mirror of
https://github.com/allaunthefox/Research-Stack.git
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7884fd074b
Lean: - OptimizedRoute.lean: 2-opt route shorter than exactishRoute optimizedRoute cost: 345147 vs exactishRoute: 401666 (14.1% shorter) Proofs: optimizedRoute_length, optimizedRoute_shorter, costSavings_positive All via native_decide. lake build: 3571 jobs, 0 errors. Python: - scale_space_solver.py: replaced Gaussian cost smoothing with cluster-based multi-scale optimization. Single-linkage clustering at each sigma, reduced TSP on representatives, expand + 2-opt polish. Fixed voltage/scale mapping.
643 lines
20 KiB
Python
643 lines
20 KiB
Python
"""
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Multi-scale optimization using scale space theory.
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Implements coarse-to-fine optimization via cluster-based route optimization
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at multiple scales, with Q16.16 fixed-point arithmetic for FPGA compatibility.
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Scale mapping:
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σ₃ (1.0): coarse — merge nearby nodes, solve small problem
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σ₂ (0.75): medium — tighter clustering
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σ₁ (0.5): fine — minimal clustering, warm-started
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σ₀ (0.25): formal verification target — full problem, 2-opt polish
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The Gaussian kernel is used for route-space smoothing, NOT cost matrix
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smoothing. At each scale σ, nodes whose pairwise cost is below σ·max_cost
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are clustered together. The reduced problem is solved, then expanded back.
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"""
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import math
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from typing import Optional
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try:
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import numpy as np
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HAS_NUMPY = True
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except ImportError:
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HAS_NUMPY = False
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# ---------------------------------------------------------------------------
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# Q16.16 fixed-point arithmetic
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# ---------------------------------------------------------------------------
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Q16_SCALE = 65536 # 2^16
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Q16_MAX = 2147483647 # 2^31 - 1
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Q16_MIN = -2147483648 # -2^31
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def q16_clamp(v: int) -> int:
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"""Clamp integer to Q16.16 representable range."""
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return max(Q16_MIN, min(Q16_MAX, v))
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def q16_from_float(f: float) -> int:
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"""Convert float to Q16.16 fixed-point."""
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return q16_clamp(round(f * Q16_SCALE))
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def q16_to_float(q: int) -> float:
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"""Convert Q16.16 fixed-point to float."""
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return q / Q16_SCALE
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def q16_multiply(a: int, b: int) -> int:
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"""Q16.16 multiplication: (a * b) >> 16."""
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return q16_clamp((a * b) >> 16)
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def q16_exp(x_q16: int) -> int:
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"""Q16.16 exponential: exp(x) where x is in Q16.16.
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Uses Python math.exp internally, converts to Q16.16.
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For FPGA, this would use a LUT-based approximation.
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"""
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x_float = q16_to_float(x_q16)
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return q16_from_float(math.exp(x_float))
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# ---------------------------------------------------------------------------
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# Gaussian kernel in Q16.16
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# ---------------------------------------------------------------------------
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def gaussian_kernel_q16(sigma: float, size: int = 256) -> list[int]:
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"""Compute 1D Gaussian kernel in Q16.16 fixed-point.
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G(x) = exp(-x²/(2σ²)) * 65536 (Q16.16 scale)
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Args:
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sigma: Standard deviation of the Gaussian (in normalized coords).
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size: Number of kernel taps. Must be odd for symmetry.
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Returns:
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List of Q16.16 kernel values, normalized so they sum to Q16_SCALE.
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"""
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if size % 2 == 0:
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size += 1 # Ensure odd for symmetry
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half = size // 2
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two_sigma_sq = 2.0 * sigma * sigma
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# Compute unnormalized kernel
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kernel_raw = []
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for i in range(size):
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x = (i - half) / half # Map to [-1, 1]
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g = math.exp(-(x * x) / two_sigma_sq)
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kernel_raw.append(q16_from_float(g))
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# Normalize so kernel sums to Q16_SCALE (1.0 in Q16.16)
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raw_sum = sum(kernel_raw)
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if raw_sum == 0:
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# Degenerate: delta function
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kernel_raw[half] = Q16_SCALE
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else:
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# Scale to sum to 65536
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kernel_raw = [q16_clamp(round(v * Q16_SCALE / raw_sum))
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for v in kernel_raw]
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return kernel_raw
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def gaussian_kernel_2d_q16(sigma: float, size: int = 16) -> list[list[int]]:
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"""Compute 2D Gaussian kernel in Q16.16.
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Args:
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sigma: Standard deviation.
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size: Kernel dimension (size x size).
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Returns:
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2D list of Q16.16 values.
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"""
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half = size // 2
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two_sigma_sq = 2.0 * sigma * sigma
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kernel = []
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total = 0
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for y in range(size):
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row = []
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for x in range(size):
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dx = (x - half) / half
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dy = (y - half) / half
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g = math.exp(-(dx * dx + dy * dy) / two_sigma_sq)
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v = q16_from_float(g)
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row.append(v)
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total += v
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kernel.append(row)
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# Normalize
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if total > 0:
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kernel = [[q16_clamp(round(v * Q16_SCALE / total))
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for v in row] for row in kernel]
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return kernel
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# ---------------------------------------------------------------------------
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# Voltage ↔ scale mapping
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# ---------------------------------------------------------------------------
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# Voltage range: 0.6V → σ=1.0 (coarse), 1.2V → σ=0.25 (fine)
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_VOLTAGE_MIN = 0.6
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_VOLTAGE_MAX = 1.2
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_SIGMA_AT_VMIN = 1.0
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_SIGMA_AT_VMAX = 0.25 # Matches finest scale in default sigmas
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def voltage_to_scale(voltage_mv: float) -> float:
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"""Map millivolt voltage to scale parameter σ.
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Range: 0.6V (600mV, σ=1.0) to 1.2V (1200mV, σ=0.25).
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Linear mapping: σ = 1.0 - (V - 0.6) / 0.6 * 0.75
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Args:
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voltage_mv: Voltage in millivolts.
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Returns:
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Scale parameter σ.
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"""
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voltage_v = voltage_mv / 1000.0
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# Clamp to range
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voltage_v = max(_VOLTAGE_MIN, min(_VOLTAGE_MAX, voltage_v))
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# Linear interpolation: σ = 1.0 - (V - 0.6) / 0.6 * 0.75
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t = (voltage_v - _VOLTAGE_MIN) / (_VOLTAGE_MAX - _VOLTAGE_MIN)
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sigma = _SIGMA_AT_VMIN + t * (_SIGMA_AT_VMAX - _SIGMA_AT_VMIN)
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return max(0.01, sigma)
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def scale_to_voltage(sigma: float) -> float:
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"""Map scale parameter σ to millivolt voltage.
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Args:
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sigma: Scale parameter.
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Returns:
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Voltage in millivolts.
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"""
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sigma = max(_SIGMA_AT_VMAX, min(_SIGMA_AT_VMIN, sigma))
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# Inverse of voltage_to_scale
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t = (sigma - _SIGMA_AT_VMIN) / (_SIGMA_AT_VMAX - _SIGMA_AT_VMIN)
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voltage_v = _VOLTAGE_MIN + t * (_VOLTAGE_MAX - _VOLTAGE_MIN)
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return voltage_v * 1000.0 # Return in mV
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# ---------------------------------------------------------------------------
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# Cluster-based multi-scale solver
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# ---------------------------------------------------------------------------
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def _single_linkage_clusters(cost_matrix: list[list[float]],
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threshold: float) -> list[list[int]]:
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"""Cluster nodes using single-linkage clustering.
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Merge nodes whose minimum pairwise cost is below the threshold.
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Args:
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cost_matrix: n×n cost matrix.
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threshold: Cost threshold for merging.
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Returns:
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List of clusters, where each cluster is a list of node indices.
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"""
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n = len(cost_matrix)
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if n == 0:
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return []
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# Union-find for single-linkage
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parent = list(range(n))
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def find(x: int) -> int:
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while parent[x] != x:
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parent[x] = parent[parent[x]]
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x = parent[x]
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return x
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def union(x: int, y: int) -> None:
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rx, ry = find(x), find(y)
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if rx != ry:
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parent[rx] = ry
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# Merge pairs whose cost is below threshold
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for i in range(n):
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for j in range(i + 1, n):
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if cost_matrix[i][j] < threshold:
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union(i, j)
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# Group nodes by cluster root
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clusters: dict[int, list[int]] = {}
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for i in range(n):
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root = find(i)
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if root not in clusters:
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clusters[root] = []
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clusters[root].append(i)
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return list(clusters.values())
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def _build_reduced_cost_matrix(cost_matrix: list[list[float]],
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clusters: list[list[int]]) -> list[list[float]]:
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"""Build a reduced cost matrix for cluster representatives.
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The cost between two clusters is the minimum cost between any pair of
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nodes across the two clusters.
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Args:
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cost_matrix: Original n×n cost matrix.
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clusters: List of clusters (each a list of node indices).
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Returns:
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Reduced k×k cost matrix where k = number of clusters.
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"""
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k = len(clusters)
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reduced = [[0.0] * k for _ in range(k)]
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for ci in range(k):
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for cj in range(ci + 1, k):
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# Minimum cost across cluster boundaries
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min_cost = float('inf')
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for ni in clusters[ci]:
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for nj in clusters[cj]:
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if cost_matrix[ni][nj] < min_cost:
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min_cost = cost_matrix[ni][nj]
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reduced[ci][cj] = min_cost
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reduced[cj][ci] = min_cost
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return reduced
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def _expand_tour(cluster_tour: list[int],
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clusters: list[list[int]],
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cost_matrix: list[list[float]]) -> list[int]:
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"""Expand a cluster-level tour back to individual nodes.
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For each cluster in the tour, we need to enter and exit through specific
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nodes. We pick the entry/exit nodes that minimize the inter-cluster edges.
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Args:
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cluster_tour: Tour over cluster indices.
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clusters: List of clusters (each a list of node indices).
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cost_matrix: Original cost matrix.
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Returns:
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Tour over original node indices.
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"""
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if len(cluster_tour) <= 1:
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# Single cluster — order nodes greedily within cluster
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nodes = clusters[cluster_tour[0]]
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if len(nodes) <= 1:
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return nodes
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return _greedy_tour_subgraph(nodes, cost_matrix)[0]
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# For each consecutive pair of clusters, find the best entry/exit nodes
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k = len(cluster_tour)
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entry_node = [0] * k # Which node in cluster i we enter through
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exit_node = [0] * k # Which node in cluster i we exit through
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for i in range(k):
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ci = cluster_tour[i]
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cj = cluster_tour[(i + 1) % k]
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# Find the pair of nodes (one in ci, one in cj) with minimum cost
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best_cost = float('inf')
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best_exit = clusters[ci][0]
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best_entry = clusters[cj][0]
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for ni in clusters[ci]:
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for nj in clusters[cj]:
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c = cost_matrix[ni][nj]
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if c < best_cost:
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best_cost = c
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best_exit = ni
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best_entry = nj
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exit_node[i] = best_exit
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entry_node[(i + 1) % k] = best_entry
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# Build the full tour by visiting each cluster's nodes between entry/exit
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full_tour = []
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for i in range(k):
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ci = cluster_tour[i]
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cluster_nodes = clusters[ci]
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entry = entry_node[i]
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exit_nd = exit_node[i]
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if len(cluster_nodes) == 1:
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full_tour.append(cluster_nodes[0])
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else:
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# Build a path through the cluster from entry to exit
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# Use greedy nearest-neighbor within the cluster, starting at entry
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path = _build_cluster_path(cluster_nodes, entry, exit_nd, cost_matrix)
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full_tour.extend(path)
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return full_tour
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def _build_cluster_path(nodes: list[int], start: int, end: int,
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cost_matrix: list[list[float]]) -> list[int]:
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"""Build a path through cluster nodes from start to end.
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Uses nearest-neighbor heuristic constrained to the cluster.
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"""
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if len(nodes) <= 2:
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# Just return all nodes, start first
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result = [start]
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for n in nodes:
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if n != start:
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result.append(n)
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return result
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visited = {start}
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path = [start]
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current = start
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# Visit all nodes except the end node
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remaining = set(nodes) - {start, end}
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while remaining:
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best_next: int = nodes[0] # will be overwritten
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best_cost = float('inf')
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for n in remaining:
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c = cost_matrix[current][n]
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if c < best_cost:
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best_cost = c
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best_next = n
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path.append(best_next)
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visited.add(best_next)
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remaining.remove(best_next)
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current = best_next
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# End at the exit node
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if end != start:
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path.append(end)
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return path
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def _greedy_tour_subgraph(nodes: list[int],
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cost_matrix: list[list[float]]) -> tuple[list[int], float]:
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"""Greedy nearest-neighbor tour on a subset of nodes."""
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if len(nodes) <= 1:
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return nodes, 0.0
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visited = {nodes[0]}
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tour = [nodes[0]]
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current = nodes[0]
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total_cost = 0.0
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for _ in range(len(nodes) - 1):
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best_j = -1
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best_c = float('inf')
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for j in nodes:
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if j not in visited and cost_matrix[current][j] < best_c:
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best_c = cost_matrix[current][j]
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best_j = j
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tour.append(best_j)
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visited.add(best_j)
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total_cost += best_c
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current = best_j
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total_cost += cost_matrix[current][tour[0]]
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return tour, total_cost
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def _greedy_tour(cost_matrix: list[list[float]]) -> tuple[list[int], float]:
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"""Nearest-neighbor heuristic for TSP."""
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n = len(cost_matrix)
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if n == 0:
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return [], 0.0
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visited = {0}
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tour = [0]
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current = 0
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total_cost = 0.0
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while len(tour) < n:
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best_j = -1
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best_c = float('inf')
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for j in range(n):
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if j not in visited and cost_matrix[current][j] < best_c:
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best_c = cost_matrix[current][j]
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best_j = j
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tour.append(best_j)
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visited.add(best_j)
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total_cost += best_c
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current = best_j
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total_cost += cost_matrix[current][tour[0]]
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return tour, total_cost
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def _2opt_improve(tour: list[int],
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cost_matrix: list[list[float]]) -> tuple[list[int], float]:
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"""2-opt local search improvement."""
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n = len(tour)
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if n < 4:
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cost = sum(cost_matrix[tour[i]][tour[(i + 1) % n]] for i in range(n))
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return tour, cost
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improved = True
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while improved:
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improved = False
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for i in range(1, n - 1):
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for j in range(i + 1, n):
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# Cost of current edges
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d1 = (cost_matrix[tour[i - 1]][tour[i]] +
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cost_matrix[tour[j]][tour[(j + 1) % n]])
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# Cost of reversed segment edges
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d2 = (cost_matrix[tour[i - 1]][tour[j]] +
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cost_matrix[tour[i]][tour[(j + 1) % n]])
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if d2 < d1 - 1e-10:
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tour[i:j + 1] = reversed(tour[i:j + 1])
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improved = True
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cost = sum(cost_matrix[tour[i]][tour[(i + 1) % n]] for i in range(n))
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return tour, cost
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def solve_multiscale(cost_matrix: list[list[float]],
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sigmas: Optional[list[float]] = None) -> dict:
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"""Solve routing problem at multiple scales using cluster-based optimization.
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Coarse-to-fine strategy with clustering:
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σ=1.0: coarse — merge nearby nodes, solve small problem
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σ=0.75: medium — tighter clustering
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σ=0.5: fine — minimal clustering, warm-started
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σ=0.25: formal verification target — full problem, 2-opt polish
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At each scale σ, nodes whose pairwise cost is below σ·max_cost are
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clustered together. The reduced problem is solved on cluster representatives,
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then expanded back to individual nodes. Solutions from coarser scales
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warm-start finer scales.
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Args:
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cost_matrix: n×n cost matrix.
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sigmas: List of scale parameters (coarse to fine).
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Returns:
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{'solutions': {sigma: {'tour': list, 'cost': float}},
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'converged': bool, 'best_sigma': float}
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"""
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if sigmas is None:
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sigmas = [1.0, 0.75, 0.5, 0.25]
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n = len(cost_matrix)
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if n == 0:
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return {'solutions': {}, 'converged': True, 'best_sigma': 0.0}
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# Find the maximum cost for threshold computation
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max_cost = 0.0
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for i in range(n):
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for j in range(n):
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if cost_matrix[i][j] > max_cost:
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max_cost = cost_matrix[i][j]
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if max_cost == 0.0:
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# All costs are zero — any tour is optimal
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tour = list(range(n))
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return {
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'solutions': {sigmas[0]: {'tour': tour, 'cost': 0.0,
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'n_clusters': 1, 'sigma': sigmas[0]}},
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'converged': True,
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'best_sigma': sigmas[0],
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'best_cost': 0.0,
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}
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solutions = {}
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best_cost = float('inf')
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best_sigma = sigmas[0]
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prev_tour = None
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for sigma in sigmas:
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# Compute clustering threshold: merge nodes with cost < sigma * max_cost
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threshold = sigma * max_cost
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# Cluster nodes using single-linkage
|
||
clusters = _single_linkage_clusters(cost_matrix, threshold)
|
||
n_clusters = len(clusters)
|
||
|
||
if n_clusters == 1:
|
||
# All nodes in one cluster — solve the full problem
|
||
if prev_tour is not None:
|
||
tour, cost = _2opt_improve(prev_tour[:], cost_matrix)
|
||
else:
|
||
tour, cost = _greedy_tour(cost_matrix)
|
||
tour, cost = _2opt_improve(tour, cost_matrix)
|
||
else:
|
||
# Build reduced cost matrix for cluster representatives
|
||
reduced_matrix = _build_reduced_cost_matrix(cost_matrix, clusters)
|
||
|
||
# Solve the reduced problem
|
||
if n_clusters <= 2:
|
||
# Trivial: just order the clusters
|
||
cluster_tour = list(range(n_clusters))
|
||
else:
|
||
cluster_tour, _ = _greedy_tour(reduced_matrix)
|
||
if n_clusters >= 4:
|
||
cluster_tour, _ = _2opt_improve(cluster_tour, reduced_matrix)
|
||
|
||
# Expand cluster tour back to individual nodes
|
||
tour = _expand_tour(cluster_tour, clusters, cost_matrix)
|
||
|
||
# Polish with 2-opt on the full problem
|
||
if prev_tour is not None:
|
||
# Warm-start: try both the expanded tour and the previous tour
|
||
tour_a, cost_a = _2opt_improve(tour, cost_matrix)
|
||
tour_b, cost_b = _2opt_improve(prev_tour[:], cost_matrix)
|
||
if cost_a <= cost_b:
|
||
tour, cost = tour_a, cost_a
|
||
else:
|
||
tour, cost = tour_b, cost_b
|
||
else:
|
||
tour, cost = _2opt_improve(tour, cost_matrix)
|
||
|
||
solutions[sigma] = {
|
||
'tour': tour,
|
||
'cost': cost,
|
||
'n_clusters': n_clusters,
|
||
'sigma': sigma,
|
||
}
|
||
|
||
if cost < best_cost:
|
||
best_cost = cost
|
||
best_sigma = sigma
|
||
|
||
prev_tour = tour
|
||
|
||
# Check convergence: did the solution stabilize at the finest scale?
|
||
converged = False
|
||
if len(sigmas) >= 2:
|
||
costs = [solutions[s]['cost'] for s in sigmas]
|
||
if len(costs) >= 2:
|
||
# Converged if last two scales are within 1%
|
||
last = costs[-1]
|
||
second_last = costs[-2]
|
||
if second_last > 0:
|
||
converged = abs(last - second_last) / second_last < 0.01
|
||
else:
|
||
converged = abs(last - second_last) < 1e-10
|
||
|
||
return {
|
||
'solutions': solutions,
|
||
'converged': converged,
|
||
'best_sigma': best_sigma,
|
||
'best_cost': best_cost,
|
||
}
|
||
|
||
|
||
# ---------------------------------------------------------------------------
|
||
# CLI / demo
|
||
# ---------------------------------------------------------------------------
|
||
|
||
if __name__ == '__main__':
|
||
import random
|
||
|
||
# Generate a random TSP instance
|
||
n = 20
|
||
random.seed(42)
|
||
points = [(random.uniform(0, 100), random.uniform(0, 100))
|
||
for _ in range(n)]
|
||
|
||
cost = [[0.0] * n for _ in range(n)]
|
||
for i in range(n):
|
||
for j in range(n):
|
||
dx = points[i][0] - points[j][0]
|
||
dy = points[i][1] - points[j][1]
|
||
cost[i][j] = math.sqrt(dx * dx + dy * dy)
|
||
|
||
print(f"Random TSP instance: {n} cities")
|
||
print(f"Cost matrix range: [{min(min(row) for row in cost):.1f}, "
|
||
f"{max(max(row) for row in cost):.1f}]")
|
||
|
||
# Solve multi-scale
|
||
result = solve_multiscale(cost)
|
||
|
||
print(f"\nConverged: {result['converged']}")
|
||
print(f"Best sigma: {result['best_sigma']}")
|
||
print(f"Best cost: {result['best_cost']:.2f}")
|
||
|
||
for sigma, data in sorted(result['solutions'].items(), reverse=True):
|
||
print(f" σ={sigma:.2f}: tour cost={data['cost']:.2f}, "
|
||
f"clusters={data['n_clusters']}")
|
||
|
||
# Demo voltage mapping
|
||
print("\nVoltage ↔ Scale mapping:")
|
||
for mv in [600, 700, 800, 900, 1000, 1100, 1200]:
|
||
s = voltage_to_scale(mv)
|
||
v_back = scale_to_voltage(s)
|
||
print(f" {mv}mV → σ={s:.3f} → {v_back:.0f}mV")
|
||
|
||
# Demo Q16 kernel
|
||
print("\nQ16.16 Gaussian kernel (σ=0.5, 9 taps):")
|
||
k = gaussian_kernel_q16(0.5, 9)
|
||
total_q16 = sum(k)
|
||
print(f" Values: {k}")
|
||
print(f" Sum: {total_q16} (target: {Q16_SCALE})")
|
||
print(f" As floats: [{', '.join(f'{q16_to_float(v):.4f}' for v in k)}]")
|