# Note: Taken from https://code.activestate.com/recipes/577197-sortedcollection/.
#
# Copyright 2010 Raymond Hettinger
#
# Permission is hereby granted, free of charge, to any person obtaining a copy of
# this software and associated documentation files (the "Software"), to deal in the
# Software without restriction, including without limitation the rights to use, copy,
# modify, merge, publish, distribute, sublicense, and/or sell copies of the Software,
# and to permit persons to whom the Software is furnished to do so, subject to the
# following conditions:
#
# The above copyright notice and this permission notice shall be included in all copies
# or substantial portions of the Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
# INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR
# PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE
# FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
# OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
# DEALINGS IN THE SOFTWARE.

from bisect import bisect_left, bisect_right


class SortedCollection(object):
    def __init__(self, iterable=(), key=None):
        self._given_key = key
        key = (lambda x: x) if key is None else key
        decorated = sorted((key(item), item) for item in iterable)
        self._keys = [k for k, item in decorated]
        self._items = [item for k, item in decorated]
        self._key = key

    def _getkey(self):
        return self._key

    def _setkey(self, key):
        if key is not self._key:
            self.__init__(self._items, key=key)

    def _delkey(self):
        self._setkey(None)

    key = property(_getkey, _setkey, _delkey, 'key function')

    def clear(self):
        self.__init__([], self._key)

    def copy(self):
        return self.__class__(self, self._key)

    def __len__(self):
        return len(self._items)

    def __getitem__(self, i):
        return self._items[i]

    def __iter__(self):
        return iter(self._items)

    def __reversed__(self):
        return reversed(self._items)

    def __repr__(self):
        return '%s(%r, key=%s)' % (
            self.__class__.__name__,
            self._items,
            getattr(self._given_key, '__name__', repr(self._given_key))
        )

    def __reduce__(self):
        return self.__class__, (self._items, self._given_key)

    def __contains__(self, item):
        k = self._key(item)
        i = bisect_left(self._keys, k)
        j = bisect_right(self._keys, k)
        return item in self._items[i:j]

    def index(self, item):
        'Find the position of an item.  Raise ValueError if not found.'
        k = self._key(item)
        i = bisect_left(self._keys, k)
        j = bisect_right(self._keys, k)
        return self._items[i:j].index(item) + i

    def count(self, item):
        'Return number of occurrences of item'
        k = self._key(item)
        i = bisect_left(self._keys, k)
        j = bisect_right(self._keys, k)
        return self._items[i:j].count(item)

    def insert(self, item):
        'Insert a new item.  If equal keys are found, add to the left'
        k = self._key(item)
        i = bisect_left(self._keys, k)
        self._keys.insert(i, k)
        self._items.insert(i, item)

    def insert_right(self, item):
        'Insert a new item.  If equal keys are found, add to the right'
        k = self._key(item)
        i = bisect_right(self._keys, k)
        self._keys.insert(i, k)
        self._items.insert(i, item)

    def remove(self, item):
        'Remove first occurence of item.  Raise ValueError if not found'
        i = self.index(item)
        del self._keys[i]
        del self._items[i]

    def find(self, k):
        'Return first item with a key == k.  Raise ValueError if not found.'
        i = bisect_left(self._keys, k)
        if i != len(self) and self._keys[i] == k:
            return self._items[i]
        raise ValueError('No item found with key equal to: %r' % (k,))

    def find_le(self, k):
        'Return last item with a key <= k.  Raise ValueError if not found.'
        i = bisect_right(self._keys, k)
        if i:
            return self._items[i-1]
        raise ValueError('No item found with key at or below: %r' % (k,))

    def find_lt(self, k):
        'Return last item with a key < k.  Raise ValueError if not found.'
        i = bisect_left(self._keys, k)
        if i:
            return self._items[i-1]
        raise ValueError('No item found with key below: %r' % (k,))

    def find_ge(self, k):
        'Return first item with a key >= equal to k.  Raise ValueError if not found'
        i = bisect_left(self._keys, k)
        if i != len(self):
            return self._items[i]
        raise ValueError('No item found with key at or above: %r' % (k,))

    def find_gt(self, k):
        'Return first item with a key > k.  Raise ValueError if not found'
        i = bisect_right(self._keys, k)
        if i != len(self):
            return self._items[i]
        raise ValueError('No item found with key above: %r' % (k,))