!L-Drawings of Directed Graphs

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ROMA
TRE
UNIVERSITÀ DEGLI STUDI
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0
1
0
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0
1
1
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0
1
0
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0
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1
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0
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1
1
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0
1
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3
7
0
2
9
4
5
8
1
6
Pp2`HQ/2/ P`i?Q;QMH
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G@.`rBM;b Q7 .B`2+i2/ :`T?b
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0
1
0
0
0
1
1
1
0
0
0
0
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1
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3
7
0
2
9
4
5
8
1
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6
Pp2`HQ/2/ P`i?Q;QMH
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oBM+2MxQ _Qb2HHB
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Pp2`HQ/2/ P`i?Q;QMH .`rBM;b, UEQ`M`QTQmHQb hQHHBb ǶRRV
3
7
0
2
9
QM2 p2`i2t T2` +QHmKMf`Qr
2/;2b +M Qp2`HT
2t+iHv QM2 #2M/ T2` 2/;2
/BbiBM;mBb?2/ 7`QK +`QbbBM;b
#v bKHH +B`+H2b
4
5
2/;2b `Qmi2/ iQT@`B;?i
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8
1
6
oBM+2MxQ _Qb2HHB
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G@.`rBM;b Q7 .B`2+i2/ :`T?b
QM2 p2`i2t T2` +QHmKMf`Qr
T`iBH 2/;2 Qp2`HT
QM2 #2M/ T2` 2/;2
2/;2b 2tBi p2`iB+HHv- +M
im`M BM #Qi? /B`2+iBQMb
n × n `2
*QKTH2i2Hv /2b+`B#2/ #v,
πH = {ÇÇÇǜ}
πV = {ÇÇǜÇ}
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
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JBMBKmK@AMF G@.`rBM;b
oBM+2MxQ _Qb2HHB
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JBMBKmK@AMF G@.`rBM;b
AMF +QMbmKTiBQM
2
2
1
3
2
2
2
2
1
3
ink(Γ) = 20
oBM+2MxQ _Qb2HHB
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G@.`rBM;b Q7 .B`2+i2/ :`T?b
AMF *QMbmKTiBQM
6+ib
P#b2`piBQM
ink(Γ) = ink(πH ) + ink(πV )
G2KK
πH M/ πV `2 BM/2T2M/2Mi- b r2HH b ink(πH ) M/ ink(πV )
oBM+2MxQ _Qb2HHB
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G@.`rBM;b Q7 .B`2+i2/ :`T?b
AMF *QMbmKTiBQM
6+ib
P#b2`piBQM
ink(Γ) = ink(πH ) + ink(πV )
G2KK
πH M/ πV `2 BM/2T2M/2Mi- b r2HH b ink(πH ) M/ ink(πV )
2
3
1
3
2
2
2
2
2
2
3
2
3
2
1
1
3
3
oBM+2MxQ _Qb2HHB
1
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
AMF *QMbmKTiBQM
6+ib
P#b2`piBQM
ink(Γ) = ink(πH ) + ink(πV )
G2KK
πH M/ πV `2 BM/2T2M/2Mi- b r2HH b ink(πH ) M/ ink(πV )
G2KK
Mv G@/`rBM; Γ Q7 Kn QM i?2 nƓn ;`B/ mb2b 2n(nƐ1) BMFX
1
3
2
3
1
1
3
3
2
2
2
2
3
2
1
2
1
2
1
1
1
3
3
oBM+2MxQ _Qb2HHB
3
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S`Q#H2K,
JBMBKmK@AMF@G@.`rBM; UJAG.V
AMbiM+2,
/B`2+i2/ ;`T? G = (V, E) M/ M BMi2;2` kX
Zm2biBQM,
.Q2b G /KBi M G@/`rBM; Γ bm+? i?i ink(Γ) ≤ k\
oBM+2MxQ _Qb2HHB
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JBMBKmK@AMF G@.`rBM;b
S`Q#H2K,
JBMBKmK@AMF@G@.`rBM; UJAG.V
AMbiM+2,
/B`2+i2/ ;`T? G = (V, E) M/ M BMi2;2` kX
Zm2biBQM,
.Q2b G /KBi M G@/`rBM; Γ bm+? i?i ink(Γ) ≤ k\
LS@*QKTH2i2
oBM+2MxQ _Qb2HHB
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AMbiM+2,
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Zm2biBQM,
.Q2b G /KBi M G@/`rBM; Γ bm+? i?i ink(Γ) ≤ k\
LS@*QKTH2i2
JAG. ∈ LS,
+?2+F HH +QK#BMiBQMb Q7 TQbbB#H2 πH M/ πV
oBM+2MxQ _Qb2HHB
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S`Q#H2K,
S`Q7BH2
LS@*QKTH2i2 U.őx 2i HX kyykV
AMbiM+2,
;`T? G = (V, E) M/ M BMi2;2` kX
Zm2biBQM,
.Q2b i?2`2 2tBbi
M Q`/2`BM; π 7Q` i?2#p2`iB+2b Q7 V
!"
bm+? i?i
π(u) − KBM π(v) ≤ k \
u∈V
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v∈N (u)∪u
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AMbiM+2 Q7 S`Q7BH2 QM n = 4 p2`iB+2b
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_2/m+iBQM 7`QK S`Q}H2, AMbiM+2 Q7 JBH/
Kp0
Kp00
AMb2`i irQ +QTB2b Q7 Kp - rBi? p = 52 n2 + 92 n + 1
oBM+2MxQ _Qb2HHB
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Kp0
u
Kp00
w
∀ v ∈ G- // 2/;2b (v, u)- (v, w)- M/ (w, v)
rBi? u ∈ Kp′ - w ∈ Kp′′
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Kp0
u
Kp00
w
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Kp0
u
Kp′ M/ Kp′′ +MMQi
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M/ #2HQr G
Kp00
w
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G@.`rBM;b Q7 .B`2+i2/ :`T?b
JBMBKmK@AMF G@.`rBM;b
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Kp0
u
2/;2b (v, u) M/ (v, w) +Qp2` i?2
p2`iB+H b2;K2Mib Q7 i?2 2/;2b Q7 i?2
;`T? U 4⇒ ink(πV ) = 0 V
2/;2b (w, v) +Qp2`
i?2 ?Q`BxQMiH
U#H+FV b2;K2Mib
2Mi2`BM; 7`QK i?2
`B;?i
Kp′ M/ Kp′′ +MMQi
#2 /`rM bKHH2`M/ Kmbi #2 #Qp2
M/ #2HQr G
Kp00
w
oBM+2MxQ _Qb2HHB
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JBMBKmK@AMF G@.`rBM;b
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Kp0
u
2/;2b (v, u) M/ (v, w) +Qp2` i?2
p2`iB+H b2;K2Mib Q7 i?2 2/;2b Q7 i?2
;`T? U 4⇒ ink(πV ) = 0 V
2/;2b (w, v) +Qp2`
i?2 ?Q`BxQMiH
U#H+FV b2;K2Mib
2Mi2`BM; 7`QK i?2
`B;?i
Kp′ M/ Kp′′ +MMQi
#2 /`rM bKHH2`M/ Kmbi #2 #Qp2
M/ #2HQr G
Kp00
πH bXiX H2M;i?b Q7 b2;K2Mib 2Mi2`BM;
7`QK i?2 H27i bmK mT iQ ≤ k, S`Q7BH2
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
w
G@.`rBM;b Q7 .B`2+i2/ :`T?b
_2/m+2/@+Qbi G@.`rBM;b
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
_2/m+2/@+Qbi G@.`rBM;b
AM+`2K2MiH SQHvMQKBH H;Q`Bi?K
A/2
bbmK2 bm#;`T? ?b H`2/v #22M /`rM- // M2r p2`i2t,
F22TBM; QH/ p2`iB+2b i i?2B` U`2HiBp2V TQbBiBQMb
M2r `Qr M/ M2r +QHmKM rBHH #2 M22/2/
KBMBKBxBM; i?2 BMF `2[mB`2/ #v i?2 BMb2`iBQM
P`/2`BM;
"6a bQ i?i i 2+? bi2T i?2 ;`T? Bb +QMM2+i2/
"
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
//BiBQMH AMF JBMBKBxiBQM
_2K`F
πH M/ πV `2 BM/2T2M/2Mi- M/ bQ `2 ink(π H ) M/ ink(πV )
A/2
+
AM/2T2M/2MiHv QTiBKBx2 inkH
(Γ, v) M/ inkV+ (Γ, v)
P#b2`piBQM
⎧
(
)
+
⎪
⎨inkH
(Γ, v) = KBM
ai`H (Γ, i) + AMH (Γ, i) + PmiH (Γ, i)
i=1,...,n+1 (
)
O(n)
+
⎪
ink
(Γ,
v)
=
KBM
ai`
(Γ,
j)
+
AM
(Γ,
j)
+
Pmi
(Γ,
j)
⎩ V
V
V
V
j=1,...,n+1
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
//BiBQMH AMF JBMBKBxiBQM
_2K`F
πH M/ πV `2 BM/2T2M/2Mi- M/ bQ `2 ink(π H ) M/ ink(πV )
A/2
+
AM/2T2M/2MiHv QTiBKBx2 inkH
(Γ, v) M/ inkV+ (Γ, v)
P#b2`piBQM
⎧
(
)
+
⎪
⎨inkH
(Γ, v) = KBM
ai`H (Γ, i) + AMH (Γ, i) + PmiH (Γ, i)
i=1,...,n+1 (
)
O(n)
+
⎪
ink
(Γ,
v)
=
KBM
ai`
(Γ,
j)
+
AM
(Γ,
j)
+
Pmi
(Γ,
j)
⎩ V
V
V
V
j=1,...,n+1
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
1tT2`BK2MiH MHvbBb
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
1tT2`BK2MiH MHvbBb
h2bi R, PP.- &_M/QK- AM+`2K2MiH- PTiBKmK@AMF' G@.`rBM;b
8- Ry- R8 p2`iB+2b
RyW- kyW- jyW- dyW 2/;2 /2MbBiv
PTiBKmK@AMF G@.`rBM;b +QKTmi2/ pB AMi2;2` GBM2`
S`Q;`KKBM;
h2bi k, PP.- &_M/QK- AM+`2K2MiH' G@.`rBM;b
Ryy- kyy- jyy- 9yy- 8yy p2`iB+2b
RyW- kyW- jyW- dyW 2/;2 /2MbBiv
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
AMi2;2` GBM2` S`Q;`K 7Q` JBH/
2
1
o2`iB+2b +M #2 +?`;2/ Q7 i?2
H2M;?i Q7 i?2B` HQM;2bi BM+B/2Mi
b2;K2Mi BM 2+? /B`2+iBQM
3
2
2
2
2
2
1
3
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
AMi2;2` GBM2` S`Q;`K 7Q` JBH/
o`B#H2b,
∀i, j = 1, . . . , n,
xij , yij =
*
1 p2`i2t vi HB2b QM +QHmKMf`Qr j
0 Qi?2`rBb2
∀i = 1, . . . , n, Ei , Wi , Ni , Si ≥ 0 7m`i?2bi #2M/ Q7 2/;2b BM+B/2Mi iQ vi
*QMbi`BMib,
+n
+n
∀i = 1, . . . , n, j=1 xij = 1c j=1 yij = 1
+n
+n
∀i, j = 1, . . . , n, xi = j=1 xij · jc yi = j=1 yij · j
QM2 p2`i2t T2` +QHmKMf`Qr
+QQ`/BMi2b Q7 vi
*
#2M/ TH+2K2Mi
∀i = 1, . . . , n, Ei ≥ xi c Wi ≤ xi c Ni ≥ yi c Si ≤ yi
∀(vi , vj ) ∈ G, Ej ≥ xi c Wj ≤ xi c Ni ≥ yj c Si ≤ yj
P#D2+iBp2 6mM+iBQM,
KBM
+n
i=1 (Ni
oBM+2MxQ _Qb2HHB
− Si + Ei − Wi )
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
1tT2`BK2MiH MHvbBb
h2bi R, R8 p2`iB+2b- RyW- kyW- jyW- dyW 2/;2b
450
400
350
optimal
incremental
ood
random
Used ink
300
250
200
150
100
50
0
0.1
0.2
0.3
0.7
Edge density
oBM+2MxQ _Qb2HHB
_QK h`2 lMBp2`bBiv
G@.`rBM;b Q7 .B`2+i2/ :`T?b
1tT2`BK2MiH MHvbBb
Saved ink
h2bi k, Ryy- kyy- jyy- 9yy- 8yy p2`iB+2b- jyW 2/;2b
10000
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
incremental
ood
random
100
200
300
400
500
19800 79600 179400 319200 499000
Number of vertices / Theoretical maximum ink
P#b2`piBQM
AM+`2bBM; i?2 MmK#2` Q7 p2`iB+2b 4⇒ [m/`iB+ BM+`2b2 Q7 2/;2b
M/ BMF- #mi AM+`2K2MiH bp2b BMF HBM2`Hv
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