In-Place Sorting Viliam Geffert P.J. Š af á rik University , K ošice, Slovakia J o zef Gajdo š
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In-Place Sorting In-Place Sorting Viliam Geffert Viliam Geffert P.J. P.J. Š Š af af á á rik University rik University , , K K ošice, Slovakia ošice, Slovakia J J o o zef Gajdo zef Gajdo š š College of International Business ISM College of International Business ISM Slovakia Slovakia Prešov Prešov , Slovakia , Slovakia The Problem
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In-Place Sorting Viliam Geffert P.J. Š af á rik University , K ošice, Slovakia J o zef Gajdo š College of International Business ISM Slovakia Prešov , Slovakia. The Problem. n elements. n elements. n elements. 1 hole. 1. n.log n + Δ (n) comparisons O(n) element moves - PowerPoint PPT Presentation
Transcript of In-Place Sorting Viliam Geffert P.J. Š af á rik University , K ošice, Slovakia J o zef Gajdo š
In-Place SortingIn-Place Sorting
Viliam GeffertViliam GeffertP.J.P.J.ŠŠafafáárik Universityrik University, , KKošice, Slovakiaošice, Slovakia
JJoozef Gajdozef Gajdošš College of International Business ISM College of International Business ISM
SlovakiaSlovakiaPrešovPrešov, Slovakia, Slovakia
The Problem
n elements
n elements
n elements
1.n.log n + Δ(n) comparisons
O(n) element moves
O(1) index variables, of log n bits each
1 hole
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
n.log n n.log n nmergesort
[Kn1973]
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
n.log n +O(n.loglog n) O(n.log n/loglog n)O(1)mergesortADV
[KaPa1999]
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
n.log n +O(n.loglog n) O(n.log n/loglog n)O(1)mergesortADV
[KaPa1999]
Θ(n2) O(n)O(1) selectsort
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
n.log n +O(n.loglog n) O(n.log n/loglog n)O(1)mergesortADV
[KaPa1999]
Θ(n1+) O(n)O(1) n1/k-heapsort
[MuRa1992]
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
n.log n +O(n.loglog n) O(n.log n/loglog n)O(1)mergesortADV
[KaPa1999]
Θ(n1+) O(n)O(1) n1/k-heapsort
[MuRa1992]
n.log n O(n)Θ(n) tablesort
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
n.log n +O(n.loglog n) O(n.log n/loglog n)O(1)mergesortADV
[KaPa1999]
Θ(n1+) O(n)O(1) n1/k-heapsort
[MuRa1992]
Θ(n.log n) O(n)Θ(n) samplesortADV
[MuRa1992]
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
n.log n +O(n.loglog n) O(n.log n/loglog n)O(1)mergesortADV
[KaPa1999]
Θ(n1+) O(n)O(1) n1/k-heapsort
[MuRa1992]
Θ(n.log n) O(n)Θ(n) samplesortADV
[MuRa1992]
1991-2005: conjectured that an algorithm matching
the lower bounds for all criteria does not exist
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
n.log n +O(n.loglog n) O(n.log n/loglog n)O(1)mergesortADV
[KaPa1999]
Θ(n1+) O(n)O(1) n1/k-heapsort
[MuRa1992]
Θ(n.log n) O(n)Θ(n) samplesortADV
[MuRa1992]
log n! +Θ(n.log n) (13+ε).nO(1) [GeFr2005]
Comparisons MovesStorage
log n! ≈ n.log n–1.44n 3/2.n
O(1) lower bound
n.log n +O(n.loglog n) O(n.log n/loglog n)O(1)mergesortADV
[KaPa1999]
Θ(n1+) O(n)O(1) n1/k-heapsort
[MuRa1992]
Θ(n.log n) O(n)Θ(n) samplesortADV
[MuRa1992]
log n! +Θ(n.log n) (13+ε).nO(1) [GeFr2005]
log n! +Θ(n.loglog n) (26+ε).nO(1) our result
1. Initial Preparation
2n/log2 n largest elements
2n/log2 n largest elements
Modified heapsort
-- branching degree log n instead of 2
2n/log2 n largest elements
Modified heapsort
-- branching degree log n instead of 2-- halts after extracting 2n/log2 n largest elements
Com: O(n)Mov : O(n/log n)
2n/log2 n largest elements
Com: O(n)Mov : O(n/log n)
2n/log2 n smallest elements 2n/log2 n largest elements
Symmetrically, by modified heapsort
Com: O(n)Mov : O(n/log n)
2n/log2 n smallest elements 2n/log2 n largest elements
X Y
2n/log2 n smallest elements 2n/log2 n largest elements
Com: 1Mov: --Com: 1Mov: --
If X=Y, we are done, the array is sorted
2n/log2 n smallest elements 2n/log2 n largest elements
X Y
If X<Y, continue
2n/log2 n smallest elements 2n/log2 n largest elements
X Y
Pointer memory
2n/log2 n smallest elements 2n/log2 n largest elements
0
0
0
0
0
0
Pointer memory
2n/log2 n smallest elements 2n/log2 n largest elements
1
0
0
0
0
0
Pointer memory
2n/log2 n smallest elements 2n/log2 n largest elements
1
0
1
0
0
0
Pointer memory
2n/log2 n smallest elements 2n/log2 n largest elements
1
1
1
1
1
1
2n/log2 n smallest elements 2n/log2 n largest elements
2n/log2 n element pairs ≡
2n/log3 n integer variables, of log n bits each
2. Partition-Based Sorting
bL R
ni elementsan element of rank ¼.ni
L R
ni elements
b
an element of rank ¼.ni
L R
ni elements
b
Com1: O( ni )Mov1 : . ni
an element of rank ¼.ni
L R
ni elements
b
an element of rank ¼.ni
L R
ni elements
b
A<
to be sortedB≥
buffer memory
an element of rank ¼.ni
L Rb
A<
to be sortedB≥
buffer memory
ni,< ≤ ¼.ni
ni,≥ ≥ 3.ni,<
an element of rank ¼.ni
L R
an element of rank ¼.ni
b
A<
to be sortedB≥
buffer memory
ni,< ≤ ¼.ni
ni,≥ ≥ 3.ni,<
Com1: O( ni )Mov1 : 2.ni,<
L R
an element of rank ¼.ni
b
A<
to be sortedB≥
buffer memory
ni,< ≤ ¼.ni
ni,≥ ≥ 3.ni,<
Using pointer memory L+R and buffer memory B≥ ,sort A<
L R
an element of rank ¼.ni
b
A<
to be sortedB≥
buffer memory
ni,< ≤ ¼.ni
ni,≥ ≥ 3.ni,<
Using pointer memory L+R and buffer memory B≥ ,sort A<
L R
an element of rank ¼.ni
b
A< B≥
ni,< ≤ ¼.ni
ni,≥ ≥ 3.ni,<
Using pointer memory L+R and buffer memory B≥ ,sort A<
Pointer memory ≥ 2*2m/log2 mBuffer memory ≥ 3m
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L R
an element of rank ¼.ni
b
ni,< ≤ ¼.ni
ni,≥ ≥ 3.ni,<
B≥A<
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L R
an element of rank ¼.ni
b
ni,< ≤ ¼.ni
ni,≥ ≥ 3.ni,<
B≥A<
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L R
an element of rank ¼.ni
b
ni,< ≤ ¼.ni
ni,≥ ≥ 3.ni,<
B≥A<
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L R
an element of rank ¼.ni
b
ni,< ≤ ¼.ni
ni,≥ ≥ 3.ni,<
B≥A<
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L R
an element of rank ¼.ni
b
ni,< ≤ ¼.ni
ni,> ≤ ¾.ni
B≥A<
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L R
an element of rank ¼.ni
b
B≥A<Com1:ni
Mov1:
3.ni,=
ni,> ≤ ¾.ni
ni,<≤¼ ni,=
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni
ni,> ≤ ¾.ni
ni,<≤¼ ni,=
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni
ni+1 ≤ ¾.ni
ni,<≤¼ ni,=
“new” ni
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+1 ≤ ¾.ni
ni,<≤¼ ni,=
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+1 ≤ ¾.ni
ni,<≤¼ ni,=
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+1 ≤ ¾.ni
ni,<≤¼ ni,=
A< B≥
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+1 ≤ ¾.ni
ni,<≤¼ ni,=
A< B≥
sort
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+1 ≤ ¾.ni
ni,<≤¼ ni,=
A< B≥
sort
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+1 ≤ ¾.ni
ni,<≤¼ ni,=
A< B≥
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+1 ≤ ¾.ni
ni,<≤¼ ni,=
A< B≥
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+1 ≤ ¾.ni
ni,<≤¼ ni,=
A< B≥
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+1 ≤ ¾.ni
ni,<≤¼ ni,= ni+1,< ni+1,=
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+2 ≤ ¾.ni+1
ni,<≤¼ ni,= ni+1,< ni+1,=
“new” ni
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
sortedL R
an element of rank ¼.ni+1
ni+2 ≤ ¾.ni+1
ni,<≤¼ ni,= ni+1,< ni+1,=
… … …
iterate
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L R… … …
iterate
ni+1 ≤ ¾.ni
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L R
ni+1 ≤ ¾.ni
restore order
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L R
ni+1 ≤ ¾.ni
restore order
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
L RCom: O(n/log2 n)Mov: O(n/log2 n)
ni+1 ≤ ¾.ni
restore order
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
Total comparisons: Total moves:
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
Total comparisons: Total moves:
Creating/clearing pointer memory: O(n) O(n/log n)
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
Total comparisons: Total moves:
Creating/clearing pointer memory: O(n) O(n/log n)Processing one block, of length ni: O(ni) + .ni +
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
Total comparisons: Total moves:
Creating/clearing pointer memory: O(n) O(n/log n)Processing one block, of length ni: O(ni) + .ni + ni,<
.log n + ni,<.O(loglog n) (26+).ni,< +
26.ni,=
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
Total comparisons: Total moves:
Creating/clearing pointer memory: O(n) O(n/log n)Processing one block, of length ni: O(ni) + .ni + ni,<
.log n + ni,<.O(loglog n) (26+).ni,< +
26.ni,=
Block lengths: n1,< + n1,= + n2,< + n2,= + … ≤ O(n) ⇐ ni+1 ≤ ¾.ni
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
Total comparisons: Total moves:
Creating/clearing pointer memory: O(n) O(n/log n)Processing one block, of length ni: O(ni) + .ni + ni,<
.log n + ni,<.O(loglog n) (26+).ni,< +
26.ni,=
Block lengths: n1,< + n1,= + n2,< + n2,= + … ≤ O(n) ⇐ ni+1 ≤ ¾.ni n1,< + n1,= + n2,< + n2,= + … ≤ n ⇐ disjoint Com?: m.log
m+ O(m.loglog
m)Mov?: (24+).m
Total comparisons: Total moves:
Creating/clearing pointer memory: O(n) O(n/log n)Processing one block, of length ni: O(ni) + .ni + ni,<
.log n + ni,<.O(loglog n) (26+).ni,< +
26.ni,=
Block lengths: n1,< + n1,= + n2,< + n2,= + … ≤ O(n) n1,< + n1,= + n2,< + n2,= + … ≤ n
Total comparisons: n.log n + n.O(loglog n) + O(n)
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
Total comparisons: Total moves:
Creating/clearing pointer memory: O(n) O(n/log n)Processing one block, of length ni: O(ni) + .ni + ni,<
.log n + ni,<.O(loglog n) (26+).ni,< +
26.ni,=
Block lengths: n1,< + n1,= + n2,< + n2,= + … ≤ O(n) n1,< + n1,= + n2,< + n2,= + … ≤ n
Total comparisons: n.log n + n.O(loglog n)
Total moves: (26+).n
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
Total comparisons: Total moves:
Creating/clearing pointer memory: O(n) O(n/log n)Processing one block, of length ni: O(ni) + .ni + ni,<
.log n + ni,<.O(loglog n) (26+).ni,< +
26.ni,=
Block lengths: n1,< + n1,= + n2,< + n2,= + … ≤ O(n) n1,< + n1,= + n2,< + n2,= + … ≤ nProvided that:
Total comparisons: n.log n + n.O(loglog n)
Total moves: (26+).n
Com?: m.log m
+ O(m.loglog m)Mov?: (24+).m
3.Sorting A with AuxiliaryBuffer and Pointer Memories
Basic Idea
A
B
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
B
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
evenly distributed frame of sizem/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
a2 ≤ a1
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
a1 … a2
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
a2 … a3
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
a3 … a4
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
af ≤ a4
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
frame growsdynamically
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
frame growsdynamically
median
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
frame growsdynamically
≤ ≥
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
frame growsdynamically
≤ ≥
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
≤ ≥
m/log4 m … 2m/log4 m
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
segments sorted by[Geffert & Franceschini
2005]
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
segments sorted by[Geffert & Franceschini
2005]
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
segments sorted by[Geffert & Franceschini
2005]
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
segments sorted by[Geffert & Franceschini
2005]
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
segments sorted by[Geffert & Franceschini
2005]
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
segments sorted by[Geffert & Franceschini
2005]
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
segments of size½.log4 m … log4 m
m/log4 m … 2m/log4 m
A
a1
a2
a3
af
Ϭ0 Ϭ1 Ϭ2 Ϭ3 Ϭf
B
A
B
Data Structure
to be sortedA
melements
to be sortedA
melements
:
pointer memory
2* 2m/log3 m* log m
to be sortedA
melements
:
pointer memory
2* 2m/log3 m* log m
buffer memory B
size 3m
m
:
pointer memory
2* 2m/log3 m* log m
3mto be sorted
A
:
pointer memory
2* 2m/log3 m* log m
3m
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
b -- smallest buffer element
hole
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
b -- smallest buffer element
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
b -- smallest buffer element
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
b -- smallest buffer element
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
b -- smallest buffer element
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
b -- smallest buffer element
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
b -- smallest buffer element
m
to be sortedA
:
pointer memory
2* 2m/log3 m* log m
3m
b -- smallest buffer element
Comparisons Comparisons
Moves 2 * Moves
3m free locations + 1 free location
+ 3m buffer elements
m
to be sortedA
b
frameR
2m/log3m
≤ m/2
:
pointer memory
2* 2m/log3 m* log m
m
to be sortedA
b
frameR
2m/log3m
≤ m/2
:
pointer memory
2* 2m/log3 m* log m
m
to be sortedA
b
to store only2m/log4 m elements
frameR
2m/log3m
≤ m/2
cacheC
2m/log2m
≤ m/2
:
pointer memory
2* 2m/log3 m* log m
m
to be sortedA
b
frameR
2m/log3m
≤ m/2
cacheC
2m/log2m
≤ m/2
segmentsS
≤ 2m/log4m * log4m
≤ 2m
:
pointer memory
2* 2m/log3 m* log m
m
to be sortedA
b
to be sortedA
frameR
cacheC
segmentsS
R:
:
2m/log4 m * log m
b
to be sortedA
frameR
cacheC
segmentsS
R:
:
2m/log4 m * log m
b
sorted frame
to be sortedA
frameR
cacheC
segmentsS
R:
:
sorted frame
buffer elementsnot sorted,larger than active elements
2m/log4 m * log m
b
to be sortedA
frameR
cacheC
segmentsS
R:
:
a
a
2m/log4 m * log m
b
a
to be sortedA
frameR
cacheC
segmentsS
R:
:
a
binary search with a:
log(2m/log4 m) ≤ log m
a
2m/log4 m * log m
b
a
to be sortedA
frameR
cacheC
segmentsS
R:
:
a
2m/log4 m * log m
b
a
to be sortedA
frameR
cacheC
segmentsS
R:
:
a
binary search with a:loglog m
a
2m/log4 m * log m
b
a
to be sortedA
frameR
cacheC
segmentsS
R:
:
C:
a
2m/log4 m * log m
2m/log3 m * log m
b
ak … ak+1
a
to be sortedA
frameR
cacheC
segmentsS
R:
:
C:k.log m
akak+1
a
2m/log3 m * log m
b
2m/log4 m * log m
a
to be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
a
not sortedbuffer elements not sorted,larger than active elements
a2m/log3 m * log m
b
a
to be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
a
b
a2m/log3 m * log m
b
a
to be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
a
a
bbinary search with
b:loglog m
2m/log3 m * log m
b
bto be sortedA
frameR
cacheC
segmentsS
R:
:
2m/log3 m * log mC:
akak+1
2m/log4 m * log m
S: … …
≤ 2m/log4 m * log4 m
k.log m
bto be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
S: … …
k.log m
k.log m k.log m
≤ 2m/log4 m * log4 m
2m/log4 m * log m
bto be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
S: … …
k.log m
k.log m k.log m
≤ 2m/log4 m * log4 m
bto be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
S: … …
k.log m
k.log m
decoding segment address:
log m comparisons
≤ 2m/log4 m * log4 m
bto be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
S: … …ak … ak+1
≤ 2m/log4 m * log4 m
≤ 2m/log4 m * log4 m
bto be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
S: … …
not sortednot sorted
bto be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
S: … …
b
≤ 2m/log4 m * log4 m
bto be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
S: … …
b
binary search with b:
log(log4 m) ≤ 4.loglog m
≤ 2m/log4 m * log4 m
bto be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
S: … …
≤ 2m/log4 m * log4 m
2m/log3 m * log m
2m/log4 m * log m
bto be sortedA
frameR
cacheC
segmentsS
R:
:
C:
akak+1
S: … …
≤ 2m/log4 m * log4 m
2m/log3 m * log m
2m/log4 m * log m
Com: m.log m +O(m.loglog m)Mov : O(m)
Final Remarks
Our result
Comparisons: log n! +Θ(n.loglog n)
Moves: (26+ε).n
Storage: O(1)
Our result[GeFr2005]
Comparisons: log n! +Θ(n.loglog n) log n! +Θ(n.log n)
Moves: (26+ε).n(13+ε).n
Storage: O(1) O(1)
Our result[GeFr2005]
Comparisons: log n! +Θ(n.loglog n) log n! +Θ(n.log n)
Moves: (26+ε).n(13+ε).n
Storage: O(1) O(1)
cache memory
short segments
sorted by [GeFr2005]
Our result
Comparisons: log n! +Θ(n.loglog n)
Moves: (26+ε).n
Storage: O(1)
- mainly of theoretical interest, too complex to implement
Our result
Comparisons: log n! +Θ(n.loglog n)
Moves: (26+ε).n
Storage: O(1)
- mainly of theoretical interest, too complex to implement- however, several new techniques and data structures can be utilized in other applications
Our result
Comparisons: log n! +Θ(n.loglog n)
Moves: (26+ε).n
Storage: O(1)
- mainly of theoretical interest, too complex to implement- however, several new techniques and data structures can be utilized in other applications
? probably, (26+ε).n moves can be substantially improved
[proceedings: (34+ε).n moves]
Our result
Comparisons: log n! +Θ(n.loglog n)
Moves: (26+ε).n
Storage: O(1)
- mainly of theoretical interest, too complex to implement- however, several new techniques and data structures can be utilized in other applications
? probably, (26+ε).n moves can be substantially improved
[proceedings: (34+ε).n moves]? log n! +O(n) comparisons
Our result
Comparisons: log n! +Θ(n.loglog n)
Moves: (26+ε).n
Storage: O(1)
- mainly of theoretical interest, too complex to implement- however, several new techniques and data structures can be utilized in other applications
? probably, (26+ε).n moves can be substantially improved
[proceedings: (34+ε).n moves]? log n! +O(n) comparisons? stable version (equal buffer elements are mixed up)
Thank You for Your Attention
