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