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)BTLFMMͰΞϧΰϦζϜΛநԽ͢Δ
ؔܕݴޠͰڝٕϓϩάϥϛϯά
/BPZB*UP !OBPZB@JUP
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ࣗݾհ
• גࣜձࣾ Ұٳ ϗςϧɺϨετϥϯͷ8FC༧ࣄۀͷձࣾ
Ͱಇ͍͍ͯ·͢
• झຯͰڝٕϓϩάϥϛϯάΛᅂΜͰ͓Γɺ)BTLFMMͰࢀઓ͍ͯ͠·͢
• ݱࡏͷϨʔτਫ৭
Ͱ͢
– ڭՊॻʹग़ͯ͘ΔྨͷΞϧΰϦζϜͳ͍͘͜ͳͤΔϨϕϧ
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͍ͨ͜͠ͱ
• )BTLFMM ͰڝٕϓϩάϥϛϯάʹࢀՃ͍ͯ͠ΔɻΞϧΰϦζϜΛநԽ͢Δͱɺͦͷ࣮͕
ࣅͨΑ͏ͳΠϯλϑΣʔεʹͳΔ͜ͱΛܦݧͨ͠
– ߴ֊ؔͱͯࣸ͠૾ Gͱ݁߹ͷೋ߲ԋࢉ͕ύϥϝʔλͱͯ͠Α͘දΕΔ
• ͦΕۮવͳͷ͔ɺ͍ɺۮવͰͳ͍
– )BTLFMMͷࠜఈʹ͋Δ࠶ؼσʔλߏɺͦΕʹ࠷దͳૢ࡞Ͱ͋Δࣸ૾ͱΈࠐΈ NBQSFEVDF
• ࣸ૾ͱΈࠐΈʹΑΔؔͷΠϯλϑΣʔε͔ΒΞϧΰϦζϜͷతߏ͕ݟ͑Δ
• ؔܕϓϩάϥϛϯάɺ)BTLFMMΞϧΰϦζϜΛΑΓநߴ͘ଊ͑ΔࢥߟΛิॿͯ͘͠
ΕΔ
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࣍
• ࣸ૾ NBQ
ͰΛղ͘
• ྦྷੵ uuu GPMETDBOMʹΑΔʮΈࠐΈʯ
• ΞϧΰϦζϜΛநԽ͢Δ uuu ೋ୳ࡧΛߴ֊ؔͰநԽ͢Δ
• #'4ΛநԽ͢Δuuu ؔ GΛʮ͋Δঢ়ଶ͔Β࣍ͷঢ়ଶͷঢ়ଶભҠʯͱଊ͑Δ
• ಈతܭը๏Λ )BTLFMMͰΠϛϡʔλϒϧʹղ͘ uuuঢ়ଶۭؒͷΈࠐΈɺͦͯ͠తߏ
͕ݟ͑Δ
• ͳͥΞϧΰϦζϜΛநԽ͢ΔͱɺNBQSFEVDF GPME
తߏ͕ uuu ؔܕϓϩά
ϥϛϯάɺ)BTLFMM͕ͦΕΛଅ͢
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ΩʔʹͳΔεϥΠυΛઌʹ
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໋ྩܕ ໋σʔλ
uuuҰ࣌తͳதؒঢ়ଶΛ܁Γฦ͠ॻ͖͑ͯ͑ʹ͔͏
ؔܕ Ӭଓσʔλ
uuuঢ়ଶΛ࣍ͷঢ়ଶࣸ͢
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ࣸ૾ͱ݁߹ ू
uuu NBQSFEVDF GPME
• ؔܕϓϩάϥϛϯάͰʮࣸ૾ͱ݁߹ ू
ʯͷϝϯλϧϞσϧͰΛղ͘
• ࣸ૾ͱ݁߹ ू
uuu NBQSFEVDF GPME
• ͳͥ NBQSFEVDF͔
– )BTLFMMͷෆมσʔλߏతσʔλܕʹΑΔ࠶ؼσʔλߏ
– ࠶ؼσʔλߏͷૢ࡞ʹ NBQSFEVDF
– -*41ͷײ͔Βʜ ͋ͱͰ
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#'4ʹ͍ͯ͠Δؔ GͷҙຯΛߟ͑ͯΈΔ
let dist = bfs f (+) (-1) bounds [(1, 0 :: Int)]
• f Λهड़͢Δ͚ͩͰ͕ղ͚Δ͕ɺf Կऀ?
• (v -> [(v, e)])
• (ঢ়ଶ -> [(࣍ͷঢ়ଶ, ભҠίετ)]) ͱ͍͏ঢ়ଶભҠͷࣸ૾ ঢ়ଶ͔ΒભҠީิͷࣸ૾
ͱߟ͑Δ͜ͱ͕Ͱ͖Δ
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݁߹ͷํͱঢ়ଶભҠͷؔ ͋ͱॳظঢ়ଶ
Λએݴ͢Εɺ%1͕ղ͚Δ
main = do
[n, wx] <- getInts
items <- replicateM n getTuple
let dp = accumArrayDP @UArray f max 0 (0, wx) [(0, 0)] items
where
f (w, acc) (wi, vi) = [(w, acc), (w + wi, acc + vi)]
print $ maximum (elems dp)
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͜Εۮવͳͷ͔
• )BTLFMMͰΞϧΰϦζϜΛநԽ͢Δͱɺ݁߹ ೋ߲
ԋࢉɺࣸ૾ ͋ͱॳظ
ঢ়ଶ
͕ύϥϝʔλԽ͞ΕͯΠϯλϑΣʔεʹදΕΔ
• NBQSFEVDF
• ͦͯ͠ΠϯλϑΣʔε͕ʮతߏΈʯΛଳͼΔ
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ͳͥͦΕ͕ɺతߏͱ݁ͼͭ͘ͷ͔
• લड़ͷ௨Γ
࠶ؼతσʔλߏࣸ૾ͱ݁߹ʹࣗવʹؐݩͰ͖Δ
• ঢ়ଶͷʮൖʯͱʮूʯతੑ࣭Λඞཁͱ͢Δ
– ʮൖʯ֤ঢ়ଶΛ৽͍͠ঢ়ଶʹࣸ͢ ˠࣸ૾
– ʮूʯෳͷ݁ՌΛҰͭʹ·ͱΊΔ ˠ݁߹
– ͜Ε͕݁߹తͰͳ͚Ε࠶ؼɾޮԽ͕͍͠ ˠతߏͷཁ
ΞϧΰϦζϜΛநԽ͢Δͱࣸ૾ͱ݁߹͕ܭࢉͷຊ࣭తͳߏཁૉͱͯ͠ݱΕΔ
ঢ়ଶۭؒΛू߹ͱΈͳ͢ͱɺతߏʢ'VODUPS
.POPJE
4FNJSJOHʣʹରԠ͢Δ
)BTLFMMͷܕͱߴ֊ؔɺͦͷߏΛࣗવʹΠϯλϑΣʔεͱͯ͠࿐ग़ͤ͞Δ
நԽʮܭࢉͷʯΛՄࢹԽ͢ΔաఔͰ͋Δ
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ࣸ૾ NBQ
ͰΛղ͘
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https://atcoder.jp/contests/abc081/tasks/abc081_b
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8 12 40
4 6 20
2 3 10
ۮ͔͠ͳ͍ͷͰૢ࡞͢Δ
·ͩۮ͔͠ͳ͍ͷͰૢ࡞͢Δ
ح͕ग़͖ͯͨͷͰऴΘΓ
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໋ྩܕϓϩάϥϛϯάͰ࣮ uuu ໋σʔλߏΛ܁Γฦ͠ॻ͖͑ͯ͑ΛಘΔ
count = 0
while all(a % 2 == 0 for a in as):
for i in range(n):
as[i] //= 2
count += 1
print(count)
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ؔܕ Ӭଓσʔλϓϩάϥϛϯά
ͳΒʮࣸ૾ NBQ
Λ܁Γฦ͠ద༻͢Δʯͱߟ͑Δ
let res = takeWhile (all even) $ iterate (map (`div` 2)) as
print $ length res
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ؔద༻ʹΑͬͯΛಘΔ ؔʹΑͬͯΛม͢Δɾࣸ͢
ࣜඞͣΓΛฦ͢ɻҾͷ͔ΒΓʹΛมͨ͠ɺࣸͨ͠ɺͱݟΔ͜ͱͰ͖Δ
𝑓
ؔܕϓϩάϥϛϯάͱܕγεςϜͷϝϯλϧϞσϧ
https://speakerdeck.com/naoya/guan-shu-xing-puroguramingutoxing-sisutemunomentarumoderu
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໋ྩܕ ໋σʔλ
uuuҰ࣌తͳதؒঢ়ଶΛ܁Γฦ͠ॻ͖͑ͯ͑ʹ͔͏
ؔܕ Ӭଓσʔλ
uuuঢ়ଶΛ࣍ͷঢ়ଶࣸ͢
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)BTLFMMͰڝٕϓϩάϥϛϯάΛ͍ͬͯ͘ɺͱ͍͏͜ͱ
• ෆมͳɺෆมͳσʔλߏ ӬଓσʔλɺӬଓσʔλߏ
Λલఏʹ͍ͬͯ͘
• ෆมͳσʔλͷૢ࡞ମܥ uuu NBQSFEVDF GPME
– ৄ͋͘͠ͱͰ
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ྦྷੵ uuu GPMETDBOMʹΑΔʮΈࠐΈʯ
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https://atcoder.jp/contests/abc305/tasks/abc305_b
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۠ؒʹྦྷੵΛ͏ͱΑ͍
let ds = [3, 1, 4, 1, 5, 9] :: [Int]
s = scanl' (+) 0 ds -- [0, 3, 4, 8, 9, 14, 23]
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໋ྩతʹ࣮͢Δ
p, q = input().split()
ds = [3, 1, 4, 1, 5, 9]
# 累積和
s = [0] * 7
for i in range(6):
s[i + 1] = s[i] + ds[i]
pi = ord(p) - ord('A')
qi = ord(q) - ord('A')
print(abs(s[qi] - s[pi]))
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໋ྩܕ uuu Ұ࣌తͳঢ়ଶΛ܁Γฦ͠ॻ͖͑ͯɺྦྷੵͱ͢Δ
# 累積和
s = [0] * 7
for i in range(6):
s[i + 1] = s[i] + ds[i]
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ؔܕͰ࣮͢Δ
main :: IO ()
main = do
(p, q) <- auto @(Char, Char)
let ds = [3, 1, 4, 1, 5, 9] :: [Int]
s = listArray @UArray ('A', 'G') $ scanl' (+) 0 ds
ans = s ! q - s ! p
print $ abs ans
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݁߹
ͷ࠶ؼతͳؔద༻ uuu ʮΈࠐΈʯ
foldl (+) 0 [3, 1, 4, 1, 5, 9]
==> ((((((0 + 3) + 1) + 4) + 1) + 5) + 9)
scanl (+) 0 [3, 1, 4, 1, 5, 9]
[ 0
, (0 + 3)
, ((0 + 3) + 1)
, (((0 + 3) + 1) + 4)
, ((((0 + 3) + 1) + 4) + 1)
, (((((0 + 3) + 1) + 4) + 1) + 5)
, ((((((0 + 3) + 1) + 4) + 1) + 5) + 9)
]
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վΊͯ )BTLFMMͰͷྦྷੵ uuu TDBOMΛΈͯΈΔ
let ds = [3, 1, 4, 1, 5, 9] :: [Int]
s = scanl' (+) 0 ds -- [0, 3, 4, 8, 9, 14, 23
scanl' :: (b -> a -> b) -> b -> [a] -> [b]
• ࠶ؼతʹ݁߹Λద༻͢Δɺͱ͍͏ྦྷੵܭࢉ ΈࠐΈ
ΛநԽͨ͠ͷ͕ TDBOM
• TDBOMͷΠϯλϑΣʔεʹ݁߹ԋࢉ͕ύϥϝʔλͱͯ͠දΕ͍ͯΔ
–
ͳΒྦྷੵ
– NBYͳΒྦྷੵ NBY
– NJOͳΒྦྷੵ NJO
– YPSͳΒྦྷੵ YPS
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ࣸ૾ͱ݁߹ ू
uuu NBQSFEVDF GPME
• ؔܕϓϩάϥϛϯάͰʮࣸ૾ͱ݁߹ ू
ʯͷϝϯλϧϞσϧͰΛղ͘
• ࣸ૾ͱ݁߹ ू
NBQSFEVDF GPME
• ͳͥ NBQSFEVDF͔
– )BTLFMMͷෆมσʔλߏతσʔλܕʹΑΔ࠶ؼσʔλߏ
– ࠶ؼσʔλߏͷૢ࡞ʹ NBQSFEVDF
– -*41ͷײ͔Βʜ ͋ͱͰ
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ΞϧΰϦζϜΛநԽ͢Δ uuu ೋ୳ࡧΛߴ֊ؔͰநԽ͢Δ
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https://atcoder.jp/contests/abc409/tasks/abc409_b
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No content
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୯ௐੑ͕͋ΔͷͰɺڥքΛఆΊΔ͜ͱ͕Ͱ͖Δ
Ҏ্ uuu ճ
Ҏ্ uuuճ
Ҏ্ uuu ճ
Ҏ্ uuuճ
Ҏ্ uuu ճ
Ҏ্ uuu ճ
͜͜·Ͱ݅Λຬͨ͢ɺҎ߱ຬͨ͞ͳ͍ YҎ্ͷཁૉ͕ YճҎ্දΕΔ͔
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ೋ୳ࡧͰղ͚·͢ɻͰɺίϯςετதʹೋ୳ࡧΛॻ͘ͱόάΒ͕ͤͪ
main :: IO ()
main = do
_ <- getInt
as <- getInts
let binSearch (ok, ng)
| abs (ng - ok) == 1 = ok
| otherwise = do
let m = (ok + ng) `div` 2
count = length $ filter (>= m) as
if count >= m
then binSearch (m, ng)
else binSearch (ok, m)
print $ binSearch (0, 10 ^ 18)
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ຖճॻ͔ͳ͍͍ͯ͘Α͏ɺநԽ͠·͢ ˠߴ֊ؔ G͕දΕΔ
-- | 左が true / 右が false で境界を引く
bisect2 :: (Integral a) => (a, a) -> (a -> Bool) -> (a, a)
bisect2 (ok, ng) f
| abs (ng - ok) == 1 = (ok, ng)
| f m = bisect2 (m, ng) f
| otherwise = bisect2 (ok, m) f
where
m = (ok + ng) `div` 2
let (ok, _) = bisect2 (0, 10 ^ 18) (¥x -> countBy (>= x) as >= x)
print ok
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݅ؔ Gࣸ૾ͱΈͳͤΔ
• ೋ୳ࡧʮڥքΛҾ݅͘ʯΛؔ GͰ͢Α͏நԽͰ͖Δ
• (a -> Bool) Λʮ
͔Β ݅Λຬ͔ͨ͢Ͳ͏͔ͷࣸ૾ʯͱଊ͑Δ͜ͱ͕Ͱ͖Δ
• ΞϧΰϦζϜΛநԽͨ͠Βͦ͜ʹࣗવͱࣸ૾ G͕දΕͨ
bisect2 :: (Integral a) => (a, a) -> (a -> Bool) -> (a, a)
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#'4 ෯༏ઌ୳ࡧ
ΛநԽ͢Δ
ؔ GΛʮ͋Δঢ়ଶ͔Β࣍ͷঢ়ଶͷঢ়ଶભҠʯͱଊ͑Δ
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https://atcoder.jp/contests/tessoku-book/tasks/math_and_algorithm_an
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1
2
4
3
5
6
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άϥϑΛʮྡϦετʯʹߏԽͯ͠ #'4 ෯༏ઌ୳ࡧ
͢Δ
main = do
[n, m] <- getInts
uvs <- replicateM m getTuple
-- 1 => [(2, 1), (4, 1)] という構造の「隣接リスト」にする
let g = accumArray @Array (flip (:)) [] (1, n)
$ concatMap (¥(u, v) -> [(u, (v, 1)), (v, (u, 1))]) uvs
-- 幅優先探索する
dist = bfs g [(1, 0)]
for_ [1 .. n] $ ¥s ->
print $ dist ! s
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ࢀߟ
ઌͷ #'4 ͷ࣮
bfs :: (Ix v, IArray a [(v, e)], Num e, Eq e) => a v [(v, e)] -> [(v, e)] -> Array v e
bfs g v0s = runSTArray $ do
dist <- newArray (bounds g) (-1)
for_ v0s $ ¥(v0, d0) -> do
writeArray dist v0 d0
aux (Seq.fromList [v0 | (v0, _) <- v0s]) dist
return dist
where
aux Empty _ = return ()
aux (v :<| queue) dist = do
acc <- readArray dist v
us <-
filterM
( ¥(u, _) ->
(fmap (== -1) . readArray dist) u
)
(g ! v)
queue' <- foldForM queue us $ ¥q (u, w) -> do
writeArray dist u $! acc + w
return $ q |> u
aux queue' dist
༨ஊ
ࢲ͜Ε͙Β͍ʹͳΔͱίϯςετ
தʹॻ͖ͨ͘ͳ͍ͷͰϥΠϒϥϦԽ͠·
͕͢ɺ$ 1ZUIPOͰͬͯΔਓຖ
ճ#'4ΛιϥͰهड़ͯͨ͠Γ͠·͢
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ʮάϥϑͷྡϦετʯͷͨΊͷ࣮ʹͳͬͯ͠·͍ͬͯΔ
let dist = bfs g [(1, 0)]
• #'4άϥϑͷྡϦετͷͨΊ͚ͩͷ୳ࡧΞϧΰϦζϜͰͳ͍
• ྡϦετͷߏʹґଘ͍ͯ͠ΔՕॴΛύϥϝʔλԽ͠ɺநΛ্͛Α͏
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நΛ্͛Δ uuu άϥϑͷભҠΛ֎෦͔ΒೖͰ͖ΔΑ͏ʹ͢Δ
let dist = bfs (g !) (+) (-1) (bounds g) [(1, 0)]
• (g !) uuuʮW͔ΒḷΕΔ ʯͱ͍͏ࣸ૾ G
– ʮ ͔Βɺ ̎ ͱ ɺͦΕͧΕॏΈ ͰભҠͰ͖ΔʯΛهड़͢Δؔ
– f v = g ! v
• ڑͷ݁߹ԋࢉ (+)
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ೋ࣍ݩάϦουͷ୳ࡧΛղ͍ͯΈΔ uuu ಉ࣮͡Ͱղ͚Δ͔
https://atcoder.jp/contests/abc308/tasks/abc308_d
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No content
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#'4ͷ࣮Λม͑Δ͜ͱͳ͘ɺؔ GΛهड़͢Δ͚ͩͰ͕ղ͚Δ
main = do
[h, w] <- getInts
grid <- getCharGrid ((1, 1), (h, w))
let dist = bfs f (+) (-1) (bounds grid) [((1, 1), 0)]
where
f v =
[ (u, 1)
| d <- [(1, 0), (0, 1), (-1, 0), (0, -1)],
let u = v + d,
inRange (bounds grid) u,
case (grid ! v, grid ! u) of
('s', 'n') -> True
('n', 'u') -> True
('u', 'k') -> True
('k', 'e') -> True
('e', 's') -> True
_ -> False
]
printYn $ dist ! (h, w) /= -1
ʮϚε WˠW͔Β౸ୡͰ͖ΔϚε VͷϦετʯ
ͱ͍͏ࣸ૾
άϦουͷ্Լࠨӈʹ݅Λຬͨ͢ભҠઌ͕͋ΔͳΒ
ͦΕΛྻڍ
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https://atcoder.jp/contests/abc235/tasks/abc235_d
Ұݟ͢ΔͱάϥϑͰάϦουͳ͍ɻछྨͷૢ࡞Λ܁Γฦͯ͠ /ʹͰ͖Δ
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B
/ ͷ߹
1 3 9 27
81
72
243
18
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main = do
[a, n] <- getInts
let bounds = (1, 10 ^ 6 + 1 :: Int)
let dist = bfs f (+) (-1) bounds [(1, 0 :: Int)]
where
f x
| x >= 10 && x `mod` 10 /= 0 = [(x', 1) | x' <- [rotate x, a * x], inRange bounds x']
| otherwise = [(a * x, 1) | inRange bounds (a * x)]
rotate y =
let ys = toDigits 10 y
in fromDigits 10 (last ys : init ys)
print $ dist ! n
Γؔ GΛهड़͢Δ͜ͱͰ͕ղ͚Δ
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վΊͯ #'4ʹ͍ͯ͠Δؔ GͷҙຯΛߟ͑ͯΈΔ
let dist = bfs f (+) (-1) bounds [(1, 0 :: Int)]
• f Λهड़͢Δ͚ͩͰ͕ղ͚Δ͕ɺf Կऀ?
• (v -> [(v, e)])
• (ঢ়ଶ -> [(࣍ͷঢ়ଶ, ભҠίετ)]) ͱ͍͏ঢ়ଶભҠͷࣸ૾ ঢ়ଶ͔ΒભҠީิͷࣸ૾
ͱߟ͑Δ͜ͱ͕Ͱ͖Δ
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ࣸ૾ Gʹ͍ͭͯߟ͑Δ͏ͪɺ#'4ͷϝϯλϧϞσϧ͕Ξοϓσʔτ͞ΕΔ
• #'4άϥϑΛ୳ࡧ͢ΔΞϧΰϦζϜ
• #'4άϦουΛ୳ࡧ͢ΔΞϧΰϦζϜ uuu άϦουάϥϑͷҰछ
• #'4ঢ়ଶΛ୳ࡧ͢ΔΞϧΰϦζϜ uuu ঢ়ଶભҠΛάϥϑભҠͱߟ͍͑ͯͬͨΓ͖ͨΓ
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༨ஊ
)BTLFMM Ҏ֎Ͱ͜Μͳ෩ʹ #'4ΛநԽͰ͖Δ
• Θ͔Βͳ͍͚Ͳɺ؆୯Ͱͳͦ͞͏
• ભҠΛʮঢ়ଶભҠͷؔʯͱ͢Δ͜ͱͰ͖Δ
• ঢ়ଶۭؒͷߏ ࣍ݩ
ΛύϥϝʔλԽ͢Δ͜ͱͱɺύϑΥʔϚϯεΛཱ྆͢Δͷ͕͍͔͠
– )BTLFMMͷ "SSBZ࣍ݩΛύϥϝʔλԽͰ͖Δ
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ಈతܭը๏Λ )BTLFMMͰΠϛϡʔλϒϧʹղ͘
ঢ়ଶۭؒͷΈࠐΈɺͦͯ͠తߏ͕ݟ͑Δ
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No content
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φοϓβοΫͷΑ͋͘ΔϞσϧ uuu ೋ࣍ݩͷάϦου ೋ࣍ݩྻ
Ͱߟ͑Δ
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for (int j = 0; j <= W; ++j) dp[0][j] = 0;
for (int i = 0; i < N; ++i) {
for (int j = 0; j <= W; ++j) {
if (j >= w[i]) dp[i+1][j] = max(dp[i][j-w[i]] + v[i], dp[i][j]);
else dp[i+1][j] = dp[i][j];
}
}
cout << dp[N][W] << endl;
໋ྩతͳ࣮ uuu Մมͳೋ࣍ݩྻΛ༻ҙͯ͠ॻ͖͑Δ
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%1Λ͜ͷϞσϧͰߟ͑ͨͱ͖ͷ՝
• ೋ࣍ݩͰ͋Ε಄ͰΠϝʔδͰ͖Δ͕ɺࡾ࣍ݩʹͳΔͱ͍͜͠ɻ࢛࣍ݩҎ্Πϝʔδ͕͍͠
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%1ͷܭࢉͷಛੑ
• ෦ॏෳੑͱ෦ߏ࠷దੑ
• ࡶʹ͍͏ͱ
– Ұͭલͷঢ়ଶ͚͔ͩΒ࣍ͷঢ়ଶ͕ܭࢉͰ͖Δ ܭࢉաఔͰ࠷ॳͷํͷঢ়ଶ͍Βͳ͍
– ͦΕΛ܁Γฦͤ࠷దղ͕ٻ·Δ
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͜ͷߦ ঢ়ଶ
Λܭࢉ͢Δͷ
ʹɺҰͭલͷߦ Ұͭલͷ
ঢ়ଶ
͚ͩΘ͔Ε͍͍
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ʮೋ࣍ݩྻΛॻ͖͑ΔʯͰͳ͘ʮঢ়ଶۭ͕ؒ࣍ͷঢ়ଶʹભҠ͢Δʯͱߟ͑Δ
ঢ়ଶۭؒ
ભҠ
ભҠ
ભҠ
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Ұͭલͷঢ়ଶΛ࣍ͷঢ়ଶʹભҠͤ͞ΔɻͦΕΛ࠶ؼతʹ uuu GPME͡ΌΜ
main = do
[n, wx] <- getInts
items <- replicateM n getTuple
-- 初期状態の状態空間
let dp0 = accumArray @UArray max minBound (0, wx) [(0, 0 :: Int)]
-- 状態空間を、再帰的に状態遷移させる
let dp' = foldl' f dp0 items
where
f dp (wi, vi) = accum max dp [(w + wi, acc + vi) | (w, acc) <- assocs dp, inRange (bounds
dp) (w + wi)]
print $ maximum (elems dp)
ՄมྻΛߋ৽͢Δͱ͍͏໋
ྩతͳ࣮Ͱͳ͘ɺෆมͳ
ΛΈࠐΜͰ͍͘ɺͱ͍͏
)BTLFMMΒ͍࣮͠ʹ
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ʹݻ༗ͷ෦ΛύϥϝʔλԽ͠ɺGPMEʹΑΔ %1ܭࢉΛநԽ͢Δ
-- 畳み込みDP
-- ex) accumArrayDP @UArray f max minBound (0, wx) [(0, 0)] wvs
accumArrayDP ::
( IArray a e,
Ix v,
Eq e,
Show e,
Show v,
Show (a v e),
Foldable t
) =>
((v, e) -> x -> [(v, e')]) -> -- 状態遷移関数 f v / x をみて v の次の遷移可能性を返す
(e -> e' -> e) -> -- 緩和の二項演算
e -> -- 初期値 (0, minBound, maxBound, False など)
(v, v) -> -- 状態空間の下界、上界
[(v, e')] -> -- 開始時点の状態
t x -> -- 入力 (時間遷移)
a v e -- Array or UArray
accumArrayDP f combine initial (l, u) v0s xs =
let dp = accumArray combine initial (l, u) v0s
in foldl' transition dp xs
where
transition dp x =
accumArray combine initial (l, u)
$ concatMap (filter (inRange (bounds dp) . fst) . (`f` x)) (assocs dp)
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φοϓβοΫ͜͏ղ͚Δ
main = do
[n, wx] <- getInts
items <- replicateM n getTuple
let dp = accumArrayDP @UArray f max 0 (0, wx) [(0, 0)] items
where
f (w, acc) (wi, vi) = [(w, acc), (w + wi, acc + vi)]
print $ maximum (elems dp)
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݁߹ͷํͱঢ়ଶભҠͷؔ ͋ͱॳظঢ়ଶ
Λએݴ͢Εɺ%1͕ղ͚Δ
main = do
[n, wx] <- getInts
items <- replicateM n getTuple
let dp = accumArrayDP @UArray f max 0 (0, wx) [(0, 0)] items
where
f (w, acc) (wi, vi) = [(w, acc), (w + wi, acc + vi)]
print $ maximum (elems dp)
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%1ɺநԽͨ͠Β݁߹ͱࣸ૾͕ύϥϝʔλԽ͞Εͨ
• ͷྲྀΕతʹೋ࣍ݩྻ ˠঢ়ଶۭؒͷΈࠐΈɺͱ͍͏ϞσϧͷసΛઌʹߟ͕͑ͨ
• ࣮ࡍʹɺ%1ͷநԽΛࢼΈͨΒ GPMEͷ࣮ʹͳΓɺޙ͚ͰҙຯΛߟ͑ͨΒʮঢ়ଶۭؒ
ͷΈࠐΈʯͱ͍͏Ϟσϧʹؾ͕͍ͭͨ
• ʮ%1 NBQ SFEVDF GPME.BQ
͡Όͳ͍͔ ʜʯ
– ࣸ૾ uuu ঢ়ଶͷભҠ
– ݁߹ uuu ෳܦ࿏͔Βಉ͡ঢ়ଶʹྲྀೖͨ͠ͱ͖ɺΛҰͭʹू
˞ͨͩ͠ɺ͜ͷϞσϧͰղ͚ΔͷφοϓβοΫ%1ɻ֝ͼ%1͜ͷϞσϧͰղ͚ͳ͍
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ΠϯλϑΣʔεΛோΊΔͱతߏݟ͑ͯདྷΔ uuu %1తߏΛ࣋ͭ
let dp = accumArrayDP @UArray f (+) 0 (0, n) [(0, 0)] items
f ((v, e) -> x -> [(v, e')]) ঢ়ଶ (v, e) ʹೖྗ x Λ࡞༻ͤ͞ෳͷભҠઌ (v, e') Λੜ͢Δ (࡞༻
తͳ๏)
(+) (e -> e' -> e) ಉ͡ঢ়ଶʹෳભҠ͕ू·Δͱ͖ɺ·ͱΊΔ (Ճ๏)
0 e ॳظ (Ճ๏ͷ୯Ґݩ)
• %1ɺ͋Δঢ়ଶ্ۭؒʹ͓͚ΔϞϊΠυʢࣸ૾ʣ࡞༻ͱɺ܈తͳूΛΈ߹Θͤͨܭࢉ
• GPME.BQ.POPJEΛ͏͕ɺͦΕΛΑΓҰൠԽͯ͠4FNJSJOHʹ֦ுͨ͠ͷ͕%1Ͱ͋ΔͱΈͳͤ
Δ
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நͷߏ ΠϯλϑΣʔε
͕ྑ͘ࣅ͍ͯΔ
• #'4 %1ɺநతʹঢ়ଶۭؒʹΛΓɾूΊΔૢ࡞ͷ্ʹߏங͞ΕͨϞσϧ
• #'4తߏ ·ͨͦͷ֦ு
Λ࣋ͭ ·ͩͪΌΜͱߟ͑ΒΕ͍ͯ·ͤΜ
let dp = accumArrayDP @UArray f (+) 0 (0, n) [(0, 0)] items
let dist = bfs f (+) (-1) (0, n) [(1, 0)]
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f
f
f
f
f
f
(+), max, min, …
NBQ
SFEVDF
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ͳͥΞϧΰϦζϜΛநԽ͢ΔͱɺNBQSFEVDF GPME
తߏ͕
ؔܕϓϩάϥϛϯάɺ)BTLFMM͕ͦΕΛଅ͢
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͠Ό͘ͱΓ๏
• (+) (-) 0 ··· ݁߹ɺٯݩɺ୯Ґݩ → ՄϞϊΠυ
• ʮՄϞϊΠυ্ͷ෦۠ؒͷྦྷੵʯʹରͯ͠ʮ݅ f Λຬͨ͢࠷େ۠ؒΛ୳ࡧ͢Δʯ
– f ··· ϞϊΠυͷΈࠐΈʹରͯ͠ఆΛ͚ͨʮ͖݅ࣸ૾ʯ
let secs = shakutori f (+) (-) 0 as
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͜Εۮવͳͷ͔
• )BTLFMMͰΞϧΰϦζϜΛநԽ͢Δͱɺ݁߹ ೋ߲
ԋࢉɺࣸ૾ ͋ͱॳظ
ঢ়ଶ
͕ύϥϝʔλԽ͞ΕͯΠϯλϑΣʔεʹදΕΔ
• NBQSFEVDF
• ͦͯ͠ΠϯλϑΣʔε͕ʮతߏΈʯΛଳͼΔ
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༨ஊ
(PPHMFͷ .BQ3FEVDF
.BQͱ3FEVDFͷΈ߹ΘͤͰ
ଟ͘ͷΛղ͘͜ͱ͕Ͱ͖Δ͜ͱʹج͍ͮͨฒྻࢄܭࢉγεςϜ
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)BTLFMM࠶ؼతσʔλߏΛσʔλߏͷجຊͱ͍ͯ͠Δ
data List a = Nil | Cons a (List a)
• )BTLFMMͷσʔλߏ uuu తσʔλܕʹΑΔ࠶ؼతσʔλߏ
• ࠶ؼతσʔλߏʹదͨ͠࠷খͷૢ࡞͕ NBQͱ SFEVDF GPME
– NBQ a -> b Λ֤ཁૉʹద༻͠ɺߏΛอͬͨ··ม ࠶ؼߏʹର͢Δؔద༻ͷ࠷খ୯Ґ
– SFEVDF GPME
ͦͷߏΛյ͠ͳ͕ΒҰͭͷʹΈࠐΉ ߏΛͨͨΉ࠶ؼతύλʔϯͷ࠷খ୯Ґ
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ʮࣸ૾ͱΈࠐΈʯͲ͔͜Βདྷͨͷ͔
• -JTQ ʙ
͕Ϧετʹର͢Δ NBQSFEVDFͰɺଟ͘ͷΛهड़Ͱ͖Δ͜ͱΛܦݧ
తʹൃݟͨ͠
• ؔܕݴޠͷൃలʹΑΓɺNBQSFEVDF GPME
ϦετʹݶΒͳ͍࠶ؼతσʔλߏʹద
ͨ͠ૢ࡞Ͱ͋ΔͱҰൠԽ͞Εͨ
• )BTLFMM ʙ
ͦΕΒΛཧ͚ͮͯ࠶ߏ͠ɺܕΫϥε֊ͱͱʹऔΓࠐΜͩ
• 'VODUPS
'PMEBCMF
"QQMJDBUJWF
.POBE
5SBWFSTBCMF
• NBQSFEVDF GPME
)BTLFMMͷʮ࠶ؼʯͱʮߏԽʯΛ؏֩͘ɺݴޠઃܭͷத৺ʹ͋Δ
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https://wiki.haskell.org/wikiupload/8/85/TMR-Issue13.pdf
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)BTLFMMͷෆมͳσʔλߏ NBQSFEVDF GPME
Ͱૢ࡞͢Δ
• Ϧετɺ4FUɺ.BQɺ"SSBZ ͳͲͷෆมσʔλߏ͍ͣΕ 'PMEBCMFɻ·ͨ 'VODUPSͰ
͋ͬͨΓɺͦ͏Ͱͳͯ͘ NBQΛ͍࣋ͬͯΔ
• ͦΕΒΛجૅʹΞϧΰϦζϜΛ࣮͠ɺ۩ମΛύϥϝʔλԽ͍ͯ͘͠ͱࣗવͱ NBQ GPME
ͷҾ͕ύϥϝʔλͱͯ͠දΕΔ
͜Ε͕ΞϧΰϦζϜͷநʹG
ͷΑ͏ͳ͕͍ؔͭදΕΔཧ༝
ӬଓσʔλϓϩάϥϛϯάΛࢤͨ͠߹ͷඞવ
˞ඞͣ͜͠ͷߏʹԊΘͳ͍ΞϧΰϦζϜσʔλߏɺͪΖΜ͋Γ·͢
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ͳͥͦΕ͕ɺతߏͱ݁ͼͭ͘ͷ͔
• લड़ͷ௨Γ
࠶ؼతσʔλߏࣸ૾ͱ݁߹ʹࣗવʹؐݩͰ͖Δ
• ঢ়ଶͷʮൖʯͱʮूʯతੑ࣭Λඞཁͱ͢Δ
– ʮൖʯ֤ঢ়ଶΛ৽͍͠ঢ়ଶʹࣸ͢ ˠࣸ૾
– ʮूʯෳͷ݁ՌΛҰͭʹ·ͱΊΔ ˠ݁߹
– ͜Ε͕݁߹తͰͳ͚Ε࠶ؼɾޮԽ͕͍͠ ˠతߏͷཁ
ΞϧΰϦζϜΛநԽ͢Δͱࣸ૾ͱ݁߹͕ܭࢉͷຊ࣭తͳߏཁૉͱͯ͠ݱΕΔ
ঢ়ଶۭؒΛू߹ͱΈͳ͢ͱɺతߏʢ'VODUPS
.POPJE
4FNJSJOHʣʹରԠ͢Δ
)BTLFMMͷܕͱߴ֊ؔɺͦͷߏΛࣗવʹΠϯλϑΣʔεͱͯ͠࿐ग़ͤ͞Δ
நԽʮܭࢉͷʯΛՄࢹԽ͢ΔաఔͰ͋Δ
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)BTLFMMͷநͷɺڝٕϓϩάϥϛϯάʹ͓͚ΔҖྗ
• ΞϧΰϦζϜͷநʹग़͖ͯͨࣸ૾݁߹ʮάϥϑͷભҠʯʮঢ়ଶભҠͷࣸ૾ʯͳͲҙຯ
͚ͮ͢Δ͜ͱ͕Ͱ͖Δ
– ͨͩͷύϥϝʔλͰͳ͘ɺղऍΛՃ͑Δ͜ͱ͕Ͱ͖Δ
– ͦͯͦ͠ΕΛతߏͷͰΈͯɺֶతੑ࣭Λҙࣝ͢Δ͜ͱͰ͖Δ
• ڝٕϓϩάϥϛϯάͷ͕ɺͦͷҙຯʹैͬͯߟ͢Δ͜ͱͰղ͚Δ
– ෦Λղ͚ͩ͘Ͱɺશମ͕ղ͚Δ
– ͜ͷࢥߟͷϑϨʔϜϫʔΫಛʹ %1άϥϑ୳ࡧͳͲʹ͓͍ͯڧྗͳثʹͳΔ
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໋ྩܕͰΞϧΰϦζϜΛॻ͘ͱ͖ uuuʮͲ͏ಈ͔͔͢ʯ
• ؔ৺ʮखଓ͖ʯʮঢ়ଶͷมԽʯʹ͘
– ྻΛߋ৽͠ɺมΛมߋ͠ͳ͕ΒʮͲ͏ͬͯಈ͔͔͢ʯΛߟ͑Δ
• Ұ࣌తͳதؒঢ়ଶ ໋ͳม
Λͬͯʮ࠷ऴঢ়ଶʹͲ͏͍͔ͬͯ͘ʯʹूத͢Δ
• ΞϧΰϦζϜҰ࿈ͷ໋ྩͱͯ͠هड़͞ΕΔϨγϐ
🧠 ࢥߟʮ࣮ߦͷॱংʯʹԊͬͯਐΉɻಈ࡞ΛͳͧΔ͜ͱͰҙຯΛѲ͢Δɻ
ϓϩηεࢤɺྲྀΕʹॏ͖Λஔ͘ࢥߟ
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for (int j = 0; j <= W; ++j) dp[0][j] = 0;
for (int i = 0; i < N; ++i) {
for (int j = 0; j <= W; ++j) {
if (j >= w[i]) dp[i+1][j] = max(dp[i][j-w[i]] + v[i], dp[i][j]);
else dp[i+1][j] = dp[i][j];
}
}
cout << dp[N][W] << endl;
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ؔܕͰΞϧΰϦζϜΛॻ͘ͱ͖ uuuʮԿΛҙຯ͍ͯ͠Δ͔ʯ
• ؔ৺ʮߏʯʮҙຯʯʹ͘
– ˠ͜ͷ NBQԿΛ͍ࣸͯ͠Δ ͜ͷ GPMEͲΜͳԋࢉΛू͍ͯ͠Δ
• ܭࢉෆมͳσʔλߏʹର͢Δࣸ૾ͱΈࠐΈͱͯ͠ߏ͞ΕΔ
• ΞϧΰϦζϜࣸ૾ͱ݁߹ͷΈ͋Θͤͱͯ͠هड़͞ΕΔࣜ
🧠ࢥߟʮߏͱҙຯʯʹԊͬͯਐΉɻࣜͷߏΛோΊͯશମ૾Λଊ͑Δɻߏࢤɺؔʹॏ͖Λஔ͘ࢥߟɻ
ܭࢉΛྲྀΕͱͯ͠͏ͷͰͳ͘ɺҙຯͷ͋Δߏ NBQ
GPME
ؔ߹
ͱͯ͠ଊ͑Δɻ
Ұ࿈ͷૢ࡞ΛʮԿΛ͔ͨ͠ʯͰͳ͘ɺʮͲͷΑ͏ͳҙຯͷ͋Δࣸ૾݁߹Λఆ͍ٛͯ͠Δ͔ʯͱݟΔ
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let dp = accumArrayDP @UArray f max 0 (0, wx) [(0, 0)] items
where
f (w, acc) (wi, vi) = [(w, acc), (w + wi, acc + vi)]
print $ maximum (elems dp)
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நԽָ͕͍͠ͷΘ͔͚ͬͨͲɺڝϓϩΛ )BTLFMMͰΔϝϦοτ
• ͋Γ·͢ΑͰͦΜͳ͜ͱͲ͏Ͱ͍͍͡Όͳ͍Ͱ͔͢ɻϩϚϯͰ͢
– ϝϦοτϝϦοτ͏ΔͤʔΜͩΑ
– )BTLFMMͰղ͚ͳ͍ʹग़ձͬͨ͜ͱͳ͍
– ʮ)BTLFMMʹΑΔந͕ิॿྠʹͳΔʯ͜Ε͚ͩͰेͳϝϦοτ
• ΑΓ۩ମతͳϝϦοτྫ͑uuu
– )BTLFMMͷ࠶ؼσʔλܕ Ӭଓσʔλߏ ˠ ӬଓσʔλߏΛલఏʹͨ͠ͷղ
– ʮӬଓσʔλϓϩάϥϛϯάͱڝٕϓϩάϥϛϯά ʙ )BTLFMMͰ͕ΜΔڝϓϩʯ
• IUUQT[FOOEFWOBPZB@JUPBSUJDMFTBCEDBBBC
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)BTLFMM͛͢ʔͳͱࢥ͏͜ͱ
• ࠓͯ͠ΔΑ͏ͳ͜ͱΛ͔ͬͯͯ )BTLFMMΛͬͯདྷͨΘ͚Ͱͳ͍
• )BTLFMMΛ͍ͬͯͨΒࣗવͱࠓͨ͠Α͏ͳநԽͷ࣮ʹͳͬͨ
• ͦͷநԽͷҙຯΛߟ͍͑ͯΔ͏ͪʹࠓͨ͠Α͏ͳߟʹࢸͬͨ
• )BTLFMMΛͬͯΔͱ͜͏͍͏ܦݧΛԿ͢Δ
– ·ΔͰϓϩάϥϛϯάݴޠͱ͍͏ಓ۩͕ɺΑΓྑ͍நͱීวͷಓےΛࢦࣔͯ͘͠͠Ε͍ͯΔΑ͏ͳײ͡
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ͨ͜͠ͱ
• )BTLFMM ͰڝٕϓϩάϥϛϯάʹࢀՃ͍ͯ͠ΔɻΞϧΰϦζϜΛநԽ͢Δͱɺͦͷ࣮͕
ࣅͨΑ͏ͳΠϯλϑΣʔεʹͳΔ͜ͱΛܦݧͨ͠
– ߴ֊ؔͱͯࣸ͠૾ Gͱ݁߹ͷೋ߲ԋࢉ͕ύϥϝʔλͱͯ͠Α͘දΕΔ
• ͦΕۮવͳͷ͔ɺ͍ɺۮવͰͳ͍
– )BTLFMMͷࠜఈʹ͋Δ࠶ؼσʔλߏɺͦΕʹ࠷దͳૢ࡞Ͱ͋Δࣸ૾ͱΈࠐΈ NBQSFEVDF
• ࣸ૾ͱΈࠐΈʹΑΔؔͷΠϯλϑΣʔε͔ΒΞϧΰϦζϜͷతߏ͕ݟ͑Δ
• ؔܕϓϩάϥϛϯάɺ)BTLFMMΞϧΰϦζϜΛΑΓநߴ͘ଊ͑ΔࢥߟΛิॿͯ͘͠
ΕΔ
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͓·͚
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NJTD
• ෆมͳσʔλߏͱ NBQSFEVDFͰશ෦͍͚Δͷ
– ͍͍͑ɻύϑΥʔϚϯεͷ؍͔Β .VUBCMF"SSBZ͕ඞཁͳ໘͋Γ·͢
• 6OJPO'JOEɺηάϝϯτɺԆධՁηάϝϯτͳͲͷ࣮
– .VUBCMF"SSBZͷૢ࡞Ϟφυͷݟ͔ΒݟΕ NBQSFEVDFతͳͷͱͯ͠ଊ͑Δ͜ͱͰ
͖Δ͕ɺͦΕΛѻ͏ਓؒͷϝϯλϧϞσϧͱ໋ͯ͠ྩܕతͳૢ࡞
– .VUBCMF"SSBZඇ࠶ؼతͳݻఆαΠζͷઢܗߏͰ͋ΓɺతσʔλܕͰએݴ͞ΕΔ࠶ؼత
σʔλߏͱҟͳΔͷ
• ೖग़ྗཚಉ༷ͷɺ֎ք 3FBM8PSME
ͷӨڹԼʹ͋Δ෭࡞༻తͳଘࡏ ֎෦ࢿݯ ϝϞϦ
ͱͷڮ͠ஔ
• .POPJEͱ͔ 4FNJHSPVQͱ͔ͷܕ࣮ͬͯ͠ͳ͍ͷ
– ࢲͬͯ·ͤΜ͕ɺͬͯ៉ྷͳநΛ࣮͍ͯ͠Δ "U$PEFSͷઌഐํ͍·͢
– ࢲ׳Εͱهड़ͷָ͞Λ༏ઌ͍ͯ͠·͢
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ܕγεςϜϓϩάϥϛϯάݴޠͷૢ࡞ମܥදݱྗͦͷͷΛܗ࡞Δϝλߏ
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