# Path: opt_einsum/helpers.py
# @overload
# def compute_size_by_dict(indices: Iterable[int], idx_dict: List[int]) -> int:
#     ...
# 
# def flop_count(
#     idx_contraction: Collection[str],
#     inner: bool,
#     num_terms: int,
#     size_dictionary: Dict[str, int],
# ) -> int:
#     """
#     Computes the number of FLOPS in the contraction.
# 
#     Parameters
#     ----------
#     idx_contraction : iterable
#         The indices involved in the contraction
#     inner : bool
#         Does this contraction require an inner product?
#     num_terms : int
#         The number of terms in a contraction
#     size_dictionary : dict
#         The size of each of the indices in idx_contraction
# 
#     Returns
#     -------
#     flop_count : int
#         The total number of FLOPS required for the contraction.
# 
#     Examples
#     --------
# 
#     >>> flop_count('abc', False, 1, {'a': 2, 'b':3, 'c':5})
#     30
# 
#     >>> flop_count('abc', True, 2, {'a': 2, 'b':3, 'c':5})
#     60
# 
#     """
# 
#     overall_size = compute_size_by_dict(idx_contraction, size_dictionary)
#     op_factor = max(1, num_terms - 1)
#     if inner:
#         op_factor += 1
# 
#     return overall_size * op_factor
# 
# Path: opt_einsum/typing.py



import functools
import heapq
import itertools
import operator
import random
import re
import numpy as np
from collections import Counter, OrderedDict, defaultdict
from typing import Any, Callable
from typing import Counter as CounterType
from typing import Dict, FrozenSet, Generator, List, Optional, Sequence, Set, Tuple, Union
from .helpers import compute_size_by_dict, flop_count
from .typing import ArrayIndexType, PathType
    from .path_random import random_greedy_128
    result_cache: Dict[Tuple[ArrayIndexType, ArrayIndexType], Tuple[FrozenSet[str], int]] = {}

    def _optimal_iterate(path, remaining, inputs, flops):

        # reached end of path (only ever get here if flops is best found so far)
        if len(remaining) == 1:
            best_flops["flops"] = flops
            best_ssa_path["ssa_path"] = path
            return

        # check all possible remaining paths
        for i, j in itertools.combinations(remaining, 2):
            if i > j:
                i, j = j, i
            key = (inputs[i], inputs[j])
            try:
                k12, flops12 = result_cache[key]
            except KeyError:
                k12, flops12 = result_cache[key] = calc_k12_flops(inputs, output_set, remaining, i, j, size_dict)

            # sieve based on current best flops
            new_flops = flops + flops12
            if new_flops >= best_flops["flops"]:
                continue

            # sieve based on memory limit
            if memory_limit not in _UNLIMITED_MEM:
                try:
                    size12 = size_cache[k12]
                except KeyError:

# Path: opt_einsum/helpers.py
# @overload
# def compute_size_by_dict(indices: Iterable[int], idx_dict: List[int]) -> int:
#     ...
# 
# def flop_count(
#     idx_contraction: Collection[str],
#     inner: bool,
#     num_terms: int,
#     size_dictionary: Dict[str, int],
# ) -> int:
#     """
#     Computes the number of FLOPS in the contraction.
# 
#     Parameters
#     ----------
#     idx_contraction : iterable
#         The indices involved in the contraction
#     inner : bool
#         Does this contraction require an inner product?
#     num_terms : int
#         The number of terms in a contraction
#     size_dictionary : dict
#         The size of each of the indices in idx_contraction
# 
#     Returns
#     -------
#     flop_count : int
#         The total number of FLOPS required for the contraction.
# 
#     Examples
#     --------
# 
#     >>> flop_count('abc', False, 1, {'a': 2, 'b':3, 'c':5})
#     30
# 
#     >>> flop_count('abc', True, 2, {'a': 2, 'b':3, 'c':5})
#     60
# 
#     """
# 
#     overall_size = compute_size_by_dict(idx_contraction, size_dictionary)
#     op_factor = max(1, num_terms - 1)
#     if inner:
#         op_factor += 1
# 
#     return overall_size * op_factor
# 
# Path: opt_einsum/typing.py



# Path: opt_einsum/paths.py
import functools
import heapq
import itertools
import operator
import random
import re
import numpy as np
from collections import Counter, OrderedDict, defaultdict
from typing import Any, Callable
from typing import Counter as CounterType
from typing import Dict, FrozenSet, Generator, List, Optional, Sequence, Set, Tuple, Union
from .helpers import compute_size_by_dict, flop_count
from .typing import ArrayIndexType, PathType
    from .path_random import random_greedy_128

    result_cache: Dict[Tuple[ArrayIndexType, ArrayIndexType], Tuple[FrozenSet[str], int]] = {}

    def _optimal_iterate(path, remaining, inputs, flops):

        # reached end of path (only ever get here if flops is best found so far)
        if len(remaining) == 1:
            best_flops["flops"] = flops
            best_ssa_path["ssa_path"] = path
            return

        # check all possible remaining paths
        for i, j in itertools.combinations(remaining, 2):
            if i > j:
                i, j = j, i
            key = (inputs[i], inputs[j])
            try:
                k12, flops12 = result_cache[key]
            except KeyError:
                k12, flops12 = result_cache[key] = calc_k12_flops(inputs, output_set, remaining, i, j, size_dict)

            # sieve based on current best flops
            new_flops = flops + flops12
            if new_flops >= best_flops["flops"]:
                continue

            # sieve based on memory limit
            if memory_limit not in _UNLIMITED_MEM:
                try:
                    size12 = size_cache[k12]
                except KeyError:

                    size12 = size_cache[k12] = compute_size_by_dict(k12, size_dict)