diff --git a/Data/BinaryIndexedTree.hs b/Data/BinaryIndexedTree.hs new file mode 100644 --- /dev/null +++ b/Data/BinaryIndexedTree.hs @@ -0,0 +1,78 @@ +{- | +Module : Data.BinaryIndexedTree +Description : Binary Indexed Trees (a.k.a. Fenwick Trees) +Copyright : (c) 2012 Maxwell Sayles. +License : LGPL + +Maintainer : maxwellsayles@gmail.com +Stability : stable +Portability : portable + +Implements persistent binary indexed trees (or Fenwick Trees) +in /O(logn)/ for increment and lookup and /O(n)/ for creation. + +Index /i/ in the tree represents the sum of all values of +indexes /j<=i/ for some array. + +The idea is that for /k/ bits, we parse the index /i/ from msb +to lsb and move left\/right on the tree for 0\/1. + +For a read, we accumulate the values in the tree where the +binary representation of the index contains a 1. (The +technique is similar to binary exponentiation.) + +For an increment, we should increment parent nodes +in the tree whose corresponding binary index representation +is />=/ than the index /i/. + +/Note: I was unable to find the algorithm used here in the literature./ +-} + +module Data.BinaryIndexedTree + (BinaryIndexedTree, new, (!), increment) +where + +import Data.Bits + +{-| A Binary indexed tree. -} +data BinaryIndexedTree a = BinaryIndexedTree Int (Tree a) + +data Tree a = Empty | Node a (Tree a) (Tree a) + +{-| +Construct a binary indexed tree on k bits. Takes O(n). +-} +new :: Num a => Int -> BinaryIndexedTree a +new k = BinaryIndexedTree k $ f k + where f 0 = Empty + f k = Node 0 (f (k - 1)) (f (k - 1)) + +{-| +Lookup the sum of all values from index 1 to index i. Takes O(logn). +-} +(!) :: Num a => BinaryIndexedTree a -> Int -> a +(!) (BinaryIndexedTree k root) i = f root (k - 1) 0 + where f Empty _ acc = acc + f (Node x l r) j acc + | i `testBit` j = acc' `seq` f l j' acc' + | otherwise = f r j' acc + where j' = j - 1 + acc' = acc + x + +{-| +Increment the value at index i by amount x. Takes O(logn). +-} +increment :: Num a => Int -> a -> BinaryIndexedTree a -> BinaryIndexedTree a +increment i x (BinaryIndexedTree k root) = + BinaryIndexedTree k $ f root (k - 1) 0 + where f (Node y l r) j acc + | i `testBit` j = + if acc' == i + then y' `seq` Node y' l r + else acc' `seq` Node y (f l j' acc') r + | otherwise = y' `seq` Node y' l (f r j' acc) + where y' = x + y + j' = j - 1 + acc' = acc `setBit` j + + diff --git a/Data/BinaryIndexedTree/ST.hs b/Data/BinaryIndexedTree/ST.hs new file mode 100644 --- /dev/null +++ b/Data/BinaryIndexedTree/ST.hs @@ -0,0 +1,60 @@ +{-| +Module : Data.BinaryIndexedTree.ST +Description : Binary Indexed Trees (a.k.a. Fenwick Trees) +Copyright : (c) 2012 Maxwell Sayles. +License : LGPL + +Maintainer : maxwellsayles@gmail.com +Stability : stable +Portability : portable + +Implements mutable binary indexed trees (a.k.a. Fenwick Trees) +in O(logn) for increment and lookup and O(n) for creation. + +Original concept from Peter M. Fenwick (1994) +\"/A new data structure for cumulative frequency tables/\" +. + +Index i in the tree represents the sum of all values of +indexes j<=i for some array. + +Indexes start at 1. +-} + +module Data.BinaryIndexedTree.ST + (BinaryIndexedTree, new, (!), increment) +where + +import Control.Applicative +import Control.Monad +import Control.Monad.ST +import Data.Array.MArray +import Data.Array.ST +import Data.Bits + +{-| Binary Indexed Tree -} +type BinaryIndexedTree s = STUArray s Int Int + +{-| Construct a new binary indexed tree on the indexes 1 through n. -} +new :: Int -> ST s (BinaryIndexedTree s) +new n = newListArray (1, n) $ repeat 0 + +{-| Compute the sum of all indexes 1 through i, inclusive. Takes O(logn). -} +(!) :: BinaryIndexedTree s -> Int -> ST s Int +(!) bit i = f i 0 + where f i acc + | i < 1 = return acc + | otherwise = + do acc' <- (+acc) <$> readArray bit i + let i' = i - (i .&. (-i)) + f i' acc' + +{-| Increment the value at index i. Takes O(logn). -} +increment :: Int -> Int -> BinaryIndexedTree s -> ST s () +increment i v bit = + do (_, u) <- getBounds bit + let f i = when (i <= u) $ do + writeArray bit i . (+v) =<< readArray bit i + let i' = i + (i .&. (-i)) + f i' + f i diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,165 @@ + GNU LESSER GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + + This version of the GNU Lesser General Public License incorporates +the terms and conditions of version 3 of the GNU General Public +License, supplemented by the additional permissions listed below. + + 0. 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They allow you to compute the sum + of all values at indexes 1 through i in O(logn) and to + increase the value at index i in O(logn). + . + This implements binary indexed trees, both + as an immutable data structure in pure code + and as a mutable data structure using the ST Monad. + . + Both the immutable and mutable version have the + same runtime complexity, but the mutable version + has a smaller constant. + . + Written by Maxwell Sayles (2012). +License: LGPL +License-file: LICENSE +Author: Maxwell Sayles +Maintainer: Maxwell Sayles +Category: Data +Build-type: Simple +Stability: Stable +Cabal-version: >= 1.8 + +Library + Exposed-modules: Data.BinaryIndexedTree, + Data.BinaryIndexedTree.ST + Build-depends: base >= 3 && < 5, + array >= 0.3 + +