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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE CPP #-}</span>
<a name="line-2"></a><span class='hs-comment'>{-# LANGUAGE MultiParamTypeClasses #-}</span>
<a name="line-3"></a><span class='hs-cpp'>#if __GLASGOW_HASKELL__ >= 702</span>
<a name="line-4"></a><span class='hs-comment'>{-# LANGUAGE Safe #-}</span>
<a name="line-5"></a><span class='hs-cpp'>#endif</span>
<a name="line-6"></a><span class='hs-cpp'>#if __GLASGOW_HASKELL__ >= 710</span>
<a name="line-7"></a><span class='hs-comment'>{-# LANGUAGE AutoDeriveTypeable #-}</span>
<a name="line-8"></a><span class='hs-cpp'>#endif</span>
<a name="line-9"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-10"></a><span class='hs-comment'>-- |</span>
<a name="line-11"></a><span class='hs-comment'>-- Module : Data.PriorityQueue.FingerTree</span>
<a name="line-12"></a><span class='hs-comment'>-- Copyright : (c) Ross Paterson 2008</span>
<a name="line-13"></a><span class='hs-comment'>-- License : BSD-style</span>
<a name="line-14"></a><span class='hs-comment'>-- Maintainer : R.Paterson@city.ac.uk</span>
<a name="line-15"></a><span class='hs-comment'>-- Stability : experimental</span>
<a name="line-16"></a><span class='hs-comment'>-- Portability : non-portable (MPTCs and functional dependencies)</span>
<a name="line-17"></a><span class='hs-comment'>--</span>
<a name="line-18"></a><span class='hs-comment'>-- Interval maps implemented using the 'FingerTree' type, following</span>
<a name="line-19"></a><span class='hs-comment'>-- section 4.8 of</span>
<a name="line-20"></a><span class='hs-comment'>--</span>
<a name="line-21"></a><span class='hs-comment'>-- * Ralf Hinze and Ross Paterson,</span>
<a name="line-22"></a><span class='hs-comment'>-- \"Finger trees: a simple general-purpose data structure\",</span>
<a name="line-23"></a><span class='hs-comment'>-- /Journal of Functional Programming/ 16:2 (2006) pp 197-217.</span>
<a name="line-24"></a><span class='hs-comment'>-- <<a href="http://staff.city.ac.uk/~ross/papers/FingerTree.html">http://staff.city.ac.uk/~ross/papers/FingerTree.html</a>></span>
<a name="line-25"></a><span class='hs-comment'>--</span>
<a name="line-26"></a><span class='hs-comment'>-- An amortized running time is given for each operation, with /n/</span>
<a name="line-27"></a><span class='hs-comment'>-- referring to the size of the priority queue. These bounds hold even</span>
<a name="line-28"></a><span class='hs-comment'>-- in a persistent (shared) setting.</span>
<a name="line-29"></a><span class='hs-comment'>--</span>
<a name="line-30"></a><span class='hs-comment'>-- /Note/: Many of these operations have the same names as similar</span>
<a name="line-31"></a><span class='hs-comment'>-- operations on lists in the "Prelude". The ambiguity may be resolved</span>
<a name="line-32"></a><span class='hs-comment'>-- using either qualification or the @hiding@ clause.</span>
<a name="line-33"></a><span class='hs-comment'>--</span>
<a name="line-34"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-35"></a>
<a name="line-36"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>IntervalMap</span><span class='hs-varop'>.</span><span class='hs-conid'>FingerTree</span> <span class='hs-layout'>(</span>
<a name="line-37"></a> <span class='hs-comment'>-- * Intervals</span>
<a name="line-38"></a> <span class='hs-conid'>Interval</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>point</span><span class='hs-layout'>,</span>
<a name="line-39"></a> <span class='hs-comment'>-- * Interval maps</span>
<a name="line-40"></a> <span class='hs-conid'>IntervalMap</span><span class='hs-layout'>,</span> <span class='hs-varid'>empty</span><span class='hs-layout'>,</span> <span class='hs-varid'>singleton</span><span class='hs-layout'>,</span> <span class='hs-varid'>insert</span><span class='hs-layout'>,</span> <span class='hs-varid'>union</span><span class='hs-layout'>,</span>
<a name="line-41"></a> <span class='hs-comment'>-- * Searching</span>
<a name="line-42"></a> <span class='hs-varid'>search</span><span class='hs-layout'>,</span> <span class='hs-varid'>intersections</span><span class='hs-layout'>,</span> <span class='hs-varid'>dominators</span>
<a name="line-43"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-44"></a>
<a name="line-45"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>FingerTree</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>FT</span>
<a name="line-46"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>FingerTree</span> <span class='hs-layout'>(</span><span class='hs-conid'>FingerTree</span><span class='hs-layout'>,</span> <span class='hs-conid'>Measured</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>ViewL</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'><|</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>><</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-47"></a>
<a name="line-48"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Applicative</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varop'><$></span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-49"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Traversable</span> <span class='hs-layout'>(</span><span class='hs-conid'>Traversable</span><span class='hs-layout'>(</span><span class='hs-varid'>traverse</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-50"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Foldable</span> <span class='hs-layout'>(</span><span class='hs-conid'>Foldable</span><span class='hs-layout'>(</span><span class='hs-varid'>foldMap</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-51"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Monoid</span>
<a name="line-52"></a>
<a name="line-53"></a><span class='hs-comment'>----------------------------------</span>
<a name="line-54"></a><span class='hs-comment'>-- 4.8 Application: interval trees</span>
<a name="line-55"></a><span class='hs-comment'>----------------------------------</span>
<a name="line-56"></a>
<a name="line-57"></a><a name="Interval"></a><span class='hs-comment'>-- | A closed interval. The lower bound should be less than or equal</span>
<a name="line-58"></a><a name="Interval"></a><span class='hs-comment'>-- to the higher bound.</span>
<a name="line-59"></a><a name="Interval"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Interval</span> <span class='hs-layout'>{</span> <span class='hs-varid'>low</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>high</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>v</span> <span class='hs-layout'>}</span>
<a name="line-60"></a> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-61"></a>
<a name="line-62"></a><a name="point"></a><span class='hs-comment'>-- | An interval in which the lower and upper bounds are equal.</span>
<a name="line-63"></a><span class='hs-definition'>point</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span>
<a name="line-64"></a><span class='hs-definition'>point</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span> <span class='hs-varid'>v</span>
<a name="line-65"></a>
<a name="line-66"></a><a name="Node"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Node</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Node</span> <span class='hs-layout'>(</span><span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span>
<a name="line-67"></a>
<a name="line-68"></a><a name="instance%20Functor%20(Node%20v)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Functor</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-69"></a> <span class='hs-varid'>fmap</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span>
<a name="line-70"></a>
<a name="line-71"></a><a name="instance%20Foldable%20(Node%20v)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Foldable</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-72"></a> <span class='hs-varid'>foldMap</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>x</span>
<a name="line-73"></a>
<a name="line-74"></a><a name="instance%20Traversable%20(Node%20v)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Traversable</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-75"></a> <span class='hs-varid'>traverse</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-varop'><$></span> <span class='hs-varid'>f</span> <span class='hs-varid'>x</span>
<a name="line-76"></a>
<a name="line-77"></a><a name="IntInterval"></a><span class='hs-comment'>-- rightmost interval (including largest lower bound) and largest upper bound.</span>
<a name="line-78"></a><a name="IntInterval"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>IntInterval</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>NoInterval</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>IntInterval</span> <span class='hs-layout'>(</span><span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>v</span>
<a name="line-79"></a>
<a name="line-80"></a><a name="instance%20Monoid%20(IntInterval%20v)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Monoid</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntInterval</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-81"></a> <span class='hs-varid'>mempty</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>NoInterval</span>
<a name="line-82"></a> <span class='hs-conid'>NoInterval</span> <span class='hs-varop'>`mappend`</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>i</span>
<a name="line-83"></a> <span class='hs-varid'>i</span> <span class='hs-varop'>`mappend`</span> <span class='hs-conid'>NoInterval</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>i</span>
<a name="line-84"></a> <span class='hs-conid'>IntInterval</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>hi1</span> <span class='hs-varop'>`mappend`</span> <span class='hs-conid'>IntInterval</span> <span class='hs-varid'>int2</span> <span class='hs-varid'>hi2</span> <span class='hs-keyglyph'>=</span>
<a name="line-85"></a> <span class='hs-conid'>IntInterval</span> <span class='hs-varid'>int2</span> <span class='hs-layout'>(</span><span class='hs-varid'>max</span> <span class='hs-varid'>hi1</span> <span class='hs-varid'>hi2</span><span class='hs-layout'>)</span>
<a name="line-86"></a>
<a name="line-87"></a><a name="instance%20Measured%20(IntInterval%20v)%20(Node%20v%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Measured</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntInterval</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-88"></a> <span class='hs-varid'>measure</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>IntInterval</span> <span class='hs-varid'>i</span> <span class='hs-layout'>(</span><span class='hs-varid'>high</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span>
<a name="line-89"></a>
<a name="line-90"></a><a name="IntervalMap"></a><span class='hs-comment'>-- | Map of closed intervals, possibly with duplicates.</span>
<a name="line-91"></a><a name="IntervalMap"></a><span class='hs-comment'>-- The 'Foldable' and 'Traversable' instances process the intervals in</span>
<a name="line-92"></a><a name="IntervalMap"></a><span class='hs-comment'>-- lexicographical order.</span>
<a name="line-93"></a><a name="IntervalMap"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span>
<a name="line-94"></a> <span class='hs-conid'>IntervalMap</span> <span class='hs-layout'>(</span><span class='hs-conid'>FingerTree</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntInterval</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-95"></a><span class='hs-comment'>-- ordered lexicographically by interval</span>
<a name="line-96"></a>
<a name="line-97"></a><a name="instance%20Functor%20(IntervalMap%20v)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Functor</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-98"></a> <span class='hs-varid'>fmap</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>IntervalMap</span> <span class='hs-layout'>(</span><span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>unsafeFmap</span> <span class='hs-layout'>(</span><span class='hs-varid'>fmap</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-99"></a>
<a name="line-100"></a><a name="instance%20Foldable%20(IntervalMap%20v)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Foldable</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-101"></a> <span class='hs-varid'>foldMap</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldMap</span> <span class='hs-layout'>(</span><span class='hs-varid'>foldMap</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-varid'>t</span>
<a name="line-102"></a>
<a name="line-103"></a><a name="instance%20Traversable%20(IntervalMap%20v)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Traversable</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-104"></a> <span class='hs-varid'>traverse</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-105"></a> <span class='hs-conid'>IntervalMap</span> <span class='hs-varop'><$></span> <span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>unsafeTraverse</span> <span class='hs-layout'>(</span><span class='hs-varid'>traverse</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-varid'>t</span>
<a name="line-106"></a>
<a name="line-107"></a><a name="instance%20Monoid%20(IntervalMap%20v%20a)"></a><span class='hs-comment'>-- | 'empty' and 'union'.</span>
<a name="line-108"></a><a name="instance%20Monoid%20(IntervalMap%20v%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Monoid</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-109"></a> <span class='hs-varid'>mempty</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>empty</span>
<a name="line-110"></a> <span class='hs-varid'>mappend</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>union</span>
<a name="line-111"></a>
<a name="line-112"></a><a name="empty"></a><span class='hs-comment'>-- | /O(1)/. The empty interval map.</span>
<a name="line-113"></a><span class='hs-definition'>empty</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span>
<a name="line-114"></a><span class='hs-definition'>empty</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>IntervalMap</span> <span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>empty</span>
<a name="line-115"></a>
<a name="line-116"></a><a name="singleton"></a><span class='hs-comment'>-- | /O(1)/. Interval map with a single entry.</span>
<a name="line-117"></a><span class='hs-definition'>singleton</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span>
<a name="line-118"></a><span class='hs-definition'>singleton</span> <span class='hs-varid'>i</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>IntervalMap</span> <span class='hs-layout'>(</span><span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>singleton</span> <span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-119"></a>
<a name="line-120"></a><a name="insert"></a><span class='hs-comment'>-- | /O(log n)/. Insert an interval into a map.</span>
<a name="line-121"></a><span class='hs-comment'>-- The map may contain duplicate intervals; the new entry will be inserted</span>
<a name="line-122"></a><span class='hs-comment'>-- before any existing entries for the same interval.</span>
<a name="line-123"></a><span class='hs-definition'>insert</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span>
<a name="line-124"></a><span class='hs-definition'>insert</span> <span class='hs-layout'>(</span><span class='hs-conid'>Interval</span> <span class='hs-varid'>lo</span> <span class='hs-varid'>hi</span><span class='hs-layout'>)</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>lo</span> <span class='hs-varop'>></span> <span class='hs-varid'>hi</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>m</span>
<a name="line-125"></a><span class='hs-definition'>insert</span> <span class='hs-varid'>i</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>IntervalMap</span> <span class='hs-layout'>(</span><span class='hs-varid'>l</span> <span class='hs-varop'>><</span> <span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-varid'>x</span> <span class='hs-varop'><|</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span>
<a name="line-126"></a> <span class='hs-keyword'>where</span>
<a name="line-127"></a> <span class='hs-layout'>(</span><span class='hs-varid'>l</span><span class='hs-layout'>,</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>split</span> <span class='hs-varid'>larger</span> <span class='hs-varid'>t</span>
<a name="line-128"></a> <span class='hs-varid'>larger</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntInterval</span> <span class='hs-varid'>k</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>k</span> <span class='hs-varop'>>=</span> <span class='hs-varid'>i</span>
<a name="line-129"></a> <span class='hs-varid'>larger</span> <span class='hs-conid'>NoInterval</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"larger NoInterval"</span>
<a name="line-130"></a>
<a name="line-131"></a><a name="union"></a><span class='hs-comment'>-- | /O(m log (n/\//m))/. Merge two interval maps.</span>
<a name="line-132"></a><span class='hs-comment'>-- The map may contain duplicate intervals; entries with equal intervals</span>
<a name="line-133"></a><span class='hs-comment'>-- are kept in the original order.</span>
<a name="line-134"></a><span class='hs-definition'>union</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span>
<a name="line-135"></a><span class='hs-definition'>union</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>IntervalMap</span> <span class='hs-layout'>(</span><span class='hs-varid'>merge1</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span>
<a name="line-136"></a> <span class='hs-keyword'>where</span>
<a name="line-137"></a> <span class='hs-varid'>merge1</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>bs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>viewl</span> <span class='hs-keyword'>as</span> <span class='hs-keyword'>of</span>
<a name="line-138"></a> <span class='hs-conid'>EmptyL</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>bs</span>
<a name="line-139"></a> <span class='hs-varid'>a</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-conop'>:<</span> <span class='hs-varid'>as'</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>l</span> <span class='hs-varop'>><</span> <span class='hs-varid'>a</span> <span class='hs-varop'><|</span> <span class='hs-varid'>merge2</span> <span class='hs-varid'>as'</span> <span class='hs-varid'>r</span>
<a name="line-140"></a> <span class='hs-keyword'>where</span>
<a name="line-141"></a> <span class='hs-layout'>(</span><span class='hs-varid'>l</span><span class='hs-layout'>,</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>split</span> <span class='hs-varid'>larger</span> <span class='hs-varid'>bs</span>
<a name="line-142"></a> <span class='hs-varid'>larger</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntInterval</span> <span class='hs-varid'>k</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>k</span> <span class='hs-varop'>>=</span> <span class='hs-varid'>i</span>
<a name="line-143"></a> <span class='hs-varid'>larger</span> <span class='hs-conid'>NoInterval</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"larger NoInterval"</span>
<a name="line-144"></a> <span class='hs-varid'>merge2</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>bs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>viewl</span> <span class='hs-varid'>bs</span> <span class='hs-keyword'>of</span>
<a name="line-145"></a> <span class='hs-conid'>EmptyL</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyword'>as</span>
<a name="line-146"></a> <span class='hs-varid'>b</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-conop'>:<</span> <span class='hs-varid'>bs'</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>l</span> <span class='hs-varop'>><</span> <span class='hs-varid'>b</span> <span class='hs-varop'><|</span> <span class='hs-varid'>merge1</span> <span class='hs-varid'>r</span> <span class='hs-varid'>bs'</span>
<a name="line-147"></a> <span class='hs-keyword'>where</span>
<a name="line-148"></a> <span class='hs-layout'>(</span><span class='hs-varid'>l</span><span class='hs-layout'>,</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>split</span> <span class='hs-varid'>larger</span> <span class='hs-keyword'>as</span>
<a name="line-149"></a> <span class='hs-varid'>larger</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntInterval</span> <span class='hs-varid'>k</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>k</span> <span class='hs-varop'>></span> <span class='hs-varid'>i</span>
<a name="line-150"></a> <span class='hs-varid'>larger</span> <span class='hs-conid'>NoInterval</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"larger NoInterval"</span>
<a name="line-151"></a>
<a name="line-152"></a><a name="intersections"></a><span class='hs-comment'>-- | /O(k log (n/\//k))/. All intervals that intersect with the given</span>
<a name="line-153"></a><span class='hs-comment'>-- interval, in lexicographical order.</span>
<a name="line-154"></a><span class='hs-definition'>intersections</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-155"></a><span class='hs-definition'>intersections</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>inRange</span> <span class='hs-layout'>(</span><span class='hs-varid'>low</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>high</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span>
<a name="line-156"></a>
<a name="line-157"></a><a name="dominators"></a><span class='hs-comment'>-- | /O(k log (n/\//k))/. All intervals that contain the given interval,</span>
<a name="line-158"></a><span class='hs-comment'>-- in lexicographical order.</span>
<a name="line-159"></a><span class='hs-definition'>dominators</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-160"></a><span class='hs-definition'>dominators</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>inRange</span> <span class='hs-layout'>(</span><span class='hs-varid'>high</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>low</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span>
<a name="line-161"></a>
<a name="line-162"></a><a name="search"></a><span class='hs-comment'>-- | /O(k log (n/\//k))/. All intervals that contain the given point,</span>
<a name="line-163"></a><span class='hs-comment'>-- in lexicographical order.</span>
<a name="line-164"></a><span class='hs-definition'>search</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-165"></a><span class='hs-definition'>search</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>inRange</span> <span class='hs-varid'>p</span> <span class='hs-varid'>p</span>
<a name="line-166"></a>
<a name="line-167"></a><a name="inRange"></a><span class='hs-comment'>-- | /O(k log (n/\//k))/. All intervals that intersect with the given</span>
<a name="line-168"></a><span class='hs-comment'>-- interval, in lexicographical order.</span>
<a name="line-169"></a><span class='hs-definition'>inRange</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>v</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>Interval</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-170"></a><span class='hs-definition'>inRange</span> <span class='hs-varid'>lo</span> <span class='hs-varid'>hi</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntervalMap</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>matches</span> <span class='hs-layout'>(</span><span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>takeUntil</span> <span class='hs-layout'>(</span><span class='hs-varid'>greater</span> <span class='hs-varid'>hi</span><span class='hs-layout'>)</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-171"></a> <span class='hs-keyword'>where</span>
<a name="line-172"></a> <span class='hs-varid'>matches</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>viewl</span> <span class='hs-layout'>(</span><span class='hs-conid'>FT</span><span class='hs-varop'>.</span><span class='hs-varid'>dropUntil</span> <span class='hs-layout'>(</span><span class='hs-varid'>atleast</span> <span class='hs-varid'>lo</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyword'>of</span>
<a name="line-173"></a> <span class='hs-conid'>EmptyL</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>[]</span>
<a name="line-174"></a> <span class='hs-conid'>Node</span> <span class='hs-varid'>i</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:<</span> <span class='hs-varid'>xs'</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>matches</span> <span class='hs-varid'>xs'</span>
<a name="line-175"></a>
<a name="line-176"></a><a name="atleast"></a><span class='hs-definition'>atleast</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntInterval</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-177"></a><span class='hs-definition'>atleast</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntInterval</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>hi</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>k</span> <span class='hs-varop'><=</span> <span class='hs-varid'>hi</span>
<a name="line-178"></a><span class='hs-definition'>atleast</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>NoInterval</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"atleast NoInterval"</span>
<a name="line-179"></a>
<a name="line-180"></a><a name="greater"></a><span class='hs-definition'>greater</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>IntInterval</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Bool</span>
<a name="line-181"></a><span class='hs-definition'>greater</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>IntInterval</span> <span class='hs-varid'>i</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>low</span> <span class='hs-varid'>i</span> <span class='hs-varop'>></span> <span class='hs-varid'>k</span>
<a name="line-182"></a><span class='hs-definition'>greater</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>NoInterval</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"greater NoInterval"</span>
<a name="line-183"></a>
<a name="line-184"></a><span class='hs-comment'>{-
<a name="line-185"></a>-- Examples
<a name="line-186"></a>
<a name="line-187"></a>mkMap :: (Ord v) => [(v, v, a)] -> IntervalMap v a
<a name="line-188"></a>mkMap = foldr ins empty
<a name="line-189"></a> where
<a name="line-190"></a> ins (lo, hi, n) = insert (Interval lo hi) n
<a name="line-191"></a>
<a name="line-192"></a>composers :: IntervalMap Int String
<a name="line-193"></a>composers = mkMap [
<a name="line-194"></a> (1685, 1750, "Bach"),
<a name="line-195"></a> (1685, 1759, "Handel"),
<a name="line-196"></a> (1732, 1809, "Haydn"),
<a name="line-197"></a> (1756, 1791, "Mozart"),
<a name="line-198"></a> (1770, 1827, "Beethoven"),
<a name="line-199"></a> (1782, 1840, "Paganini"),
<a name="line-200"></a> (1797, 1828, "Schubert"),
<a name="line-201"></a> (1803, 1869, "Berlioz"),
<a name="line-202"></a> (1810, 1849, "Chopin"),
<a name="line-203"></a> (1833, 1897, "Brahms"),
<a name="line-204"></a> (1838, 1875, "Bizet")]
<a name="line-205"></a>
<a name="line-206"></a>mathematicians :: IntervalMap Int String
<a name="line-207"></a>mathematicians = mkMap [
<a name="line-208"></a> (1642, 1727, "Newton"),
<a name="line-209"></a> (1646, 1716, "Leibniz"),
<a name="line-210"></a> (1707, 1783, "Euler"),
<a name="line-211"></a> (1736, 1813, "Lagrange"),
<a name="line-212"></a> (1777, 1855, "Gauss"),
<a name="line-213"></a> (1811, 1831, "Galois")]
<a name="line-214"></a>-}</span>
</pre></body>
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