r""" Static order of nodes in dask graph Dask makes decisions on what tasks to prioritize both * Dynamically at runtime * Statically before runtime Dynamically we prefer to run tasks that were just made available. However when several tasks become available at the same time we have an opportunity to break ties in an intelligent way d | b c \ / a For example after we finish ``a`` we can choose to run either ``b`` or ``c`` next. Making small decisions like this can greatly affect our performance, especially because the order in which we run tasks affects the order in which we can release memory, which operationally we find to have a large affect on many computation. We want to run tasks in such a way that we keep only a small amount of data in memory at any given time. Static Ordering --------------- And so we create a total ordering over all nodes to serve as a tie breaker. We represent this ordering with a dictionary mapping keys to integer values. Lower scores have higher priority. These scores correspond to the order in which a sequential scheduler would visit each node. {'a': 0, 'c': 1, 'd': 2, 'b': 3} There are several ways in which we might order our keys. This is a nuanced process that has to take into account many different kinds of workflows, and operate efficiently in linear time. We strongly recommend that readers look at the docstrings of tests in dask/tests/test_order.py. These tests usually have graph types laid out very carefully to show the kinds of situations that often arise, and the order we would like to be determined. Policy ------ Work towards *small goals* with *big steps*. 1. **Small goals**: prefer tasks that have few total dependents and whose final dependents have few total dependencies. We prefer to prioritize those tasks that help branches of computation that can terminate quickly. With more detail, we compute the total number of dependencies that each task depends on (both its own dependencies, and the dependencies of its dependencies, and so on), and then we choose those tasks that drive towards results with a low number of total dependencies. We choose to prioritize tasks that work towards finishing shorter computations first. 2. **Big steps**: prefer tasks with many dependents However, many tasks work towards the same final dependents. Among those, we choose those tasks with the most work left to do. We want to finish the larger portions of a sub-computation before we start on the smaller ones. 3. **Name comparison**: break ties with key name Often graphs are made with regular keynames. When no other structural difference exists between two keys, use the key name to break ties. This relies on the regularity of graph constructors like dask.array to be a good proxy for ordering. This is usually a good idea and a sane default. """ from collections import defaultdict from math import log from .core import get_dependencies, get_deps, getcycle, reverse_dict # noqa: F401 from .utils_test import add, inc # noqa: F401 def order(dsk, dependencies=None): """Order nodes in dask 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 -------- >>> 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) ) # Single root nodes that depend on everything. These cause issues for # the current ordering algorithm, since we often hit the root node # and fell back to the key tie-breaker to choose which immediate dependency # to finish next, rather than finishing off subtrees. # So under the special case of a single root node that depends on the entire # tree, we skip processing it normally. # See https://github.com/dask/dask/issues/6745 root_nodes = {k for k, v in dependents.items() if not v} skip_root_node = len(root_nodes) == 1 and len(dsk) > 1 # 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]: if skip_root_node and item in root_nodes: continue 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 1 < len(now_keys): 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 type(init_stack) is 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 if skip_root_node and item in root_nodes: item = init_stack_pop() while item in result: item = init_stack_pop() inner_stacks_append([item]) return result def graph_metrics(dependencies, dependents, total_dependencies): r"""Useful measures of a graph used by ``dask.order.order`` Example DAG (a1 has no dependencies; b2 and c1 are root nodes): c1 | b1 b2 \ / a1 For each key we return: 1. **total_dependents**: The number of keys that can only be run after this key is run. The root nodes have value 1 while deep child nodes will have larger values. 1 | 2 1 \ / 4 2. **min_dependencies**: The minimum value of the total number of dependencies of all final dependents (see module-level comment for more). In other words, the minimum of ``ndependencies`` of root nodes connected to the current node. 3 | 3 2 \ / 2 3. **max_dependencies**: The maximum value of the total number of dependencies of all final dependents (see module-level comment for more). In other words, the maximum of ``ndependencies`` of root nodes connected to the current node. 3 | 3 2 \ / 3 4. **min_height**: The minimum height from a root node 0 | 1 0 \ / 1 5. **max_height**: The maximum height from a root node 0 | 1 0 \ / 2 Examples -------- >>> dsk = {'a1': 1, 'b1': (inc, 'a1'), 'b2': (inc, 'a1'), 'c1': (inc, 'b1')} >>> dependencies, dependents = get_deps(dsk) >>> _, total_dependencies = ndependencies(dependencies, dependents) >>> metrics = graph_metrics(dependencies, dependents, total_dependencies) >>> sorted(metrics.items()) [('a1', (4, 2, 3, 1, 2)), ('b1', (2, 3, 3, 1, 1)), ('b2', (1, 2, 2, 0, 0)), ('c1', (1, 3, 3, 0, 0))] Returns ------- metrics: Dict[key, Tuple[int, int, int, int, int]] """ result = {} num_needed = {k: len(v) for k, v in dependents.items() if v} current = [] current_pop = current.pop current_append = current.append for key, deps in dependents.items(): if not deps: val = total_dependencies[key] result[key] = (1, val, val, 0, 0) for child in dependencies[key]: num_needed[child] -= 1 if not num_needed[child]: current_append(child) while current: key = current_pop() parents = dependents[key] if len(parents) == 1: (parent,) = parents ( total_dependents, min_dependencies, max_dependencies, min_heights, max_heights, ) = result[parent] result[key] = ( 1 + total_dependents, min_dependencies, max_dependencies, 1 + min_heights, 1 + max_heights, ) else: ( total_dependents, min_dependencies, max_dependencies, min_heights, max_heights, ) = zip(*(result[parent] for parent in dependents[key])) result[key] = ( 1 + sum(total_dependents), min(min_dependencies), max(max_dependencies), 1 + min(min_heights), 1 + max(max_heights), ) for child in dependencies[key]: num_needed[child] -= 1 if not num_needed[child]: current_append(child) return result def ndependencies(dependencies, dependents): """Number of total data elements on which this key depends For each key we return the number of tasks that must be run for us to run this task. Examples -------- >>> dsk = {'a': 1, 'b': (inc, 'a'), 'c': (inc, 'b')} >>> dependencies, dependents = get_deps(dsk) >>> num_dependencies, total_dependencies = ndependencies(dependencies, dependents) >>> sorted(total_dependencies.items()) [('a', 1), ('b', 2), ('c', 3)] Returns ------- num_dependencies: Dict[key, int] total_dependencies: Dict[key, int] """ num_needed = {} result = {} for k, v in dependencies.items(): num_needed[k] = len(v) if not v: result[k] = 1 num_dependencies = num_needed.copy() current = [] current_pop = current.pop current_append = current.append for key in result: for parent in dependents[key]: num_needed[parent] -= 1 if not num_needed[parent]: current_append(parent) while current: key = current_pop() result[key] = 1 + sum(result[child] for child in dependencies[key]) for parent in dependents[key]: num_needed[parent] -= 1 if not num_needed[parent]: current_append(parent) return num_dependencies, result class StrComparable: """Wrap object so that it defaults to string comparison When comparing two objects of different types Python fails >>> 'a' < 1 Traceback (most recent call last): ... TypeError: '<' not supported between instances of 'str' and 'int' This class wraps the object so that, when this would occur it instead compares the string representation >>> StrComparable('a') < StrComparable(1) False """ __slots__ = ("obj",) def __init__(self, obj): self.obj = obj def __lt__(self, other): try: return self.obj < other.obj except Exception: return str(self.obj) < str(other.obj)