@@ -320,14 +320,15 @@ class HoloMap(UniformNdMapping, Overlayable):
                 ysamples = [(ly+uy)/2.0 for ly,uy in zip(yedges[:-1], yedges[1:])]
 
                 Y,X = np.meshgrid(ysamples, xsamples)
-                linsamples = zip(X.flat, Y.flat)
+                linsamples = list(zip(X.flat, Y.flat))
             else:
                 raise NotImplementedError("Regular sampling not implemented "
                                           "for high-dimensional Views.")
 
             samples = list(util.unique_iterator(self.last.closest(linsamples)))
 
-        sampled = self.clone([(k, view.sample(samples, **sample_values))
+        sampled = self.clone([(k, view.sample(samples, closest=False,
+                                              **sample_values))
                               for k, view in self.data.items()])
         return sampled.table()
 
import itertools
import types
from numbers import Number
from itertools import groupby
from functools import partial

import numpy as np
import param

from . import traversal, util
from .dimension import OrderedDict, Dimension, ViewableElement
from .layout import Layout, AdjointLayout, NdLayout
from .ndmapping import UniformNdMapping, NdMapping, item_check
from .overlay import Overlay, CompositeOverlay, NdOverlay, Overlayable
from .options import Store, StoreOptions

class HoloMap(UniformNdMapping, Overlayable):
    """
    A HoloMap can hold any number of DataLayers indexed by a list of
    dimension values. It also has a number of properties, which can find
    the x- and y-dimension limits and labels.
    """

    data_type = (ViewableElement, NdMapping, Layout)

    def overlay(self, dimensions=None, **kwargs):
        """
        Splits the UniformNdMapping along a specified number of dimensions and
        overlays items in the split out Maps.

        Shows all HoloMap data When no dimensions are specified.
        """
        dimensions = self._valid_dimensions(dimensions)
        if len(dimensions) == self.ndims:
            with item_check(False):
                return NdOverlay(self, **kwargs).reindex(dimensions)
        else:
            dims = [d for d in self.kdims if d not in dimensions]
            return self.groupby(dims, group_type=NdOverlay, **kwargs)


    def grid(self, dimensions=None, **kwargs):
        """
        GridSpace takes a list of one or two dimensions, and lays out the containing
        Views along these axes in a GridSpace.

        Shows all HoloMap data When no dimensions are specified.
        """
        dimensions = self._valid_dimensions(dimensions)
        if len(dimensions) == self.ndims:
            with item_check(False):
                return GridSpace(self, **kwargs).reindex(dimensions)
        return self.groupby(dimensions, container_type=GridSpace, **kwargs)


    def layout(self, dimensions=None, **kwargs):
        """
        GridSpace takes a list of one or two dimensions, and lays out the containing
        Views along these axes in a GridSpace.

        Shows all HoloMap data When no dimensions are specified.
        """
        dimensions = self._valid_dimensions(dimensions)
        if len(dimensions) == self.ndims:
            with item_check(False):
                return NdLayout(self, **kwargs).reindex(dimensions)
        return self.groupby(dimensions, container_type=NdLayout, **kwargs)


    def split_overlays(self):
        """
        Given a UniformNdMapping of Overlays of N layers, split out the layers into
        N separate Maps.
        """
        if not issubclass(self.type, CompositeOverlay):
            return None, self.clone()

        item_maps = OrderedDict()
        for k, overlay in self.data.items():
            for key, el in overlay.items():
                if key not in item_maps:
                    item_maps[key] = [(k, el)]
                else:
                    item_maps[key].append((k, el))

        maps, keys = [], []
        for k, layermap in item_maps.items():
            maps.append(self.clone(layermap))
            keys.append(k)
        return keys, maps


    def _dimension_keys(self):
        """
        Helper for __mul__ that returns the list of keys together with
        the dimension labels.
        """
        return [tuple(zip([d.name for d in self.kdims], [k] if self.ndims == 1 else k))
                for k in self.keys()]


    def _dynamic_mul(self, dimensions, other, keys):
        """
        Implements dynamic version of overlaying operation overlaying
        DynamicMaps and HoloMaps where the key dimensions of one is
        a strict superset of the other.
        """
        # If either is a HoloMap compute Dimension values
        if not isinstance(self, DynamicMap) or not isinstance(other, DynamicMap):
            keys = sorted((d, v) for k in keys for d, v in k)
            grouped =  dict([(g, [v for _, v in group])
                             for g, group in groupby(keys, lambda x: x[0])])
            dimensions = [d(values=grouped[d.name]) for d in dimensions]
            map_obj = None

        map_obj = self if isinstance(self, DynamicMap) else other

        def dynamic_mul(*key, **kwargs):
            layers = []
            try:
                if isinstance(self, DynamicMap):
                    safe_key = () if not self.kdims else key
                    _, self_el = util.get_dynamic_item(self, dimensions, safe_key)
                    if self_el is not None:
                        layers.append(self_el)
                else:
                    layers.append(self[key])
            except KeyError:
                pass
            try:
                if isinstance(other, DynamicMap):
                    safe_key = () if not other.kdims else key
                    _, other_el = util.get_dynamic_item(other, dimensions, safe_key)
                    if other_el is not None:
                        layers.append(other_el)
                else:
                    layers.append(other[key])
            except KeyError:
                pass
            return Overlay(layers)
        callback = Callable(dynamic_mul, inputs=[self, other])
        if map_obj:
            return map_obj.clone(callback=callback, shared_data=False,
                                 kdims=dimensions, streams=[])
        else:
            return DynamicMap(callback=callback, kdims=dimensions)


    def __mul__(self, other):
        """
        The mul (*) operator implements overlaying of different Views.
        This method tries to intelligently overlay Maps with differing
        keys. If the UniformNdMapping is mulled with a simple
        ViewableElement each element in the UniformNdMapping is
        overlaid with the ViewableElement. If the element the
        UniformNdMapping is mulled with is another UniformNdMapping it
        will try to match up the dimensions, making sure that items
        with completely different dimensions aren't overlaid.
        """
        if isinstance(other, HoloMap):
            self_set = {d.name for d in self.kdims}
            other_set = {d.name for d in other.kdims}

            # Determine which is the subset, to generate list of keys and
            # dimension labels for the new view
            self_in_other = self_set.issubset(other_set)
            other_in_self = other_set.issubset(self_set)
            dims = [other.kdims, self.kdims] if self_in_other else [self.kdims, other.kdims]
            dimensions = util.merge_dimensions(dims)

            if self_in_other and other_in_self: # superset of each other
                keys = self._dimension_keys() + other._dimension_keys()
                super_keys = util.unique_iterator(keys)
            elif self_in_other: # self is superset
                dimensions = other.kdims
                super_keys = other._dimension_keys()
            elif other_in_self: # self is superset
                super_keys = self._dimension_keys()
            else: # neither is superset
                raise Exception('One set of keys needs to be a strict subset of the other.')

            if isinstance(self, DynamicMap) or isinstance(other, DynamicMap):
                return self._dynamic_mul(dimensions, other, super_keys)

            items = []
            for dim_keys in super_keys:
                # Generate keys for both subset and superset and sort them by the dimension index.
                self_key = tuple(k for p, k in sorted(
                    [(self.get_dimension_index(dim), v) for dim, v in dim_keys
                     if dim in self.kdims]))
                other_key = tuple(k for p, k in sorted(
                    [(other.get_dimension_index(dim), v) for dim, v in dim_keys
                     if dim in other.kdims]))
                new_key = self_key if other_in_self else other_key
                # Append SheetOverlay of combined items
                if (self_key in self) and (other_key in other):
                    items.append((new_key, self[self_key] * other[other_key]))
                elif self_key in self:
                    items.append((new_key, Overlay([self[self_key]])))
                else:
                    items.append((new_key, Overlay([other[other_key]])))
            return self.clone(items, kdims=dimensions, label=self._label, group=self._group)
        elif isinstance(other, self.data_type):
            if isinstance(self, DynamicMap):
                def dynamic_mul(*args, **kwargs):
                    element = self[args]
                    return element * other
                callback = Callable(dynamic_mul, inputs=[self, other])
                return self.clone(shared_data=False, callback=callback,
                                  streams=[])
            items = [(k, v * other) for (k, v) in self.data.items()]
            return self.clone(items, label=self._label, group=self._group)
        else:
            return NotImplemented


    def __add__(self, obj):
        return Layout.from_values([self, obj])


    def __lshift__(self, other):
        if isinstance(other, (ViewableElement, UniformNdMapping)):
            return AdjointLayout([self, other])
        elif isinstance(other, AdjointLayout):
            return AdjointLayout(other.data+[self])
        else:
            raise TypeError('Cannot append {0} to a AdjointLayout'.format(type(other).__name__))


    def collate(self, merge_type=None, drop=[], drop_constant=False):
        """
        Collation allows collapsing nested HoloMaps by merging
        their dimensions. In the simple case a HoloMap containing
        other HoloMaps can easily be joined in this way. However
        collation is particularly useful when the objects being
        joined are deeply nested, e.g. you want to join multiple
        Layouts recorded at different times, collation will return
        one Layout containing HoloMaps indexed by Time. Changing
        the merge_type will allow merging the outer Dimension
        into any other UniformNdMapping type.

        Specific dimensions may be dropped if they are redundant
        by supplying them in a list. Enabling drop_constant allows
        ignoring any non-varying dimensions during collation.
        """
        from .element import Collator
        merge_type=merge_type if merge_type else self.__class__
        return Collator(self, merge_type=merge_type, drop=drop,
                        drop_constant=drop_constant)()


    def collapse(self, dimensions=None, function=None, spreadfn=None, **kwargs):
        """
        Allows collapsing one of any number of key dimensions
        on the HoloMap. Homogenous Elements may be collapsed by
        supplying a function, inhomogenous elements are merged.
        """
        from .operation import MapOperation
        if not dimensions:
            dimensions = self.kdims
        if not isinstance(dimensions, list): dimensions = [dimensions]
        if self.ndims > 1 and len(dimensions) != self.ndims:
            groups = self.groupby([dim for dim in self.kdims
                                   if dim not in dimensions])
        elif all(d in self.kdims for d in dimensions):
            groups = HoloMap([(0, self)])
        else:
            raise KeyError("Supplied dimensions not found.")

        collapsed = groups.clone(shared_data=False)
        for key, group in groups.items():
            if isinstance(function, MapOperation):
                collapsed[key] = function(group, **kwargs)
            else:
                group_data = [el.data for el in group]
                args = (group_data, function, group.last.kdims)
                if hasattr(group.last, 'interface'):
                    col_data = group.type(group.table().aggregate(group.last.kdims, function, spreadfn, **kwargs))

                else:
                    data = group.type.collapse_data(*args, **kwargs)
                    col_data = group.last.clone(data)
                collapsed[key] = col_data
        return collapsed if self.ndims > 1 else collapsed.last


    def sample(self, samples=[], bounds=None, **sample_values):
        """
        Sample each Element in the UniformNdMapping by passing either a list of
        samples or a tuple specifying the number of regularly spaced
        samples per dimension. Alternatively, a single sample may be
        requested using dimension-value pairs. Optionally, the bounds
        argument can be used to specify the bounding extent from which
        the coordinates are to regularly sampled. Regular sampling
        assumes homogenous and regularly sampled data.

        For 1D sampling, the shape is simply as the desired number of
        samples (and not a tuple). The bounds format for 1D sampling
        is the tuple (lower, upper) and the tuple (left, bottom,
        right, top) for 2D sampling.
        """
        dims = self.last.ndims
        if isinstance(samples, tuple) or np.isscalar(samples):
            if dims == 1:
                xlim = self.last.range(0)
                lower, upper = (xlim[0], xlim[1]) if bounds is None else bounds
                edges = np.linspace(lower, upper, samples+1)
                linsamples = [(l+u)/2.0 for l,u in zip(edges[:-1], edges[1:])]
            elif dims == 2:
                (rows, cols) = samples
                if bounds:
                    (l,b,r,t) = bounds
                else:
                    l, r = self.last.range(0)
                    b, t = self.last.range(1)

                xedges = np.linspace(l, r, cols+1)
                yedges = np.linspace(b, t, rows+1)
                xsamples = [(lx+ux)/2.0 for lx,ux in zip(xedges[:-1], xedges[1:])]
                ysamples = [(ly+uy)/2.0 for ly,uy in zip(yedges[:-1], yedges[1:])]

                Y,X = np.meshgrid(ysamples, xsamples)
                linsamples = zip(X.flat, Y.flat)
            else:
                raise NotImplementedError("Regular sampling not implemented "
                                          "for high-dimensional Views.")

            samples = list(util.unique_iterator(self.last.closest(linsamples)))

        sampled = self.clone([(k, view.sample(samples, **sample_values))
                              for k, view in self.data.items()])
        return sampled.table()


    def reduce(self, dimensions=None, function=None, **reduce_map):
        """
        Reduce each Element in the HoloMap using a function supplied
        via the kwargs, where the keyword has to match a particular
        dimension in the Elements.
        """
        from ..element import Table
        reduced_items = [(k, v.reduce(dimensions, function, **reduce_map))
                         for k, v in self.items()]
        if not isinstance(reduced_items[0][1], Table):
            params = dict(util.get_param_values(self.last),
                          kdims=self.kdims, vdims=self.last.vdims)
            return Table(reduced_items, **params)
        return self.clone(reduced_items).table()


    def relabel(self, label=None, group=None, depth=1):
        # Identical to standard relabel method except for default depth of 1
        return super(HoloMap, self).relabel(label=label, group=group, depth=depth)


    def hist(self, num_bins=20, bin_range=None, adjoin=True, individually=True, **kwargs):
        histmaps = [self.clone(shared_data=False) for _ in
                    kwargs.get('dimension', range(1))]

        if individually:
            map_range = None
        else:
            if 'dimension' not in kwargs:
                raise Exception("Please supply the dimension to compute a histogram for.")
            map_range = self.range(kwargs['dimension'])
        bin_range = map_range if bin_range is None else bin_range
        style_prefix = 'Custom[<' + self.name + '>]_'
        if issubclass(self.type, (NdOverlay, Overlay)) and 'index' not in kwargs:
            kwargs['index'] = 0
        for k, v in self.data.items():
            hists = v.hist(adjoin=False, bin_range=bin_range,
                           individually=individually, num_bins=num_bins,
                           style_prefix=style_prefix, **kwargs)
            if isinstance(hists, Layout):
                for i, hist in enumerate(hists):
                    histmaps[i][k] = hist
            else:
                histmaps[0][k] = hists

        if adjoin:
            layout = self
            for hist in histmaps:
                layout = (layout << hist)
            if issubclass(self.type, (NdOverlay, Overlay)):
                layout.main_layer = kwargs['index']
            return layout
        else:
            if len(histmaps) > 1:
                return Layout.from_values(histmaps)
            else:
                return histmaps[0]


class Callable(param.Parameterized):
    """
    Callable allows wrapping callbacks on one or more DynamicMaps
    allowing their inputs (and in future outputs) to be defined.
    This makes it possible to wrap DynamicMaps with streams and
    makes it possible to traverse the graph of operations applied
    to a DynamicMap. Additionally a Callable will memoize the last
    returned value based on the arguments to the function and the
    state of all streams on its inputs, to avoid calling the function
    unnecessarily.

    A Callable may also specify a stream_mapping which specifies the
    objects that are associated with interactive (i.e linked) streams
    when composite objects such as Layouts are returned from the
    callback. This is required for building interactive, linked
    visualizations (for the backends that support them) when returning
    Layouts, NdLayouts or GridSpace objects.

    The mapping should map from an appropriate key to a list of
    streams associated with the selected object. The appropriate key
    may be a type[.group][.label] specification for Layouts, an
    integer index or a suitable NdLayout/GridSpace key. For more
    information see the DynamicMap tutorial at holoviews.org.
    """

    callable = param.Callable(default=None, doc="""
         The callable function being wrapped.""")

    inputs = param.List(default=[], doc="""
         The list of inputs the callable function is wrapping.""")

    stream_mapping = param.Dict(default={}, doc="""
         Defines how streams should be mapped to objects returned by
         the Callable, e.g. when it returns a Layout.""")

    def __init__(self, callable, **params):
        super(Callable, self).__init__(callable=callable, **params)
        self._memoized = {}

    def __call__(self, *args, **kwargs):
        inputs = [i for i in self.inputs if isinstance(i, DynamicMap)]
        streams = [s for i in inputs for s in get_nested_streams(i)]
        values = tuple(tuple(sorted(s.contents.items())) for s in streams)
        key = args + tuple(sorted(kwargs.items())) + values

        hashed_key = util.deephash(key)
        ret = self._memoized.get(hashed_key, None)
        if hashed_key and ret is None:
            ret = self.callable(*args, **kwargs)
            self._memoized = {hashed_key : ret}
        return ret


def get_nested_streams(dmap):
    """
    Get all (potentially nested) streams from DynamicMap with Callable
    callback.
    """
    layer_streams = list(dmap.streams)
    if not isinstance(dmap.callback, Callable):
        return list(set(layer_streams))
    for o in dmap.callback.inputs:
        if isinstance(o, DynamicMap):
            layer_streams += get_nested_streams(o)
    return list(set(layer_streams))


class DynamicMap(HoloMap):
    """
    A DynamicMap is a type of HoloMap where the elements are dynamically
    generated by a callable. The callable is invoked with values
    associated with the key dimensions or with values supplied by stream
    parameters.
    """

    # Declare that callback is a positional parameter (used in clone)
    __pos_params = ['callback']

    callback = param.ClassSelector(class_=Callable, doc="""
        The callable used to generate the elements. The arguments to the
        callable includes any number of declared key dimensions as well
        as any number of stream parameters defined on the input streams.

        If the callable is an instance of Callable it will be used
        directly, otherwise it will be automatically wrapped in one.""")

    streams = param.List(default=[], doc="""
       List of Stream instances to associate with the DynamicMap. The
       set of parameter values across these streams will be supplied as
       keyword arguments to the callback when the events are received,
       updating the streams.""" )

    cache_size = param.Integer(default=500, doc="""
       The number of entries to cache for fast access. This is an LRU
       cache where the least recently used item is overwritten once
       the cache is full.""")

    sampled = param.Boolean(default=False, doc="""
       Allows defining a DynamicMap without defining the dimension
       bounds or values. The DynamicMap may then be explicitly sampled
       via getitem or the sampling is determined during plotting by a
       HoloMap with fixed sampling.
       """)

    def __init__(self, callback, initial_items=None, **params):
        if not isinstance(callback, Callable):
            callback = Callable(callback)
        super(DynamicMap, self).__init__(initial_items, callback=callback, **params)

        # Set source to self if not already specified
        for stream in self.streams:
            if stream.source is None:
                stream.source = self

    def _initial_key(self):
        """
        Construct an initial key for based on the lower range bounds or
        values on the key dimensions.
        """
        key = []
        undefined = []
        for kdim in self.kdims:
            if kdim.values:
                key.append(kdim.values[0])
            elif kdim.range:
                key.append(kdim.range[0])
            else:
                undefined.append(kdim)
        if undefined:
            raise KeyError('dimensions do not specify a range or values, '
                           'cannot supply initial key' % ', '.join(undefined))
        return tuple(key)


    def _validate_key(self, key):
        """
        Make sure the supplied key values are within the bounds
        specified by the corresponding dimension range and soft_range.
        """
        key = util.wrap_tuple(key)
        assert len(key) == len(self.kdims)
        for ind, val in enumerate(key):
            kdim = self.kdims[ind]
            low, high = util.max_range([kdim.range, kdim.soft_range])
            if low is not np.NaN:
                if val < low:
                    raise StopIteration("Key value %s below lower bound %s"
                                        % (val, low))
            if high is not np.NaN:
                if val > high:
                    raise StopIteration("Key value %s above upper bound %s"
                                        % (val, high))

    def event(self, trigger=True, **kwargs):
        """
        This method allows any of the available stream parameters
        (renamed as appropriate) to be updated in an event.
        """
        stream_params = set(util.stream_parameters(self.streams))
        for k in stream_params - set(kwargs.keys()):
            raise KeyError('Key %r does not correspond to any stream parameter')

        updated_streams = []
        for stream in self.streams:
            applicable_kws = {k:v for k,v in kwargs.items()
                              if k in set(stream.contents.keys())}
            rkwargs = util.rename_stream_kwargs(stream, applicable_kws, reverse=True)
            stream.update(**dict(rkwargs, trigger=False))
            updated_streams.append(stream)

        if updated_streams and trigger:
            updated_streams[0].trigger(updated_streams)


    def _style(self, retval):
        """
        Use any applicable OptionTree of the DynamicMap to apply options
        to the return values of the callback.
        """
        if self.id not in Store.custom_options():
            return retval
        spec = StoreOptions.tree_to_dict(Store.custom_options()[self.id])
        return retval(spec)


    def _execute_callback(self, *args):
        """
        Execute the callback, validating both the input key and output
        key where applicable.
        """
        self._validate_key(args)      # Validate input key

        # Additional validation needed to ensure kwargs don't clash
        kdims = [kdim.name for kdim in self.kdims]
        kwarg_items = [s.contents.items() for s in self.streams]
        flattened = [(k,v) for kws in kwarg_items for (k,v) in kws
                     if k not in kdims]
        retval = self.callback(*args, **dict(flattened))
        return self._style(retval)


    def clone(self, data=None, shared_data=True, new_type=None, *args, **overrides):
        """
        Clone method to adapt the slightly different signature of
        DynamicMap that also overrides Dimensioned clone to avoid
        checking items if data is unchanged.
        """
        if data is None and shared_data:
            data = self.data
        return super(UniformNdMapping, self).clone(overrides.pop('callback', self.callback),
                                                   shared_data, new_type,
                                                   *(data,) + args, **overrides)

    def reset(self):
        """
        Return a cleared dynamic map with a cleared cached
        """
        self.data = OrderedDict()
        return self


    def _cross_product(self, tuple_key, cache, data_slice):
        """
        Returns a new DynamicMap if the key (tuple form) expresses a
        cross product, otherwise returns None. The cache argument is a
        dictionary (key:element pairs) of all the data found in the
        cache for this key.

        Each key inside the cross product is looked up in the cache
        (self.data) to check if the appropriate element is
        available. Otherwise the element is computed accordingly.

        The data_slice may specify slices into each value in the
        the cross-product.
        """
        if not any(isinstance(el, (list, set)) for el in tuple_key):
            return None
        if len(tuple_key)==1:
            product = tuple_key[0]
        else:
            args = [set(el) if isinstance(el, (list,set))
                    else set([el]) for el in tuple_key]
            product = itertools.product(*args)

        data = []
        for inner_key in product:
            key = util.wrap_tuple(inner_key)
            if key in cache:
                val = cache[key]
            else:
                val = self._execute_callback(*key)
            if data_slice:
                val = self._dataslice(val, data_slice)
            data.append((key, val))
        product = self.clone(data)

        if data_slice:
            from ..util import Dynamic
            return Dynamic(product, operation=lambda obj: obj[data_slice],
                           shared_data=True)
        return product


    def _slice_bounded(self, tuple_key, data_slice):
        """
        Slices bounded DynamicMaps by setting the soft_ranges on
        key dimensions and applies data slice to cached and dynamic
        values.
        """
        slices = [el for el in tuple_key if isinstance(el, slice)]
        if any(el.step for el in slices):
            raise Exception("DynamicMap slices cannot have a step argument")
        elif len(slices) not in [0, len(tuple_key)]:
            raise Exception("Slices must be used exclusively or not at all")
        elif not slices:
            return None

        sliced = self.clone(self)
        for i, slc in enumerate(tuple_key):
            (start, stop) = slc.start, slc.stop
            if start is not None and start < sliced.kdims[i].range[0]:
                raise Exception("Requested slice below defined dimension range.")
            if stop is not None and stop > sliced.kdims[i].range[1]:
                raise Exception("Requested slice above defined dimension range.")
            sliced.kdims[i].soft_range = (start, stop)
        if data_slice:
            if not isinstance(sliced, DynamicMap):
                return self._dataslice(sliced, data_slice)
            else:
                from ..util import Dynamic
                if len(self):
                    slices = [slice(None) for _ in range(self.ndims)] + list(data_slice)
                    sliced = super(DynamicMap, sliced).__getitem__(tuple(slices))
                return Dynamic(sliced, operation=lambda obj: obj[data_slice],
                               shared_data=True)
        return sliced


    def __getitem__(self, key):
        """
        Return an element for any key chosen key. Also allows for usual
        deep slicing semantics by slicing values in the cache and
        applying the deep slice to newly generated values.
        """
        # Split key dimensions and data slices
        sample = False
        if key is Ellipsis:
            return self
        elif isinstance(key, (list, set)) and all(isinstance(v, tuple) for v in key):
            map_slice, data_slice = key, ()
            sample = True
        else:
            map_slice, data_slice = self._split_index(key)
        tuple_key = util.wrap_tuple_streams(map_slice, self.kdims, self.streams)

        # Validation
        if not sample:
            sliced = self._slice_bounded(tuple_key, data_slice)
            if sliced is not None:
                return sliced

        # Cache lookup
        try:
            dimensionless = util.dimensionless_contents(get_nested_streams(self),
                                                        self.kdims, no_duplicates=False)
            if dimensionless:
                raise KeyError('Using dimensionless streams disables DynamicMap cache')
            cache = super(DynamicMap,self).__getitem__(key)
        except KeyError as e:
            cache = None

        # If the key expresses a cross product, compute the elements and return
        product = self._cross_product(tuple_key, cache.data if cache else {}, data_slice)
        if product is not None:
            return product

        # Not a cross product and nothing cached so compute element.
        if cache is not None: return cache
        val = self._execute_callback(*tuple_key)
        if data_slice:
            val = self._dataslice(val, data_slice)
        self._cache(tuple_key, val)
        return val


    def select(self, selection_specs=None, **kwargs):
        """
        Allows slicing or indexing into the DynamicMap objects by
        supplying the dimension and index/slice as key value
        pairs. Select descends recursively through the data structure
        applying the key dimension selection and applies to dynamically
        generated items by wrapping the callback.

        The selection may also be selectively applied to specific
        objects by supplying the selection_specs as an iterable of
        type.group.label specs, types or functions.
        """
        if selection_specs is not None and not isinstance(selection_specs, (list, tuple)):
            selection_specs = [selection_specs]
        selection = super(DynamicMap, self).select(selection_specs, **kwargs)
        def dynamic_select(obj):
            if selection_specs is not None:
                matches = any(obj.matches(spec) for spec in selection_specs)
            else:
                matches = True
            if matches:
                return obj.select(**kwargs)
            return obj

        if not isinstance(selection, DynamicMap):
            return dynamic_select(selection)
        else:
            from ..util import Dynamic
            return Dynamic(selection, operation=dynamic_select,
                           shared_data=True)


    def _cache(self, key, val):
        """
        Request that a key/value pair be considered for caching.
        """
        cache_size = (1 if util.dimensionless_contents(self.streams, self.kdims)
                      else self.cache_size)
        if len(self) >= cache_size:
            first_key = next(k for k in self.data)
            self.data.pop(first_key)
        self.data[key] = val


    def relabel(self, label=None, group=None, depth=1):
        """
        Assign a new label and/or group to an existing LabelledData
        object, creating a clone of the object with the new settings.
        """
        relabelled = super(DynamicMap, self).relabel(label, group, depth)
        if depth > 0:
            from ..util import Dynamic
            def dynamic_relabel(obj):
                return obj.relabel(group=group, label=label, depth=depth-1)
            return Dynamic(relabelled, shared_data=True, operation=dynamic_relabel)
        return relabelled


    def redim(self, specs=None, **dimensions):
        """
        Replaces existing dimensions in an object with new dimensions
        or changing specific attributes of a dimensions. Dimension
        mapping should map between the old dimension name and a
        dictionary of the new attributes, a completely new dimension
        or a new string name.
        """
        redimmed = super(DynamicMap, self).redim(specs, **dimensions)
        from ..util import Dynamic
        def dynamic_redim(obj):
            return obj.redim(specs, **dimensions)
        return Dynamic(redimmed, shared_data=True, operation=dynamic_redim)


    def collate(self):
        """
        Collation allows reorganizing DynamicMaps with invalid nesting
        hierarchies. This is particularly useful when defining
        DynamicMaps returning an (Nd)Layout or GridSpace
        types. Collating will split the DynamicMap into individual
        DynamicMaps for each item in the container. Note that the
        composite object has to be of consistent length and types for
        this to work correctly.
        """
        # Initialize
        if self.last is not None:
            initialized = self
        else:
            initialized = self.clone()
            initialized[initialized._initial_key()]

        if not isinstance(initialized.last, (Layout, NdLayout, GridSpace)):
            return self

        container = initialized.last.clone(shared_data=False)

        # Get stream mapping from callback
        remapped_streams = []
        streams = self.callback.stream_mapping
        for i, (k, v) in enumerate(initialized.last.data.items()):
            vstreams = streams.get(i, [])
            if not vstreams:
                if isinstance(initialized.last, Layout):
                    for l in range(len(k)):
                        path = '.'.join(k[:l])
                        if path in streams:
                            vstreams = streams[path]
                            break
                else:
                    vstreams = streams.get(k, [])
            if any(s in remapped_streams for s in vstreams):
                raise ValueError(
                    "The stream_mapping supplied on the Callable "
                    "is ambiguous please supply more specific Layout "
                    "path specs.")
            remapped_streams += vstreams

            # Define collation callback
            def collation_cb(*args, **kwargs):
                return self[args][kwargs['selection_key']]
            callback = Callable(partial(collation_cb, selection_key=k),
                                inputs=[self])
            vdmap = self.clone(callback=callback, shared_data=False,
                               streams=vstreams)

            # Remap source of streams
            for stream in vstreams:
                if stream.source is self:
                    stream.source = vdmap
            container[k] = vdmap

        unmapped_streams = [repr(stream) for stream in self.streams
                            if (stream.source is self) and
                            (stream not in remapped_streams)
                            and stream.linked]
        if unmapped_streams:
            raise ValueError(
                'The following streams are set to be automatically '
                'linked to a plot, but no stream_mapping specifying '
                'which item in the (Nd)Layout to link it to was found:\n%s'
                % ', '.join(unmapped_streams)
            )
        return container


    def groupby(self, dimensions=None, container_type=None, group_type=None, **kwargs):
        """
        Implements a dynamic version of a groupby, which will
        intelligently expand either the inner or outer dimensions
        depending on whether the container_type or group_type is dynamic.

        To apply a groupby to a DynamicMap the dimensions, which are
        expanded into a non-dynamic type must define a fixed sampling
        via the values attribute.

        Using the dynamic groupby makes it incredibly easy to generate
        dynamic views into a high-dimensional space while taking
        advantage of the capabilities of NdOverlay, GridSpace and
        NdLayout types to visualize more than one Element at a time.
        """
        if dimensions is None:
            dimensions = self.kdims
        if not isinstance(dimensions, (list, tuple)):
            dimensions = [dimensions]

        container_type = container_type if container_type else type(self)
        group_type = group_type if group_type else type(self)

        outer_kdims = [self.get_dimension(d) for d in dimensions]
        inner_kdims = [d for d in self.kdims if not d in outer_kdims]

        outer_dynamic = issubclass(container_type, DynamicMap)
        inner_dynamic = issubclass(group_type, DynamicMap)

        if ((not outer_dynamic and any(not d.values for d in outer_kdims)) or
            (not inner_dynamic and any(not d.values for d in inner_kdims))):
            raise Exception('Dimensions must specify sampling via '
                            'values to apply a groupby')

        if outer_dynamic:
            def outer_fn(*outer_key):
                if inner_dynamic:
                    def inner_fn(*inner_key):
                        outer_vals = zip(outer_kdims, util.wrap_tuple(outer_key))
                        inner_vals = zip(inner_kdims, util.wrap_tuple(inner_key))
                        inner_sel = [(k.name, v) for k, v in inner_vals]
                        outer_sel = [(k.name, v) for k, v in outer_vals]
                        return self.select(**dict(inner_sel+outer_sel))
                    return self.clone([], callback=inner_fn, kdims=inner_kdims)
                else:
                    dim_vals = [(d.name, d.values) for d in inner_kdims]
                    dim_vals += [(d.name, [v]) for d, v in
                                   zip(outer_kdims, util.wrap_tuple(outer_key))]
                    return group_type(self.select(**dict(dim_vals))).reindex(inner_kdims)
            if outer_kdims:
                return self.clone([], callback=outer_fn, kdims=outer_kdims)
            else:
                return outer_fn(())
        else:
            outer_product = itertools.product(*[self.get_dimension(d).values
                                                for d in dimensions])
            groups = []
            for outer in outer_product:
                outer_vals = [(d.name, [o]) for d, o in zip(outer_kdims, outer)]
                if inner_dynamic or not inner_kdims:
                    def inner_fn(outer_vals, *key):
                        inner_dims = zip(inner_kdims, util.wrap_tuple(key))
                        inner_vals = [(d.name, k) for d, k in inner_dims]
                        return self.select(**dict(outer_vals+inner_vals)).last
                    if inner_kdims:
                        group = self.clone(callback=partial(inner_fn, outer_vals),
                                           kdims=inner_kdims)
                    else:
                        group = inner_fn(outer_vals, ())
                    groups.append((outer, group))
                else:
                    inner_vals = [(d.name, self.get_dimension(d).values)
                                     for d in inner_kdims]
                    group = group_type(self.select(**dict(outer_vals+inner_vals)).reindex(inner_kdims))
                    groups.append((outer, group))
            return container_type(groups, kdims=outer_kdims)


    def grid(self, dimensions=None, **kwargs):
        return self.groupby(dimensions, container_type=GridSpace, **kwargs)


    def layout(self, dimensions=None, **kwargs):
        return self.groupby(dimensions, container_type=NdLayout, **kwargs)


    def overlay(self, dimensions=None, **kwargs):
        if dimensions is None:
            dimensions = self.kdims
        else:
            if not isinstance(dimensions, (list, tuple)):
                dimensions = [dimensions]
            dimensions = [self.get_dimension(d, strict=True)
                          for d in dimensions]
        dims = [d for d in self.kdims if d not in dimensions]
        return self.groupby(dims, group_type=NdOverlay)


    def hist(self, num_bins=20, bin_range=None, adjoin=True, individually=True, **kwargs):
        """
        Computes a histogram from the object and adjoins it by
        default.  By default the histogram is computed for the bottom
        layer, which can be overriden by supplying an ``index`` and
        for the first value dimension, which may be overridden by
        supplying an explicit ``dimension``.
        """
        def dynamic_hist(obj):
            if isinstance(obj, (NdOverlay, Overlay)):
                index = kwargs.get('index', 0)
                obj = obj.get(index)
            return obj.hist(num_bins=num_bins, bin_range=bin_range,
                            adjoin=False, **kwargs)

        from ..util import Dynamic
        hist = Dynamic(self, operation=dynamic_hist)
        if adjoin:
            return self << hist
        else:
            return hist


    # For Python 2 and 3 compatibility
    __next__ = next



class GridSpace(UniformNdMapping):
    """
    Grids are distinct from Layouts as they ensure all contained
    elements to be of the same type. Unlike Layouts, which have
    integer keys, Grids usually have floating point keys, which
    correspond to a grid sampling in some two-dimensional space. This
    two-dimensional space may have to arbitrary dimensions, e.g. for
    2D parameter spaces.
    """

    kdims = param.List(default=[Dimension("X"), Dimension("Y")], bounds=(1,2))

    def __init__(self, initial_items=None, **params):
        super(GridSpace, self).__init__(initial_items, **params)
        if self.ndims > 2:
            raise Exception('Grids can have no more than two dimensions.')


    def __mul__(self, other):
        if isinstance(other, GridSpace):
            if set(self.keys()) != set(other.keys()):
                raise KeyError("Can only overlay two ParameterGrids if their keys match")
            zipped = zip(self.keys(), self.values(), other.values())
            overlayed_items = [(k, el1 * el2) for (k, el1, el2) in zipped]
            return self.clone(overlayed_items)
        elif isinstance(other, UniformNdMapping) and len(other) == 1:
            view = other.last
        elif isinstance(other, UniformNdMapping) and len(other) != 1:
            raise Exception("Can only overlay with HoloMap of length 1")
        else:
            view = other

        overlayed_items = [(k, el * view) for k, el in self.items()]
        return self.clone(overlayed_items)


    def __lshift__(self, other):
        if isinstance(other, (ViewableElement, UniformNdMapping)):
            return AdjointLayout([self, other])
        elif isinstance(other, AdjointLayout):
            return AdjointLayout(other.data+[self])
        else:
            raise TypeError('Cannot append {0} to a AdjointLayout'.format(type(other).__name__))


    def _transform_indices(self, key):
        """
        Transforms indices by snapping to the closest value if
        values are numeric, otherwise applies no transformation.
        """
        ndims = self.ndims
        if all(not (isinstance(el, slice) or callable(el)) for el in key):
            dim_inds = []
            for dim in self.kdims:
                dim_type = self.get_dimension_type(dim)
                if isinstance(dim_type, type) and issubclass(dim_type, Number):
                    dim_inds.append(self.get_dimension_index(dim))
            str_keys = iter(key[i] for i in range(self.ndims)
                            if i not in dim_inds)
            num_keys = []
            if len(dim_inds):
                keys = list({tuple(k[i] if ndims > 1 else k for i in dim_inds)
                             for k in self.keys()})
                q = np.array([tuple(key[i] if ndims > 1 else key for i in dim_inds)])
                idx = np.argmin([np.inner(q - np.array(x), q - np.array(x))
                                 if len(dim_inds) == 2 else np.abs(q-x)
                                     for x in keys])
                num_keys = iter(keys[idx])
            key = tuple(next(num_keys) if i in dim_inds else next(str_keys)
                        for i in range(self.ndims))
        elif any(not (isinstance(el, slice) or callable(el)) for el in key):
            index_inds = [idx for idx, el in enumerate(key)
                         if not isinstance(el, (slice, str))]
            if len(index_inds):
                index_ind = index_inds[0]
                dim_keys = np.array([k[index_ind] for k in self.keys()])
                snapped_val = dim_keys[np.argmin(np.abs(dim_keys-key[index_ind]))]
                key = list(key)
                key[index_ind] = snapped_val
                key = tuple(key)
        return key


    def keys(self, full_grid=False):
        """
        Returns a complete set of keys on a GridSpace, even when GridSpace isn't fully
        populated. This makes it easier to identify missing elements in the
        GridSpace.
        """
        keys = super(GridSpace, self).keys()
        if self.ndims == 1 or not full_grid:
            return keys
        dim1_keys = sorted(set(k[0] for k in keys))
        dim2_keys = sorted(set(k[1] for k in keys))
        return [(d1, d2) for d1 in dim1_keys for d2 in dim2_keys]


    @property
    def last(self):
        """
        The last of a GridSpace is another GridSpace
        constituted of the last of the individual elements. To access
        the elements by their X,Y position, either index the position
        directly or use the items() method.
        """
        if self.type == HoloMap:
            last_items = [(k, v.last if isinstance(v, HoloMap) else v)
                          for (k, v) in self.data.items()]
        else:
            last_items = self.data
        return self.clone(last_items)


    def __len__(self):
        """
        The maximum depth of all the elements. Matches the semantics
        of __len__ used by Maps. For the total number of elements,
        count the full set of keys.
        """
        return max([(len(v) if hasattr(v, '__len__') else 1) for v in self.values()] + [0])


    def __add__(self, obj):
        return Layout.from_values([self, obj])


    @property
    def shape(self):
        keys = self.keys()
        if self.ndims == 1:
            return (len(keys), 1)
        return len(set(k[0] for k in keys)), len(set(k[1] for k in keys))



class GridMatrix(GridSpace):
    """
    GridMatrix is container type for heterogeneous Element types
    laid out in a grid. Unlike a GridSpace the axes of the Grid
    must not represent an actual coordinate space, but may be used
    to plot various dimensions against each other. The GridMatrix
    is usually constructed using the gridmatrix operation, which
    will generate a GridMatrix plotting each dimension in an
    Element against each other.
    """


    def _item_check(self, dim_vals, data):
        if not traversal.uniform(NdMapping([(0, self), (1, data)])):
            raise ValueError("HoloMaps dimensions must be consistent in %s." %
                             type(self).__name__)
        NdMapping._item_check(self, dim_vals, data)

Guess this isn't tested? Wouldn't there have been Python 3 issues if it had been?