AIStoryBoard/python/system_venv/Lib/site-packages/opt_einsum/paths.py

"""
Contains the path technology behind opt_einsum in addition to several path helpers
"""

import functools
import heapq
import itertools
import random
from collections import Counter, OrderedDict, defaultdict

import numpy as np

from . import helpers

__all__ = [
    "optimal", "BranchBound", "branch", "greedy", "auto", "auto_hq", "get_path_fn", "DynamicProgramming",
    "dynamic_programming"
]

_UNLIMITED_MEM = {-1, None, float('inf')}


class PathOptimizer(object):
    """Base class for different path optimizers to inherit from.

    Subclassed optimizers should define a call method with signature::

        def __call__(self, inputs, output, size_dict, memory_limit=None):
            \"\"\"
            Parameters
            ----------
            inputs : list[set[str]]
                The indices of each input array.
            outputs : set[str]
                The output indices
            size_dict : dict[str, int]
                The size of each index
            memory_limit : int, optional
                If given, the maximum allowed memory.
            \"\"\"
            # ... compute path here ...
            return path

    where ``path`` is a list of int-tuples specifiying a contraction order.
    """

    def _check_args_against_first_call(self, inputs, output, size_dict):
        """Utility that stateful optimizers can use to ensure they are not
        called with different contractions across separate runs.
        """
        args = (inputs, output, size_dict)
        if not hasattr(self, '_first_call_args'):
            # simply set the attribute as currently there is no global PathOptimizer init
            self._first_call_args = args
        elif args != self._first_call_args:
            raise ValueError("The arguments specifiying the contraction that this path optimizer "
                             "instance was called with have changed - try creating a new instance.")

    def __call__(self, inputs, output, size_dict, memory_limit=None):
        raise NotImplementedError


def ssa_to_linear(ssa_path):
    """
    Convert a path with static single assignment ids to a path with recycled
    linear ids. For example::

        >>> ssa_to_linear([(0, 3), (2, 4), (1, 5)])
        [(0, 3), (1, 2), (0, 1)]
    """
    ids = np.arange(1 + max(map(max, ssa_path)), dtype=np.int32)
    path = []
    for ssa_ids in ssa_path:
        path.append(tuple(int(ids[ssa_id]) for ssa_id in ssa_ids))
        for ssa_id in ssa_ids:
            ids[ssa_id:] -= 1
    return path


def linear_to_ssa(path):
    """
    Convert a path with recycled linear ids to a path with static single
    assignment ids. For example::

        >>> linear_to_ssa([(0, 3), (1, 2), (0, 1)])
        [(0, 3), (2, 4), (1, 5)]
    """
    num_inputs = sum(map(len, path)) - len(path) + 1
    linear_to_ssa = list(range(num_inputs))
    new_ids = itertools.count(num_inputs)
    ssa_path = []
    for ids in path:
        ssa_path.append(tuple(linear_to_ssa[id_] for id_ in ids))
        for id_ in sorted(ids, reverse=True):
            del linear_to_ssa[id_]
        linear_to_ssa.append(next(new_ids))
    return ssa_path


def calc_k12_flops(inputs, output, remaining, i, j, size_dict):
    """
    Calculate the resulting indices and flops for a potential pairwise
    contraction - used in the recursive (optimal/branch) algorithms.

    Parameters
    ----------
    inputs : tuple[frozenset[str]]
        The indices of each tensor in this contraction, note this includes
        tensors unavaiable to contract as static single assignment is used ->
        contracted tensors are not removed from the list.
    output : frozenset[str]
        The set of output indices for the whole contraction.
    remaining : frozenset[int]
        The set of indices (corresponding to ``inputs``) of tensors still
        available to contract.
    i : int
        Index of potential tensor to contract.
    j : int
        Index of potential tensor to contract.
    size_dict dict[str, int]
        Size mapping of all the indices.

    Returns
    -------
    k12 : frozenset
        The resulting indices of the potential tensor.
    cost : int
        Estimated flop count of operation.
    """
    k1, k2 = inputs[i], inputs[j]
    either = k1 | k2
    shared = k1 & k2
    keep = frozenset.union(output, *map(inputs.__getitem__, remaining - {i, j}))

    k12 = either & keep
    cost = helpers.flop_count(either, shared - keep, 2, size_dict)

    return k12, cost


def _compute_oversize_flops(inputs, remaining, output, size_dict):
    """
    Compute the flop count for a contraction of all remaining arguments. This
    is used when a memory limit means that no pairwise contractions can be made.
    """
    idx_contraction = frozenset.union(*map(inputs.__getitem__, remaining))
    inner = idx_contraction - output
    num_terms = len(remaining)
    return helpers.flop_count(idx_contraction, inner, num_terms, size_dict)


def optimal(inputs, output, size_dict, memory_limit=None):
    """
    Computes all possible pair contractions in a depth-first recursive manner,
    sieving results based on ``memory_limit`` and the best path found so far.
    Returns the lowest cost path. This algorithm scales factoriallly with
    respect to the elements in the list ``input_sets``.

    Parameters
    ----------
    inputs : list
        List of sets that represent the lhs side of the einsum subscript.
    output : set
        Set that represents the rhs side of the overall einsum subscript.
    size_dict : dictionary
        Dictionary of index sizes.
    memory_limit : int
        The maximum number of elements in a temporary array.

    Returns
    -------
    path : list
        The optimal contraction order within the memory limit constraint.

    Examples
    --------
    >>> isets = [set('abd'), set('ac'), set('bdc')]
    >>> oset = set('')
    >>> idx_sizes = {'a': 1, 'b':2, 'c':3, 'd':4}
    >>> optimal(isets, oset, idx_sizes, 5000)
    [(0, 2), (0, 1)]
    """
    inputs = tuple(map(frozenset, inputs))
    output = frozenset(output)

    best = {'flops': float('inf'), 'ssa_path': (tuple(range(len(inputs))), )}
    size_cache = {}
    result_cache = {}

    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
            best['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, remaining, i, j, size_dict)

            # sieve based on current best flops
            new_flops = flops + flops12
            if new_flops >= best['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] = helpers.compute_size_by_dict(k12, size_dict)

                # possibly terminate this path with an all-terms einsum
                if size12 > memory_limit:
                    new_flops = flops + _compute_oversize_flops(inputs, remaining, output, size_dict)
                    if new_flops < best['flops']:
                        best['flops'] = new_flops
                        best['ssa_path'] = path + (tuple(remaining), )
                    continue

            # add contraction and recurse into all remaining
            _optimal_iterate(path=path + ((i, j), ),
                             inputs=inputs + (k12, ),
                             remaining=remaining - {i, j} | {len(inputs)},
                             flops=new_flops)

    _optimal_iterate(path=(), inputs=inputs, remaining=set(range(len(inputs))), flops=0)

    return ssa_to_linear(best['ssa_path'])


# functions for comparing which of two paths is 'better'


def better_flops_first(flops, size, best_flops, best_size):
    return (flops, size) < (best_flops, best_size)


def better_size_first(flops, size, best_flops, best_size):
    return (size, flops) < (best_size, best_flops)


_BETTER_FNS = {
    'flops': better_flops_first,
    'size': better_size_first,
}


def get_better_fn(key):
    return _BETTER_FNS[key]


# functions for assigning a heuristic 'cost' to a potential contraction


def cost_memory_removed(size12, size1, size2, k12, k1, k2):
    """The default heuristic cost, corresponding to the total reduction in
    memory of performing a contraction.
    """
    return size12 - size1 - size2


def cost_memory_removed_jitter(size12, size1, size2, k12, k1, k2):
    """Like memory-removed, but with a slight amount of noise that breaks ties
    and thus jumbles the contractions a bit.
    """
    return random.gauss(1.0, 0.01) * (size12 - size1 - size2)


_COST_FNS = {
    'memory-removed': cost_memory_removed,
    'memory-removed-jitter': cost_memory_removed_jitter,
}


class BranchBound(PathOptimizer):
    """
    Explores possible pair contractions in a depth-first recursive manner like
    the ``optimal`` approach, but with extra heuristic early pruning of branches
    as well sieving by ``memory_limit`` and the best path found so far. Returns
    the lowest cost path. This algorithm still scales factorially with respect
    to the elements in the list ``input_sets`` if ``nbranch`` is not set, but it
    scales exponentially like ``nbranch**len(input_sets)`` otherwise.

    Parameters
    ----------
    nbranch : None or int, optional
        How many branches to explore at each contraction step. If None, explore
        all possible branches. If an integer, branch into this many paths at
        each step. Defaults to None.
    cutoff_flops_factor : float, optional
        If at any point, a path is doing this much worse than the best path
        found so far was, terminate it. The larger this is made, the more paths
        will be fully explored and the slower the algorithm. Defaults to 4.
    minimize : {'flops', 'size'}, optional
        Whether to optimize the path with regard primarily to the total
        estimated flop-count, or the size of the largest intermediate. The
        option not chosen will still be used as a secondary criterion.
    cost_fn : callable, optional
        A function that returns a heuristic 'cost' of a potential contraction
        with which to sort candidates. Should have signature
        ``cost_fn(size12, size1, size2, k12, k1, k2)``.
    """
    def __init__(self, nbranch=None, cutoff_flops_factor=4, minimize='flops', cost_fn='memory-removed'):
        self.nbranch = nbranch
        self.cutoff_flops_factor = cutoff_flops_factor
        self.minimize = minimize
        self.cost_fn = _COST_FNS.get(cost_fn, cost_fn)

        self.better = get_better_fn(minimize)
        self.best = {'flops': float('inf'), 'size': float('inf')}
        self.best_progress = defaultdict(lambda: float('inf'))

    @property
    def path(self):
        return ssa_to_linear(self.best['ssa_path'])

    def __call__(self, inputs, output, size_dict, memory_limit=None):
        """

        Parameters
        ----------
        input_sets : list
            List of sets that represent the lhs side of the einsum subscript
        output_set : set
            Set that represents the rhs side of the overall einsum subscript
        idx_dict : dictionary
            Dictionary of index sizes
        memory_limit : int
            The maximum number of elements in a temporary array

        Returns
        -------
        path : list
            The contraction order within the memory limit constraint.

        Examples
        --------
        >>> isets = [set('abd'), set('ac'), set('bdc')]
        >>> oset = set('')
        >>> idx_sizes = {'a': 1, 'b':2, 'c':3, 'd':4}
        >>> optimal(isets, oset, idx_sizes, 5000)
        [(0, 2), (0, 1)]
        """
        self._check_args_against_first_call(inputs, output, size_dict)

        inputs = tuple(map(frozenset, inputs))
        output = frozenset(output)

        size_cache = {k: helpers.compute_size_by_dict(k, size_dict) for k in inputs}
        result_cache = {}

        def _branch_iterate(path, inputs, remaining, flops, size):

            # reached end of path (only ever get here if flops is best found so far)
            if len(remaining) == 1:
                self.best['size'] = size
                self.best['flops'] = flops
                self.best['ssa_path'] = path
                return

            def _assess_candidate(k1, k2, i, j):
                # find resulting indices and flops
                try:
                    k12, flops12 = result_cache[k1, k2]
                except KeyError:
                    k12, flops12 = result_cache[k1, k2] = calc_k12_flops(inputs, output, remaining, i, j, size_dict)

                try:
                    size12 = size_cache[k12]
                except KeyError:
                    size12 = size_cache[k12] = helpers.compute_size_by_dict(k12, size_dict)

                new_flops = flops + flops12
                new_size = max(size, size12)

                # sieve based on current best i.e. check flops and size still better
                if not self.better(new_flops, new_size, self.best['flops'], self.best['size']):
                    return None

                # compare to how the best method was doing as this point
                if new_flops < self.best_progress[len(inputs)]:
                    self.best_progress[len(inputs)] = new_flops
                # sieve based on current progress relative to best
                elif new_flops > self.cutoff_flops_factor * self.best_progress[len(inputs)]:
                    return None

                # sieve based on memory limit
                if (memory_limit not in _UNLIMITED_MEM) and (size12 > memory_limit):
                    # terminate path here, but check all-terms contract first
                    new_flops = flops + _compute_oversize_flops(inputs, remaining, output, size_dict)
                    if new_flops < self.best['flops']:
                        self.best['flops'] = new_flops
                        self.best['ssa_path'] = path + (tuple(remaining), )
                    return None

                # set cost heuristic in order to locally sort possible contractions
                size1, size2 = size_cache[inputs[i]], size_cache[inputs[j]]
                cost = self.cost_fn(size12, size1, size2, k12, k1, k2)

                return cost, flops12, new_flops, new_size, (i, j), k12

            # check all possible remaining paths
            candidates = []
            for i, j in itertools.combinations(remaining, 2):
                if i > j:
                    i, j = j, i
                k1, k2 = inputs[i], inputs[j]

                # initially ignore outer products
                if k1.isdisjoint(k2):
                    continue

                candidate = _assess_candidate(k1, k2, i, j)
                if candidate:
                    heapq.heappush(candidates, candidate)

            # assess outer products if nothing left
            if not candidates:
                for i, j in itertools.combinations(remaining, 2):
                    if i > j:
                        i, j = j, i
                    k1, k2 = inputs[i], inputs[j]
                    candidate = _assess_candidate(k1, k2, i, j)
                    if candidate:
                        heapq.heappush(candidates, candidate)

            # recurse into all or some of the best candidate contractions
            bi = 0
            while (self.nbranch is None or bi < self.nbranch) and candidates:
                _, _, new_flops, new_size, (i, j), k12 = heapq.heappop(candidates)
                _branch_iterate(path=path + ((i, j), ),
                                inputs=inputs + (k12, ),
                                remaining=(remaining - {i, j}) | {len(inputs)},
                                flops=new_flops,
                                size=new_size)
                bi += 1

        _branch_iterate(path=(), inputs=inputs, remaining=set(range(len(inputs))), flops=0, size=0)

        return self.path


def branch(inputs, output, size_dict, memory_limit=None, **optimizer_kwargs):
    optimizer = BranchBound(**optimizer_kwargs)
    return optimizer(inputs, output, size_dict, memory_limit)


branch_all = functools.partial(branch, nbranch=None)
branch_2 = functools.partial(branch, nbranch=2)
branch_1 = functools.partial(branch, nbranch=1)


def _get_candidate(output, sizes, remaining, footprints, dim_ref_counts, k1, k2, cost_fn):
    either = k1 | k2
    two = k1 & k2
    one = either - two
    k12 = (either & output) | (two & dim_ref_counts[3]) | (one & dim_ref_counts[2])
    cost = cost_fn(helpers.compute_size_by_dict(k12, sizes), footprints[k1], footprints[k2], k12, k1, k2)
    id1 = remaining[k1]
    id2 = remaining[k2]
    if id1 > id2:
        k1, id1, k2, id2 = k2, id2, k1, id1
    cost = cost, id2, id1  # break ties to ensure determinism
    return cost, k1, k2, k12


def _push_candidate(output, sizes, remaining, footprints, dim_ref_counts, k1, k2s, queue, push_all, cost_fn):
    candidates = (_get_candidate(output, sizes, remaining, footprints, dim_ref_counts, k1, k2, cost_fn) for k2 in k2s)
    if push_all:
        # want to do this if we e.g. are using a custom 'choose_fn'
        for candidate in candidates:
            heapq.heappush(queue, candidate)
    else:
        heapq.heappush(queue, min(candidates))


def _update_ref_counts(dim_to_keys, dim_ref_counts, dims):
    for dim in dims:
        count = len(dim_to_keys[dim])
        if count <= 1:
            dim_ref_counts[2].discard(dim)
            dim_ref_counts[3].discard(dim)
        elif count == 2:
            dim_ref_counts[2].add(dim)
            dim_ref_counts[3].discard(dim)
        else:
            dim_ref_counts[2].add(dim)
            dim_ref_counts[3].add(dim)


def _simple_chooser(queue, remaining):
    """Default contraction chooser that simply takes the minimum cost option.
    """
    cost, k1, k2, k12 = heapq.heappop(queue)
    if k1 not in remaining or k2 not in remaining:
        return None  # candidate is obsolete
    return cost, k1, k2, k12


def ssa_greedy_optimize(inputs, output, sizes, choose_fn=None, cost_fn='memory-removed'):
    """
    This is the core function for :func:`greedy` but produces a path with
    static single assignment ids rather than recycled linear ids.
    SSA ids are cheaper to work with and easier to reason about.
    """
    if len(inputs) == 1:
        # Perform a single contraction to match output shape.
        return [(0, )]

    # set the function that assigns a heuristic cost to a possible contraction
    cost_fn = _COST_FNS.get(cost_fn, cost_fn)

    # set the function that chooses which contraction to take
    if choose_fn is None:
        choose_fn = _simple_chooser
        push_all = False
    else:
        # assume chooser wants access to all possible contractions
        push_all = True

    # A dim that is common to all tensors might as well be an output dim, since it
    # cannot be contracted until the final step. This avoids an expensive all-pairs
    # comparison to search for possible contractions at each step, leading to speedup
    # in many practical problems where all tensors share a common batch dimension.
    inputs = list(map(frozenset, inputs))
    output = frozenset(output) | frozenset.intersection(*inputs)

    # Deduplicate shapes by eagerly computing Hadamard products.
    remaining = {}  # key -> ssa_id
    ssa_ids = itertools.count(len(inputs))
    ssa_path = []
    for ssa_id, key in enumerate(inputs):
        if key in remaining:
            ssa_path.append((remaining[key], ssa_id))
            remaining[key] = next(ssa_ids)
        else:
            remaining[key] = ssa_id

    # Keep track of possible contraction dims.
    dim_to_keys = defaultdict(set)
    for key in remaining:
        for dim in key - output:
            dim_to_keys[dim].add(key)

    # Keep track of the number of tensors using each dim; when the dim is no longer
    # used it can be contracted. Since we specialize to binary ops, we only care about
    # ref counts of >=2 or >=3.
    dim_ref_counts = {
        count: set(dim for dim, keys in dim_to_keys.items() if len(keys) >= count) - output
        for count in [2, 3]
    }

    # Compute separable part of the objective function for contractions.
    footprints = {key: helpers.compute_size_by_dict(key, sizes) for key in remaining}

    # Find initial candidate contractions.
    queue = []
    for dim, keys in dim_to_keys.items():
        keys = sorted(keys, key=remaining.__getitem__)
        for i, k1 in enumerate(keys[:-1]):
            k2s = keys[1 + i:]
            _push_candidate(output, sizes, remaining, footprints, dim_ref_counts, k1, k2s, queue, push_all, cost_fn)

    # Greedily contract pairs of tensors.
    while queue:

        con = choose_fn(queue, remaining)
        if con is None:
            continue  # allow choose_fn to flag all candidates obsolete
        cost, k1, k2, k12 = con

        ssa_id1 = remaining.pop(k1)
        ssa_id2 = remaining.pop(k2)
        for dim in k1 - output:
            dim_to_keys[dim].remove(k1)
        for dim in k2 - output:
            dim_to_keys[dim].remove(k2)
        ssa_path.append((ssa_id1, ssa_id2))
        if k12 in remaining:
            ssa_path.append((remaining[k12], next(ssa_ids)))
        else:
            for dim in k12 - output:
                dim_to_keys[dim].add(k12)
        remaining[k12] = next(ssa_ids)
        _update_ref_counts(dim_to_keys, dim_ref_counts, k1 | k2 - output)
        footprints[k12] = helpers.compute_size_by_dict(k12, sizes)

        # Find new candidate contractions.
        k1 = k12
        k2s = set(k2 for dim in k1 for k2 in dim_to_keys[dim])
        k2s.discard(k1)
        if k2s:
            _push_candidate(output, sizes, remaining, footprints, dim_ref_counts, k1, k2s, queue, push_all, cost_fn)

    # Greedily compute pairwise outer products.
    queue = [(helpers.compute_size_by_dict(key & output, sizes), ssa_id, key) for key, ssa_id in remaining.items()]
    heapq.heapify(queue)
    _, ssa_id1, k1 = heapq.heappop(queue)
    while queue:
        _, ssa_id2, k2 = heapq.heappop(queue)
        ssa_path.append((min(ssa_id1, ssa_id2), max(ssa_id1, ssa_id2)))
        k12 = (k1 | k2) & output
        cost = helpers.compute_size_by_dict(k12, sizes)
        ssa_id12 = next(ssa_ids)
        _, ssa_id1, k1 = heapq.heappushpop(queue, (cost, ssa_id12, k12))

    return ssa_path


def greedy(inputs, output, size_dict, memory_limit=None, choose_fn=None, cost_fn='memory-removed'):
    """
    Finds the path by a three stage algorithm:

    1. Eagerly compute Hadamard products.
    2. Greedily compute contractions to maximize ``removed_size``
    3. Greedily compute outer products.

    This algorithm scales quadratically with respect to the
    maximum number of elements sharing a common dim.

    Parameters
    ----------
    inputs : list
        List of sets that represent the lhs side of the einsum subscript
    output : set
        Set that represents the rhs side of the overall einsum subscript
    size_dict : dictionary
        Dictionary of index sizes
    memory_limit : int
        The maximum number of elements in a temporary array
    choose_fn : callable, optional
        A function that chooses which contraction to perform from the queu
    cost_fn : callable, optional
        A function that assigns a potential contraction a cost.

    Returns
    -------
    path : list
        The contraction order (a list of tuples of ints).

    Examples
    --------
    >>> isets = [set('abd'), set('ac'), set('bdc')]
    >>> oset = set('')
    >>> idx_sizes = {'a': 1, 'b':2, 'c':3, 'd':4}
    >>> greedy(isets, oset, idx_sizes)
    [(0, 2), (0, 1)]
    """
    if memory_limit not in _UNLIMITED_MEM:
        return branch(inputs, output, size_dict, memory_limit, nbranch=1, cost_fn=cost_fn)

    ssa_path = ssa_greedy_optimize(inputs, output, size_dict, cost_fn=cost_fn, choose_fn=choose_fn)
    return ssa_to_linear(ssa_path)


def _tree_to_sequence(c):
    """
    Converts a contraction tree to a contraction path as it has to be
    returned by path optimizers. A contraction tree can either be an int
    (=no contraction) or a tuple containing the terms to be contracted. An
    arbitrary number (>= 1) of terms can be contracted at once. Note that
    contractions are commutative, e.g. (j, k, l) = (k, l, j). Note that in
    general, solutions are not unique.

    Parameters
    ----------
    c : tuple or int
        Contraction tree

    Returns
    -------
    path : list[set[int]]
        Contraction path

    Examples
    --------
    >>> _tree_to_sequence(((1,2),(0,(4,5,3))))
    [(1, 2), (1, 2, 3), (0, 2), (0, 1)]
    """

    # ((1,2),(0,(4,5,3))) --> [(1, 2), (1, 2, 3), (0, 2), (0, 1)]
    #
    # 0     0         0           (1,2)       --> ((1,2),(0,(3,4,5)))
    # 1     3         (1,2)   --> (0,(3,4,5))
    # 2 --> 4     --> (3,4,5)
    # 3     5
    # 4     (1,2)
    # 5
    #
    # this function iterates through the table shown above from right to left;

    if type(c) == int:
        return []

    c = [c]  # list of remaining contractions (lower part of columns shown above)
    t = []  # list of elementary tensors (upper part of colums)
    s = []  # resulting contraction sequence

    while len(c) > 0:
        j = c.pop(-1)
        s.insert(0, tuple())

        for i in sorted([i for i in j if type(i) == int]):
            s[0] += (sum(1 for q in t if q < i), )
            t.insert(s[0][-1], i)

        for i in [i for i in j if type(i) != int]:
            s[0] += (len(t) + len(c), )
            c.append(i)

    return s


def _find_disconnected_subgraphs(inputs, output):
    """
    Finds disconnected subgraphs in the given list of inputs. Inputs are
    connected if they share summation indices. Note: Disconnected subgraphs
    can be contracted independently before forming outer products.

    Parameters
    ----------
    inputs : list[set]
        List of sets that represent the lhs side of the einsum subscript
    output : set
        Set that represents the rhs side of the overall einsum subscript

    Returns
    -------
    subgraphs : list[set[int]]
        List containing sets of indices for each subgraph

    Examples
    --------
    >>> _find_disconnected_subgraphs([set("ab"), set("c"), set("ad")], set("bd"))
    [{0, 2}, {1}]

    >>> _find_disconnected_subgraphs([set("ab"), set("c"), set("ad")], set("abd"))
    [{0}, {1}, {2}]
    """

    subgraphs = []
    unused_inputs = set(range(len(inputs)))

    i_sum = set.union(*inputs) - output  # all summation indices

    while len(unused_inputs) > 0:
        g = set()
        q = [unused_inputs.pop()]
        while len(q) > 0:
            j = q.pop()
            g.add(j)
            i_tmp = i_sum & inputs[j]
            n = {k for k in unused_inputs if len(i_tmp & inputs[k]) > 0}
            q.extend(n)
            unused_inputs.difference_update(n)

        subgraphs.append(g)

    return subgraphs


def _bitmap_select(s, seq):
    """Select elements of ``seq`` which are marked by the bitmap set ``s``.

    E.g.:

        >>> list(_bitmap_select(0b11010, ['A', 'B', 'C', 'D', 'E']))
        ['B', 'D', 'E']
    """
    return (x for x, b in zip(seq, bin(s)[:1:-1]) if b == '1')


def _dp_calc_legs(g, all_tensors, s, inputs, i1_cut_i2_wo_output, i1_union_i2):
    """Calculates the effective outer indices of the intermediate tensor
    corresponding to the subgraph ``s``.
    """
    # set of remaining tensors (=g-s)
    r = g & (all_tensors ^ s)
    # indices of remaining indices:
    if r:
        i_r = set.union(*_bitmap_select(r, inputs))
    else:
        i_r = set()
    # contraction indices:
    i_contract = i1_cut_i2_wo_output - i_r
    return i1_union_i2 - i_contract


def _dp_compare_flops(cost1, cost2, i1_union_i2, size_dict, cost_cap, s1, s2, xn, g, all_tensors, inputs,
                      i1_cut_i2_wo_output, memory_limit, cntrct1, cntrct2):
    """Performs the inner comparison of whether the two subgraphs (the bitmaps
    ``s1`` and ``s2``) should be merged and added to the dynamic programming
    search. Will skip for a number of reasons:

    1. If the number of operations to form ``s = s1 | s2`` including previous
       contractions is above the cost-cap.
    2. If we've already found a better way of making ``s``.
    3. If the intermediate tensor corresponding to ``s`` is going to break the
       memory limit.
    """
    cost = cost1 + cost2 + helpers.compute_size_by_dict(i1_union_i2, size_dict)
    if cost <= cost_cap:
        s = s1 | s2
        if s not in xn or cost < xn[s][1]:
            i = _dp_calc_legs(g, all_tensors, s, inputs, i1_cut_i2_wo_output, i1_union_i2)
            mem = helpers.compute_size_by_dict(i, size_dict)
            if memory_limit is None or mem <= memory_limit:
                xn[s] = (i, cost, (cntrct1, cntrct2))


def _dp_compare_size(cost1, cost2, i1_union_i2, size_dict, cost_cap, s1, s2, xn, g, all_tensors, inputs,
                     i1_cut_i2_wo_output, memory_limit, cntrct1, cntrct2):
    """Like ``_dp_compare_flops`` but sieves the potential contraction based
    on the size of the intermediate tensor created, rather than the number of
    operations, and so calculates that first.
    """
    s = s1 | s2
    i = _dp_calc_legs(g, all_tensors, s, inputs, i1_cut_i2_wo_output, i1_union_i2)
    mem = helpers.compute_size_by_dict(i, size_dict)
    cost = max(cost1, cost2, mem)
    if cost <= cost_cap:
        if s not in xn or cost < xn[s][1]:
            if memory_limit is None or mem <= memory_limit:
                xn[s] = (i, cost, (cntrct1, cntrct2))


def simple_tree_tuple(seq):
    """Make a simple left to right binary tree out of iterable ``seq``.

        >>> tuple_nest([1, 2, 3, 4])
        (((1, 2), 3), 4)

    """
    return functools.reduce(lambda x, y: (x, y), seq)


def _dp_parse_out_single_term_ops(inputs, all_inds, ind_counts):
    """Take ``inputs`` and parse for single term index operations, i.e. where
    an index appears on one tensor and nowhere else.

    If a term is completely reduced to a scalar in this way it can be removed
    to ``inputs_done``. If only some indices can be summed then add a 'single
    term contraction' that will perform this summation.
    """
    i_single = {i for i, c in enumerate(all_inds) if ind_counts[c] == 1}
    inputs_parsed, inputs_done, inputs_contractions = [], [], []
    for j, i in enumerate(inputs):
        i_reduced = i - i_single
        if not i_reduced:
            # input reduced to scalar already - remove
            inputs_done.append((j, ))
        else:
            # if the input has any index reductions, add single contraction
            inputs_parsed.append(i_reduced)
            inputs_contractions.append((j, ) if i_reduced != i else j)

    return inputs_parsed, inputs_done, inputs_contractions


class DynamicProgramming(PathOptimizer):
    """
    Finds the optimal path of pairwise contractions without intermediate outer
    products based a dynamic programming approach presented in
    Phys. Rev. E 90, 033315 (2014) (the corresponding preprint is publically
    available at https://arxiv.org/abs/1304.6112). This method is especially
    well-suited in the area of tensor network states, where it usually
    outperforms all the other optimization strategies.

    This algorithm shows exponential scaling with the number of inputs
    in the worst case scenario (see example below). If the graph to be
    contracted consists of disconnected subgraphs, the algorithm scales
    linearly in the number of disconnected subgraphs and only exponentially
    with the number of inputs per subgraph.

    Parameters
    ----------
    minimize : {'flops', 'size'}, optional
        Whether to find the contraction that minimizes the number of
        operations or the size of the largest intermediate tensor.
    cost_cap : {True, False, int}, optional
        How to implement cost-capping:

            * True - iteratively increase the cost-cap
            * False - implement no cost-cap at all
            * int - use explicit cost cap

    search_outer : bool, optional
        In rare circumstances the optimal contraction may involve an outer
        product, this option allows searching such contractions but may well
        slow down the path finding considerably on all but very small graphs.
    """
    def __init__(self, minimize='flops', cost_cap=True, search_outer=False):

        # set whether inner function minimizes against flops or size
        self.minimize = minimize
        self._check_contraction = {
            'flops': _dp_compare_flops,
            'size': _dp_compare_size,
        }[self.minimize]

        # set whether inner function considers outer products
        self.search_outer = search_outer
        self._check_outer = {
            False: lambda x: x,
            True: lambda x: True,
        }[self.search_outer]

        self.cost_cap = cost_cap

    def __call__(self, inputs, output, size_dict, memory_limit=None):
        """
        Parameters
        ----------
        inputs : list
            List of sets that represent the lhs side of the einsum subscript
        output : set
            Set that represents the rhs side of the overall einsum subscript
        size_dict : dictionary
            Dictionary of index sizes
        memory_limit : int
            The maximum number of elements in a temporary array

        Returns
        -------
        path : list
            The contraction order (a list of tuples of ints).

        Examples
        --------
        >>> n_in = 3  # exponential scaling
        >>> n_out = 2 # linear scaling
        >>> s = dict()
        >>> i_all = []
        >>> for _ in range(n_out):
        >>>     i = [set() for _ in range(n_in)]
        >>>     for j in range(n_in):
        >>>         for k in range(j+1, n_in):
        >>>             c = oe.get_symbol(len(s))
        >>>             i[j].add(c)
        >>>             i[k].add(c)
        >>>             s[c] = 2
        >>>     i_all.extend(i)
        >>> o = DynamicProgramming()
        >>> o(i_all, set(), s)
        [(1, 2), (0, 4), (1, 2), (0, 2), (0, 1)]
        """
        ind_counts = Counter(itertools.chain(*inputs, output))
        all_inds = tuple(ind_counts)

        # convert all indices to integers (makes set operations ~10 % faster)
        symbol2int = {c: j for j, c in enumerate(all_inds)}
        inputs = [set(symbol2int[c] for c in i) for i in inputs]
        output = set(symbol2int[c] for c in output)
        size_dict = {symbol2int[c]: v for c, v in size_dict.items() if c in symbol2int}
        size_dict = [size_dict[j] for j in range(len(size_dict))]

        inputs, inputs_done, inputs_contractions = _dp_parse_out_single_term_ops(inputs, all_inds, ind_counts)

        if not inputs:
            # nothing left to do after single axis reductions!
            return _tree_to_sequence(simple_tree_tuple(inputs_done))

        # a list of all neccessary contraction expressions for each of the
        # disconnected subgraphs and their size
        subgraph_contractions = inputs_done
        subgraph_contractions_size = [1] * len(inputs_done)

        if self.search_outer:
            # optimize everything together if we are considering outer products
            subgraphs = [set(range(len(inputs)))]
        else:
            subgraphs = _find_disconnected_subgraphs(inputs, output)

        # the bitmap set of all tensors is computed as it is needed to
        # compute set differences: s1 - s2 transforms into
        # s1 & (all_tensors ^ s2)
        all_tensors = (1 << len(inputs)) - 1

        for g in subgraphs:

            # dynamic programming approach to compute x[n] for subgraph g;
            # x[n][set of n tensors] = (indices, cost, contraction)
            # the set of n tensors is represented by a bitmap: if bit j is 1,
            # tensor j is in the set, e.g. 0b100101 = {0,2,5}; set unions
            # (intersections) can then be computed by bitwise or (and);
            x = [None] * 2 + [dict() for j in range(len(g) - 1)]
            x[1] = OrderedDict((1 << j, (inputs[j], 0, inputs_contractions[j])) for j in g)

            # convert set of tensors g to a bitmap set:
            g = functools.reduce(lambda x, y: x | y, (1 << j for j in g))

            # try to find contraction with cost <= cost_cap and increase
            # cost_cap successively if no such contraction is found;
            # this is a major performance improvement; start with product of
            # output index dimensions as initial cost_cap
            subgraph_inds = set.union(*_bitmap_select(g, inputs))
            if self.cost_cap is True:
                cost_cap = helpers.compute_size_by_dict(subgraph_inds & output, size_dict)
            elif self.cost_cap is False:
                cost_cap = float('inf')
            else:
                cost_cap = self.cost_cap
            # set the factor to increase the cost by each iteration (ensure > 1)
            cost_increment = max(min(map(size_dict.__getitem__, subgraph_inds)), 2)

            while len(x[-1]) == 0:
                for n in range(2, len(x[1]) + 1):
                    xn = x[n]

                    # try to combine solutions from x[m] and x[n-m]
                    for m in range(1, n // 2 + 1):
                        for s1, (i1, cost1, cntrct1) in x[m].items():
                            for s2, (i2, cost2, cntrct2) in x[n - m].items():

                                # can only merge if s1 and s2 are disjoint
                                # and avoid e.g. s1={0}, s2={1} and s1={1}, s2={0}
                                if (not s1 & s2) and (m != n - m or s1 < s2):
                                    i1_cut_i2_wo_output = (i1 & i2) - output

                                    # maybe ignore outer products:
                                    if self._check_outer(i1_cut_i2_wo_output):

                                        i1_union_i2 = i1 | i2
                                        self._check_contraction(cost1, cost2, i1_union_i2, size_dict, cost_cap, s1, s2,
                                                                xn, g, all_tensors, inputs, i1_cut_i2_wo_output,
                                                                memory_limit, cntrct1, cntrct2)

                # increase cost cap for next iteration:
                cost_cap = cost_increment * cost_cap

            i, cost, contraction = list(x[-1].values())[0]
            subgraph_contractions.append(contraction)
            subgraph_contractions_size.append(helpers.compute_size_by_dict(i, size_dict))

        # sort the subgraph contractions by the size of the subgraphs in
        # ascending order (will give the cheapest contractions); note that
        # outer products should be performed pairwise (to use BLAS functions)
        subgraph_contractions = [
            subgraph_contractions[j]
            for j in sorted(range(len(subgraph_contractions_size)), key=subgraph_contractions_size.__getitem__)
        ]

        # build the final contraction tree
        tree = simple_tree_tuple(subgraph_contractions)
        return _tree_to_sequence(tree)


def dynamic_programming(inputs, output, size_dict, memory_limit=None, **kwargs):
    optimizer = DynamicProgramming(**kwargs)
    return optimizer(inputs, output, size_dict, memory_limit)


_AUTO_CHOICES = {}
for i in range(1, 5):
    _AUTO_CHOICES[i] = optimal
for i in range(5, 7):
    _AUTO_CHOICES[i] = branch_all
for i in range(7, 9):
    _AUTO_CHOICES[i] = branch_2
for i in range(9, 15):
    _AUTO_CHOICES[i] = branch_1


def auto(inputs, output, size_dict, memory_limit=None):
    """Finds the contraction path by automatically choosing the method based on
    how many input arguments there are.
    """
    N = len(inputs)
    return _AUTO_CHOICES.get(N, greedy)(inputs, output, size_dict, memory_limit)


_AUTO_HQ_CHOICES = {}
for i in range(1, 6):
    _AUTO_HQ_CHOICES[i] = optimal
for i in range(6, 17):
    _AUTO_HQ_CHOICES[i] = dynamic_programming


def auto_hq(inputs, output, size_dict, memory_limit=None):
    """Finds the contraction path by automatically choosing the method based on
    how many input arguments there are, but targeting a more generous
    amount of search time than ``'auto'``.
    """
    from .path_random import random_greedy_128

    N = len(inputs)
    return _AUTO_HQ_CHOICES.get(N, random_greedy_128)(inputs, output, size_dict, memory_limit)


_PATH_OPTIONS = {
    'auto': auto,
    'auto-hq': auto_hq,
    'optimal': optimal,
    'branch-all': branch_all,
    'branch-2': branch_2,
    'branch-1': branch_1,
    'greedy': greedy,
    'eager': greedy,
    'opportunistic': greedy,
    'dp': dynamic_programming,
    'dynamic-programming': dynamic_programming
}


def register_path_fn(name, fn):
    """Add path finding function ``fn`` as an option with ``name``.
    """
    if name in _PATH_OPTIONS:
        raise KeyError("Path optimizer '{}' already exists.".format(name))

    _PATH_OPTIONS[name.lower()] = fn


def get_path_fn(path_type):
    """Get the correct path finding function from str ``path_type``.
    """
    if path_type not in _PATH_OPTIONS:
        raise KeyError("Path optimizer '{}' not found, valid options are {}.".format(
            path_type, set(_PATH_OPTIONS.keys())))

    return _PATH_OPTIONS[path_type]