cds2.combinatorial
Discrete optimization: TSP heuristics, knapsack, assignment, LCS.
Classic discrete optimization: TSP heuristics, 0/1 knapsack, assignment and LCS.
TourResult
dataclass
Closed tour (first city repeated at the end) with its total cost.
Source code in src/cds2/combinatorial.py
| @dataclass(frozen=True)
class TourResult:
"""Closed tour (first city repeated at the end) with its total cost."""
tour: IntArray
cost: float
improved_from: float | None = None
|
KnapsackResult
dataclass
Best 0/1 knapsack packing with the ascending indices of chosen items.
Source code in src/cds2/combinatorial.py
| @dataclass(frozen=True)
class KnapsackResult:
"""Best 0/1 knapsack packing with the ascending indices of chosen items."""
total_value: float
chosen_indices: IntArray
total_weight: float
|
AssignmentResult
dataclass
Optimal assignment cost together with the row-to-column mapping.
Source code in src/cds2/combinatorial.py
| @dataclass(frozen=True)
class AssignmentResult:
"""Optimal assignment cost together with the row-to-column mapping."""
cost: float
row_to_col: IntArray
|
nearest_neighbor_tsp(distance_matrix, start=0)
Greedy nearest-neighbour heuristic for the travelling-salesman problem.
Source code in src/cds2/combinatorial.py
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111 | def nearest_neighbor_tsp(
distance_matrix: FloatArray | Sequence[Sequence[float]],
start: int = 0,
) -> TourResult:
"""Greedy nearest-neighbour heuristic for the travelling-salesman problem."""
matrix = _as_distance_matrix(distance_matrix)
city_count = matrix.shape[0]
if not 0 <= start < city_count:
msg = "start city out of range"
raise ValueError(msg)
visited = np.zeros(city_count, dtype=bool)
visited[start] = True
order = [start]
current = start
total = 0.0
for _ in range(city_count - 1):
best_city = -1
best_distance = math.inf
for candidate in range(city_count):
if visited[candidate]:
continue
candidate_distance = float(matrix[current, candidate])
if candidate_distance < best_distance:
best_distance = candidate_distance
best_city = candidate
visited[best_city] = True
order.append(best_city)
total += best_distance
current = best_city
total += float(matrix[order[-1], start])
order.append(start)
tour: IntArray = np.asarray(order, dtype=np.int64)
return TourResult(tour=tour, cost=float(total))
|
two_opt(tour, distance_matrix)
Improve a closed tour with 2-opt edge exchanges until no swap helps.
Source code in src/cds2/combinatorial.py
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154 | def two_opt(
tour: IntArray | Sequence[int],
distance_matrix: FloatArray | Sequence[Sequence[float]],
) -> TourResult:
"""Improve a closed tour with 2-opt edge exchanges until no swap helps."""
matrix = _as_distance_matrix(distance_matrix)
city_count = matrix.shape[0]
nodes: IntArray = np.asarray(tour, dtype=np.int64)
if nodes[0] != nodes[-1]:
msg = "tour must start and end at the same city"
raise ValueError(msg)
if int(nodes.min()) < 0 or int(nodes.max()) >= city_count:
msg = "tour contains city indices outside the distance matrix"
raise ValueError(msg)
initial_cost = _tour_cost(nodes, matrix)
best_cost = initial_cost
improved = True
while improved:
improved = False
for i in range(1, city_count - 1):
for j in range(i + 1, city_count):
removed = float(matrix[int(nodes[i - 1]), int(nodes[i])]) + float(
matrix[int(nodes[j]), int(nodes[j + 1])]
)
added = float(matrix[int(nodes[i - 1]), int(nodes[j])]) + float(
matrix[int(nodes[i]), int(nodes[j + 1])]
)
if added - removed < -_EPSILON:
nodes[i : j + 1] = nodes[i : j + 1][::-1]
best_cost += added - removed
improved = True
break
if improved:
break
return TourResult(
tour=nodes,
cost=best_cost,
improved_from=initial_cost if best_cost < initial_cost else None,
)
|
knapsack_01(weights, values, capacity)
Exact 0/1 knapsack via dynamic programming over integer capacity units.
Source code in src/cds2/combinatorial.py
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211 | def knapsack_01(
weights: Sequence[float] | FloatArray,
values: Sequence[float] | FloatArray,
capacity: float,
) -> KnapsackResult:
"""Exact 0/1 knapsack via dynamic programming over integer capacity units."""
weight_array = np.asarray(weights, dtype=float)
value_array = np.asarray(values, dtype=float)
if (
weight_array.ndim != 1
or value_array.ndim != 1
or weight_array.size != value_array.size
or weight_array.size == 0
):
msg = "weights and values must be equal-length non-empty"
raise ValueError(msg)
capacity_value = float(capacity)
weight_values = [float(item) for item in weight_array.tolist()]
value_values = [float(item) for item in value_array.tolist()]
if not all(item.is_integer() for item in (*weight_values, capacity_value)):
msg = "weights and capacity must be integer-valued for exact DP"
raise ValueError(msg)
capacity_units = int(round(capacity_value))
if capacity_units <= 0:
msg = "capacity must be positive"
raise ValueError(msg)
item_count = weight_array.size
integer_weights = [int(round(item)) for item in weight_values]
table = np.zeros((item_count + 1, capacity_units + 1), dtype=float)
for item in range(1, item_count + 1):
item_weight = integer_weights[item - 1]
item_value = value_values[item - 1]
for remaining in range(capacity_units + 1):
best = float(table[item - 1, remaining])
if item_weight <= remaining:
candidate = float(table[item - 1, remaining - item_weight]) + item_value
if candidate > best:
best = candidate
table[item, remaining] = best
chosen: list[int] = []
remaining = capacity_units
for item in range(item_count, 0, -1):
if float(table[item, remaining]) != float(table[item - 1, remaining]):
chosen.append(item - 1)
remaining -= integer_weights[item - 1]
chosen.reverse()
return KnapsackResult(
total_value=float(table[item_count, capacity_units]),
chosen_indices=np.asarray(chosen, dtype=np.int64),
total_weight=float(sum(weight_values[index] for index in chosen)),
)
|
assign_min_cost(cost_matrix)
Minimum-cost bipartite assignment over a (possibly rectangular) matrix.
Source code in src/cds2/combinatorial.py
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225 | def assign_min_cost(
cost_matrix: FloatArray | Sequence[Sequence[float]],
) -> AssignmentResult:
"""Minimum-cost bipartite assignment over a (possibly rectangular) matrix."""
matrix: FloatArray = np.asarray(cost_matrix, dtype=float)
if matrix.ndim != 2 or min(matrix.shape) < 1:
msg = "cost matrix must be 2-D with at least one row and one column"
raise ValueError(msg)
rows, columns = linear_sum_assignment(matrix)
mapping: IntArray = np.asarray(columns, dtype=np.int64)
total = float(np.asarray(matrix[rows, columns], dtype=float).sum())
return AssignmentResult(cost=total, row_to_col=mapping)
|
longest_common_subsequence(a, b)
One longest common subsequence of a and b via dynamic programming.
Source code in src/cds2/combinatorial.py
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249 | def longest_common_subsequence(a: str, b: str) -> str:
"""One longest common subsequence of ``a`` and ``b`` via dynamic programming."""
table = [[0] * (len(b) + 1) for _ in range(len(a) + 1)]
for i in range(1, len(a) + 1):
for j in range(1, len(b) + 1):
if a[i - 1] == b[j - 1]:
table[i][j] = table[i - 1][j - 1] + 1
else:
table[i][j] = max(table[i - 1][j], table[i][j - 1])
characters: list[str] = []
i, j = len(a), len(b)
while i > 0 and j > 0:
if a[i - 1] == b[j - 1]:
characters.append(a[i - 1])
i -= 1
j -= 1
elif table[i - 1][j] >= table[i][j - 1]:
i -= 1
else:
j -= 1
return "".join(reversed(characters))
|