# Exercise 14.3.4

Given an interval tree $T$ and an interval $i$, describe how to list all intervals in $T$ that overlap $i$ in $\O(\min(n, k \lg n))$ time, where $k$ is the number of intervals in the output list. (Hint: One simple method makes several queries, modifying the tree between queries. A slightly more complicated method does not modify the tree).

The simple method would be to remove the interval and search again, until an interval is no longer present.

I struggle to formally establish an upper bound. Intuitively, if $k = n$, that is, all intervals overlap with the one that is being searched for, it will always be the root, and removing the root would be a constant operating, establishing $\O(n)$ complexity. I'm not sure how to reason about this when $k < n$ but $n < k \lg n$. Should pan out, but no idea why.

An algorithm that does not modify the tree is doable, but the upper bound still evades me. Let's explore it nonetheless.

Once we visit a node, we can check whether it overlaps and add its interval to the result if it does. We're then in a situation in which we may have to explore both branches of the tree. Letting $x$ be the node, we have the following constraints to work with:

• $x.left$ will not contain overlapping intervals if $x.left.max < i.low$.
• $x.right$ will not contain overlapping intervals unless $[x.int.low, x.right.max]$ overlaps with $i$.

We can use this to avoid visiting some of the nodes in the tree. The resulting algorithm will certainly be $\O(n)$, because it does not visit a node more than once. Whether it is $\O(k \lg n)$, I have no idea.

def search(tree, interval):
result = []

def collect(node):
if node.interval.overlaps(interval):
result.append(node.interval)

if node.left and interval.low <= node.left.max:
collect(node.left)

if node.right and Interval(node.interval.low,
node.right.max).overlaps(interval):
collect(node.right)

collect(tree.root)

return result


### Python code

from enum import Enum
from collections import deque

class Interval:
def __init__(self, low, high):
assert low <= high
self.low = low
self.high = high

def __eq__(self, other):
return isinstance(other, Interval) and self.low == other.low and \
self.high == other.high

def __hash__(self):
return hash((self.low, self.high))

def __contains__(self, n):
return self.low <= n <= self.high

def __repr__(self):
return f"Interval({self.low}, {self.high})"

__str__ = __repr__

def overlaps(self, other):
return self.low <= other.high and other.low <= self.high

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)

def max_maybe(*args):
return max([arg for arg in args if arg is not None])

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

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

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

return f"{color}({self.interval}, max={self.max}, {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.interval 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

self.max = max_maybe(
self.interval.high,
self.left.max if self.left else None,
self.right.max if self.right else None,
)

child.max = max_maybe(
child.interval.high,
child.left.max if child.left else None,
child.right.max if child.right else None,
)

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

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

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

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

def find(self, interval):
node = self.root

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

return None

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

while node:
if interval.overlaps(node.interval):
return node
elif node.left and node.left.max >= interval.low:
node = node.left
else:
node = node.right

return None

def search_all(self, interval):
result = []

def collect(node):
if node.interval.overlaps(interval):
result.append(node.interval)

if node.left and interval.low <= node.left.max:
collect(node.left)

if node.right and Interval(node.interval.low,
node.right.max).overlaps(interval):
collect(node.right)

collect(self.root)

return result

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 insert(self, interval):
new = Node(Color.RED, interval, None, None, None, interval.high, self)
parent = nil
node = self.root
while node:
parent = node
if new.interval.low < node.interval.low:
node = node.left
else:
node = node.right

new.parent = parent

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

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

self.max_fixup(parent)

self.insert_fixup(new)

def max_fixup(self, node):
while node:
node.max = max_maybe(
node.interval.high,
node.left.max if node.left else None,
node.right.max if node.right else None
)

node = node.parent

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, interval):
deleted = self.find(interval)
y = deleted
y_original_color = y.color

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

todo = None

if y.parent is deleted:
extra_black.parent = y
else:
todo = y.parent
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

self.max_fixup(todo or y)

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