def PBD_from_TD(k,t,u):
r"""
Return a `(kt,\{k,t\})`-PBD if `u=0` and a `(kt+u,\{k,k+1,t,u\})`-PBD otherwise.
This is theorem 23 from [ClaytonSmith]_. The PBD is obtained from the blocks
a truncated `TD(k+1,t)`, to which are added the blocks corresponding to the
groups of the TD. When `u=0`, a `TD(k,t)` is used instead.
INPUT:
- ``k,t,u`` -- integers such that `0\leq u \leq t`.
EXAMPLES::
sage: from sage.combinat.designs.bibd import PBD_from_TD
sage: from sage.combinat.designs.bibd import is_pairwise_balanced_design
sage: PBD = PBD_from_TD(2,2,1); PBD
[[0, 2, 4], [0, 3], [1, 2], [1, 3, 4], [0, 1], [2, 3]]
sage: is_pairwise_balanced_design(PBD,2*2+1,[2,3])
True
"""
from orthogonal_arrays import transversal_design
TD = transversal_design(k+bool(u),t, check=False)
TD = [[x for x in X if x<k*t+u] for X in TD]
for i in range(k):
TD.append(range(t*i,t*i+t))
if u>=2:
TD.append(range(k*t,k*t+u))
return TD
def PBD_from_TD(k, t, u):
r"""
Return a `(kt,\{k,t\})`-PBD if `u=0` and a `(kt+u,\{k,k+1,t,u\})`-PBD otherwise.
This is theorem 23 from [ClaytonSmith]_. The PBD is obtained from the blocks
a truncated `TD(k+1,t)`, to which are added the blocks corresponding to the
groups of the TD. When `u=0`, a `TD(k,t)` is used instead.
INPUT:
- ``k,t,u`` -- integers such that `0\leq u \leq t`.
EXAMPLES::
sage: from sage.combinat.designs.bibd import PBD_from_TD
sage: from sage.combinat.designs.bibd import is_pairwise_balanced_design
sage: PBD = PBD_from_TD(2,2,1); PBD
[[0, 2, 4], [0, 3], [1, 2], [1, 3, 4], [0, 1], [2, 3]]
sage: is_pairwise_balanced_design(PBD,2*2+1,[2,3])
True
"""
from orthogonal_arrays import transversal_design
TD = transversal_design(k + bool(u), t, check=False)
TD = [[x for x in X if x < k * t + u] for X in TD]
for i in range(k):
TD.append(range(t * i, t * i + t))
if u >= 2:
TD.append(range(k * t, k * t + u))
return TD
def PBD_4_5_8_9_12(v, check=True):
"""
Return a `(v,\{4,5,8,9,12\})`-PBD on `v` elements.
A `(v,\{4,5,8,9,12\})`-PBD exists if and only if `v\equiv 0,1 \pmod 4`. The
construction implemented here appears page 168 in [Stinson2004]_.
INPUT:
- ``v`` -- an integer congruent to `0` or `1` modulo `4`.
- ``check`` (boolean) -- whether to check that output is correct before
returning it. As this is expected to be useless (but we are cautious
guys), you may want to disable it whenever you want speed. Set to ``True``
by default.
EXAMPLES::
sage: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest
[[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 10],
[0, 5, 7, 11], [0, 13, 26, 39], [0, 14, 25, 28],
[0, 15, 27, 38], [0, 16, 22, 32], [0, 17, 23, 34],
...
Check that :trac:`16476` is fixed::
sage: from sage.combinat.designs.bibd import PBD_4_5_8_9_12
sage: for v in (0,1,4,5,8,9,12,13,16,17,20,21,24,25):
....: _ = PBD_4_5_8_9_12(v)
"""
if not v%4 in [0,1]:
raise ValueError
if v <= 1:
PBD = []
elif v <= 12:
PBD = [range(v)]
elif v == 13 or v == 28:
PBD = v_4_1_BIBD(v, check=False)
elif v == 29:
TD47 = transversal_design(4,7)._blocks
four_more_sets = [[28]+[i*7+j for j in range(7)] for i in range(4)]
PBD = TD47 + four_more_sets
elif v == 41:
TD59 = transversal_design(5,9)
PBD = ([[x for x in X if x<41] for X in TD59]
+[[i*9+j for j in range(9)] for i in range(4)]
+[[36,37,38,39,40]])
elif v == 44:
TD59 = transversal_design(5,9)
PBD = ([[x for x in X if x<44] for X in TD59]
+[[i*9+j for j in range(9)] for i in range(4)]
+[[36,37,38,39,40,41,42,43]])
elif v == 45:
TD59 = transversal_design(5,9)._blocks
PBD = (TD59+[[i*9+j for j in range(9)] for i in range(5)])
elif v == 48:
TD4_12 = transversal_design(4,12)._blocks
PBD = (TD4_12+[[i*12+j for j in range(12)] for i in range(4)])
elif v == 49:
# Lemma 7.16 : A (49,{4,13})-PBD
TD4_12 = transversal_design(4,12)._blocks
# Replacing the block of size 13 with a BIBD
BIBD_13_4 = v_4_1_BIBD(13)
for i in range(4):
for B in BIBD_13_4:
TD4_12.append([i*12+x if x != 12 else 48
for x in B])
PBD = TD4_12
else:
t,u = _get_t_u(v)
TD = transversal_design(5,t)
TD = [[x for x in X if x<4*t+u] for X in TD]
for B in [range(t*i,t*(i+1)) for i in range(4)]:
TD.extend(_PBD_4_5_8_9_12_closure([B]))
if u > 1:
TD.extend(_PBD_4_5_8_9_12_closure([range(4*t,4*t+u)]))
PBD = TD
if check:
assert is_pairwise_balanced_design(PBD,v,[4,5,8,9,12])
return PBD
def BIBD_from_TD(v,k,existence=False):
r"""
Return a BIBD through TD-based constructions.
INPUT:
- ``v,k`` (integers) -- computes a `(v,k,1)`-BIBD.
- ``existence`` (boolean) -- instead of building the design, return:
- ``True`` -- meaning that Sage knows how to build the design
- ``Unknown`` -- meaning that Sage does not know how to build the
design, but that the design may exist (see :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the design does not exist.
This method implements three constructions:
- If there exists a `TD(k,v)` and a `(v,k,1)`-BIBD then there exists a
`(kv,k,1)`-BIBD.
The BIBD is obtained from all blocks of the `TD`, and from the blocks of
the `(v,k,1)`-BIBDs defined over the `k` groups of the `TD`.
- If there exists a `TD(k,v)` and a `(v+1,k,1)`-BIBD then there exists a
`(kv+1,k,1)`-BIBD.
The BIBD is obtained from all blocks of the `TD`, and from the blocks of
the `(v+1,k,1)`-BIBDs defined over the sets `V_1\cup \infty,\dots,V_k\cup
\infty` where the `V_1,\dots,V_k` are the groups of the TD.
- If there exists a `TD(k,v)` and a `(v+k,k,1)`-BIBD then there exists a
`(kv+k,k,1)`-BIBD.
The BIBD is obtained from all blocks of the `TD`, and from the blocks of
the `(v+k,k,1)`-BIBDs defined over the sets `V_1\cup
\{\infty_1,\dots,\infty_k\},\dots,V_k\cup \{\infty_1,\dots,\infty_k\}`
where the `V_1,\dots,V_k` are the groups of the TD. By making sure that
all copies of the `(v+k,k,1)`-BIBD contain the block
`\{\infty_1,\dots,\infty_k\}`, the result is also a BIBD.
These constructions can be found in
`<http://www.argilo.net/files/bibd.pdf>`_.
EXAMPLES:
First construction::
sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(25,5,existence=True)
True
sage: _ = BlockDesign(25,BIBD_from_TD(25,5))
Second construction::
sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(21,5,existence=True)
True
sage: _ = BlockDesign(21,BIBD_from_TD(21,5))
Third construction::
sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(85,5,existence=True)
True
sage: _ = BlockDesign(85,BIBD_from_TD(85,5))
No idea::
sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(20,5,existence=True)
Unknown
sage: BIBD_from_TD(20,5)
Traceback (most recent call last):
...
NotImplementedError: I do not know how to build a (20,5,1)-BIBD!
"""
# First construction
if (v%k == 0 and
balanced_incomplete_block_design(v//k,k,existence=True) and
transversal_design(k,v//k,existence=True)):
if existence:
return True
v = v//k
BIBDvk = balanced_incomplete_block_design(v,k)._blocks
TDkv = transversal_design(k,v,check=False)
BIBD = TDkv._blocks
for i in range(k):
BIBD.extend([[x+i*v for x in B] for B in BIBDvk])
# Second construction
elif ((v-1)%k == 0 and
balanced_incomplete_block_design((v-1)//k+1,k,existence=True) and
transversal_design(k,(v-1)//k,existence=True)):
if existence:
return True
v = (v-1)//k
BIBDv1k = balanced_incomplete_block_design(v+1,k)._blocks
TDkv = transversal_design(k,v,check=False)._blocks
inf = v*k
BIBD = TDkv
for i in range(k):
BIBD.extend([[inf if x == v else x+i*v for x in B] for B in BIBDv1k])
# Third construction
elif ((v-k)%k == 0 and
balanced_incomplete_block_design((v-k)//k+k,k,existence=True) and
transversal_design(k,(v-k)//k,existence=True)):
if existence:
return True
v = (v-k)//k
BIBDvpkk = balanced_incomplete_block_design(v+k,k)
TDkv = transversal_design(k,v,check=False)._blocks
inf = v*k
BIBD = TDkv
# makes sure that [v,...,v+k-1] is a block of BIBDvpkk. Then, we remove it.
BIBDvpkk = _relabel_bibd(BIBDvpkk,v+k)
BIBDvpkk = [B for B in BIBDvpkk if min(B) < v]
for i in range(k):
BIBD.extend([[(x-v)+inf if x >= v else x+i*v for x in B] for B in BIBDvpkk])
BIBD.append(range(k*v,v*k+k))
# No idea ...
else:
if existence:
return Unknown
else:
raise NotImplementedError("I do not know how to build a ({},{},1)-BIBD!".format(v,k))
return BIBD
def PBD_4_5_8_9_12(v, check=True):
"""
Returns a `(v,\{4,5,8,9,12\})-PBD` on `v` elements.
A `(v,\{4,5,8,9,12\})`-PBD exists if and only if `v\equiv 0,1 \pmod 4`. The
construction implemented here appears page 168 in [Stinson2004]_.
INPUT:
- ``v`` (integer)
- ``check`` (boolean) -- whether to check that output is correct before
returning it. As this is expected to be useless (but we are cautious
guys), you may want to disable it whenever you want speed. Set to ``True``
by default.
EXAMPLES::
sage: designs.BalancedIncompleteBlockDesign(40,4).blocks() # indirect doctest
[[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 11], [0, 5, 7, 10],
[0, 13, 26, 39], [0, 14, 28, 38], [0, 15, 25, 27],
[0, 16, 32, 35], [0, 17, 34, 37], [0, 18, 33, 36],
...
"""
if not v % 4 in [0, 1]:
raise ValueError
if v == 0:
return []
if v == 13:
PBD = v_4_1_BIBD(v, check=False)
elif v == 28:
PBD = v_4_1_BIBD(v, check=False)
elif v == 29:
TD47 = transversal_design(4, 7)
four_more_sets = [[28] + [i * 7 + j for j in range(7)]
for i in range(4)]
PBD = TD47 + four_more_sets
elif v == 41:
TD59 = transversal_design(5, 9)
PBD = ([[x for x in X if x < 41]
for X in TD59] + [[i * 9 + j for j in range(9)]
for i in range(4)] + [[36, 37, 38, 39, 40]])
elif v == 44:
TD59 = transversal_design(5, 9)
PBD = ([[x for x in X if x < 44] for X in TD59] +
[[i * 9 + j for j in range(9)]
for i in range(4)] + [[36, 37, 38, 39, 40, 41, 42, 43]])
elif v == 45:
TD59 = transversal_design(5, 9)
PBD = (TD59 + [[i * 9 + j for j in range(9)] for i in range(5)])
elif v == 48:
TD4_12 = transversal_design(4, 12)
PBD = (TD4_12 + [[i * 12 + j for j in range(12)] for i in range(4)])
elif v == 49:
# Lemma 7.16 : A (49,{4,13})-PBD
TD4_12 = transversal_design(4, 12)
# Replacing the block of size 13 with a BIBD
BIBD_13_4 = v_4_1_BIBD(13)
for i in range(4):
for B in BIBD_13_4:
TD4_12.append([i * 12 + x if x != 12 else 48 for x in B])
PBD = TD4_12
else:
t, u = _get_t_u(v)
TD = transversal_design(5, t)
TD = [[x for x in X if x < 4 * t + u] for X in TD]
for B in [range(t * i, t * (i + 1)) for i in range(4)]:
TD.extend(_PBD_4_5_8_9_12_closure([B]))
if u > 1:
TD.extend(_PBD_4_5_8_9_12_closure([range(4 * t, 4 * t + u)]))
PBD = TD
if check:
_check_pbd(PBD, v, [4, 5, 8, 9, 12])
return PBD
def PBD_4_5_8_9_12(v, check=True):
"""
Return a `(v,\{4,5,8,9,12\})`-PBD on `v` elements.
A `(v,\{4,5,8,9,12\})`-PBD exists if and only if `v\equiv 0,1 \pmod 4`. The
construction implemented here appears page 168 in [Stinson2004]_.
INPUT:
- ``v`` -- an integer congruent to `0` or `1` modulo `4`.
- ``check`` (boolean) -- whether to check that output is correct before
returning it. As this is expected to be useless (but we are cautious
guys), you may want to disable it whenever you want speed. Set to ``True``
by default.
EXAMPLES::
sage: designs.balanced_incomplete_block_design(40,4).blocks() # indirect doctest
[[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 10],
[0, 5, 7, 11], [0, 13, 26, 39], [0, 14, 25, 28],
[0, 15, 27, 38], [0, 16, 22, 32], [0, 17, 23, 34],
...
Check that :trac:`16476` is fixed::
sage: from sage.combinat.designs.bibd import PBD_4_5_8_9_12
sage: for v in (0,1,4,5,8,9,12,13,16,17,20,21,24,25):
....: _ = PBD_4_5_8_9_12(v)
"""
if not v % 4 in [0, 1]:
raise ValueError
if v <= 1:
PBD = []
elif v <= 12:
PBD = [range(v)]
elif v == 13 or v == 28:
PBD = v_4_1_BIBD(v, check=False)
elif v == 29:
TD47 = transversal_design(4, 7)._blocks
four_more_sets = [[28] + [i * 7 + j for j in range(7)]
for i in range(4)]
PBD = TD47 + four_more_sets
elif v == 41:
TD59 = transversal_design(5, 9)
PBD = ([[x for x in X if x < 41]
for X in TD59] + [[i * 9 + j for j in range(9)]
for i in range(4)] + [[36, 37, 38, 39, 40]])
elif v == 44:
TD59 = transversal_design(5, 9)
PBD = ([[x for x in X if x < 44] for X in TD59] +
[[i * 9 + j for j in range(9)]
for i in range(4)] + [[36, 37, 38, 39, 40, 41, 42, 43]])
elif v == 45:
TD59 = transversal_design(5, 9)._blocks
PBD = (TD59 + [[i * 9 + j for j in range(9)] for i in range(5)])
elif v == 48:
TD4_12 = transversal_design(4, 12)._blocks
PBD = (TD4_12 + [[i * 12 + j for j in range(12)] for i in range(4)])
elif v == 49:
# Lemma 7.16 : A (49,{4,13})-PBD
TD4_12 = transversal_design(4, 12)._blocks
# Replacing the block of size 13 with a BIBD
BIBD_13_4 = v_4_1_BIBD(13)
for i in range(4):
for B in BIBD_13_4:
TD4_12.append([i * 12 + x if x != 12 else 48 for x in B])
PBD = TD4_12
else:
t, u = _get_t_u(v)
TD = transversal_design(5, t)
TD = [[x for x in X if x < 4 * t + u] for X in TD]
for B in [range(t * i, t * (i + 1)) for i in range(4)]:
TD.extend(_PBD_4_5_8_9_12_closure([B]))
if u > 1:
TD.extend(_PBD_4_5_8_9_12_closure([range(4 * t, 4 * t + u)]))
PBD = TD
if check:
assert is_pairwise_balanced_design(PBD, v, [4, 5, 8, 9, 12])
return PBD
def BIBD_from_TD(v, k, existence=False):
r"""
Return a BIBD through TD-based constructions.
INPUT:
- ``v,k`` (integers) -- computes a `(v,k,1)`-BIBD.
- ``existence`` (boolean) -- instead of building the design, return:
- ``True`` -- meaning that Sage knows how to build the design
- ``Unknown`` -- meaning that Sage does not know how to build the
design, but that the design may exist (see :mod:`sage.misc.unknown`).
- ``False`` -- meaning that the design does not exist.
This method implements three constructions:
- If there exists a `TD(k,v)` and a `(v,k,1)`-BIBD then there exists a
`(kv,k,1)`-BIBD.
The BIBD is obtained from all blocks of the `TD`, and from the blocks of
the `(v,k,1)`-BIBDs defined over the `k` groups of the `TD`.
- If there exists a `TD(k,v)` and a `(v+1,k,1)`-BIBD then there exists a
`(kv+1,k,1)`-BIBD.
The BIBD is obtained from all blocks of the `TD`, and from the blocks of
the `(v+1,k,1)`-BIBDs defined over the sets `V_1\cup \infty,\dots,V_k\cup
\infty` where the `V_1,\dots,V_k` are the groups of the TD.
- If there exists a `TD(k,v)` and a `(v+k,k,1)`-BIBD then there exists a
`(kv+k,k,1)`-BIBD.
The BIBD is obtained from all blocks of the `TD`, and from the blocks of
the `(v+k,k,1)`-BIBDs defined over the sets `V_1\cup
\{\infty_1,\dots,\infty_k\},\dots,V_k\cup \{\infty_1,\dots,\infty_k\}`
where the `V_1,\dots,V_k` are the groups of the TD. By making sure that
all copies of the `(v+k,k,1)`-BIBD contain the block
`\{\infty_1,\dots,\infty_k\}`, the result is also a BIBD.
These constructions can be found in
`<http://www.argilo.net/files/bibd.pdf>`_.
EXAMPLES:
First construction::
sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(25,5,existence=True)
True
sage: _ = designs.BlockDesign(25,BIBD_from_TD(25,5))
Second construction::
sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(21,5,existence=True)
True
sage: _ = designs.BlockDesign(21,BIBD_from_TD(21,5))
Third construction::
sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(85,5,existence=True)
True
sage: _ = designs.BlockDesign(85,BIBD_from_TD(85,5))
No idea::
sage: from sage.combinat.designs.bibd import BIBD_from_TD
sage: BIBD_from_TD(20,5,existence=True)
Unknown
sage: BIBD_from_TD(20,5)
Traceback (most recent call last):
...
NotImplementedError: I do not know how to build a (20,5,1)-BIBD!
"""
# First construction
if (v % k == 0
and balanced_incomplete_block_design(v // k, k, existence=True)
and transversal_design(k, v // k, existence=True)):
if existence:
return True
v = v // k
BIBDvk = balanced_incomplete_block_design(v, k)._blocks
TDkv = transversal_design(k, v, check=False)
BIBD = TDkv._blocks
for i in range(k):
BIBD.extend([[x + i * v for x in B] for B in BIBDvk])
# Second construction
elif ((v - 1) % k == 0 and balanced_incomplete_block_design(
(v - 1) // k + 1, k, existence=True)
and transversal_design(k, (v - 1) // k, existence=True)):
if existence:
return True
v = (v - 1) // k
BIBDv1k = balanced_incomplete_block_design(v + 1, k)._blocks
TDkv = transversal_design(k, v, check=False)._blocks
inf = v * k
BIBD = TDkv
for i in range(k):
BIBD.extend([[inf if x == v else x + i * v for x in B]
for B in BIBDv1k])
# Third construction
elif ((v - k) % k == 0 and balanced_incomplete_block_design(
(v - k) // k + k, k, existence=True)
and transversal_design(k, (v - k) // k, existence=True)):
if existence:
return True
v = (v - k) // k
BIBDvpkk = balanced_incomplete_block_design(v + k, k)
TDkv = transversal_design(k, v, check=False)._blocks
inf = v * k
BIBD = TDkv
# makes sure that [v,...,v+k-1] is a block of BIBDvpkk. Then, we remove it.
BIBDvpkk = _relabel_bibd(BIBDvpkk, v + k)
BIBDvpkk = [B for B in BIBDvpkk if min(B) < v]
for i in range(k):
BIBD.extend([[(x - v) + inf if x >= v else x + i * v for x in B]
for B in BIBDvpkk])
BIBD.append(range(k * v, v * k + k))
# No idea ...
else:
if existence:
return Unknown
else:
raise NotImplementedError(
"I do not know how to build a ({},{},1)-BIBD!".format(v, k))
return BIBD
def PBD_4_5_8_9_12(v, check=True):
"""
Returns a `(v,\{4,5,8,9,12\})-PBD` on `v` elements.
A `(v,\{4,5,8,9,12\})`-PBD exists if and only if `v\equiv 0,1 \pmod 4`. The
construction implemented here appears page 168 in [Stinson2004]_.
INPUT:
- ``v`` (integer)
- ``check`` (boolean) -- whether to check that output is correct before
returning it. As this is expected to be useless (but we are cautious
guys), you may want to disable it whenever you want speed. Set to ``True``
by default.
EXAMPLES::
sage: designs.BalancedIncompleteBlockDesign(40,4).blocks() # indirect doctest
[[0, 1, 2, 12], [0, 3, 6, 9], [0, 4, 8, 11], [0, 5, 7, 10],
[0, 13, 26, 39], [0, 14, 28, 38], [0, 15, 25, 27],
[0, 16, 32, 35], [0, 17, 34, 37], [0, 18, 33, 36],
...
"""
if not v%4 in [0,1]:
raise ValueError
if v == 0:
return []
if v == 13:
PBD = v_4_1_BIBD(v, check=False)
elif v == 28:
PBD = v_4_1_BIBD(v, check=False)
elif v == 29:
TD47 = transversal_design(4,7)
four_more_sets = [[28]+[i*7+j for j in range(7)] for i in range(4)]
PBD = TD47 + four_more_sets
elif v == 41:
TD59 = transversal_design(5,9)
PBD = ([[x for x in X if x<41] for X in TD59]
+[[i*9+j for j in range(9)] for i in range(4)]
+[[36,37,38,39,40]])
elif v == 44:
TD59 = transversal_design(5,9)
PBD = ([[x for x in X if x<44] for X in TD59]
+[[i*9+j for j in range(9)] for i in range(4)]
+[[36,37,38,39,40,41,42,43]])
elif v == 45:
TD59 = transversal_design(5,9)
PBD = (TD59+[[i*9+j for j in range(9)] for i in range(5)])
elif v == 48:
TD4_12 = transversal_design(4,12)
PBD = (TD4_12+[[i*12+j for j in range(12)] for i in range(4)])
elif v == 49:
# Lemma 7.16 : A (49,{4,13})-PBD
TD4_12 = transversal_design(4,12)
# Replacing the block of size 13 with a BIBD
BIBD_13_4 = v_4_1_BIBD(13)
for i in range(4):
for B in BIBD_13_4:
TD4_12.append([i*12+x if x != 12 else 48
for x in B])
PBD = TD4_12
else:
t,u = _get_t_u(v)
TD = transversal_design(5,t)
TD = [[x for x in X if x<4*t+u] for X in TD]
for B in [range(t*i,t*(i+1)) for i in range(4)]:
TD.extend(_PBD_4_5_8_9_12_closure([B]))
if u > 1:
TD.extend(_PBD_4_5_8_9_12_closure([range(4*t,4*t+u)]))
PBD = TD
if check:
_check_pbd(PBD,v,[4,5,8,9,12])
return PBD