"""Sequence.py

Doubly-linked circular list for maintaining a sequence of items
subject to insertions and deletions.

D. Eppstein, November 2003.
"""

from __future__ import generators
import math
import sys

class SequenceError(Exception): pass

class Sequence:
    """Maintain a sequence of items subject to insertions and removals.
    All sequence operations take constant time except indexing, which
    takes time proportional to the index.
    """

    def __init__(self, iterable=[], key=None):
        """We represent the sequence as a doubly-linked circular linked list,
        stored in two dictionaries, self._next and self._prev.  We also store
        a pointer self._first to the first item in the sequence.  If key is
        supplied, key(x) is used in place of x to look up item positions;
        e.g. using key=id allows sequences of lists or sets.
        """
        self._key = key
        self._items = {}
        self._next = {}
        self._prev = {}
        self._first = None
        for x in iterable:
            self.append(x)

    def __iter__(self):
        """Iterate through the objects in the sequence.
        May give unpredictable results if sequence changes mid-iteration.
        """
        item = self._first
        while self._next:
            yield self._items.get(item,item)
            item = self._next[item]
            if item == self._first:
                return

    def __getitem__(self,i):
        """Return the ith item in the sequence."""
        item = self._first
        while i:
            item = self._next[item]
            if item == self._first:
                raise IndexError("Index out of range")
            i -= 1
        return self._items.get(item,item)

    def __len__(self):
        """Number of items in the sequence."""
        return len(self._next)

    def __repr__(self):
        """Printable representation of the sequence."""
        output = []
        for x in self:
            output.append(repr(x))
        return 'Sequence([' + ','.join(output) + '])'

    def key(self,x):
        """Apply supplied key function."""
        if not self._key:
            return x
        key = self._key(x)
        self._items[key] = x
        return key

    def _insafter(self,x,y):
        """Unkeyed version of insertAfter."""
        if y in self._next:
            raise SequenceError("Item already in sequence: "+repr(y))
        self._next[y] = z = self._next[x]
        self._next[x] = self._prev[z] = y
        self._prev[y] = x

    def append(self,x):
        """Add x to the end of the sequence."""
        x = self.key(x)
        if not self._next:  # add to empty sequence
            self._next = {x:x}
            self._prev = {x:x}
            self._first = x
        else:
            self._insafter(self._prev[self._first],x)

    def remove(self,x):
        """Remove x from the sequence."""
        x = self.key(x)
        prev = self._prev[x]
        self._next[prev] = next = self._next[x]
        self._prev[next] = prev
        if x == self._first:
            self._first = next
        del self._next[x], self._prev[x]

    def insertAfter(self,x,y):
        """Add y after x in the sequence."""
        y = self.key(y)
        x = self.key(x)
        self._insafter(x,y)

    def insertBefore(self,x,y):
        """Add y before x in the sequence."""
        y = self.key(y)
        x = self.key(x)
        self._insafter(self._prev[x],y)
        if self._first == x:
            self._first = y