SortedList.py

# Copyright 2014 Grant Jenks
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
#
# Copied from https://github.com/grantjenks/sorted_containers
#
# -*- coding: utf-8 -*-
#
# Sorted list implementation.

from bisect import bisect_left, bisect_right, insort
from itertools import chain, repeat, starmap
from collections import Sequence, MutableSequence
import operator as op
from operator import iadd, add
from functools import wraps
from math import log

from functools import reduce
try:
    from _thread import get_ident
except ImportError:
    from _dummy_thread import get_ident

def recursive_repr(func):
    """Decorator to prevent infinite repr recursion."""
    repr_running = set()

    @wraps(func)
    def wrapper(self):
        key = id(self), get_ident()

        if key in repr_running:
            return '...'

        repr_running.add(key)

        try:
            return func(self)
        finally:
            repr_running.discard(key)

    return wrapper


class SortedList(MutableSequence):
    """
    SortedList provides most of the same methods as a list but keeps the items
    in sorted order.
    """

    def __init__(self, iterable=None, load=1000):
        """
        SortedList provides most of the same methods as a list but keeps the
        items in sorted order.

        An optional *iterable* provides an initial series of items to populate
        the SortedList.

        An optional *load* specifies the load-factor of the list. The default
        load factor of '1000' works well for lists from tens to tens of millions
        of elements.  Good practice is to use a value that is the cube root of
        the list size.  With billions of elements, the best load factor depends
        on your usage.  It's best to leave the load factor at the default until
        you start benchmarking.
        """
        self._len, self._maxes, self._lists, self._index = 0, [], [], []
        self._load, self._twice, self._half = load, load * 2, load >> 1
        self._offset = 0

        if iterable is not None:
            self._update(iterable)

    def __new__(cls, iterable=None, key=None, load=1000):
        """
        SortedList provides most of the same methods as a list but keeps the
        items in sorted order.

        An optional *iterable* provides an initial series of items to populate
        the SortedList.

        An optional *key* argument will return an instance of subtype
        SortedListWithKey.

        An optional *load* specifies the load-factor of the list. The default
        load factor of '1000' works well for lists from tens to tens of millions
        of elements.  Good practice is to use a value that is the cube root of
        the list size.  With billions of elements, the best load factor depends
        on your usage.  It's best to leave the load factor at the default until
        you start benchmarking.
        """
        if key is None:
            return object.__new__(cls)
        else:
            if cls is SortedList:
                return SortedListWithKey(iterable=iterable, key=key, load=load)
            else:
                raise TypeError('inherit SortedListWithKey for key argument')

    def clear(self):
        """Remove all the elements from the list."""
        self._len = 0
        del self._maxes[:]
        del self._lists[:]
        del self._index[:]

    _clear = clear

    def add(self, val):
        """Add the element *val* to the list."""
        _maxes, _lists = self._maxes, self._lists

        if _maxes:
            pos = bisect_right(_maxes, val)

            if pos == len(_maxes):
                pos -= 1
                _maxes[pos] = val
                _lists[pos].append(val)
            else:
                insort(_lists[pos], val)

            self._expand(pos)
        else:
            _maxes.append(val)
            _lists.append([val])

        self._len += 1

    def _expand(self, pos):
        """Splits sublists that are more than double the load level.

        Updates the index when the sublist length is less than double the load
        level. This requires incrementing the nodes in a traversal from the leaf
        node to the root. For an example traversal see self._loc.
        """
        _lists, _index = self._lists, self._index

        if len(_lists[pos]) > self._twice:
            _maxes, _load = self._maxes, self._load
            half = _lists[pos][_load:]
            del _lists[pos][_load:]
            _maxes[pos] = _lists[pos][-1]
            _maxes.insert(pos + 1, half[-1])
            _lists.insert(pos + 1, half)
            del _index[:]
        else:
            if _index:
                child = self._offset + pos
                while child > 0:
                    _index[child] += 1
                    child = (child - 1) >> 1
                _index[0] += 1

    def update(self, iterable):
        """Update the list by adding all elements from *iterable*."""
        _maxes, _lists = self._maxes, self._lists
        values = sorted(iterable)

        if _maxes:
            if len(values) * 4 >= self._len:
                values.extend(chain.from_iterable(_lists))
                values.sort()
                self._clear()
            else:
                _add = self.add
                for val in values:
                    _add(val)
                return

        _load, _index = self._load, self._index
        _lists.extend(values[pos:(pos + _load)]
                      for pos in range(0, len(values), _load))
        _maxes.extend(sublist[-1] for sublist in _lists)
        self._len = len(values)
        del _index[:]

    _update = update

    def __contains__(self, val):
        """Return True if and only if *val* is an element in the list."""
        _maxes = self._maxes

        if not _maxes:
            return False

        pos = bisect_left(_maxes, val)

        if pos == len(_maxes):
            return False

        _lists = self._lists
        idx = bisect_left(_lists[pos], val)
        return _lists[pos][idx] == val

    def discard(self, val):
        """
        Remove the first occurrence of *val*.

        If *val* is not a member, does nothing.
        """
        _maxes = self._maxes

        if not _maxes:
            return

        pos = bisect_left(_maxes, val)

        if pos == len(_maxes):
            return

        _lists = self._lists
        idx = bisect_left(_lists[pos], val)
        if _lists[pos][idx] == val:
            self._delete(pos, idx)

    def remove(self, val):
        """
        Remove first occurrence of *val*.

        Raises ValueError if *val* is not present.
        """
        _maxes = self._maxes

        if not _maxes:
            raise ValueError('{0} not in list'.format(repr(val)))

        pos = bisect_left(_maxes, val)

        if pos == len(_maxes):
            raise ValueError('{0} not in list'.format(repr(val)))

        _lists = self._lists
        idx = bisect_left(_lists[pos], val)
        if _lists[pos][idx] == val:
            self._delete(pos, idx)
        else:
            raise ValueError('{0} not in list'.format(repr(val)))

    def _delete(self, pos, idx):
        """Delete the item at the given (pos, idx).

        Combines lists that are less than half the load level.

        Updates the index when the sublist length is more than half the load
        level. This requires decrementing the nodes in a traversal from the leaf
        node to the root. For an example traversal see self._loc.
        """
        _maxes, _lists, _index = self._maxes, self._lists, self._index

        lists_pos = _lists[pos]

        del lists_pos[idx]
        self._len -= 1

        len_lists_pos = len(lists_pos)

        if len_lists_pos > self._half:

            _maxes[pos] = lists_pos[-1]

            if _index:
                child = self._offset + pos
                while child > 0:
                    _index[child] -= 1
                    child = (child - 1) >> 1
                _index[0] -= 1

        elif len(_lists) > 1:

            if not pos:
                pos += 1

            prev = pos - 1
            _lists[prev].extend(_lists[pos])
            _maxes[prev] = _lists[prev][-1]

            del _maxes[pos]
            del _lists[pos]
            del _index[:]

            self._expand(prev)

        elif len_lists_pos:

            _maxes[pos] = lists_pos[-1]

        else:

            del _maxes[pos]
            del _lists[pos]
            del _index[:]

    def _loc(self, pos, idx):
        """Convert an index pair (alpha, beta) into a single index that corresponds to
        the position of the value in the sorted list.

        Most queries require the index be built. Details of the index are
        described in self._build_index.

        Indexing requires traversing the tree from a leaf node to the root. The
        parent of each node is easily computable at (pos - 1) // 2.

        Left-child nodes are always at odd indices and right-child nodes are
        always at even indices.

        When traversing up from a right-child node, increment the total by the
        left-child node.

        The final index is the sum from traversal and the index in the sublist.

        For example, using the index from self._build_index:

        _index = 14 5 9 3 2 4 5
        _offset = 3

        Tree:

                 14
              5      9
            3   2  4   5

        Converting index pair (2, 3) into a single index involves iterating like
        so:

        1. Starting at the leaf node: offset + alpha = 3 + 2 = 5. We identify
           the node as a left-child node. At such nodes, we simply traverse to
           the parent.

        2. At node 9, position 2, we recognize the node as a right-child node
           and accumulate the left-child in our total. Total is now 5 and we
           traverse to the parent at position 0.

        3. Iteration ends at the root.

        Computing the index is the sum of the total and beta: 5 + 3 = 8.
        """
        if not pos:
            return idx

        _index = self._index

        if not len(_index):
            self._build_index()

        total = 0

        # Increment pos to point in the index to len(self._lists[pos]).

        pos += self._offset

        # Iterate until reaching the root of the index tree at pos = 0.

        while pos:

            # Right-child nodes are at odd indices. At such indices
            # account the total below the left child node.

            if not (pos & 1):
                total += _index[pos - 1]

            # Advance pos to the parent node.

            pos = (pos - 1) >> 1

        return total + idx

    def _pos(self, idx):
        """Convert an index into a pair (alpha, beta) that can be used to access
        the corresponding _lists[alpha][beta] position.

        Most queries require the index be built. Details of the index are
        described in self._build_index.

        Indexing requires traversing the tree to a leaf node. Each node has
        two children which are easily computable. Given an index, pos, the
        left-child is at pos * 2 + 1 and the right-child is at pos * 2 + 2.

        When the index is less than the left-child, traversal moves to the
        left sub-tree. Otherwise, the index is decremented by the left-child
        and traversal moves to the right sub-tree.

        At a child node, the indexing pair is computed from the relative
        position of the child node as compared with the offset and the remaining
        index.

        For example, using the index from self._build_index:

        _index = 14 5 9 3 2 4 5
        _offset = 3

        Tree:

                 14
              5      9
            3   2  4   5

        Indexing position 8 involves iterating like so:

        1. Starting at the root, position 0, 8 is compared with the left-child
           node (5) which it is greater than. When greater the index is
           decremented and the position is updated to the right child node.

        2. At node 9 with index 3, we again compare the index to the left-child
           node with value 4. Because the index is the less than the left-child
           node, we simply traverse to the left.

        3. At node 4 with index 3, we recognize that we are at a leaf node and
           stop iterating.

        4. To compute the sublist index, we subtract the offset from the index
           of the leaf node: 5 - 3 = 2. To compute the index in the sublist, we
           simply use the index remaining from iteration. In this case, 3.

        The final index pair from our example is (2, 3) which corresponds to
        index 8 in the sorted list.
        """
        if idx < 0:
            last_len = len(self._lists[-1])
            if (-idx) <= last_len:
                return len(self._lists) - 1, last_len + idx
            idx += self._len
            if idx < 0:
                raise IndexError('list index out of range')
        elif idx >= self._len:
            raise IndexError('list index out of range')

        if idx < len(self._lists[0]):
            return 0, idx

        _index = self._index

        if not _index:
            self._build_index()

        pos = 0
        child = 1
        len_index = len(_index)

        while child < len_index:
            index_child = _index[child]

            if idx < index_child:
                pos = child
            else:
                idx -= index_child
                pos = child + 1

            child = (pos << 1) + 1

        return (pos - self._offset, idx)

    def _build_index(self):
        """Build an index for indexing the sorted list.

        Indexes are represented as binary trees in a dense array notation
        similar to a binary heap.

        For example, given a _lists representation storing integers:

        [0]: 1 2 3
        [1]: 4 5
        [2]: 6 7 8 9
        [3]: 10 11 12 13 14

        The first transformation maps the sub-lists by their length. The
        first row of the index is the length of the sub-lists.

        [0]: 3 2 4 5

        Each row after that is the sum of consecutive pairs of the previous row:

        [1]: 5 9
        [2]: 14

        Finally, the index is built by concatenating these lists together:

        _index = 14 5 9 3 2 4 5

        An offset storing the start of the first row is also stored:

        _offset = 3

        When built, the index can be used for efficient indexing into the list.
        See the comment and notes on self._pos for details.
        """
        row0 = list(map(len, self._lists))

        if len(row0) == 1:
            self._index[:] = row0
            self._offset = 0
            return

        head = iter(row0)
        tail = iter(head)
        row1 = list(starmap(add, zip(head, tail)))

        if len(row0) & 1:
            row1.append(row0[-1])

        if len(row1) == 1:
            self._index[:] = row1 + row0
            self._offset = 1
            return

        size = 2 ** (int(log(len(row1) - 1, 2)) + 1)
        row1.extend(repeat(0, size - len(row1)))
        tree = [row0, row1]

        while len(tree[-1]) > 1:
            head = iter(tree[-1])
            tail = iter(head)
            row = list(starmap(add, zip(head, tail)))
            tree.append(row)

        reduce(iadd, reversed(tree), self._index)
        self._offset = size * 2 - 1

    def __delitem__(self, idx):
        """Remove the element at *idx*. Supports slicing."""
        if isinstance(idx, slice):
            start, stop, step = idx.indices(self._len)

            if step == 1 and start < stop:
                if start == 0 and stop == self._len:
                    return self._clear()
                elif self._len <= 8 * (stop - start):
                    values = self._getitem(slice(None, start))
                    if stop < self._len:
                        values += self._getitem(slice(stop, None))
                    self._clear()
                    return self._update(values)

            indices = range(start, stop, step)

            # Delete items from greatest index to least so
            # that the indices remain valid throughout iteration.

            if step > 0:
                indices = reversed(indices)

            _pos, _delete = self._pos, self._delete

            for index in indices:
                pos, idx = _pos(index)
                _delete(pos, idx)
        else:
            pos, idx = self._pos(idx)
            self._delete(pos, idx)

    _delitem = __delitem__

    def __getitem__(self, idx):
        """Return the element at *idx*. Supports slicing."""
        _lists = self._lists

        if isinstance(idx, slice):
            start, stop, step = idx.indices(self._len)

            if step == 1 and start < stop:
                if start == 0 and stop == self._len:
                    return reduce(iadd, self._lists, [])

                start_pos, start_idx = self._pos(start)

                if stop == self._len:
                    stop_pos = len(_lists) - 1
                    stop_idx = len(_lists[stop_pos])
                else:
                    stop_pos, stop_idx = self._pos(stop)

                if start_pos == stop_pos:
                    return _lists[start_pos][start_idx:stop_idx]

                prefix = _lists[start_pos][start_idx:]
                middle = _lists[(start_pos + 1):stop_pos]
                result = reduce(iadd, middle, prefix)
                result += _lists[stop_pos][:stop_idx]

                return result

            if step == -1 and start > stop:
                result = self._getitem(slice(stop + 1, start + 1))
                result.reverse()
                return result

            # Return a list because a negative step could
            # reverse the order of the items and this could
            # be the desired behavior.

            indices = range(start, stop, step)
            return list(self._getitem(index) for index in indices)
        else:
            if self._len:
                if idx == 0:
                    return _lists[0][0]
                elif idx == -1:
                    return _lists[-1][-1]
            else:
                raise IndexError('list index out of range')

            if 0 <= idx < len(_lists[0]):
                return _lists[0][idx]

            len_last = len(_lists[-1])

            if -len_last < idx < 0:
                return _lists[-1][len_last + idx]

            pos, idx = self._pos(idx)
            return _lists[pos][idx]

    _getitem = __getitem__

    def _check_order(self, idx, val):
        _lists, _len = self._lists, self._len

        pos, loc = self._pos(idx)

        if idx < 0:
            idx += _len

        # Check that the inserted value is not less than the
        # previous value.

        if idx > 0:
            idx_prev = loc - 1
            pos_prev = pos

            if idx_prev < 0:
                pos_prev -= 1
                idx_prev = len(_lists[pos_prev]) - 1

            if _lists[pos_prev][idx_prev] > val:
                msg = '{0} not in sort order at index {1}'.format(repr(val), idx)
                raise ValueError(msg)

        # Check that the inserted value is not greater than
        # the previous value.

        if idx < (_len - 1):
            idx_next = loc + 1
            pos_next = pos

            if idx_next == len(_lists[pos_next]):
                pos_next += 1
                idx_next = 0

            if _lists[pos_next][idx_next] < val:
                msg = '{0} not in sort order at index {1}'.format(repr(val), idx)
                raise ValueError(msg)

    def __setitem__(self, index, value):
        """
        Replace the item at position *index* with *value*.

        Supports slice notation. Raises a :exc:`ValueError` if the sort order
        would be violated. When used with a slice and iterable, the
        :exc:`ValueError` is raised before the list is mutated if the sort order
        would be violated by the operation.
        """
        _maxes, _lists, _pos = self._maxes, self._lists, self._pos
        _check_order = self._check_order

        if isinstance(index, slice):
            start, stop, step = index.indices(self._len)
            indices = range(start, stop, step)

            if step != 1:
                if not hasattr(value, '__len__'):
                    value = list(value)

                indices = list(indices)

                if len(value) != len(indices):
                    raise ValueError(
                        'attempt to assign sequence of size {0}'
                        ' to extended slice of size {1}'
                        .format(len(value), len(indices)))

                # Keep a log of values that are set so that we can
                # roll back changes if ordering is violated.

                log = []
                _append = log.append

                for idx, val in zip(indices, value):
                    pos, loc = _pos(idx)
                    _append((idx, _lists[pos][loc], val))
                    _lists[pos][loc] = val
                    if len(_lists[pos]) == (loc + 1):
                        _maxes[pos] = val

                try:
                    # Validate ordering of new values.

                    for idx, oldval, newval in log:
                        _check_order(idx, newval)

                except ValueError:

                    # Roll back changes from log.

                    for idx, oldval, newval in log:
                        pos, loc = _pos(idx)
                        _lists[pos][loc] = oldval
                        if len(_lists[pos]) == (loc + 1):
                            _maxes[pos] = oldval

                    raise
            else:
                if start == 0 and stop == self._len:
                    self._clear()
                    return self._update(value)

                # Test ordering using indexing. If the given value
                # isn't a Sequence, convert it to a tuple.

                if not isinstance(value, Sequence):
                    value = tuple(value)

                # Check that the given values are ordered properly.

                iterator = range(1, len(value))

                if not all(value[pos - 1] <= value[pos] for pos in iterator):
                    raise ValueError('given sequence not in sort order')

                # Check ordering in context of sorted list.

                if not start or not len(value):
                    # Nothing to check on the lhs.
                    pass
                else:
                    if self._getitem(start - 1) > value[0]:
                        msg = '{0} not in sort order at index {1}'.format(repr(value[0]), start)
                        raise ValueError(msg)

                if stop == len(self) or not len(value):
                    # Nothing to check on the rhs.
                    pass
                else:
                    # "stop" is exclusive so we don't need
                    # to add one for the index.
                    if self._getitem(stop) < value[-1]:
                        msg = '{0} not in sort order at index {1}'.format(repr(value[-1]), stop)
                        raise ValueError(msg)

                # Delete the existing values.

                self._delitem(index)

                # Insert the new values.

                _insert = self.insert
                for idx, val in enumerate(value):
                    _insert(start + idx, val)
        else:
            pos, loc = _pos(index)
            _check_order(index, value)
            _lists[pos][loc] = value
            if len(_lists[pos]) == (loc + 1):
                _maxes[pos] = value

    def __iter__(self):
        """
        Return an iterator over the Sequence.

        Iterating the Sequence while adding or deleting values may raise a
        `RuntimeError` or fail to iterate over all entries.
        """
        return chain.from_iterable(self._lists)

    def __reversed__(self):
        """
        Return an iterator to traverse the Sequence in reverse.

        Iterating the Sequence while adding or deleting values may raise a
        `RuntimeError` or fail to iterate over all entries.
        """
        return chain.from_iterable(map(reversed, reversed(self._lists)))

    def islice(self, start=None, stop=None, reverse=False):
        """
        Returns an iterator that slices `self` from `start` to `stop` index,
        inclusive and exclusive respectively.

        When `reverse` is `True`, values are yielded from the iterator in
        reverse order.

        Both `start` and `stop` default to `None` which is automatically
        inclusive of the beginning and end.
        """
        _len = self._len

        if not _len:
            return iter(())

        start, stop, step = slice(start, stop).indices(self._len)

        if start >= stop:
            return iter(())

        _pos = self._pos

        min_pos, min_idx = _pos(start)

        if stop == _len:
            max_pos = len(self._lists) - 1
            max_idx = len(self._lists[-1])
        else:
            max_pos, max_idx = _pos(stop)

        return self._islice(min_pos, min_idx, max_pos, max_idx, reverse)

    def _islice(self, min_pos, min_idx, max_pos, max_idx, reverse):
        """
        Returns an iterator that slices `self` using two index pairs,
        `(min_pos, min_idx)` and `(max_pos, max_idx)`; the first inclusive
        and the latter exclusive. See `_pos` for details on how an index
        is converted to an index pair.

        When `reverse` is `True`, values are yielded from the iterator in
        reverse order.
        """
        _lists = self._lists

        if min_pos > max_pos:
            return iter(())
        elif min_pos == max_pos and not reverse:
            return iter(_lists[min_pos][min_idx:max_idx])
        elif min_pos == max_pos and reverse:
            return reversed(_lists[min_pos][min_idx:max_idx])
        elif min_pos + 1 == max_pos and not reverse:
            return chain(_lists[min_pos][min_idx:], _lists[max_pos][:max_idx])
        elif min_pos + 1 == max_pos and reverse:
            return chain(
                reversed(_lists[max_pos][:max_idx]),
                reversed(_lists[min_pos][min_idx:]),
            )
        elif not reverse:
            return chain(
                _lists[min_pos][min_idx:],
                chain.from_iterable(_lists[(min_pos + 1):max_pos]),
                _lists[max_pos][:max_idx],
            )
        else:
            temp = map(reversed, reversed(_lists[(min_pos + 1):max_pos]))
            return chain(
                reversed(_lists[max_pos][:max_idx]),
                chain.from_iterable(temp),
                reversed(_lists[min_pos][min_idx:]),
            )

    def irange(self, minimum=None, maximum=None, inclusive=(True, True),
               reverse=False):
        """
        Create an iterator of values between `minimum` and `maximum`.

        `inclusive` is a pair of booleans that indicates whether the minimum
        and maximum ought to be included in the range, respectively. The
        default is (True, True) such that the range is inclusive of both
        minimum and maximum.

        Both `minimum` and `maximum` default to `None` which is automatically
        inclusive of the start and end of the list, respectively.

        When `reverse` is `True` the values are yielded from the iterator in
        reverse order; `reverse` defaults to `False`.
        """
        _maxes = self._maxes

        if not _maxes:
            return iter(())

        _lists = self._lists

        # Calculate the minimum (pos, idx) pair. By default this location
        # will be inclusive in our calculation.

        if minimum is None:
            min_pos = 0
            min_idx = 0
        else:
            if inclusive[0]:
                min_pos = bisect_left(_maxes, minimum)

                if min_pos == len(_maxes):
                    return iter(())

                min_idx = bisect_left(_lists[min_pos], minimum)
            else:
                min_pos = bisect_right(_maxes, minimum)

                if min_pos == len(_maxes):
                    return iter(())

                min_idx = bisect_right(_lists[min_pos], minimum)

        # Calculate the maximum (pos, idx) pair. By default this location
        # will be exclusive in our calculation.

        if maximum is None:
            max_pos = len(_maxes) - 1
            max_idx = len(_lists[max_pos])
        else:
            if inclusive[1]:
                max_pos = bisect_right(_maxes, maximum)

                if max_pos == len(_maxes):
                    max_pos -= 1
                    max_idx = len(_lists[max_pos])
                else:
                    max_idx = bisect_right(_lists[max_pos], maximum)
            else:
                max_pos = bisect_left(_maxes, maximum)

                if max_pos == len(_maxes):
                    max_pos -= 1
                    max_idx = len(_lists[max_pos])
                else:
                    max_idx = bisect_left(_lists[max_pos], maximum)

        return self._islice(min_pos, min_idx, max_pos, max_idx, reverse)

    def __len__(self):
        """Return the number of elements in the list."""
        return self._len

    def bisect_left(self, val):
        """
        Similar to the *bisect* module in the standard library, this returns an
        appropriate index to insert *val*. If *val* is already present, the
        insertion point will be before (to the left of) any existing entries.
        """
        _maxes = self._maxes

        if not _maxes:
            return 0

        pos = bisect_left(_maxes, val)

        if pos == len(_maxes):
            return self._len

        idx = bisect_left(self._lists[pos], val)

        return self._loc(pos, idx)

    def bisect_right(self, val):
        """
        Same as *bisect_left*, but if *val* is already present, the insertion
        point will be after (to the right of) any existing entries.
        """
        _maxes = self._maxes

        if not _maxes:
            return 0

        pos = bisect_right(_maxes, val)

        if pos == len(_maxes):
            return self._len

        idx = bisect_right(self._lists[pos], val)

        return self._loc(pos, idx)

    bisect = bisect_right
    _bisect_right = bisect_right

    def count(self, val):
        """Return the number of occurrences of *val* in the list."""
        _maxes = self._maxes

        if not _maxes:
            return 0

        pos_left = bisect_left(_maxes, val)

        if pos_left == len(_maxes):
            return 0

        _lists = self._lists
        idx_left = bisect_left(_lists[pos_left], val)
        pos_right = bisect_right(_maxes, val)

        if pos_right == len(_maxes):
            return self._len - self._loc(pos_left, idx_left)

        idx_right = bisect_right(_lists[pos_right], val)

        if pos_left == pos_right:
            return idx_right - idx_left

        right = self._loc(pos_right, idx_right)
        left = self._loc(pos_left, idx_left)

        return right - left

    def copy(self):
        """Return a shallow copy of the sorted list."""
        return self.__class__(self, load=self._load)

    __copy__ = copy

    def append(self, val):
        """
        Append the element *val* to the list. Raises a ValueError if the *val*
        would violate the sort order.
        """
        _maxes, _lists = self._maxes, self._lists

        if not _maxes:
            _maxes.append(val)
            _lists.append([val])
            self._len = 1
            return

        pos = len(_lists) - 1

        if val < _lists[pos][-1]:
            msg = '{0} not in sort order at index {1}'.format(repr(val), self._len)
            raise ValueError(msg)

        _maxes[pos] = val
        _lists[pos].append(val)
        self._len += 1
        self._expand(pos)

    def extend(self, values):
        """
        Extend the list by appending all elements from the *values*. Raises a
        ValueError if the sort order would be violated.
        """
        _maxes, _lists, _load = self._maxes, self._lists, self._load

        if not isinstance(values, list):
            values = list(values)

        if any(values[pos - 1] > values[pos]
               for pos in range(1, len(values))):
            raise ValueError('given sequence not in sort order')

        offset = 0

        if _maxes:
            if values[0] < _lists[-1][-1]:
                msg = '{0} not in sort order at index {1}'.format(repr(values[0]), self._len)
                raise ValueError(msg)

            if len(_lists[-1]) < self._half:
                _lists[-1].extend(values[:_load])
                _maxes[-1] = _lists[-1][-1]
                offset = _load

        len_lists = len(_lists)

        for idx in range(offset, len(values), _load):
            _lists.append(values[idx:(idx + _load)])
            _maxes.append(_lists[-1][-1])

        _index = self._index

        if len_lists == len(_lists):
            len_index = len(_index)
            if len_index > 0:
                len_values = len(values)
                child = len_index - 1
                while child:
                    _index[child] += len_values
                    child = (child - 1) >> 1
                _index[0] += len_values
        else:
            del _index[:]

        self._len += len(values)

    def insert(self, idx, val):
        """
        Insert the element *val* into the list at *idx*. Raises a ValueError if
        the *val* at *idx* would violate the sort order.
        """
        _maxes, _lists, _len = self._maxes, self._lists, self._len

        if idx < 0:
            idx += _len
        if idx < 0:
            idx = 0
        if idx > _len:
            idx = _len

        if not _maxes:
            # The idx must be zero by the inequalities above.
            _maxes.append(val)
            _lists.append([val])
            self._len = 1
            return

        if not idx:
            if val > _lists[0][0]:
                msg = '{0} not in sort order at index {1}'.format(repr(val), 0)
                raise ValueError(msg)
            else:
                _lists[0].insert(0, val)
                self._expand(0)
                self._len += 1
                return

        if idx == _len:
            pos = len(_lists) - 1
            if _lists[pos][-1] > val:
                msg = '{0} not in sort order at index {1}'.format(repr(val), _len)
                raise ValueError(msg)
            else:
                _lists[pos].append(val)
                _maxes[pos] = _lists[pos][-1]
                self._expand(pos)
                self._len += 1
                return

        pos, idx = self._pos(idx)
        idx_before = idx - 1
        if idx_before < 0:
            pos_before = pos - 1
            idx_before = len(_lists[pos_before]) - 1
        else:
            pos_before = pos

        before = _lists[pos_before][idx_before]
        if before <= val <= _lists[pos][idx]:
            _lists[pos].insert(idx, val)
            self._expand(pos)
            self._len += 1
        else:
            msg = '{0} not in sort order at index {1}'.format(repr(val), idx)
            raise ValueError(msg)

    def pop(self, idx=-1):
        """
        Remove and return item at *idx* (default last).  Raises IndexError if
        list is empty or index is out of range.  Negative indices are supported,
        as for slice indices.
        """
        if not self._len:
            raise IndexError('pop index out of range')

        _lists = self._lists

        if idx == 0:
            val = _lists[0][0]
            self._delete(0, 0)
            return val

        if idx == -1:
            pos = len(_lists) - 1
            loc = len(_lists[pos]) - 1
            val = _lists[pos][loc]
            self._delete(pos, loc)
            return val

        if 0 <= idx < len(_lists[0]):
            val = _lists[0][idx]
            self._delete(0, idx)
            return val

        len_last = len(_lists[-1])

        if -len_last < idx < 0:
            pos = len(_lists) - 1
            loc = len_last + idx
            val = _lists[pos][loc]
            self._delete(pos, loc)
            return val

        pos, idx = self._pos(idx)
        val = _lists[pos][idx]
        self._delete(pos, idx)

        return val

    def index(self, val, start=None, stop=None):
        """
        Return the smallest *k* such that L[k] == val and i <= k < j`.  Raises
        ValueError if *val* is not present.  *stop* defaults to the end of the
        list. *start* defaults to the beginning. Negative indices are supported,
        as for slice indices.
        """
        _len, _maxes = self._len, self._maxes

        if not _maxes:
            raise ValueError('{0} is not in list'.format(repr(val)))

        if start is None:
            start = 0
        if start < 0:
            start += _len
        if start < 0:
            start = 0

        if stop is None:
            stop = _len
        if stop < 0:
            stop += _len
        if stop > _len:
            stop = _len

        if stop <= start:
            raise ValueError('{0} is not in list'.format(repr(val)))

        stop -= 1
        pos_left = bisect_left(_maxes, val)

        if pos_left == len(_maxes):
            raise ValueError('{0} is not in list'.format(repr(val)))

        _lists = self._lists
        idx_left = bisect_left(_lists[pos_left], val)

        if _lists[pos_left][idx_left] != val:
            raise ValueError('{0} is not in list'.format(repr(val)))

        left = self._loc(pos_left, idx_left)

        if start <= left:
            if left <= stop:
                return left
        else:
            right = self._bisect_right(val) - 1

            if start <= right:
                return start

        raise ValueError('{0} is not in list'.format(repr(val)))

    def __add__(self, that):
        """
        Return a new sorted list containing all the elements in *self* and
        *that*. Elements in *that* do not need to be properly ordered with
        respect to *self*.
        """
        values = reduce(iadd, self._lists, [])
        values.extend(that)
        return self.__class__(values, load=self._load)

    def __iadd__(self, that):
        """
        Update *self* to include all values in *that*. Elements in *that* do not
        need to be properly ordered with respect to *self*.
        """
        self._update(that)
        return self

    def __mul__(self, that):
        """
        Return a new sorted list containing *that* shallow copies of each item
        in SortedList.
        """
        values = reduce(iadd, self._lists, []) * that
        return self.__class__(values, load=self._load)

    def __imul__(self, that):
        """
        Increase the length of the list by appending *that* shallow copies of
        each item.
        """
        values = reduce(iadd, self._lists, []) * that
        self._clear()
        self._update(values)
        return self

    def _make_cmp(seq_op, doc):
        def comparer(self, that):
            if not isinstance(that, Sequence):
                return NotImplemented

            self_len = self._len
            len_that = len(that)

            if self_len != len_that:
                if seq_op is op.eq:
                    return False
                if seq_op is op.ne:
                    return True

            for alpha, beta in zip(self, that):
                if alpha != beta:
                    return seq_op(alpha, beta)

            return seq_op(self_len, len_that)

        comparer.__name__ = '__{0}__'.format(seq_op.__name__)
        doc_str = 'Return `True` if and only if Sequence is {0} `that`.'
        comparer.__doc__ = doc_str.format(doc)

        return comparer

    __eq__ = _make_cmp(op.eq, 'equal to')
    __ne__ = _make_cmp(op.ne, 'not equal to')
    __lt__ = _make_cmp(op.lt, 'less than')
    __gt__ = _make_cmp(op.gt, 'greater than')
    __le__ = _make_cmp(op.le, 'less than or equal to')
    __ge__ = _make_cmp(op.ge, 'greater than or equal to')

    @recursive_repr
    def __repr__(self):
        """Return string representation of sequence."""
        temp = '{0}({1}, load={2})'
        return temp.format(
            self.__class__.__name__,
            repr(list(self)),
            repr(self._load)
        )

    def _check(self):
        try:
            # Check load parameters.

            assert self._load >= 4
            assert self._half == (self._load >> 1)
            assert self._twice == (self._load * 2)

            # Check empty sorted list case.

            if self._maxes == []:
                assert self._lists == []
                return

            assert len(self._maxes) > 0 and len(self._lists) > 0

            # Check all sublists are sorted.

            assert all(sublist[pos - 1] <= sublist[pos]
                       for sublist in self._lists
                       for pos in range(1, len(sublist)))

            # Check beginning/end of sublists are sorted.

            for pos in range(1, len(self._lists)):
                assert self._lists[pos - 1][-1] <= self._lists[pos][0]

            # Check length of _maxes and _lists match.

            assert len(self._maxes) == len(self._lists)

            # Check _maxes is a map of _lists.

            assert all(self._maxes[pos] == self._lists[pos][-1]
                       for pos in range(len(self._maxes)))

            # Check load level is less than _twice.

            assert all(len(sublist) <= self._twice for sublist in self._lists)

            # Check load level is greater than _half for all
            # but the last sublist.

            assert all(len(self._lists[pos]) >= self._half
                       for pos in range(0, len(self._lists) - 1))

            # Check length.

            assert self._len == sum(len(sublist) for sublist in self._lists)

            # Check index.

            if len(self._index):
                assert len(self._index) == self._offset + len(self._lists)
                assert self._len == self._index[0]

                def test_offset_pos(pos):
                    from_index = self._index[self._offset + pos]
                    return from_index == len(self._lists[pos])

                assert all(test_offset_pos(pos)
                           for pos in range(len(self._lists)))

                for pos in range(self._offset):
                    child = (pos << 1) + 1
                    if child >= len(self._index):
                        assert self._index[pos] == 0
                    elif child + 1 == len(self._index):
                        assert self._index[pos] == self._index[child]
                    else:
                        child_sum = self._index[child] + self._index[child + 1]
                        assert self._index[pos] == child_sum

        except:
            import sys
            import traceback

            traceback.print_exc(file=sys.stdout)

            print('len', self._len)
            print('load', self._load, self._half, self._twice)
            print('offset', self._offset)
            print('len_index', len(self._index))
            print('index', self._index)
            print('len_maxes', len(self._maxes))
            print('maxes', self._maxes)
            print('len_lists', len(self._lists))
            print('lists', self._lists)

            raise

def identity(value):
    return value

class SortedListWithKey(SortedList):
    """
    SortedListWithKey provides most of the same methods as a list but keeps
    the items in sorted order.
    """

    def __init__(self, iterable=None, key=identity, load=1000):
        """
        SortedListWithKey provides most of the same methods as a list but
        keeps the items in sorted order.

        An optional *iterable* provides an initial series of items to populate
        the SortedListWithKey.

        An optional *key* argument defines a callable that, like the `key`
        argument to Python's `sorted` function, extracts a comparison key from
        each element. The default is the identity function.

        An optional *load* specifies the load-factor of the list. The default
        load factor of '1000' works well for lists from tens to tens of millions
        of elements.  Good practice is to use a value that is the cube root of
        the list size.  With billions of elements, the best load factor depends
        on your usage.  It's best to leave the load factor at the default until
        you start benchmarking.
        """
        self._len, self._maxes, self._lists, self._keys, self._index = 0, [], [], [], []
        self._key, self._load, self._twice, self._half = key, load, load * 2, load >> 1
        self._offset = 0

        if iterable is not None:
            self._update(iterable)

    def __new__(cls, iterable=None, key=identity, load=1000):
        return object.__new__(cls)

    def clear(self):
        """Remove all the elements from the list."""
        self._len = 0
        del self._maxes[:]
        del self._lists[:]
        del self._keys[:]
        del self._index[:]

    _clear = clear

    def add(self, val):
        """Add the element *val* to the list."""
        _maxes, _lists, _keys = self._maxes, self._lists, self._keys

        key = self._key(val)

        if _maxes:
            pos = bisect_right(_maxes, key)

            if pos == len(_maxes):
                pos -= 1
                _maxes[pos] = key
                _keys[pos].append(key)
                _lists[pos].append(val)
            else:
                idx = bisect_right(_keys[pos], key)
                _keys[pos].insert(idx, key)
                _lists[pos].insert(idx, val)

            self._expand(pos)
        else:
            _maxes.append(key)
            _keys.append([key])
            _lists.append([val])

        self._len += 1

    def _expand(self, pos):
        """
        Splits sublists that are more than double the load level.

        Updates the index when the sublist length is less than double the load
        level. This requires incrementing the nodes in a traversal from the leaf
        node to the root. For an example traversal see self._loc.
        """
        _lists, _keys, _index = self._lists, self._keys, self._index

        if len(_keys[pos]) > self._twice:
            _maxes, _load = self._maxes, self._load

            half = _keys[pos][_load:]
            half_list = _lists[pos][_load:]
            del _keys[pos][_load:]
            del _lists[pos][_load:]
            _maxes[pos] = _keys[pos][-1]

            _maxes.insert(pos + 1, half[-1])
            _keys.insert(pos + 1, half)
            _lists.insert(pos + 1, half_list)

            del _index[:]
        else:
            if _index:
                child = self._offset + pos
                while child > 0:
                    _index[child] += 1
                    child = (child - 1) >> 1
                _index[0] += 1

    def update(self, iterable):
        """Update the list by adding all elements from *iterable*."""
        _maxes, _lists, _keys = self._maxes, self._lists, self._keys
        values = sorted(iterable, key=self._key)

        if _maxes:
            if len(values) * 4 >= self._len:
                values.extend(chain.from_iterable(_lists))
                values.sort(key=self._key)
                self._clear()
            else:
                _add = self.add
                for val in values:
                    _add(val)
                return

        _load, _index = self._load, self._index
        _lists.extend(values[pos:(pos + _load)]
                      for pos in range(0, len(values), _load))
        _keys.extend(list(map(self._key, _list)) for _list in _lists)
        _maxes.extend(sublist[-1] for sublist in _keys)
        self._len = len(values)
        del _index[:]

    _update = update

    def __contains__(self, val):
        """Return True if and only if *val* is an element in the list."""
        _maxes = self._maxes

        if not _maxes:
            return False

        key = self._key(val)
        pos = bisect_left(_maxes, key)

        if pos == len(_maxes):
            return False

        _keys = self._keys
        _lists = self._lists

        idx = bisect_left(_keys[pos], key)

        len_keys = len(_keys)
        len_sublist = len(_keys[pos])

        while True:
            if _keys[pos][idx] != key:
                return False
            if _lists[pos][idx] == val:
                return True
            idx += 1
            if idx == len_sublist:
                pos += 1
                if pos == len_keys:
                    return False
                len_sublist = len(_keys[pos])
                idx = 0

    def discard(self, val):
        """
        Remove the first occurrence of *val*.

        If *val* is not a member, does nothing.
        """
        _maxes = self._maxes

        if not _maxes:
            return

        key = self._key(val)
        pos = bisect_left(_maxes, key)

        if pos == len(_maxes):
            return

        _keys = self._keys
        _lists = self._lists
        idx = bisect_left(_keys[pos], key)

        len_keys = len(_keys)
        len_sublist = len(_keys[pos])

        while True:
            if _keys[pos][idx] != key:
                return
            if _lists[pos][idx] == val:
                self._delete(pos, idx)
                return
            idx += 1
            if idx == len_sublist:
                pos += 1
                if pos == len_keys:
                    return
                len_sublist = len(_keys[pos])
                idx = 0

    def remove(self, val):
        """
        Remove first occurrence of *val*.

        Raises ValueError if *val* is not present.
        """
        _maxes = self._maxes

        if not _maxes:
            raise ValueError('{0} not in list'.format(repr(val)))

        key = self._key(val)
        pos = bisect_left(_maxes, key)

        if pos == len(_maxes):
            raise ValueError('{0} not in list'.format(repr(val)))

        _keys = self._keys
        _lists = self._lists
        idx = bisect_left(_keys[pos], key)

        len_keys = len(_keys)
        len_sublist = len(_keys[pos])

        while True:
            if _keys[pos][idx] != key:
                raise ValueError('{0} not in list'.format(repr(val)))
            if _lists[pos][idx] == val:
                self._delete(pos, idx)
                return
            idx += 1
            if idx == len_sublist:
                pos += 1
                if pos == len_keys:
                    raise ValueError('{0} not in list'.format(repr(val)))
                len_sublist = len(_keys[pos])
                idx = 0

    def _delete(self, pos, idx):
        """
        Delete the item at the given (pos, idx).

        Combines lists that are less than half the load level.

        Updates the index when the sublist length is more than half the load
        level. This requires decrementing the nodes in a traversal from the leaf
        node to the root. For an example traversal see self._loc.
        """
        _maxes, _lists, _keys, _index = self._maxes, self._lists, self._keys, self._index

        keys_pos = _keys[pos]
        lists_pos = _lists[pos]

        del keys_pos[idx]
        del lists_pos[idx]
        self._len -= 1

        len_keys_pos = len(keys_pos)

        if len_keys_pos > self._half:

            _maxes[pos] = keys_pos[-1]

            if _index:
                child = self._offset + pos
                while child > 0:
                    _index[child] -= 1
                    child = (child - 1) >> 1
                _index[0] -= 1

        elif len(_keys) > 1:

            if not pos:
                pos += 1

            prev = pos - 1
            _keys[prev].extend(_keys[pos])
            _lists[prev].extend(_lists[pos])
            _maxes[prev] = _keys[prev][-1]

            del _keys[pos]
            del _lists[pos]
            del _maxes[pos]
            del _index[:]

            self._expand(prev)

        elif len_keys_pos:

            _maxes[pos] = keys_pos[-1]

        else:

            del _keys[pos]
            del _lists[pos]
            del _maxes[pos]
            del _index[:]

    def _check_order(self, idx, key, val):
        _keys, _len = self._keys, self._len

        pos, loc = self._pos(idx)

        if idx < 0:
            idx += _len

        # Check that the inserted value is not less than the
        # previous value.

        if idx > 0:
            idx_prev = loc - 1
            pos_prev = pos

            if idx_prev < 0:
                pos_prev -= 1
                idx_prev = len(_keys[pos_prev]) - 1

            if _keys[pos_prev][idx_prev] > key:
                msg = '{0} not in sort order at index {1}'.format(repr(val), idx)
                raise ValueError(msg)

        # Check that the inserted value is not greater than
        # the previous value.

        if idx < (_len - 1):
            idx_next = loc + 1
            pos_next = pos

            if idx_next == len(_keys[pos_next]):
                pos_next += 1
                idx_next = 0

            if _keys[pos_next][idx_next] < key:
                msg = '{0} not in sort order at index {1}'.format(repr(val), idx)
                raise ValueError(msg)

    def __setitem__(self, index, value):
        """
        Replace the item at position *index* with *value*.

        Supports slice notation. Raises a :exc:`ValueError` if the sort order
        would be violated. When used with a slice and iterable, the
        :exc:`ValueError` is raised before the list is mutated if the sort order
        would be violated by the operation.
        """
        _maxes, _lists, _keys, _pos = self._maxes, self._lists, self._keys, self._pos
        _check_order = self._check_order

        if isinstance(index, slice):
            start, stop, step = index.indices(self._len)
            indices = range(start, stop, step)

            if step != 1:
                if not hasattr(value, '__len__'):
                    value = list(value)

                indices = list(indices)

                if len(value) != len(indices):
                    raise ValueError(
                        'attempt to assign sequence of size {0}'
                        ' to extended slice of size {1}'
                        .format(len(value), len(indices)))

                # Keep a log of values that are set so that we can
                # roll back changes if ordering is violated.

                log = []
                _append = log.append

                for idx, val in zip(indices, value):
                    pos, loc = _pos(idx)
                    key = self._key(val)
                    _append((idx, _keys[pos][loc], key, _lists[pos][loc], val))
                    _keys[pos][loc] = key
                    _lists[pos][loc] = val
                    if len(_keys[pos]) == (loc + 1):
                        _maxes[pos] = key

                try:
                    # Validate ordering of new values.

                    for idx, oldkey, newkey, oldval, newval in log:
                        _check_order(idx, newkey, newval)

                except ValueError:

                    # Roll back changes from log.

                    for idx, oldkey, newkey, oldval, newval in log:
                        pos, loc = _pos(idx)
                        _keys[pos][loc] = oldkey
                        _lists[pos][loc] = oldval
                        if len(_keys[pos]) == (loc + 1):
                            _maxes[pos] = oldkey

                    raise
            else:
                if start == 0 and stop == self._len:
                    self._clear()
                    return self._update(value)

                # Test ordering using indexing. If the given value
                # isn't a Sequence, convert it to a tuple.

                if not isinstance(value, Sequence):
                    value = tuple(value)

                # Check that the given values are ordered properly.

                keys = tuple(map(self._key, value))
                iterator = range(1, len(keys))

                if not all(keys[pos - 1] <= keys[pos] for pos in iterator):
                    raise ValueError('given sequence not in sort order')

                # Check ordering in context of sorted list.

                if not start or not len(value):
                    # Nothing to check on the lhs.
                    pass
                else:
                    pos, loc = _pos(start - 1)
                    if _keys[pos][loc] > keys[0]:
                        msg = '{0} not in sort order at index {1}'.format(repr(value[0]), start)
                        raise ValueError(msg)

                if stop == len(self) or not len(value):
                    # Nothing to check on the rhs.
                    pass
                else:
                    # "stop" is exclusive so we don't need
                    # to add one for the index.
                    pos, loc = _pos(stop)
                    if _keys[pos][loc] < keys[-1]:
                        msg = '{0} not in sort order at index {1}'.format(repr(value[-1]), stop)
                        raise ValueError(msg)

                # Delete the existing values.

                self._delitem(index)

                # Insert the new values.

                _insert = self.insert
                for idx, val in enumerate(value):
                    _insert(start + idx, val)
        else:
            pos, loc = _pos(index)
            key = self._key(value)
            _check_order(index, key, value)
            _keys[pos][loc] = key
            _lists[pos][loc] = value
            if len(_lists[pos]) == (loc + 1):
                _maxes[pos] = key

    def irange(self, minimum=None, maximum=None, inclusive=(True, True),
               reverse=False):
        """
        Create an iterator of values between `minimum` and `maximum`.

        `inclusive` is a pair of booleans that indicates whether the minimum
        and maximum ought to be included in the range, respectively. The
        default is (True, True) such that the range is inclusive of both
        minimum and maximum.

        Both `minimum` and `maximum` default to `None` which is automatically
        inclusive of the start and end of the list, respectively.

        When `reverse` is `True` the values are yielded from the iterator in
        reverse order; `reverse` defaults to `False`.
        """
        minimum = self._key(minimum) if minimum is not None else None
        maximum = self._key(maximum) if maximum is not None else None
        return self._irange_key(
            min_key=minimum, max_key=maximum,
            inclusive=inclusive, reverse=reverse,
        )

    def irange_key(self, min_key=None, max_key=None, inclusive=(True, True),
                   reverse=False):
        """
        Create an iterator of values between `min_key` and `max_key`.

        `inclusive` is a pair of booleans that indicates whether the min_key
        and max_key ought to be included in the range, respectively. The
        default is (True, True) such that the range is inclusive of both
        `min_key` and `max_key`.

        Both `min_key` and `max_key` default to `None` which is automatically
        inclusive of the start and end of the list, respectively.

        When `reverse` is `True` the values are yielded from the iterator in
        reverse order; `reverse` defaults to `False`.
        """
        _maxes = self._maxes

        if not _maxes:
            return iter(())

        _keys = self._keys

        # Calculate the minimum (pos, idx) pair. By default this location
        # will be inclusive in our calculation.

        if min_key is None:
            min_pos = 0
            min_idx = 0
        else:
            if inclusive[0]:
                min_pos = bisect_left(_maxes, min_key)

                if min_pos == len(_maxes):
                    return iter(())

                min_idx = bisect_left(_keys[min_pos], min_key)
            else:
                min_pos = bisect_right(_maxes, min_key)

                if min_pos == len(_maxes):
                    return iter(())

                min_idx = bisect_right(_keys[min_pos], min_key)

        # Calculate the maximum (pos, idx) pair. By default this location
        # will be exclusive in our calculation.

        if max_key is None:
            max_pos = len(_maxes) - 1
            max_idx = len(_keys[max_pos])
        else:
            if inclusive[1]:
                max_pos = bisect_right(_maxes, max_key)

                if max_pos == len(_maxes):
                    max_pos -= 1
                    max_idx = len(_keys[max_pos])
                else:
                    max_idx = bisect_right(_keys[max_pos], max_key)
            else:
                max_pos = bisect_left(_maxes, max_key)

                if max_pos == len(_maxes):
                    max_pos -= 1
                    max_idx = len(_keys[max_pos])
                else:
                    max_idx = bisect_left(_keys[max_pos], max_key)

        return self._islice(min_pos, min_idx, max_pos, max_idx, reverse)

    _irange_key = irange_key

    def bisect_left(self, val):
        """
        Similar to the *bisect* module in the standard library, this returns an
        appropriate index to insert *val*. If *val* is already present, the
        insertion point will be before (to the left of) any existing entries.
        """
        return self._bisect_key_left(self._key(val))

    def bisect_right(self, val):
        """
        Same as *bisect_left*, but if *val* is already present, the insertion
        point will be after (to the right of) any existing entries.
        """
        return self._bisect_key_right(self._key(val))

    bisect = bisect_right

    def bisect_key_left(self, key):
        """
        Similar to the *bisect* module in the standard library, this returns an
        appropriate index to insert a value with a given *key*. If values with
        *key* are already present, the insertion point will be before (to the
        left of) any existing entries.
        """
        _maxes = self._maxes

        if not _maxes:
            return 0

        pos = bisect_left(_maxes, key)

        if pos == len(_maxes):
            return self._len

        idx = bisect_left(self._keys[pos], key)

        return self._loc(pos, idx)

    _bisect_key_left = bisect_key_left

    def bisect_key_right(self, key):
        """
        Same as *bisect_key_left*, but if *key* is already present, the insertion
        point will be after (to the right of) any existing entries.
        """
        _maxes = self._maxes

        if not _maxes:
            return 0

        pos = bisect_right(_maxes, key)

        if pos == len(_maxes):
            return self._len

        idx = bisect_right(self._keys[pos], key)

        return self._loc(pos, idx)

    bisect_key = bisect_key_right
    _bisect_key_right = bisect_key_right

    def count(self, val):
        """Return the number of occurrences of *val* in the list."""
        _maxes = self._maxes

        if not _maxes:
            return 0

        key = self._key(val)
        pos = bisect_left(_maxes, key)

        if pos == len(_maxes):
            return 0

        _keys = self._keys
        _lists = self._lists

        idx = bisect_left(_keys[pos], key)

        total = 0
        len_keys = len(_keys)
        len_sublist = len(_keys[pos])

        while True:
            if _keys[pos][idx] != key:
                return total
            if _lists[pos][idx] == val:
                total += 1
            idx += 1
            if idx == len_sublist:
                pos += 1
                if pos == len_keys:
                    return total
                len_sublist = len(_keys[pos])
                idx = 0

    def copy(self):
        """Return a shallow copy of the sorted list."""
        return self.__class__(self, key=self._key, load=self._load)

    __copy__ = copy

    def append(self, val):
        """
        Append the element *val* to the list. Raises a ValueError if the *val*
        would violate the sort order.
        """
        _maxes, _lists, _keys = self._maxes, self._lists, self._keys

        key = self._key(val)

        if not _maxes:
            _maxes.append(key)
            _keys.append([key])
            _lists.append([val])
            self._len = 1
            return

        pos = len(_keys) - 1

        if key < _keys[pos][-1]:
            msg = '{0} not in sort order at index {1}'.format(repr(val), self._len)
            raise ValueError(msg)

        _maxes[pos] = key
        _keys[pos].append(key)
        _lists[pos].append(val)
        self._len += 1
        self._expand(pos)

    def extend(self, values):
        """
        Extend the list by appending all elements from the *values*. Raises a
        ValueError if the sort order would be violated.
        """
        _maxes, _keys, _lists, _load = self._maxes, self._keys, self._lists, self._load

        if not isinstance(values, list):
            values = list(values)

        keys = list(map(self._key, values))

        if any(keys[pos - 1] > keys[pos]
               for pos in range(1, len(keys))):
            raise ValueError('given sequence not in sort order')

        offset = 0

        if _maxes:
            if keys[0] < _keys[-1][-1]:
                msg = '{0} not in sort order at index {1}'.format(repr(values[0]), self._len)
                raise ValueError(msg)

            if len(_keys[-1]) < self._half:
                _lists[-1].extend(values[:_load])
                _keys[-1].extend(keys[:_load])
                _maxes[-1] = _keys[-1][-1]
                offset = _load

        len_keys = len(_keys)

        for idx in range(offset, len(keys), _load):
            _lists.append(values[idx:(idx + _load)])
            _keys.append(keys[idx:(idx + _load)])
            _maxes.append(_keys[-1][-1])

        _index = self._index

        if len_keys == len(_keys):
            len_index = len(_index)
            if len_index > 0:
                len_values = len(values)
                child = len_index - 1
                while child:
                    _index[child] += len_values
                    child = (child - 1) >> 1
                _index[0] += len_values
        else:
            del _index[:]

        self._len += len(values)

    def insert(self, idx, val):
        """
        Insert the element *val* into the list at *idx*. Raises a ValueError if
        the *val* at *idx* would violate the sort order.
        """
        _maxes, _lists, _keys, _len = self._maxes, self._lists, self._keys, self._len

        if idx < 0:
            idx += _len
        if idx < 0:
            idx = 0
        if idx > _len:
            idx = _len

        key = self._key(val)

        if not _maxes:
            # The idx must be zero by the inequalities above.
            _maxes.append(key)
            _lists.append([val])
            _keys.append([key])
            self._len = 1
            return

        if not idx:
            if key > _keys[0][0]:
                msg = '{0} not in sort order at index {1}'.format(repr(val), 0)
                raise ValueError(msg)
            else:
                _keys[0].insert(0, key)
                _lists[0].insert(0, val)
                self._expand(0)
                self._len += 1
                return

        if idx == _len:
            pos = len(_keys) - 1
            if _keys[pos][-1] > key:
                msg = '{0} not in sort order at index {1}'.format(repr(val), _len)
                raise ValueError(msg)
            else:
                _keys[pos].append(key)
                _lists[pos].append(val)
                _maxes[pos] = _keys[pos][-1]
                self._expand(pos)
                self._len += 1
                return

        pos, idx = self._pos(idx)
        idx_before = idx - 1
        if idx_before < 0:
            pos_before = pos - 1
            idx_before = len(_keys[pos_before]) - 1
        else:
            pos_before = pos

        before = _keys[pos_before][idx_before]
        if before <= key <= _keys[pos][idx]:
            _lists[pos].insert(idx, val)
            _keys[pos].insert(idx, key)
            self._expand(pos)
            self._len += 1
        else:
            msg = '{0} not in sort order at index {1}'.format(repr(val), idx)
            raise ValueError(msg)

    def index(self, val, start=None, stop=None):
        """
        Return the smallest *k* such that L[k] == val and i <= k < j`.  Raises
        ValueError if *val* is not present.  *stop* defaults to the end of the
        list. *start* defaults to the beginning. Negative indices are supported,
        as for slice indices.
        """
        _len, _maxes = self._len, self._maxes

        if not _maxes:
            raise ValueError('{0} is not in list'.format(repr(val)))

        if start is None:
            start = 0
        if start < 0:
            start += _len
        if start < 0:
            start = 0

        if stop is None:
            stop = _len
        if stop < 0:
            stop += _len
        if stop > _len:
            stop = _len

        if stop <= start:
            raise ValueError('{0} is not in list'.format(repr(val)))

        stop -= 1
        key = self._key(val)
        pos = bisect_left(_maxes, key)

        if pos == len(_maxes):
            raise ValueError('{0} is not in list'.format(repr(val)))

        _keys = self._keys
        _lists = self._lists

        idx = bisect_left(_keys[pos], key)

        len_keys = len(_keys)
        len_sublist = len(_keys[pos])

        while True:
            if _keys[pos][idx] != key:
                raise ValueError('{0} is not in list'.format(repr(val)))
            if _lists[pos][idx] == val:
                loc = self._loc(pos, idx)
                if start <= loc <= stop:
                    return loc
                elif loc > stop:
                    break
            idx += 1
            if idx == len_sublist:
                pos += 1
                if pos == len_keys:
                    raise ValueError('{0} is not in list'.format(repr(val)))
                len_sublist = len(_keys[pos])
                idx = 0

        raise ValueError('{0} is not in list'.format(repr(val)))

    def __add__(self, that):
        """
        Return a new sorted list containing all the elements in *self* and
        *that*. Elements in *that* do not need to be properly ordered with
        respect to *self*.
        """
        values = reduce(iadd, self._lists, [])
        values.extend(that)
        return self.__class__(values, key=self._key, load=self._load)

    def __mul__(self, that):
        """
        Return a new sorted list containing *that* shallow copies of each item
        in SortedListWithKey.
        """
        values = reduce(iadd, self._lists, []) * that
        return self.__class__(values, key=self._key, load=self._load)

    def __imul__(self, that):
        """
        Increase the length of the list by appending *that* shallow copies of
        each item.
        """
        values = reduce(iadd, self._lists, []) * that
        self._clear()
        self._update(values)
        return self

    @recursive_repr
    def __repr__(self):
        """Return string representation of sequence."""
        temp = '{0}({1}, key={2}, load={3})'
        return temp.format(
            self.__class__.__name__,
            repr(list(self)),
            repr(self._key),
            repr(self._load)
        )

    def _check(self):
        try:
            # Check load parameters.

            assert self._load >= 4
            assert self._half == (self._load >> 1)
            assert self._twice == (self._load * 2)

            # Check empty sorted list case.

            if self._maxes == []:
                assert self._keys == []
                assert self._lists == []
                return

            assert len(self._maxes) > 0 and len(self._keys) > 0 and len(self._lists) > 0

            # Check all sublists are sorted.

            assert all(sublist[pos - 1] <= sublist[pos]
                       for sublist in self._keys
                       for pos in range(1, len(sublist)))

            # Check beginning/end of sublists are sorted.

            for pos in range(1, len(self._keys)):
                assert self._keys[pos - 1][-1] <= self._keys[pos][0]

            # Check length of _maxes and _lists match.

            assert len(self._maxes) == len(self._lists) == len(self._keys)

            # Check _keys matches _key mapped to _lists.

            assert all(len(val_list) == len(key_list)
                       for val_list, key_list in zip(self._lists, self._keys))
            assert all(self._key(val) == key for val, key in
                       zip((_val for _val_list in self._lists for _val in _val_list),
                           (_key for _key_list in self._keys for _key in _key_list)))

            # Check _maxes is a map of _keys.

            assert all(self._maxes[pos] == self._keys[pos][-1]
                       for pos in range(len(self._maxes)))

            # Check load level is less than _twice.

            assert all(len(sublist) <= self._twice for sublist in self._lists)

            # Check load level is greater than _half for all
            # but the last sublist.

            assert all(len(self._lists[pos]) >= self._half
                       for pos in range(0, len(self._lists) - 1))

            # Check length.

            assert self._len == sum(len(sublist) for sublist in self._lists)

            # Check index.

            if len(self._index):
                assert len(self._index) == self._offset + len(self._lists)
                assert self._len == self._index[0]

                def test_offset_pos(pos):
                    from_index = self._index[self._offset + pos]
                    return from_index == len(self._lists[pos])

                assert all(test_offset_pos(pos)
                           for pos in range(len(self._lists)))

                for pos in range(self._offset):
                    child = (pos << 1) + 1
                    if self._index[pos] == 0:
                        assert child >= len(self._index)
                    elif child + 1 == len(self._index):
                        assert self._index[pos] == self._index[child]
                    else:
                        child_sum = self._index[child] + self._index[child + 1]
                        assert self._index[pos] == child_sum

        except:
            import sys
            import traceback

            traceback.print_exc(file=sys.stdout)

            print('len', self._len)
            print('load', self._load, self._half, self._twice)
            print('offset', self._offset)
            print('len_index', len(self._index))
            print('index', self._index)
            print('len_maxes', len(self._maxes))
            print('maxes', self._maxes)
            print('len_keys', len(self._keys))
            print('keys', self._keys)
            print('len_lists', len(self._lists))
            print('lists', self._lists)

            raise