def order(dsk, dependencies=None):
    """Order nodes in the task graph.

    This produces an ordering over our tasks that we use to break ties when
    executing.  We do this ahead of time to reduce a bit of stress on the
    scheduler and also to assist in static analysis.

    This currently traverses the graph as a single-threaded scheduler would
    traverse it.  It breaks ties in the following ways:

    1.  Begin at a leaf node that is a dependency of a root node that has the
        largest subgraph (start hard things first)
    2.  Prefer tall branches with few dependents (start hard things first and
        try to avoid memory usage)
    3.  Prefer dependents that are dependencies of root nodes that have
        the smallest subgraph (do small goals that can terminate quickly)

    Examples
    --------
    >>> from flash.core.serve.dag.utils_test import add, inc
    >>> dsk = {'a': 1, 'b': 2, 'c': (inc, 'a'), 'd': (add, 'b', 'c')}
    >>> order(dsk)
    {'a': 0, 'c': 1, 'b': 2, 'd': 3}

    """
    if not dsk:
        return {}

    if dependencies is None:
        dependencies = {k: get_dependencies(dsk, k) for k in dsk}

    dependents = reverse_dict(dependencies)
    num_needed, total_dependencies = ndependencies(dependencies, dependents)
    metrics = graph_metrics(dependencies, dependents, total_dependencies)
    if len(metrics) != len(dsk):
        cycle = getcycle(dsk, None)
        raise RuntimeError(
            "Cycle detected between the following keys:\n  -> %s" % "\n  -> ".join(str(x) for x in cycle)
        )

    # Leaf nodes.  We choose one--the initial node--for each weakly connected subgraph.
    # Let's calculate the `initial_stack_key` as we determine `init_stack` set.
    init_stack = {
        # First prioritize large, tall groups, then prioritize the same as ``dependents_key``.
        key: (
            # at a high-level, work towards a large goal (and prefer tall and narrow)
            -max_dependencies,
            num_dependents - max_heights,
            # tactically, finish small connected jobs first
            min_dependencies,
            num_dependents - min_heights,  # prefer tall and narrow
            -total_dependents,  # take a big step
            # try to be memory efficient
            num_dependents,
            # tie-breaker
            StrComparable(key),
        )
        for key, num_dependents, (
            total_dependents,
            min_dependencies,
            max_dependencies,
            min_heights,
            max_heights,
        ) in ((key, len(dependents[key]), metrics[key]) for key, val in dependencies.items() if not val)
    }
    # `initial_stack_key` chooses which task to run at the very beginning.
    # This value is static, so we pre-compute as the value of this dict.
    initial_stack_key = init_stack.__getitem__

    def dependents_key(x):
        """Choose a path from our starting task to our tactical goal.

        This path is connected to a large goal, but focuses on completing a small goal and being memory efficient.

        """
        return (
            # Focus on being memory-efficient
            len(dependents[x]) - len(dependencies[x]) + num_needed[x],
            -metrics[x][3],  # min_heights
            # tie-breaker
            StrComparable(x),
        )

    def dependencies_key(x):
        """Choose which dependency to run as part of a reverse DFS.

        This is very similar to both ``initial_stack_key``.

        """
        num_dependents = len(dependents[x])
        (
            total_dependents,
            min_dependencies,
            max_dependencies,
            min_heights,
            max_heights,
        ) = metrics[x]
        # Prefer short and narrow instead of tall in narrow, because we're going in
        # reverse along dependencies.
        return (
            # at a high-level, work towards a large goal (and prefer short and narrow)
            -max_dependencies,
            num_dependents + max_heights,
            # tactically, finish small connected jobs first
            min_dependencies,
            num_dependents + min_heights,  # prefer short and narrow
            -total_dependencies[x],  # go where the work is
            # try to be memory efficient
            num_dependents - len(dependencies[x]) + num_needed[x],
            num_dependents,
            total_dependents,  # already found work, so don't add more
            # tie-breaker
            StrComparable(x),
        )

    def finish_now_key(x):
        """Determine the order of dependents that are ready to run and be released."""
        return -len(dependencies[x]), StrComparable(x)

    # Computing this for all keys can sometimes be relatively expensive :(
    partition_keys = {
        key: ((min_dependencies - total_dependencies[key] + 1) * (total_dependents - min_heights))
        for key, (
            total_dependents,
            min_dependencies,
            _,
            min_heights,
            _,
        ) in metrics.items()
    }

    result = {}
    i = 0

    # `inner_stask` is used to perform a DFS along dependencies.  Once emptied
    # (when traversing dependencies), this continue down a path along dependents
    # until a root node is reached.
    #
    # Sometimes, a better path along a dependent is discovered (i.e., something
    # that is easier to compute and doesn't requiring holding too much in memory).
    # In this case, the current `inner_stack` is appended to `inner_stacks` and
    # we begin a new DFS from the better node.
    #
    # A "better path" is determined by comparing `partition_keys`.
    inner_stacks = [[min(init_stack, key=initial_stack_key)]]
    inner_stacks_append = inner_stacks.append
    inner_stacks_extend = inner_stacks.extend
    inner_stacks_pop = inner_stacks.pop

    # Okay, now we get to the data structures used for fancy behavior.
    #
    # As we traverse nodes in the DFS along dependencies, we partition the dependents
    # via `partition_key`.  A dependent goes to:
    #    1) `inner_stack` if it's better than our current target,
    #    2) `next_nodes` if the partition key is lower than it's parent,
    #    3) `later_nodes` otherwise.
    # When the inner stacks are depleted, we process `next_nodes`.  If `next_nodes` is
    # empty (and `outer_stacks` is empty`), then we process `later_nodes` the same way.
    # These dicts use `partition_keys` as keys.  We process them by placing the values
    # in `outer_stack` so that the smallest keys will be processed first.
    next_nodes = defaultdict(list)
    later_nodes = defaultdict(list)

    # `outer_stack` is used to populate `inner_stacks`.  From the time we partition the
    # dependents of a node, we group them: one list per partition key per parent node.
    # This likely results in many small lists.  We do this to avoid sorting many larger
    # lists (i.e., to avoid n*log(n) behavior).  So, we have many small lists that we
    # partitioned, and we keep them in the order that we saw them (we will process them
    # in a FIFO manner).  By delaying sorting for as long as we can, we can first filter
    # out nodes that have already been computed.  All this complexity is worth it!
    outer_stack = []
    outer_stack_extend = outer_stack.extend
    outer_stack_pop = outer_stack.pop

    # Keep track of nodes that are in `inner_stack` or `inner_stacks` so we don't
    # process them again.
    seen = set()  # seen in an inner_stack (and has dependencies)
    seen_update = seen.update
    seen_add = seen.add

    # alias for speed
    set_difference = set.difference

    is_init_sorted = False
    while True:
        while inner_stacks:
            inner_stack = inner_stacks_pop()
            inner_stack_pop = inner_stack.pop
            while inner_stack:
                # Perform a DFS along dependencies until we complete our tactical goal
                item = inner_stack_pop()
                if item in result:
                    continue
                if num_needed[item]:
                    inner_stack.append(item)
                    deps = set_difference(dependencies[item], result)
                    if 1 < len(deps) < 1000:
                        inner_stack.extend(sorted(deps, key=dependencies_key, reverse=True))
                    else:
                        inner_stack.extend(deps)
                    seen_update(deps)
                    continue

                result[item] = i
                i += 1
                deps = dependents[item]

                # If inner_stack is empty, then we typically add the best dependent to it.
                # However, we don't add to it if we complete a node early via "finish_now" below
                # or if a dependent is already on an inner_stack.  In this case, we add the
                # dependents (not in an inner_stack) to next_nodes or later_nodes to handle later.
                # This serves three purposes:
                #   1. shrink `deps` so that it can be processed faster,
                #   2. make sure we don't process the same dependency repeatedly, and
                #   3. make sure we don't accidentally continue down an expensive-to-compute path.
                add_to_inner_stack = True
                if metrics[item][3] == 1:  # min_height
                    # Don't leave any dangling single nodes!  Finish all dependents that are
                    # ready and are also root nodes.
                    finish_now = {dep for dep in deps if not dependents[dep] and num_needed[dep] == 1}
                    if finish_now:
                        deps -= finish_now  # Safe to mutate
                        if len(finish_now) > 1:
                            finish_now = sorted(finish_now, key=finish_now_key)
                        for dep in finish_now:
                            result[dep] = i
                            i += 1
                        add_to_inner_stack = False

                if deps:
                    for dep in deps:
                        num_needed[dep] -= 1

                    already_seen = deps & seen
                    if already_seen:
                        if len(deps) == len(already_seen):
                            continue
                        add_to_inner_stack = False
                        deps -= already_seen

                    if len(deps) == 1:
                        # Fast path!  We trim down `deps` above hoping to reach here.
                        (dep,) = deps
                        if not inner_stack:
                            if add_to_inner_stack:
                                inner_stack = [dep]
                                inner_stack_pop = inner_stack.pop
                                seen_add(dep)
                                continue
                            key = partition_keys[dep]
                        else:
                            key = partition_keys[dep]
                            if key < partition_keys[inner_stack[0]]:
                                # Run before `inner_stack` (change tactical goal!)
                                inner_stacks_append(inner_stack)
                                inner_stack = [dep]
                                inner_stack_pop = inner_stack.pop
                                seen_add(dep)
                                continue
                        if key < partition_keys[item]:
                            next_nodes[key].append(deps)
                        else:
                            later_nodes[key].append(deps)
                    else:
                        # Slow path :(.  This requires grouping by partition_key.
                        dep_pools = defaultdict(list)
                        for dep in deps:
                            dep_pools[partition_keys[dep]].append(dep)
                        item_key = partition_keys[item]
                        if inner_stack:
                            # If we have an inner_stack, we need to look for a "better" path
                            prev_key = partition_keys[inner_stack[0]]
                            now_keys = []  # < inner_stack[0]
                            for key, vals in dep_pools.items():
                                if key < prev_key:
                                    now_keys.append(key)
                                elif key < item_key:
                                    next_nodes[key].append(vals)
                                else:
                                    later_nodes[key].append(vals)
                            if now_keys:
                                # Run before `inner_stack` (change tactical goal!)
                                inner_stacks_append(inner_stack)
                                if len(now_keys) > 1:
                                    now_keys.sort(reverse=True)
                                for key in now_keys:
                                    pool = dep_pools[key]
                                    if 1 < len(pool) < 100:
                                        pool.sort(key=dependents_key, reverse=True)
                                    inner_stacks_extend([dep] for dep in pool)
                                    seen_update(pool)
                                inner_stack = inner_stacks_pop()
                                inner_stack_pop = inner_stack.pop
                        else:
                            # If we don't have an inner_stack, then we don't need to look
                            # for a "better" path, but we do need traverse along dependents.
                            if add_to_inner_stack:
                                min_key = min(dep_pools)
                                min_pool = dep_pools.pop(min_key)
                                if len(min_pool) == 1:
                                    inner_stack = min_pool
                                    seen_update(inner_stack)
                                elif 10 * item_key > 11 * len(min_pool) * len(min_pool) * min_key:
                                    # Put all items in min_pool onto inner_stacks.
                                    # I know this is a weird comparison.  Hear me out.
                                    # Although it is often beneficial to put all of the items in `min_pool`
                                    # onto `inner_stacks` to process next, it is very easy to be overzealous.
                                    # Sometimes it is actually better to defer until `next_nodes` is handled.
                                    # We should only put items onto `inner_stacks` that we're reasonably
                                    # confident about.  The above formula is a best effort heuristic given
                                    # what we have easily available.  It is obviously very specific to our
                                    # choice of partition_key.  Dask tests take this route about 40%.
                                    if len(min_pool) < 100:
                                        min_pool.sort(key=dependents_key, reverse=True)
                                    inner_stacks_extend([dep] for dep in min_pool)
                                    inner_stack = inner_stacks_pop()
                                    seen_update(min_pool)
                                else:
                                    # Put one item in min_pool onto inner_stack and the rest into next_nodes.
                                    if len(min_pool) < 100:
                                        inner_stack = [min(min_pool, key=dependents_key)]
                                    else:
                                        inner_stack = [min_pool.pop()]
                                    next_nodes[min_key].append(min_pool)
                                    seen_update(inner_stack)

                                inner_stack_pop = inner_stack.pop
                            for key, vals in dep_pools.items():
                                if key < item_key:
                                    next_nodes[key].append(vals)
                                else:
                                    later_nodes[key].append(vals)

        if len(dependencies) == len(result):
            break  # all done!

        if next_nodes:
            for key in sorted(next_nodes, reverse=True):
                # `outer_stacks` may not be empty here--it has data from previous `next_nodes`.
                # Since we pop things off of it (onto `inner_nodes`), this means we handle
                # multiple `next_nodes` in a LIFO manner.
                outer_stack_extend(reversed(next_nodes[key]))
            next_nodes = defaultdict(list)

        while outer_stack:
            # Try to add a few items to `inner_stacks`
            deps = [x for x in outer_stack_pop() if x not in result]
            if deps:
                if 1 < len(deps) < 100:
                    deps.sort(key=dependents_key, reverse=True)
                inner_stacks_extend([dep] for dep in deps)
                seen_update(deps)
                break

        if inner_stacks:
            continue

        if later_nodes:
            # You know all those dependents with large keys we've been hanging onto to run "later"?
            # Well, "later" has finally come.
            next_nodes, later_nodes = later_nodes, next_nodes
            continue

        # We just finished computing a connected group.
        # Let's choose the first `item` in the next group to compute.
        # If we have few large groups left, then it's best to find `item` by taking a minimum.
        # If we have many small groups left, then it's best to sort.
        # If we have many tiny groups left, then it's best to simply iterate.
        if not is_init_sorted:
            prev_len = len(init_stack)
            if isinstance(init_stack, dict):
                init_stack = set(init_stack)
            init_stack = set_difference(init_stack, result)
            N = len(init_stack)
            m = prev_len - N
            # is `min` likely better than `sort`?
            if m >= N or N + (N - m) * log(N - m) < N * log(N):
                item = min(init_stack, key=initial_stack_key)
                continue

            if len(init_stack) < 10000:
                init_stack = sorted(init_stack, key=initial_stack_key, reverse=True)
            else:
                init_stack = list(init_stack)
            init_stack_pop = init_stack.pop
            is_init_sorted = True

        item = init_stack_pop()
        while item in result:
            item = init_stack_pop()
        inner_stacks_append([item])

    return result
Order nodes in the task graph.

    This produces an ordering over our tasks that we use to break ties when
    executing.  We do this ahead of time to reduce a bit of stress on the
    scheduler and also to assist in static analysis.

    This currently traverses the graph as a single-threaded scheduler would
    traverse it.  It breaks ties in the following ways:

    1.  Begin at a leaf node that is a dependency of a root node that has the
        largest subgraph (start hard things first)
    2.  Prefer tall branches with few dependents (start hard things first and
        try to avoid memory usage)
    3.  Prefer dependents that are dependencies of root nodes that have
        the smallest subgraph (do small goals that can terminate quickly)

    Examples
    --------
    >>> from flash.core.serve.dag.utils_test import add, inc
    >>> dsk = {'a': 1, 'b': 2, 'c': (inc, 'a'), 'd': (add, 'b', 'c')}
    >>> order(dsk)
    {'a': 0, 'c': 1, 'b': 2, 'd': 3}

    
src/flash/core/serve/dag/order.py