{-# LANGUAGE CPP #-}
#ifdef __GLASGOW_HASKELL__
{-# LANGUAGE Safe #-}
#endif

#include "containers.h"

-----------------------------------------------------------------------------
-- |
-- Module      :  Data.IntMap
-- Copyright   :  (c) Daan Leijen 2002
--                (c) Andriy Palamarchuk 2008
-- License     :  BSD-style
-- Maintainer  :  libraries@haskell.org
-- Portability :  portable
--
--
-- = Finite Int Maps (lazy interface)
--
-- This module re-exports the value lazy "Data.IntMap.Lazy" API.
--
-- The @'IntMap' v@ type represents a finite map (sometimes called a dictionary)
-- from keys of type @Int@ to values of type @v@.
--
-- The functions in "Data.IntMap.Strict" are careful to force values before
-- installing them in an 'IntMap'. This is usually more efficient in cases where
-- laziness is not essential. The functions in this module do not do so.
--
-- For a walkthrough of the most commonly used functions see the
-- <https://haskell-containers.readthedocs.io/en/latest/map.html maps introduction>.
--
-- This module is intended to be imported qualified, to avoid name clashes with
-- Prelude functions, e.g.
--
-- > import Data.IntMap.Lazy (IntMap)
-- > import qualified Data.IntMap.Lazy as IntMap
--
-- Note that the implementation is generally /left-biased/. Functions that take
-- two maps as arguments and combine them, such as `union` and `intersection`,
-- prefer the values in the first argument to those in the second.
--
--
-- == Implementation
--
-- The implementation is based on /big-endian patricia trees/.  This data
-- structure performs especially well on binary operations like 'union'
-- and 'intersection'. Additionally, benchmarks show that it is also
-- (much) faster on insertions and deletions when compared to a generic
-- size-balanced map implementation (see "Data.Map").
--
--    * Chris Okasaki and Andy Gill,
--      \"/Fast Mergeable Integer Maps/\",
--      Workshop on ML, September 1998, pages 77-86,
--      <https://web.archive.org/web/20150417234429/https://ittc.ku.edu/~andygill/papers/IntMap98.pdf>.
--
--    * D.R. Morrison,
--      \"/PATRICIA -- Practical Algorithm To Retrieve Information Coded In Alphanumeric/\",
--      Journal of the ACM, 15(4), October 1968, pages 514-534,
--      <https://doi.org/10.1145/321479.321481>.
--
--
-- == Performance information
--
-- Operation comments contain the operation time complexity in
-- [big-O notation](http://en.wikipedia.org/wiki/Big_O_notation), with \(n\)
-- referring to the number of entries in the map and \(W\) referring to the
-- number of bits in an 'Int' (32 or 64).
--
-- Operations like 'lookup', 'insert', and 'delete' have a worst-case
-- complexity of \(O(\min(n,W))\). This means that the operation can become
-- linear in the number of elements with a maximum of \(W\) -- the number of
-- bits in an 'Int' (32 or 64). These peculiar asymptotics are determined by the
-- depth of the Patricia trees:
--
-- * even for an extremely unbalanced tree, the depth cannot be larger than
--   the number of elements \(n\),
-- * each level of a Patricia tree determines at least one more bit
--   shared by all subelements, so there could not be more
--   than \(W\) levels.
--
-- If all \(n\) keys in the tree are between 0 and \(N\) (or, say, between \(-N\) and \(N\)),
-- the estimate can be refined to \(O(\min(n, \log N))\). If the set of keys
-- is sufficiently "dense", this becomes \(O(\min(n, \log n))\) or simply
-- the familiar \(O(\log n)\), matching balanced binary trees.
--
-- The most performant scenario for 'IntMap' are keys from a contiguous subset,
-- in which case the complexity is proportional to \(\log n\), capped by \(W\).
-- The worst scenario are exponentially growing keys \(1,2,4,\ldots,2^n\),
-- for which complexity grows as fast as \(n\) but again is capped by \(W\).
--
-- Binary set operations like 'union' and 'intersection' take
-- \(O(\min(n, m \log \frac{2^W}{m}))\) time, where \(m\) and \(n\)
-- are the sizes of the smaller and larger input maps respectively.
--
-----------------------------------------------------------------------------

module Data.IntMap
    ( module Data.IntMap.Lazy
    ) where

import Data.IntMap.Lazy
