Eliminating Amortizations PurelyFunctionalDataStrctures ?? ?6?

1
Eliminating Amortizations PurelyFunctionalDataSt
rctures ?? ?6?

• 2006/06/23
• ???

2
??????? (1)

• Imperative Queue (Ch2)
• ??????????????????snoc????
• Persistent ???
• head, tail, snoc ???????? O(1)

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2
3

4

3
??????? (2)

• Functional Queue (Ch5)
• Front??Rear??2????????
• Front 0 ????? Rear ? reverse
• Persistent ???
• ????????reverse??????? O(n)
• ??????? (persistency????????????) O(1)

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2
3

5
4

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??????? (3)

• Functional Queue via Lazy Evaluation (Ch6)
• Front??Rear??2????????
• Front1 Rear ??????? reverse
• Persistent ???
• ????????reverse??????? O(n)
• ??????? (persistency???????????) O(1)

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2
3

5
4

5
????

• Functional Real-Time Queue (Ch7)
• Persistent ???
• ????????O(1)
• ?????????? O(1)

?
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Real-Time Queues ?????

• 6??????????Queue??
• ???? ?suspension???????????
• reverse ? suspension ????????????
• ??? ?????? ??suspension?force????????????
  reverse???Rear???
• ???suspension?force???????????????????????????????
  ??
• revese?Front Rear ??????????Front ? tail
  ????force????????1??tail?1???????

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Real-Time Queues ?????

• Real-Time Queue ??
• ???? ?suspension???????????
• ? tail ??????????????????
• ????tail?????reverse?suspension?????(??????)?????
  ?

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Real-Time Queues

• OCaml ?????????
• MyRTQ.ml
• ????
• ?Front1 Rear ?????reverse???????
• ???suspension??????force???

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Amortization?? 6??Queue???

• Rear?????
• f _at_ rev r
• rev ? incremental ???(force?????????????)?????????
  ?????????????
• rotate f r
• incremental???????????
• Schedule, exec
• ???????????????????????????????????????????

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Amortization?? ???????? (1)

• Monolithic ? suspension ??????incremental
  ?????????
• ? suspension ???????????????????????????
• ? _at_rev ? rotate, ?????O(1)

?????(internal cost) suspension
????????????????????? suspension
??????????O(1)????????????????????????? (?suspend?
??????????????????)
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Amortization?? ???????? (2)

• ?suspension??????????????suspension???????suspensi
  on???????????????
• ? schedule??

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Exercise

• 7.1
• 6?? _at_rev ? rotate ???????(Scheduling???????)?snoc,
  head, tail ????????O(n) ?? O(log n) ?????????
• Ans
• Schedule???????rotate??????force?
• ??suspension???????????snoc????????????????O(log
  n)

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Exercise

• 7.2
• Real-time Queue ??????????????f r
  ????????r ? s ?????????????????????????
• Ans.
• s ???rotate????? tail?? snoc??
• r snoc??
• ???? ???rotate????? - tail?? snoc??
  s 2 r
• fr gt rs ???????????
• ?????????

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??????2

• Binomial Heaps (Ch3)
• insert, deleteMin, findMin, merge ?? O(log n)
• Binomial Heaps (Ch5)
• Physicists Method ?? persistency????????????
  insert ?? O(1) ????
• Lazy Binomial Heaps (Ch6)
• ???????????????persistency ???????insert ??
  O(1) ????
• Amortization??? (Ch7)
• insert ? ?? O(1) ????

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Binomial Heaps ????

• ?????2???????????????
• ??? 2a12a2...2an ?Heap?? ???2a1??????2a2????
  ???2an?????????

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Binomial Heaps ????
type tree T of int Elt.t tree list type
heap tree list
insert e h insTree (T(0,e,)) h insTree t h
match h with -gt t hdtl -gt if
rank t lt rank hd then th else
insTree (link t hd) tl
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??????

• ??????????suspension? ?monolithic?? incremental
  ??????
• ????????????????????
• monolithic ? insTree ? incremental, O(1) ?
• Scheduling

insert e h insTree (T(0,e,)) h insTree t h
match h with -gt t hdtl -gt if
rank t lt rank hd then th else
insTree (link t hd) tl
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1 Incremental ?

• ???2????????

type tree T of Elt.t tree list type bit
Zero One of tree type heap bit list


,
,
,
0
0
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1 Incremental ?

• ???2????????

insTree t h match h with -gt
One t Zerotl -gt (One t) tl (One
hd)tl -gt Zero insTree
(link t hd) tl
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1 Incremental ?

• ???2????????
• Lazy?

insTree t h match force h with SNil -gt
lazy (SCons(One t, lazy(SNil))) SCons(Zero,
tl) -gt lazy (SCons(One t, tl)) SCons(One
hd, tl) -gt lazy (SCons(Zero, insTree (link t
hd) tl))
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2 Scheduling

• ?insTree???suspension?????????????????suspension?
  ????????????????

insTree t h match force h with SNil -gt
lazy (SCons(One t, lazy(SNil))) SCons(Zero,
tl) -gt lazy (SCons(One t, tl)) SCons(One
hd, tl) -gt lazy (SCons(Zero, insTree (link t
hd) tl))
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2 Scheduling

• insert ??????(????????)

0
0
0
1

• ????suspension
• ????????
• ???????
• ??? bit ?0????????????????????? bit ??????

1
1
0
0
1
0
1
0
1
0
1
1
1
1
1
0
1
0
1
0
0
1
0
1
0
1
1
0
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
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2 Scheduling

• insert ??????(????????)

0
0
0
1
type schd bit stream list type heap bit
stream schd let exec (bsbsl) match force
bs with SCons(Zero, t) -gt tbsl
SCons(One _,_) -gt bsl let insert x (bs,sc)
let bs insTree ... bs in (bs, exec (exec
(bssc)))
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1
0
0
1
0
1
0
1
0
1
1
1
1
1
0
1
0
1
0
0
1
0
1
0
1
1
0
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
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2 Scheduling

• Q ??2?exec????
• A 6?????????? Amortized Cost?2??????

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2 Scheduling (Thm7.1)
Schedule

• 1??insert???2?exec????????????

• Schedule?????range???????
• ???????range??????????Zero?1????
• ??????range??????????Zero?2????

0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
1

0
0
0
0
1
0
0
1

0
0
0
1
1
0
0
1
0
0
0
1
1
0
0
1
???Heap
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2 Scheduling (Thm7.1) ??

• ??????
• ???? (empty) ??????? ? ??
• ????????? insert ?1????????????
• insert ??????Zero?One????
• ????(4??)

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??????4??
1 ??????B
1 ??????A
0
0
1
0
0
1
insert
1
1
1
0
0
1
insert
0
1
1
0
0
1
exec 2
1
1
1
0
0
1
exec 2
0,1 ??????
0
1
1
0
0
1
insert
0
0
0
0
1
1
insert
1
0
1
0
0
1
exec 2
0
0
0
0
1
1
exec 2
1
0
1
0
0
1
0,...,0,1 ??????
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??????

• deleteMin, findMin, merge
• O(log n) ????????????? schedule
  ??????????????????????
• insert ?????????? schedule ????? O(log n) ????
• MyBinHeap2.ml

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???

• Lazy Amortized Persistent ???????Lazy
  WorstCase Persistent ??????
• ????????? incremental ???
• ?????? suspension ? Schedule ?????
• ????????????????????????Schedule ???