def test_residual_0(self):
graph = happiness_upload._servermap_flow_graph(['peer0'], ['share0'],
servermap={
'peer0': ['share0'],
})
flow = [[0 for _ in graph] for _ in graph]
residual, capacity = happiness_upload.residual_network(graph, flow)
# XXX no idea if these are right; hand-verify
self.assertEqual(residual, [[1], [2], [3], []])
self.assertEqual(
capacity,
[[0, 1, 0, 0], [-1, 0, 1, 0], [0, -1, 0, 1], [0, 0, -1, 0]])
def test_residual_0(self):
graph = happiness_upload._servermap_flow_graph(
['peer0'],
['share0'],
servermap={
'peer0': ['share0'],
}
)
flow = [[0 for _ in graph] for _ in graph]
residual, capacity = happiness_upload.residual_network(graph, flow)
# XXX no idea if these are right; hand-verify
self.assertEqual(residual, [[1], [2], [3], []])
self.assertEqual(capacity, [[0, 1, 0, 0], [-1, 0, 1, 0], [0, -1, 0, 1], [0, 0, -1, 0]])
def servers_of_happiness(sharemap):
"""
I accept 'sharemap', a dict of shareid -> set(peerid) mappings. I
return the 'servers_of_happiness' number that sharemap results in.
To calculate the 'servers_of_happiness' number for the sharemap, I
construct a bipartite graph with servers in one partition of vertices
and shares in the other, and with an edge between a server s and a share t
if s is to store t. I then compute the size of a maximum matching in
the resulting graph; this is then returned as the 'servers_of_happiness'
for my arguments.
For example, consider the following layout:
server 1: shares 1, 2, 3, 4
server 2: share 6
server 3: share 3
server 4: share 4
server 5: share 2
From this, we can construct the following graph:
L = {server 1, server 2, server 3, server 4, server 5}
R = {share 1, share 2, share 3, share 4, share 6}
V = L U R
E = {(server 1, share 1), (server 1, share 2), (server 1, share 3),
(server 1, share 4), (server 2, share 6), (server 3, share 3),
(server 4, share 4), (server 5, share 2)}
G = (V, E)
Note that G is bipartite since every edge in e has one endpoint in L
and one endpoint in R.
A matching in a graph G is a subset M of E such that, for any vertex
v in V, v is incident to at most one edge of M. A maximum matching
in G is a matching that is no smaller than any other matching. For
this graph, a matching of cardinality 5 is:
M = {(server 1, share 1), (server 2, share 6),
(server 3, share 3), (server 4, share 4),
(server 5, share 2)}
Since G is bipartite, and since |L| = 5, we cannot have an M' such
that |M'| > |M|. Then M is a maximum matching in G. Intuitively, and
as long as k <= 5, we can see that the layout above has
servers_of_happiness = 5, which matches the results here.
"""
if sharemap == {}:
return 0
servermap = shares_by_server(sharemap)
graph = _flow_network_for(servermap)
# XXX this core stuff is identical to
# happiness_upload._compute_maximum_graph and we should find a way
# to share the code.
# This is an implementation of the Ford-Fulkerson method for finding
# a maximum flow in a flow network applied to a bipartite graph.
# Specifically, it is the Edmonds-Karp algorithm, since it uses a
# BFS to find the shortest augmenting path at each iteration, if one
# exists.
#
# The implementation here is an adapation of an algorithm described in
# "Introduction to Algorithms", Cormen et al, 2nd ed., pp 658-662.
dim = len(graph)
flow_function = [[0 for sh in xrange(dim)] for s in xrange(dim)]
residual_graph, residual_function = residual_network(graph, flow_function)
while augmenting_path_for(residual_graph):
path = augmenting_path_for(residual_graph)
# Delta is the largest amount that we can increase flow across
# all of the edges in path. Because of the way that the residual
# function is constructed, f[u][v] for a particular edge (u, v)
# is the amount of unused capacity on that edge. Taking the
# minimum of a list of those values for each edge in the
# augmenting path gives us our delta.
delta = min(residual_function[u][v] for (u, v) in path)
for (u, v) in path:
flow_function[u][v] += delta
flow_function[v][u] -= delta
residual_graph, residual_function = residual_network(graph,
flow_function)
num_servers = len(servermap)
# The value of a flow is the total flow out of the source vertex
# (vertex 0, in our graph). We could just as well sum across all of
# f[0], but we know that vertex 0 only has edges to the servers in
# our graph, so we can stop after summing flow across those. The
# value of a flow computed in this way is the size of a maximum
# matching on the bipartite graph described above.
return sum([flow_function[0][v] for v in xrange(1, num_servers+1)])
def servers_of_happiness(sharemap):
"""
I accept 'sharemap', a dict of shareid -> set(peerid) mappings. I
return the 'servers_of_happiness' number that sharemap results in.
To calculate the 'servers_of_happiness' number for the sharemap, I
construct a bipartite graph with servers in one partition of vertices
and shares in the other, and with an edge between a server s and a share t
if s is to store t. I then compute the size of a maximum matching in
the resulting graph; this is then returned as the 'servers_of_happiness'
for my arguments.
For example, consider the following layout:
server 1: shares 1, 2, 3, 4
server 2: share 6
server 3: share 3
server 4: share 4
server 5: share 2
From this, we can construct the following graph:
L = {server 1, server 2, server 3, server 4, server 5}
R = {share 1, share 2, share 3, share 4, share 6}
V = L U R
E = {(server 1, share 1), (server 1, share 2), (server 1, share 3),
(server 1, share 4), (server 2, share 6), (server 3, share 3),
(server 4, share 4), (server 5, share 2)}
G = (V, E)
Note that G is bipartite since every edge in e has one endpoint in L
and one endpoint in R.
A matching in a graph G is a subset M of E such that, for any vertex
v in V, v is incident to at most one edge of M. A maximum matching
in G is a matching that is no smaller than any other matching. For
this graph, a matching of cardinality 5 is:
M = {(server 1, share 1), (server 2, share 6),
(server 3, share 3), (server 4, share 4),
(server 5, share 2)}
Since G is bipartite, and since |L| = 5, we cannot have an M' such
that |M'| > |M|. Then M is a maximum matching in G. Intuitively, and
as long as k <= 5, we can see that the layout above has
servers_of_happiness = 5, which matches the results here.
"""
if sharemap == {}:
return 0
servermap = shares_by_server(sharemap)
graph = _flow_network_for(servermap)
# XXX this core stuff is identical to
# happiness_upload._compute_maximum_graph and we should find a way
# to share the code.
# This is an implementation of the Ford-Fulkerson method for finding
# a maximum flow in a flow network applied to a bipartite graph.
# Specifically, it is the Edmonds-Karp algorithm, since it uses a
# BFS to find the shortest augmenting path at each iteration, if one
# exists.
#
# The implementation here is an adapation of an algorithm described in
# "Introduction to Algorithms", Cormen et al, 2nd ed., pp 658-662.
dim = len(graph)
flow_function = [[0 for sh in range(dim)] for s in range(dim)]
residual_graph, residual_function = residual_network(graph, flow_function)
while augmenting_path_for(residual_graph):
path = augmenting_path_for(residual_graph)
# Delta is the largest amount that we can increase flow across
# all of the edges in path. Because of the way that the residual
# function is constructed, f[u][v] for a particular edge (u, v)
# is the amount of unused capacity on that edge. Taking the
# minimum of a list of those values for each edge in the
# augmenting path gives us our delta.
delta = min(residual_function[u][v] for (u, v) in path)
for (u, v) in path:
flow_function[u][v] += delta
flow_function[v][u] -= delta
residual_graph, residual_function = residual_network(
graph, flow_function)
num_servers = len(servermap)
# The value of a flow is the total flow out of the source vertex
# (vertex 0, in our graph). We could just as well sum across all of
# f[0], but we know that vertex 0 only has edges to the servers in
# our graph, so we can stop after summing flow across those. The
# value of a flow computed in this way is the size of a maximum
# matching on the bipartite graph described above.
return sum([flow_function[0][v] for v in range(1, num_servers + 1)])