# Exercise 14.1.5

Given an element $x$ in an $n$-node order-statistic tree and a natural number $i$, how can we determine the $i$th successor of $x$ in the linear order of the tree in $\O(\lg n)$ time?

Here's the code:

def nth_successor(node, i):
while i > node.right.size:
if node.parent.left is node:
i -= 1 + node.right.size
node = node.parent
else:
i += 1 + node.left.size
node = node.parent

if i == 0:
return node

return node.right.select(i)


The code first determines whether the successor is in the right subtree, or whether it's somewhere along the parents. It navigates the tree up until it identifies a node, which contains the $i$-th successor in its right subtree, while updating $i$. Once it identifies one, it uses SELECT to find the node with the specific rank.

The time is $\O(\lg n)$, because the height of the three is $\O(\lg n)$, and the algorithm traverses that length at most twice – once on the way up, and then once for SELECT.

### Python code

from enum import Enum
from collections import deque

class Color(Enum):
RED = 1
BLACK = 2

NIL_KEY = object()

def other(direction):
if direction == 'left':
return 'right'
elif direction == 'right':
return 'left'
else:
assert(False)

class Node:
def __init__(self, color, key, parent, left, right, tree, size):
self.color = color
self.key = key
self.parent = parent
self.left = left
self.right = right
self.tree = tree
self.size = size

def sexp(self):
if self.isNil():
return 'NIL'

color = 'R' if self.color == Color.RED else 'B'

return f"{color}({self.key}, {self.left}, {self.right})"

__str__ = sexp

def black_height(self):
node = self
height = 0

while node is not nil:
if node.color == Color.BLACK:
height += 1
node = node.parent

return height

def isRed(self):
return self.color == Color.RED

def isBlack(self):
return self.color == Color.BLACK

def isNil(self):
return self.key is NIL_KEY

def isNotNil(self):
return not self.isNil()

def __bool__(self):
return self.isNotNil()

def child(self, direction):
if direction == 'left':
return self.left
elif direction == 'right':
return self.right
else:
assert(False)

def set_child(self, direction, child):
if direction == 'left':
self.left = child
elif direction == 'right':
self.right = child
else:
assert(False)

__getitem__ = child
__setitem__ = set_child

def other(self, direction):
return self.child(other(direction))

def rotate(self, direction):
child = self.other(direction)
self[other(direction)] = child[direction]

if child[direction]:
child[direction].parent = self

child.parent = self.parent

if not self.parent:
self.tree.root = child
elif self is self.parent[direction]:
self.parent[direction] = child
else:
self.parent[other(direction)] = child

child[direction] = self
self.parent = child

child.size = self.size
self.size = self.left.size + self.right.size + 1

def left_rotate(self):
self.rotate('left')

def right_rotate(self):
self.rotate('right')

def transplant(self, other):
if not self.parent:
self.tree.root = other
elif self is self.parent.left:
self.parent.left = other
else:
self.parent.right = other
other.parent = self.parent

def set(self, parent=None, left=None, right=None, color=None):
if color:
self.color = color
if left is not None:
self.left = left
if right is not None:
self.right = right
if parent is not None:
self.parent = parent

def minimum(self):
node = self

while node.left:
node = node.left

return node

def select(self, i):
node = self

while node:
rank = node.left.size + 1
if i == rank:
return node
elif i < rank:
node = node.left
else:
i -= rank
node = node.right

assert(False)

def rank(self):
rank = self.left.size + 1

node = self

while node.parent:
if node == node.parent.right:
rank += node.parent.left.size + 1
node = node.parent

return rank

def key_rank(self, key):
if self.key == key:
return self.left.size + 1
elif key < self.key:
return self.left.key_rank(key)
else:
return self.left.size + 1 + self.right.key_rank(key)

def nth_successor(self, n):
node = self

while n > node.right.size:
if node.parent.left is node:
n -= 1 + node.right.size
node = node.parent
else:
n += 1 + node.left.size
node = node.parent

if n == 0:
return node

return node.right.select(n)

nil = Node(Color.BLACK, NIL_KEY, None, None, None, None, 0)
nil.parent = nil
nil.left = nil
nil.right = nil

class Tree:
def __init__(self):
self.root = nil

def __str__(self):
return self.root.sexp()

def search(self, key):
node = self.root

while node:
if node.key == key:
return node
elif key < node.key:
node = node.left
else:
node = node.right

return None

def key_rank(self, key):
return self.root.key_rank(key)

def nodes(self):
items = deque()

if self.root:
items.append(self.root)

while items:
node = items.popleft()

yield node

if node.left:
items.append(node.left)

if node.right:
items.append(node.right)

def select(self, i):
return self.root.select(i)

def insert(self, key):
new = Node(Color.RED, key, None, None, None, self, 1)
parent = nil
node = self.root
while node:
node.size += 1

parent = node
if new.key < node.key:
node = node.left
else:
node = node.right

new.parent = parent

if not parent:
self.root = new
elif new.key < parent.key:
parent.left = new
else:
parent.right = new

new.set(left=nil, right=nil, color=Color.RED)

self.insert_fixup(new)

def insert_fixup(self, node):
while node.parent.isRed():
if node.parent is node.parent.parent.left:
direction = 'left'
else:
direction = 'right'

if direction == 'left' or direction == 'right':
uncle = node.parent.parent[other(direction)]
if uncle.isRed():
node.parent.color = Color.BLACK
uncle.color = Color.BLACK
node.parent.parent.color = Color.RED
node = node.parent.parent
else:
if node is node.parent[other(direction)]:
node = node.parent
node.rotate(direction)
node.parent.color = Color.BLACK
node.parent.parent.color = Color.RED
node.parent.parent.rotate(other(direction))

self.root.color = Color.BLACK

def delete(self, key):
def decrease_ancestor_sizes(node):
while node:
node.size -= 1
node = node.parent

deleted = self.search(key)
y = deleted
y_original_color = y.color

if not deleted.left:
decrease_ancestor_sizes(deleted)
extra_black = deleted.right
deleted.transplant(deleted.right)
elif not deleted.right:
decrease_ancestor_sizes(deleted)
extra_black = deleted.left
deleted.transplant(deleted.left)
else:
y = deleted.right.minimum()
y_original_color = y.color
extra_black = y.right

decrease_ancestor_sizes(y)

if y.parent is deleted:
extra_black.parent = y
else:
y.transplant(y.right)
y.right = deleted.right
y.right.parent = y

deleted.transplant(y)
y.left = deleted.left
y.left.parent = y
y.color = deleted.color
y.size = y.left.size + y.right.size + 1

if y_original_color == Color.BLACK:
self.delete_fixup(extra_black)

def delete_fixup(self, node):
while node is not self.root and node.isBlack():
if node is node.parent.left:
direction = 'left'
else:
direction = 'right'

sibling = node.parent[other(direction)]

if sibling.isRed():
sibling.color = Color.BLACK
node.parent.color = Color.RED
node.parent.rotate(direction)
sibling = node.parent[other(direction)]

if sibling.left.isBlack() and sibling.right.isBlack():
sibling.color = Color.RED
node = node.parent
else:
if sibling[other(direction)].isBlack():
sibling[direction].color = Color.BLACK
sibling.color = Color.RED
sibling.rotate(other(direction))
sibling = node.parent[other(direction)]

sibling.color = node.parent.color
node.parent.color = Color.BLACK
sibling[other(direction)].color = Color.BLACK
sibling.parent.rotate(direction)
node = self.root

node.color = Color.BLACK