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1-----------------------------------------------------------------------------
2-- |
3-- Module : Numeric.Chain
4-- Copyright : (c) Vivian McPhail 2010
5-- License : GPL-style
6--
7-- Maintainer : Vivian McPhail <haskell.vivian.mcphail <at> gmail.com>
8-- Stability : provisional
9-- Portability : portable
10--
11-- optimisation of association order for chains of matrix multiplication
12--
13-----------------------------------------------------------------------------
14
15module Numeric.Chain (
16 optimiseMult,
17 ) where
18
19import Data.Maybe
20
21import Data.Packed.Matrix
22import Data.Packed.Numeric
23
24import qualified Data.Array.IArray as A
25
26-----------------------------------------------------------------------------
27{- |
28 Provide optimal association order for a chain of matrix multiplications
29 and apply the multiplications.
30
31 The algorithm is the well-known O(n\^3) dynamic programming algorithm
32 that builds a pyramid of optimal associations.
33
34> m1, m2, m3, m4 :: Matrix Double
35> m1 = (10><15) [1..]
36> m2 = (15><20) [1..]
37> m3 = (20><5) [1..]
38> m4 = (5><10) [1..]
39
40> >>> optimiseMult [m1,m2,m3,m4]
41
42will perform @((m1 `multiply` (m2 `multiply` m3)) `multiply` m4)@
43
44The naive left-to-right multiplication would take @4500@ scalar multiplications
45whereas the optimised version performs @2750@ scalar multiplications. The complexity
46in this case is 32 (= 4^3/2) * (2 comparisons, 3 scalar multiplications, 3 scalar additions,
475 lookups, 2 updates) + a constant (= three table allocations)
48-}
49optimiseMult :: Product t => [Matrix t] -> Matrix t
50optimiseMult = chain
51
52-----------------------------------------------------------------------------
53
54type Matrices a = A.Array Int (Matrix a)
55type Sizes = A.Array Int (Int,Int)
56type Cost = A.Array Int (A.Array Int (Maybe Int))
57type Indexes = A.Array Int (A.Array Int (Maybe ((Int,Int),(Int,Int))))
58
59update :: A.Array Int (A.Array Int a) -> (Int,Int) -> a -> A.Array Int (A.Array Int a)
60update a (r,c) e = a A.// [(r,(a A.! r) A.// [(c,e)])]
61
62newWorkSpaceCost :: Int -> A.Array Int (A.Array Int (Maybe Int))
63newWorkSpaceCost n = A.array (1,n) $ map (\i -> (i, subArray i)) [1..n]
64 where subArray i = A.listArray (1,i) (repeat Nothing)
65
66newWorkSpaceIndexes :: Int -> A.Array Int (A.Array Int (Maybe ((Int,Int),(Int,Int))))
67newWorkSpaceIndexes n = A.array (1,n) $ map (\i -> (i, subArray i)) [1..n]
68 where subArray i = A.listArray (1,i) (repeat Nothing)
69
70matricesToSizes :: [Matrix a] -> Sizes
71matricesToSizes ms = A.listArray (1,length ms) $ map (\m -> (rows m,cols m)) ms
72
73chain :: Product a => [Matrix a] -> Matrix a
74chain [] = error "chain: zero matrices to multiply"
75chain [m] = m
76chain [ml,mr] = ml `multiply` mr
77chain ms = let ln = length ms
78 ma = A.listArray (1,ln) ms
79 mz = matricesToSizes ms
80 i = chain_cost mz
81 in chain_paren (ln,ln) i ma
82
83chain_cost :: Sizes -> Indexes
84chain_cost mz = let (_,u) = A.bounds mz
85 cost = newWorkSpaceCost u
86 ixes = newWorkSpaceIndexes u
87 (_,_,i) = foldl chain_cost' (mz,cost,ixes) (order u)
88 in i
89
90chain_cost' :: (Sizes,Cost,Indexes) -> (Int,Int) -> (Sizes,Cost,Indexes)
91chain_cost' sci@(mz,cost,ixes) (r,c)
92 | c == 1 = let cost' = update cost (r,c) (Just 0)
93 ixes' = update ixes (r,c) (Just ((r,c),(r,c)))
94 in (mz,cost',ixes')
95 | otherwise = minimum_cost sci (r,c)
96
97minimum_cost :: (Sizes,Cost,Indexes) -> (Int,Int) -> (Sizes,Cost,Indexes)
98minimum_cost sci fu = foldl (smaller_cost fu) sci (fulcrum_order fu)
99
100smaller_cost :: (Int,Int) -> (Sizes,Cost,Indexes) -> ((Int,Int),(Int,Int)) -> (Sizes,Cost,Indexes)
101smaller_cost (r,c) (mz,cost,ixes) ix@((lr,lc),(rr,rc)) = let op_cost = fromJust ((cost A.! lr) A.! lc)
102 + fromJust ((cost A.! rr) A.! rc)
103 + fst (mz A.! (lr-lc+1))
104 * snd (mz A.! lc)
105 * snd (mz A.! rr)
106 cost' = (cost A.! r) A.! c
107 in case cost' of
108 Nothing -> let cost'' = update cost (r,c) (Just op_cost)
109 ixes'' = update ixes (r,c) (Just ix)
110 in (mz,cost'',ixes'')
111 Just ct -> if op_cost < ct then
112 let cost'' = update cost (r,c) (Just op_cost)
113 ixes'' = update ixes (r,c) (Just ix)
114 in (mz,cost'',ixes'')
115 else (mz,cost,ixes)
116
117
118fulcrum_order (r,c) = let fs' = zip (repeat r) [1..(c-1)]
119 in map (partner (r,c)) fs'
120
121partner (r,c) (a,b) = ((r-b, c-b), (a,b))
122
123order 0 = []
124order n = order (n-1) ++ zip (repeat n) [1..n]
125
126chain_paren :: Product a => (Int,Int) -> Indexes -> Matrices a -> Matrix a
127chain_paren (r,c) ixes ma = let ((lr,lc),(rr,rc)) = fromJust $ (ixes A.! r) A.! c
128 in if lr == rr && lc == rc then (ma A.! lr)
129 else (chain_paren (lr,lc) ixes ma) `multiply` (chain_paren (rr,rc) ixes ma)
130
131--------------------------------------------------------------------------
132
133{- TESTS -}
134
135-- optimal association is ((m1*(m2*m3))*m4)
136m1, m2, m3, m4 :: Matrix Double
137m1 = (10><15) [1..]
138m2 = (15><20) [1..]
139m3 = (20><5) [1..]
140m4 = (5><10) [1..]
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