:┤propSplit (\xs -> hullSeqNonNested xs == hullSeq xs) . splitSeq . sortByRight $ Seq.fromList ([(1,3),(2,4),(4,5),(3,6)] :: [(Int,Int)])FalseмThus, when querying against a set of intervals with nesting, you must use an К instead. Observe that non-nestedness is a quite strong property. The logical negation of the sentence 8there exist intervals i, j such that i is contained in j is for all i, j either lb i < lb j or ub i > ub j. But if lb i < lb j then also ub i < ub j- because otherwise i contains j. Likewise, ub i > ub j implies lb i > lb j⌠ otherwise i contains j. Hence a non-nested sequence of intervals can be sorted by either left of right end-point resulting in the same order. Ьforevery genNonNestedIntervalSeq $ \xs -> propSplit (\subseq -> hullSeqNonNested subseq == hullSeq subseq) (splitSeq xs) closed-intervalsкclass of search structures for interval intersection queries, returning a [ of intervals.closed-intervals%all intervalls touching the first oneclosed-intervals1all intervals properly intersecting the first one closed-intervals&does any interval touch the first one?closed-intervals3does any interval properly intersect the first one?closed-intervalsthe convex hull of the contentsclosed-intervals*dump the entire search structure's contentclosed-intervals overlapTime i i == intervalDuration iзforeveryPair genInterval $ \i j -> not (i `properlyIntersects` j) ==> overlapTime i j == 0КforeveryPair genInterval $ \i j -> overlapTime i j == (sum $ fmap intervalDuration $ maybeIntersection i j)closed-intervals0Prevailing annotation in the first time intervalсforevery genInterval $ \i c -> prevailing i (Seq.singleton (c,i)) == Just (c::Char)КforeveryPairOf genInterval genLabeledSeq $ \i js -> isJust (prevailing i js) == any (intersects i . snd) js forevery genInterval $ \i -> foreveryPair genLabeledSeq $ \js ks -> all (flip elem $ catMaybes [prevailing i js, prevailing i ks]) $ prevailing i (js<>ks)closed-intervals isJust (maybeUnion i j) ==> fromJust (maybeUnion i j) `contains` i && fromJust (maybeUnion i j) `contains` jЙforeveryPair genInterval $ \i j -> i `intersects` j ==> (maybeUnion i j >>= maybeIntersection i) == Just iclosed-intervalsаthe intersection of two intervals is either empty or an interval.ЕforeveryPair genInterval $ \i j -> i `intersects` j ==> i `contains` fromJust (maybeIntersection i j)closed-intervalsO(n) convex hullш\xs -> isJust (hull xs) ==> all (\x -> fromJust (hull xs) `contains` x) (xs :: [(Int,Int)])closed-intervalsаSet difference. The resulting list has zero, one or two elements.without' (1,5) (4,5)[(1,4)]without' (1,5) (2,3) [(1,2),(3,5)]without' (1,5) (1,5)[]without' (1,5) (0,1)[(1,5)]>foreveryPair genInterval $ \i j -> length (i `without` j) <= 24forevery genInterval $ \i -> i `without` i == []цforeveryPair genInterval $ \i j -> all (contains i) (i `without` j)сforeveryPair genInterval $ \i j -> not $ any (properlyIntersects j) (i `without` j)closed-intervals$ф is not an equivalence relation, because it is not transitive. Hence groupBy $Ч does not do what one might expect. This function does the expected and groups overlapping intervals into contiguous blocks.Бforevery genSortedIntervals $ all (\xs -> and $ List.zipWith intersects xs (tail xs)) . contiguousхforevery genSortedIntervals $ all ((1==).length.components) . contiguousclosed-intervals)Connected components of a list sorted by ), akin to groupBy $#. The precondition is not checked.ыforevery genSortedIntervals $ \xs -> all (\i -> any (flip contains i) (components xs)) xs▌forevery genSortedIntervals $ \xs -> let cs = components xs in all (\(i,j) -> i == j || not (i `intersects` j)) [(c1,c2) | c1 <- cs, c2 <- cs]closed-intervalssame as . Is there a way to unify both?Фforevery genSortedIntervals $ \xs -> componentsSeq (Seq.fromList xs) == Seq.fromList (components xs)·forevery genSortedIntervalSeq $ \xs -> let cs = componentsSeq xs in all (\(i,j) -> i == j || not (i `intersects` j)) $ do {c1 <- cs; c2 <- cs; return (c1,c2)}closed-intervals&compute the components of the part of i covered by the intervals.тforeveryPairOf genInterval genIntervalSeq $ \i js -> all (contains i) (covered i js)щforeveryPairOf genInterval genIntervalSeq $ \i js -> covered i (covered i js) == covered i js closed-intervals\ц if the first interval is completely covered by the given intervalsхforeveryPair genInterval $ \i j -> j `contains` i == i `coveredBy` [j]СforeveryPairOf genInterval genSortedIntervals $ \i js -> i `coveredBy` js ==> any (flip contains i) (components js)!closed-intervalsпpercentage of coverage of the first interval by the second sequence of intervals─foreveryPairOf genNonEmptyInterval genIntervalSeq $ \i js -> i `coveredBy` js == (fractionCovered i js >= (1::Rational))▄foreveryPairOf genNonEmptyInterval genNonEmptyIntervalSeq $ \i js -> any (properlyIntersects i) js == (fractionCovered i js > (0::Rational))"closed-intervalsOverlap ordering. Returns ] or ^! if the intervals are disjoint, _к if the intervals overlap. Note that this violates the following property:" x y == _ && " y z == _ => " x z == _ i.e., " is not transitive.лforeveryPair genInterval $ \i j -> i `intersects` j == (overlap i j == EQ)#closed-intervalsOverlap ordering. Returns ] or ^а if the intervals are disjoint or touch in end point(s) only, _т if the intervals properly overlap. Note that this violates the following property:# x y == _ && # y z == _ => # x z == _ i.e., # is not transitive.зforeveryPair genInterval $ \i j -> i `properlyIntersects` j == (properOverlap i j == EQ)$closed-intervalsintersection query.2((1,2)::(Int,Int)) `intersects` ((2,3)::(Int,Int))True┐foreveryPair genInterval $ \i j -> (lb i <= ub i && lb j <= ub j && i `intersects` j) == (max (lb i) (lb j) <= min (ub i) (ub j))%closed-intervalsproper intersection.ЩforeveryPair genInterval $ \i j -> ((i `intersects` j) && not (i `properlyIntersects` j)) == (ub i == lb j || ub j == lb i)&closed-intervalssubset containment/forevery genInterval $ \i -> i `contains` iоforeveryPair genInterval $ \i j -> (i `contains` j && j `contains` i) == (i==j)оforeveryPair genInterval $ \i j -> i `contains` j == (maybeUnion i j == Just i)'closed-intervalsproper subset containment(closed-intervalsconstruct a sorted Е sequence of intervals from a sorted sequence of bounds. Fails if the input sequence is not sorted.Чforevery genSortedList $ \xs -> (components $ toList $ fromEndPoints xs) == if length xs < 2 then [] else [(head xs, last xs)]─forevery genSortedList $ \xs -> hullSeqNonNested (fromEndPoints xs) == if length xs < 2 then Nothing else Just (head xs,last xs))closed-intervalslexicographical sort by , then inverse ⌠. If the sequence of intervals is non-nested, then in the resulting list the intervals intersecting a given interval form a contiguous sublist.┘foreveryPairOf genInterval genNonNestedIntervalSeq $ \i js -> toList (getIntersects i (FromSortedSeq js)) `isSubsequenceOf` toList jsХforevery genSortedIntervalSeq $ \xs -> propSplit (\subseq -> subseq == sortByRight subseq) (splitSeq xs)*closed-intervalsO(n)0 Extract all intervals intersecting a given one.+closed-intervalsO(n)9 Extract all intervals properly intersecting a given one.,closed-intervalsO(n) convex hull of a sorted ()├) sequence of intervals. the upper bound is guaranteed to be in the rightmost interval, but we have no guarantee of the lower bound.├forevery genSortedIntervalSeq $ \xs -> hullSeq xs == if Seq.null xs then Nothing else Just (minimum (fmap lb xs),maximum (fmap ub xs))еforevery genSortedIntervalSeq $ \xs -> hullSeq xs == hull (toList xs)-closed-intervalsЮWhen you face the problem of matching two series of intervals against each other, a streaming approach might be more efficient than transforming one of the streams into a search structure. This function drops intervals from the list until the (contiguous) block of intersecting intervals is found. This block (except intervals containing the э of the query) is removed from the stream. When used as a state transformer on a stream [i]Т of non-properly overlapping intervals, then one obtains the stream of blocks intersecting the stream of queries. вsplitIntersecting ((2,5) :: (Int,Int)) ([(0,1),(2,2),(2,3),(3,6),(6,7)] :: [(Int,Int)])#([(2,2),(2,3),(3,6)],[(3,6),(6,7)])▌foreveryPairOf genInterval genNonNestedIntervalSeq $ \i js' -> let js = toList js' in fst (splitIntersecting i js) == filter (intersects i) js▓foreveryPairOf genInterval genNonNestedIntervalSeq $ \i js' -> let js = toList js' in all (\j -> not (ub j < ub i)) (snd (splitIntersecting i js)).closed-intervalsLike -В but disregards those intervals that merely touch the query. Retains overlapping intervals properly containing the и of the query. When used as a state transformer on an ascending stream [i]Э of non-properly overlapping intervals, then one obtains the stream of blocks properly intersecting the stream of queries.ФsplitProperlyIntersecting ((2,5) :: (Int,Int)) ([(0,1),(2,3),(2,2),(3,5),(5,6),(6,7)] :: [(Int,Int)])([(2,3),(3,5)],[(5,6),(6,7)])·foreveryPairOf genInterval genNonNestedIntervalSeq $ \i js' -> let js = toList js' in fst (splitProperlyIntersecting i js) == filter (properlyIntersects i) js≥foreveryPairOf genInterval genNonNestedIntervalSeq $ \i js' -> let js = toList js' in all (not.properlyContains i) (snd (splitProperlyIntersecting i js))/closed-intervals the empty 0closed-intervals,smallest interval covering the entire tree. ` if the tree is empty.лforevery genSortedIntervalSeq $ \xs -> hullSeq xs == hullOfTree (itree 4 xs)1closed-intervals:invariant to be maintained for proper intersection queries9forevery genIntervalSeq $ \xs -> invariant . itree 4 $ xsaclosed-intervals.Intersection query. O(binsize+log(n/binsize)).ЗforeveryPairOf genInterval genIntervalSeq $ \i t -> on (==) sortByRight (getIntersects i $ itree 2 t) (i `intersecting` t)bclosed-intervals.Intersection query. O(binsize+log(n/binsize)).┬foreveryPairOf genInterval genIntervalSeq $ \i t -> on (==) sortByRight (getProperIntersects i $ itree 2 t) (i `intersectingProperly` t)cclosed-intervalsWhen the actual result of a9 is not important, only whether there are intersections.dclosed-intervalsWhen the actual result of a9 is not important, only whether there are intersections.2closed-intervals2transform the interval tree into the tree of hulls3closed-intervals!generalises Control.Monad.filterM4closed-intervals#re-assemble a split into a sequence5closed-intervalsаtest if a sequence property holds for each sequence in the split.6closed-intervals)Split a Sequence in half, needed for the о instance. prop> forevery genIntervalSeq $ is -> joinSeq (splitSeq is) == is7closed-intervals≈insert the interval at the deepest possible location into the tree. Does not change the overall structure, in particular no re-balancing is performed.8closed-intervalsшConstruct an interval tree with bins of maximal given size. The function first sorts the intervals, then splits into chunks of given size. The leftmost endpoints of the chunks define boundary points. Next, all intervals properly overlapping a boundary are removed from the chunks and kept separately. The chunks are arranged as the leaves of a binary search tree. Then the intervals overlapping boundaries are placed at internal nodes of the tree. Hence if all intervals are mutually non-overlapping properly, the resulting tree is a pure binary search tree with bins of given size as leaves.9closed-intervalsO(1)9 bounds of an ordered, non-nested sequence of intervals. `, if empty.кforevery genNonNestedIntervalSeq $ \xs -> hullSeqNonNested xs == hullSeq xseclosed-intervalsQuery an ordered Z/uence of non-nested intervals for a predicate p that has the propertyj & k && p i k ==> p i j 1and return all elements satisfying the predicate.УforeveryPairOf genInterval genNonNestedIntervalSeq $ \i js -> getIntersects i (FromSortedSeq js) == intersecting i jsfclosed-intervalsQuery an ordered Z/uence of non-nested intervals for a predicate p that has the propertyj & k && p i k ==> p i j =closed-intervalsBeware that using >>= may destroy non-nestedness.>closed-intervalsBeware that using * may destroy non-nestedness.Cclosed-intervals preserves the TimeZoneclosed-intervalsadjust lower and upper boundclosed-intervals*change both bounds using the same functionclosed-intervalslower boundclosed-intervalsupper boundclosed-intervalsend points (inclusive): !"#$%&'()*+,-./0123456789: $%&' "#!,9)(-.8/7012*+3456 Safe-Inferred;[ghijklmnО !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdecdfcdgcdh`ijklmnopqrstuvwxЫ/closed-intervals-0.2.0.1-En9h1BpbkSPFip97hOS3KF Data.IntervalPaths_closed_intervalsITreeAdjustadjustBoundsshiftTimeDifferencediffTimeaddTimeNonNestedSeq FromSortedSeqgetSeqIntersectionQuery getIntersectsgetProperIntersectssomeIntersectssomeProperlyIntersectsmaybeBoundsstoredIntervalsIntervallbub endPointsintervalDurationoverlapTime prevailing maybeUnionmaybeIntersectionhullwithout contiguous components componentsSeqcovered coveredByfractionCoveredoverlap properOverlap intersectsproperlyIntersectscontainsproperlyContains fromEndPointssortByRightintersectingintersectingProperlyhullSeqsplitIntersectingsplitProperlyIntersecting emptyITree hullOfTree invarianttoTreefilterMjoinSeq propSplitsplitSeqinsertitreehullSeqNonNested$fIntervaleIdentity$fIntervale(,)$fFiltrableNonNestedSeq$fMonadNonNestedSeq$fAlternativeNonNestedSeq$fApplicativeNonNestedSeq$fMonoidNonNestedSeq$fSemigroupNonNestedSeq#$fIntersectionQueryNonNestedSeqeSeq$fTimeDifferenceZonedTime$fTimeDifferenceLocalTime$fTimeDifferenceUTCTime$fAdjuste(,)$fFoldableITree$fFunctorITree$fIntersectionQueryITreeeSeq$fShowBlock$fIntervaleBlock $fMonoidBlock$fSemigroupBlock$fFoldableBlock$fFunctorBlock $fOrdBlock $fEqBlock$fShowSplitSeq$fEqNonNestedSeq$fOrdNonNestedSeq$fShowNonNestedSeq$fFunctorNonNestedSeq$fFoldableNonNestedSeq$fTraversableNonNestedSeqSplitSeqcontainers-0.6.5.1Data.Sequence.InternalSeqbase Data.FoldableFoldableghc-prim GHC.TypesTrueLTGTEQ GHC.MaybeNothinggetIntersectsITgetProperIntersectsITsomeIntersectsITsomeProperlyIntersectsITfindSeq existsSeqversiongetDataFileName getBinDir getLibDirgetDynLibDir getDataDir getLibexecDir getSysconfDir