quoracle/quorums/quorums.py

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# TODO(mwhittaker): We can define a set of read quorums that are not minimal.
# Does this mess things up?
from typing import (Dict, Iterator, Generic, List, Optional, Set, Tuple,
TypeVar, Union)
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import collections
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import itertools
import numpy as np
import pulp
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T = TypeVar('T')
def _min_hitting_set(sets: Iterator[Set[T]]) -> int:
x_vars: Dict[T, pulp.LpVariable] = dict()
next_id = itertools.count()
problem = pulp.LpProblem("min_hitting_set", pulp.LpMinimize)
for (i, xs) in enumerate(sets):
for x in xs:
if x not in x_vars:
id = next(next_id)
x_vars[x] = pulp.LpVariable(f'x{id}', cat=pulp.LpBinary)
problem += sum(x_vars[x] for x in xs) >= 1
problem += sum(x_vars.values())
problem.solve(pulp.apis.PULP_CBC_CMD(msg=False))
return int(sum(v.varValue for v in x_vars.values()))
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class Expr(Generic[T]):
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def __add__(self, rhs: 'Expr[T]') -> 'Expr[T]':
return _or(self, rhs)
def __mul__(self, rhs: 'Expr[T]') -> 'Expr[T]':
return _and(self, rhs)
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def quorums(self) -> Iterator[Set[T]]:
raise NotImplementedError
def is_quorum(self, xs: Set[T]) -> bool:
raise NotImplementedError
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def elements(self) -> Set[T]:
return {node.x for node in self.nodes()}
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def nodes(self) -> Set['Node[T]']:
raise NotImplementedError
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def resilience(self) -> int:
if self.dup_free():
return self._dup_free_min_failures() - 1
else:
return _min_hitting_set(self.quorums()) - 1
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def dual(self) -> 'Expr[T]':
raise NotImplementedError
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def dup_free(self) -> bool:
return len(self.nodes()) == self._num_leaves()
def _num_leaves(self) -> int:
raise NotImplementedError
def _dup_free_min_failures(self) -> int:
raise NotImplementedError
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class Node(Expr[T]):
def __init__(self,
x: T,
capacity: Optional[float] = None,
read_capacity: Optional[float] = None,
write_capacity: Optional[float] = None) -> None:
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self.x = x
# A user either specifies capacity or (read_capacity and
# write_capacity), but not both.
if (capacity is None and
read_capacity is None and
write_capacity is None):
self.read_capacity = 1.0
self.write_capacity = 1.0
elif (capacity is not None and
read_capacity is None and
write_capacity is None):
self.read_capacity = capacity
self.write_capacity = capacity
elif (capacity is None and
read_capacity is not None and
write_capacity is not None):
self.read_capacity = read_capacity
self.write_capacity = write_capacity
else:
raise ValueError('You must specify capacity or (read_capacity '
'and write_capacity)')
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def __str__(self) -> str:
return str(self.x)
def __repr__(self) -> str:
return f'Node({self.x})'
def quorums(self) -> Iterator[Set[T]]:
yield {self.x}
def is_quorum(self, xs: Set[T]) -> bool:
return self.x in xs
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def nodes(self) -> Set['Node[T]']:
return {self}
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def dual(self) -> Expr:
return self
def _num_leaves(self) -> int:
return 1
def _dup_free_min_failures(self) -> int:
return 1
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class Or(Expr[T]):
def __init__(self, es: List[Expr[T]]) -> None:
if len(es) == 0:
raise ValueError(f'Or cannot be constructed with an empty list')
self.es = es
def __str__(self) -> str:
return '(' + ' + '.join(str(e) for e in self.es) + ')'
def __repr__(self) -> str:
return f'Or({self.es})'
def quorums(self) -> Iterator[Set[T]]:
for e in self.es:
yield from e.quorums()
def is_quorum(self, xs: Set[T]) -> bool:
return any(e.is_quorum(xs) for e in self.es)
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def nodes(self) -> Set[Node[T]]:
return set.union(*[e.nodes() for e in self.es])
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def dual(self) -> Expr:
return And([e.dual() for e in self.es])
def _num_leaves(self) -> int:
return sum(e._num_leaves() for e in self.es)
def _dup_free_min_failures(self) -> int:
return sum(e._dup_free_min_failures() for e in self.es)
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class And(Expr[T]):
def __init__(self, es: List[Expr[T]]) -> None:
if len(es) == 0:
raise ValueError(f'And cannot be constructed with an empty list')
self.es = es
def __str__(self) -> str:
return '(' + ' * '.join(str(e) for e in self.es) + ')'
def __repr__(self) -> str:
return f'And({self.es})'
def quorums(self) -> Iterator[Set[T]]:
for subquorums in itertools.product(*[e.quorums() for e in self.es]):
yield set.union(*subquorums)
def is_quorum(self, xs: Set[T]) -> bool:
return all(e.is_quorum(xs) for e in self.es)
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def nodes(self) -> Set[Node[T]]:
return set.union(*[e.nodes() for e in self.es])
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def dual(self) -> Expr:
return Or([e.dual() for e in self.es])
def _num_leaves(self) -> int:
return sum(e._num_leaves() for e in self.es)
def _dup_free_min_failures(self) -> int:
return min(e._dup_free_min_failures() for e in self.es)
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class Choose(Expr[T]):
def __init__(self, k: int, es: List[Expr[T]]) -> None:
if k <= 0 or k > len(es):
raise ValueError(f'k must be in the range [1, {len(es)}]')
self.k = k
self.es = es
def __str__(self) -> str:
return f'choose{self.k}(' + ', '.join(str(e) for e in self.es) + ')'
def __repr__(self) -> str:
return f'Chose({self.k}, {self.es})'
def quorums(self) -> Iterator[Set[T]]:
for combo in itertools.combinations(self.es, self.k):
for subquorums in itertools.product(*[e.quorums() for e in combo]):
yield set.union(*subquorums)
def is_quorum(self, xs: Set[T]) -> bool:
return sum(1 if e.is_quorum(xs) else 0 for e in self.es) >= self.k
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def nodes(self) -> Set[Node[T]]:
return set.union(*[e.nodes() for e in self.es])
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def dual(self) -> Expr:
# TODO(mwhittaker): Prove that this is in fact the dual.
return Choose(len(self.es) - self.k + 1, [e.dual() for e in self.es])
def _num_leaves(self) -> int:
return sum(e._num_leaves() for e in self.es)
def _dup_free_min_failures(self) -> int:
return sum(sorted(e._dup_free_min_failures() for e in self.es)[:self.k])
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def _and(lhs: Expr[T], rhs: Expr[T]) -> 'And[T]':
if isinstance(lhs, And) and isinstance(rhs, And):
return And(lhs.es + rhs.es)
elif isinstance(lhs, And):
return And(lhs.es + [rhs])
elif isinstance(rhs, And):
return And([lhs] + rhs.es)
else:
return And([lhs, rhs])
def _or(lhs: Expr[T], rhs: Expr[T]) -> 'Or[T]':
if isinstance(lhs, Or) and isinstance(rhs, Or):
return Or(lhs.es + rhs.es)
elif isinstance(lhs, Or):
return Or(lhs.es + [rhs])
elif isinstance(rhs, Or):
return Or([lhs] + rhs.es)
else:
return Or([lhs, rhs])
def choose(k: int, es: List[Expr[T]]) -> Expr[T]:
if k == 1:
return Or(es)
elif k == len(es):
return And(es)
else:
return Choose(k, es)
def majority(es: List[Expr[T]]) -> Expr[T]:
return choose(len(es) // 2 + 1, es)
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ReadFraction = float
ReadWriteFraction = float
Weight = float
Probability = float
Distribution = Union[
# For example, 1 means 100% reads.
int,
# For example, 0.25 means 25% reads.
float,
# For example, {0.25: 1, 0.8: 2} means 25% reads one third of the time and
# 80% reads two thirds of the time.
Dict[ReadWriteFraction, Weight],
]
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def _canonicalize_distribution(d: Distribution) \
-> Dict[ReadWriteFraction, Probability]:
if isinstance(d, int):
if d < 0 or d > 1:
raise ValueError('distribution must be in the range [0, 1]')
return {float(d): 1.}
elif isinstance(d, float):
if d < 0 or d > 1:
raise ValueError('distribution must be in the range [0, 1]')
return {d: 1.}
elif isinstance(d, dict):
if len(d) == 0:
raise ValueError('distribution cannot empty')
if any(weight < 0 for weight in d.values()):
raise ValueError('distribution cannot have negative weights')
total_weight = sum(d.values())
if total_weight == 0:
raise ValueError('distribution cannot have zero weight')
return {float(f): weight / total_weight
for (f, weight) in d.items()
if weight > 0}
else:
raise ValueError('distribution must be an int, a float, a Dict[float, '
'float] or a List[Tuple[float, float]]')
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def _canonicalize_rw_distribution(read_fraction: Optional[Distribution],
write_fraction: Optional[Distribution]) \
-> Dict[ReadFraction, Probability]:
if read_fraction is None and write_fraction is None:
raise ValueError('Either read_fraction or write_fraction must be given')
elif read_fraction is not None and write_fraction is not None:
raise ValueError('Only one of read_fraction or write_fraction can be '
'given')
elif read_fraction is not None:
return _canonicalize_distribution(read_fraction)
else:
assert write_fraction is not None
return {1 - f: weight
for (f, weight) in
_canonicalize_distribution(write_fraction).items()}
class QuorumSystem(Generic[T]):
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def __init__(self, reads: Optional[Expr[T]] = None,
writes: Optional[Expr[T]] = None) -> None:
if reads is not None and writes is not None:
# TODO(mwhittaker): Think of ways to make this more efficient.
assert all(len(r & w) > 0
for (r, w) in itertools.product(reads.quorums(),
writes.quorums()))
self.reads = reads
self.writes = writes
elif reads is not None and writes is None:
self.reads = reads
self.writes = reads.dual()
elif reads is None and writes is not None:
self.reads = writes.dual()
self.writes = writes
else:
raise ValueError('A QuorumSystem must be instantiated with a set '
'of read quorums or a set of write quorums')
def __repr__(self) -> str:
return f'QuorumSystem(reads={self.reads}, writes={self.writes})'
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def read_quorums(self) -> Iterator[Set[T]]:
return self.reads.quorums()
def write_quorums(self) -> Iterator[Set[T]]:
return self.writes.quorums()
def is_read_quorum(self, xs: Set[T]) -> bool:
return self.reads.is_quorum(xs)
def is_write_quorum(self, xs: Set[T]) -> bool:
return self.writes.is_quorum(xs)
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def nodes(self) -> Set[Node[T]]:
return self.reads.nodes() | self.writes.nodes()
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def resilience(self) -> int:
return min(self.read_resilience(), self.write_resilience())
def read_resilience(self) -> int:
return self.reads.resilience()
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def write_resilience(self) -> int:
return self.writes.resilience()
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def strategy(self,
read_fraction: Optional[Distribution] = None,
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write_fraction: Optional[Distribution] = None,
f: int = 0) \
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-> 'Strategy[T]':
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if f < 0:
raise ValueError('f must be >= 0')
d = _canonicalize_rw_distribution(read_fraction, write_fraction)
if f == 0:
return self._load_optimal_strategy(
list(self.read_quorums()),
list(self.write_quorums()),
d)
else:
xs = [node.x for node in self.nodes()]
read_quorums = list(self._f_resilient_quorums(f, xs, self.reads))
write_quorums = list(self._f_resilient_quorums(f, xs, self.reads))
if len(read_quorums) == 0:
raise ValueError(f'There are no {f}-resilient read quorums')
if len(write_quorums) == 0:
raise ValueError(f'There are no {f}-resilient write quorums')
return self._load_optimal_strategy(read_quorums, write_quorums, d)
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def dup_free(self) -> bool:
return self.reads.dup_free() and self.writes.dup_free()
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def _f_resilient_quorums(self,
f: int,
xs: List[T],
e: Expr) -> Iterator[Set[T]]:
assert f >= 1
def helper(s: Set[T], i: int) -> Iterator[Set[T]]:
if all(e.is_quorum(s - set(failure))
for failure in itertools.combinations(s, min(f, len(s)))):
yield set(s)
return
for j in range(i, len(xs)):
s.add(xs[j])
yield from helper(s, j + 1)
s.remove(xs[j])
return helper(set(), 0)
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def load(self,
read_fraction: Optional[Distribution] = None,
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write_fraction: Optional[Distribution] = None,
f: int = 0) \
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-> float:
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sigma = self.strategy(read_fraction, write_fraction, f)
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return sigma.load(read_fraction, write_fraction)
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def _load_optimal_strategy(self,
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read_quorums: List[Set[T]],
write_quorums: List[Set[T]],
read_fraction: Dict[float, float]) \
-> 'Strategy[T]':
# TODO(mwhittaker): Explain f_r calculation.
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fr = sum(f * weight for (f, weight) in read_fraction.items())
nodes = self.reads.nodes() | self.writes.nodes()
read_capacity = {node.x: node.read_capacity for node in nodes}
write_capacity = {node.x: node.write_capacity for node in nodes}
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read_quorum_vars: List[pulp.LpVariable] = []
x_to_read_quorum_vars: Dict[T, List[pulp.LpVariable]] = \
collections.defaultdict(list)
for (i, read_quorum) in enumerate(read_quorums):
v = pulp.LpVariable(f'r{i}', 0, 1)
read_quorum_vars.append(v)
for x in read_quorum:
x_to_read_quorum_vars[x].append(v)
write_quorum_vars: List[pulp.LpVariable] = []
x_to_write_quorum_vars: Dict[T, List[pulp.LpVariable]] = \
collections.defaultdict(list)
for (i, write_quorum) in enumerate(write_quorums):
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v = pulp.LpVariable(f'w{i}', 0, 1)
write_quorum_vars.append(v)
for x in write_quorum:
x_to_write_quorum_vars[x].append(v)
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# Form the linear program to find the load.
problem = pulp.LpProblem("load", pulp.LpMinimize)
# If we're trying to balance the strategy, then we want to minimize the
# pairwise absolute differences between the read probabilities and the
# write probabilities.
l = pulp.LpVariable('l', 0, 1)
problem += l
problem += (sum(read_quorum_vars) == 1, 'valid read strategy')
problem += (sum(write_quorum_vars) == 1, 'valid write strategy')
for node in nodes:
x = node.x
x_load: pulp.LpAffineExpression = 0
if x in x_to_read_quorum_vars:
x_load += fr * sum(x_to_read_quorum_vars[x]) / read_capacity[x]
if x in x_to_write_quorum_vars:
x_load += ((1 - fr) * sum(x_to_write_quorum_vars[x]) /
write_capacity[x])
problem += (x_load <= l, x)
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problem.solve(pulp.apis.PULP_CBC_CMD(msg=False))
return ExplicitStrategy(nodes,
read_quorums,
[v.varValue for v in read_quorum_vars],
write_quorums,
[v.varValue for v in write_quorum_vars])
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class Strategy(Generic[T]):
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def load(self,
read_fraction: Optional[Distribution] = None,
write_fraction: Optional[Distribution] = None) \
-> float:
raise NotImplementedError
def get_read_quorum(self) -> Set[T]:
raise NotImplementedError
def get_write_quorum(self) -> Set[T]:
raise NotImplementedError
class ExplicitStrategy(Strategy[T]):
def __init__(self,
nodes: Set[Node[T]],
reads: List[Set[T]],
read_weights: List[float],
writes: List[Set[T]],
write_weights: List[float]) -> None:
self.nodes = nodes
self.read_capacity = {node.x: node.read_capacity for node in nodes}
self.write_capacity = {node.x: node.write_capacity for node in nodes}
self.reads = reads
self.read_weights = read_weights
self.writes = writes
self.write_weights = write_weights
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def __str__(self) -> str:
non_zero_reads = {tuple(r): p
for (r, p) in zip(self.reads, self.read_weights)
if p > 0}
non_zero_writes = {tuple(w): p
for (w, p) in zip(self.writes, self.write_weights)
if p > 0}
return (f'ExplicitStrategy(reads={non_zero_reads}, ' +
f'writes={non_zero_writes})')
def __repr__(self) -> str:
return (f'ExplicitStrategy(nodes={self.nodes}, '+
f'reads={self.reads}, ' +
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f'read_weights={self.read_weights},' +
f'writes={self.writes}, ' +
f'write_weights={self.write_weights})')
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def load(self,
read_fraction: Optional[Distribution] = None,
write_fraction: Optional[Distribution] = None) \
-> float:
d = _canonicalize_rw_distribution(read_fraction, write_fraction)
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fr = sum(f * weight for (f, weight) in d.items())
read_load: Dict[T, float] = collections.defaultdict(float)
for (read_quorum, weight) in zip(self.reads, self.read_weights):
for x in read_quorum:
read_load[x] += weight
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write_load: Dict[T, float] = collections.defaultdict(float)
for (write_quorum, weight) in zip(self.writes, self.write_weights):
for x in write_quorum:
write_load[x] += weight
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loads: List[float] = []
for node in self.nodes:
x = node.x
load = 0.0
if x in read_load:
load += fr * read_load[x] / self.read_capacity[x]
if x in write_load:
load += (1 - fr) * write_load[x] / self.write_capacity[x]
loads.append(load)
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return max(loads)
# TODO(mwhittaker): Add read/write load and capacity and read/write cap.
def get_read_quorum(self) -> Set[T]:
return np.random.choice(self.reads, p=self.read_weights)
def get_write_quorum(self) -> Set[T]:
return np.random.choice(self.writes, p=self.write_weights)
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# a = Node('a')
# b = Node('b')
# c = Node('c')
# d = Node('d')
# e = Node('e')
# f = Node('f')
# g = Node('g')
# h = Node('h')
# i = Node('i')
#
# walls = QuorumSystem(reads=a*b + c*d*e)
# paths = QuorumSystem(reads=a*b + a*c*e + d*e + d*c*b)
# maj = QuorumSystem(reads=majority([a, b, c, d, e]))
#
# for qs in [walls, paths, maj]:
# print(qs.dup_free())
# print(qs.resilience())
# sigma_0 = qs.strategy(read_fraction=0.5)
# sigma_1 = qs.strategy(read_fraction=0.5, f=1)
# print(sigma_0.load(read_fraction=0.5), sigma_1.load(read_fraction=0.5))
# print(sigma_1)
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#
# qs = QuorumSystem(reads = a*b + a*c)
# print(list(qs.read_quorums()))
# sigma = qs.strategy(read_fraction=0.5)
# print(list(qs.write_quorums()))
# print(sigma)
# print(1 / sigma.load(read_fraction=0.5))
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# paths = QuorumSystem(reads=a*b + a*c*e + d*e + d*c*b)
# print(paths.resilience())
# sigma = paths.strategy(read_fraction=0.5)
# print(sigma.load(read_fraction=0.5))
#
# walls = QuorumSystem(reads=a*b + c*d*e)
# print(walls.resilience())
# sigma = walls.strategy(read_fraction=0.5)
# print(sigma.load(read_fraction=0.5))
# wpaxos = QuorumSystem(reads=majority([majority([a, b, c]),
# majority([d, e, f]),
# majority([g, h, i])]))
# sigma_1 = wpaxos.strategy(read_fraction=0.1)
# sigma_5 = wpaxos.strategy(read_fraction=0.5)
# sigma_9 = wpaxos.strategy(read_fraction=0.9)
# sigma_even = wpaxos.strategy(read_fraction={0.1: 2, 0.5: 2, 0.9: 1})
# for sigma in [sigma_1, sigma_5, sigma_9, sigma_even]:
# frs = [0.1, 0.5, 0.9, {0.1: 2, 0.5: 2, 0.9: 1}]
# print([sigma.load(fr) for fr in frs])
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# - num_quorums
# - has dups?
# - optimal schedule
# - independent schedule
# - node read and write throughputs