File size: 12,345 Bytes
6fbb45f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 | """Training-free potential-field routing on the graph Laplacian, and the classical baselines.
For flow *f* with source *s* and sink *t* the potential φ solves the discrete Poisson equation
L_g φ = b, L = D − W, w_ij = capacity_ij / latency_ij (live links only)
where L_g is the Laplacian with the sink's row and column removed, i.e. the Dirichlet boundary
condition φ_t = 0: the sink is the grounded, attractive well of the field. The right-hand side
injects a unit current at the source, a small positive background at every node and a repulsive
current proportional to each node's buffer occupancy.
The grounded inverse is available in closed form from the Laplacian pseudo-inverse L⁺:
(L_g⁻¹)_ij = L⁺_ij − L⁺_it − L⁺_tj + L⁺_tt. Because the background and congestion injections are
the same for every flow, the fields of all F flows follow from one matrix–vector product with L⁺
plus O(N) work per flow, so L⁺ is formed once per topology change (dense, N ≤ 375) and every step
costs O(N·F) instead of F sparse solves. ``grounded_solve`` keeps the sparse SuperLU reference
solution for validation.
Packets follow the routing gradient. On every live directed link the current is
I_ij = w_ij (φ_i − φ_j); the *steepest* rule forwards along the largest current out of a node and
the *split* rule sprays packets over the outgoing links in proportion to their positive currents,
exactly how electrical current divides. Because b_i > 0 at every non-sink node, Σ_j I_ij = b_i > 0,
so at least one current is positive and every positive-current link leads strictly downhill:
both rules are loop-free and reach the sink in at most N − 1 hops for any congestion pattern.
The baselines — static shortest path, equal-cost multipath and queue-aware adaptive shortest
path — share one Dijkstra helper (``scipy.sparse.csgraph.dijkstra`` on the reversed graph, whose
predecessor tree is exactly the next-hop table towards each sink).
"""
from __future__ import annotations
from dataclasses import dataclass
import numpy as np
import scipy.sparse as sp
from scipy.sparse.csgraph import dijkstra, laplacian
from scipy.sparse.linalg import splu
GOLDEN = 0.6180339887498949 # low-discrepancy per-packet coordinate for multipath spraying
@dataclass(frozen=True)
class LiveGraph:
"""Directed view of the links with positive capacity at one point in time, sorted by tail node."""
n: int
n_edges: int # E of the base topology
src: np.ndarray # (M,) int32 tail of every live directed link
dst: np.ndarray # (M,) int32 head
base: np.ndarray # (M,) int32 index into the 2E directed base links: edge + E * direction
capacity: np.ndarray # (M,) int32
latency: np.ndarray # (M,) int32
conductance: np.ndarray # (M,) float64
indptr: np.ndarray # (n + 1,) segment boundaries of each tail node
dir_edge: np.ndarray # (n, n) int32 live directed link index, −1 where there is none
@property
def degree(self) -> np.ndarray:
return np.diff(self.indptr)
@property
def max_degree(self) -> int:
return int(self.degree.max())
def laplacian(self) -> np.ndarray:
weights = sp.csr_matrix((self.conductance, (self.src, self.dst)), shape=(self.n, self.n))
return laplacian(weights).toarray()
def live_graph(n: int, edges: np.ndarray, capacity: np.ndarray, latency: np.ndarray) -> LiveGraph:
n_edges = len(edges)
live = np.flatnonzero(capacity > 0)
u, v = edges[live, 0].astype(np.int32), edges[live, 1].astype(np.int32)
src = np.concatenate([u, v])
dst = np.concatenate([v, u])
base = np.concatenate([live, live + n_edges]).astype(np.int32)
order = np.lexsort((dst, src))
src, dst, base = src[order], dst[order], base[order]
cap = capacity[base % n_edges].astype(np.int32)
lat = latency[base % n_edges].astype(np.int32)
indptr = np.searchsorted(src, np.arange(n + 1))
dir_edge = np.full((n, n), -1, np.int32)
dir_edge[src, dst] = np.arange(len(src))
return LiveGraph(n, n_edges, src, dst, base, cap, lat, cap / lat, indptr, dir_edge)
class PotentialField:
"""Grounded-Laplacian Green's functions of every flow, from one dense pseudo-inverse."""
def __init__(self, g: LiveGraph, sources: np.ndarray, sinks: np.ndarray, source_injection: float):
n = g.n
pinv = np.linalg.inv(g.laplacian() + 1.0 / n) - 1.0 / n # L⁺ = (L + J/N)⁻¹ − J/N
self.pinv = 0.5 * (pinv + pinv.T) # exactly symmetric, like L⁺ itself
self.sinks = np.asarray(sinks, np.intp)
self.flow_ids = np.arange(len(self.sinks))
s, t = np.asarray(sources, np.intp), self.sinks
self.row_t = self.pinv[t] # (F, N) rows L⁺_t·
self.diag_t = self.pinv[t, t] # (F,)
# Response to the unit source injection, G^(t) e_s, zero at the sink by construction.
self.source_term = source_injection * (self.pinv[s] - self.row_t
- self.pinv[t, s][:, None] + self.diag_t[:, None])
def solve(self, injection: np.ndarray) -> np.ndarray:
"""Per-node injection shared by all flows (N,) → potentials φ (F, N) with φ[f, sink_f] = 0."""
p = self.pinv @ injection
total = injection.sum()
# (G^(t) c)_i = p_i − L⁺_it·Σc − (p_t − L⁺_tt·Σc); the injection at the grounded sink cancels.
phi = p[None, :] - self.row_t * total - (p[self.sinks] - self.diag_t * total)[:, None] + self.source_term
phi[self.flow_ids, self.sinks] = 0.0 # exact ground, free of rounding residue
return phi
def grounded_solve(g: LiveGraph, sink: int, b: np.ndarray) -> np.ndarray:
"""Reference sparse solution of L_g φ = b for one sink (SuperLU); used for validation."""
keep = np.delete(np.arange(g.n), sink)
lap = sp.csr_matrix(g.laplacian())
phi = np.zeros(g.n)
phi[keep] = splu(lap[keep][:, keep].tocsc()).solve(b[keep])
return phi
def currents(phi: np.ndarray, g: LiveGraph) -> np.ndarray:
"""I_ij = w_ij (φ_i − φ_j) on every live directed link, shape (F, M)."""
return (np.take(phi, g.src, axis=1) - np.take(phi, g.dst, axis=1)) * g.conductance
TIE_TOLERANCE = 1e-9 # currents within this relative margin of a node's largest current count as tied
def steepest_next_hops(cur: np.ndarray, g: LiveGraph) -> np.ndarray:
"""Next hop per (flow, node): the link carrying the largest current; −1 if none is positive.
Symmetric topologies produce mathematically equal currents on parallel links; ties are resolved
towards the first link in tail-node order within a relative tolerance far above rounding noise,
so the choice does not depend on the last bits of the linear algebra on a given platform.
"""
starts = g.indptr[:-1]
best = np.maximum.reduceat(cur, starts, axis=1) # (F, N)
tied = cur >= np.take(best, g.src, axis=1) * (1.0 - TIE_TOLERANCE)
first = np.where(tied, np.arange(len(g.src)), len(g.src))
k = np.minimum.reduceat(first, starts, axis=1) # (F, N)
return np.where(best > 0, g.dst[np.minimum(k, len(g.src) - 1)], -1).astype(np.int16)
def spray_next_hops(cur: np.ndarray, g: LiveGraph, node: np.ndarray, flow: np.ndarray,
packet_id: np.ndarray) -> np.ndarray:
"""Per-packet next hop drawn in proportion to the positive currents leaving the packet's node.
Packets are grouped by (flow, node); each group's outgoing shares form a cumulative
distribution, all groups are laid out as one monotone array (group index doubled plus CDF),
and every packet's low-discrepancy coordinate is placed with a single ``searchsorted``.
"""
key = flow.astype(np.int64) * g.n + node
order = np.argsort(key, kind="stable")
sorted_key = key[order]
starts = np.flatnonzero(np.r_[True, sorted_key[1:] != sorted_key[:-1]])
group = np.repeat(np.arange(len(starts)), np.diff(np.r_[starts, len(order)]))
g_flow, g_node = flow[order][starts].astype(np.int64), node[order][starts].astype(np.int64)
degree = g.degree[g_node]
width = int(degree.max())
slot = np.arange(width)
valid = slot[None, :] < degree[:, None] # (G, width)
link = np.minimum(g.indptr[g_node][:, None] + slot[None, :], len(g.src) - 1)
share = np.where(valid, np.maximum(cur[g_flow[:, None], link], 0.0), 0.0)
cdf = np.cumsum(share, axis=1)
total = cdf[:, -1]
ok = total > 0
cdf = np.where(valid & ok[:, None], cdf / np.where(ok, total, 1.0)[:, None], 1.0) # exact 1.0 at the last link
augmented = (cdf + 2.0 * np.arange(len(starts))[:, None]).ravel()
u = (packet_id[order] * GOLDEN) % 1.0
pick = np.searchsorted(augmented, 2.0 * group + u, side="right") - group * width
hop = np.where(ok[group], g.dst[g.indptr[g_node[group]] + pick], -1)
out = np.empty(len(order), np.int16)
out[order] = hop
return out
COST_QUANTUM = 1e-6 # adaptive link costs are rounded to this many steps so that path sums are exact integers
def dijkstra_next_hops(cost: np.ndarray, g: LiveGraph, sinks: np.ndarray):
"""Shortest paths to each sink under per-directed-link costs: (next_hop (D, N), dist (D, N)).
Costs must be integer-valued floats (latencies, or quantised adaptive costs) so that every
path sum is exact. Distances then do not depend on the solver's tie-breaking, and the next hop
is derived from them here — the first link, in tail-node order, that lies on a shortest path —
which keeps the tables identical across SciPy versions and platforms.
"""
reverse = sp.csr_matrix((cost, (g.dst, g.src)), shape=(g.n, g.n)) # reverse[j, i] = cost(i → j)
dist = dijkstra(reverse, directed=True, indices=np.asarray(sinks, np.intp))
starts = g.indptr[:-1]
through = np.take(dist, g.dst, axis=1) + cost # (D, M)
first = np.where(through == np.take(dist, g.src, axis=1), np.arange(len(g.src)), len(g.src))
k = np.minimum.reduceat(first, starts, axis=1) # (D, N)
next_hop = g.dst[np.minimum(k, len(g.src) - 1)].astype(np.int16)
next_hop[np.arange(len(sinks)), np.asarray(sinks, np.intp)] = -1
return next_hop, dist
class EcmpTable:
"""All equal-latency next hops per (sink, node); packets are sprayed round-robin over them."""
def __init__(self, dist: np.ndarray, g: LiveGraph):
starts = g.indptr[:-1]
equal = np.take(dist, g.dst, axis=1) + g.latency == np.take(dist, g.src, axis=1) # (D, M)
self.count = np.add.reduceat(equal.astype(np.int32), starts, axis=1) # (D, N)
csum = np.cumsum(equal, axis=1)
pos = csum - np.take(csum[:, starts] - equal[:, starts], g.src, axis=1) - 1
self.table = np.full((dist.shape[0], g.n, g.max_degree), -1, np.int16)
rows, cols = np.nonzero(equal)
self.table[rows, g.src[cols], pos[rows, cols]] = g.dst[cols]
def hops(self, sink_index: np.ndarray, node: np.ndarray, packet_id: np.ndarray) -> np.ndarray:
count = self.count[sink_index, node]
return self.table[sink_index, node, packet_id % np.maximum(count, 1)] # −1 where count is 0
def path_metrics(g: LiveGraph, sources: np.ndarray, sinks: np.ndarray):
"""Minimum hop count and minimum latency from every flow's source to its sink."""
origins, index = np.unique(np.asarray(sources, np.intp), return_inverse=True)
hops = dijkstra(sp.csr_matrix((np.ones(len(g.src)), (g.src, g.dst)), shape=(g.n, g.n)),
directed=True, indices=origins)
lat = dijkstra(sp.csr_matrix((g.latency.astype(np.float64), (g.src, g.dst)), shape=(g.n, g.n)),
directed=True, indices=origins)
return hops[index, sinks].astype(np.int16), lat[index, sinks].astype(np.int16)
|