Chapter 14 Advanced Trees © 2006 Pearson Education Inc., Upper Saddle River, NJ. All rights reserved.

Chapter 14

Advanced Trees

© 2006 Pearson Education Inc., Upper Saddle River, NJ. All rights reserved.

Overview● 14.1

– Heaps provide an efficient implementation of a new kind of queue.

● 14.2– Trees to model a cluster of disjoint sets.

● 14.3– Digital search trees provide a new way to store sets

of Strings.● 14.4

– Red-black trees, a variation on binary search trees, are guaranteed to remain balanced.

Heaps● Heap

– A binary tree.● The value at each node is less than or equal to the

values at the children of that node.● The tree is perfect or close to perfect.

– The requirement that a heap is “perfect or close to perfect” lets us use a contiguous representation somewhat analogous to the representation of multidimensional arrays from Section 12.3

– We use an ArrayList with the root at position 0, its children in the next two positions, their children in the next four.

A Sample Heap

● Find the relatives– The left child of the node at index i is at index 2i +

1.– The right child is at index 2i + 2 (e.g. nodes 5 & 10)– The parent is at index:

Heaps

Heaps

Priority Queues

● A heap is a good data structure for implementing a priority queue.– We remove something from a regular queue and get

the oldest element, because first-in first-out (FIFO).– A priority queue, we get the smallest element.– The smallest element in a heap: it's always at index 0.

Priority Queues (to add new node 3)

● Add something to a priority queue by tacking it onto the end.

● Filtering the element up toward the root until it is in a valid position.

● Even the worst case, this takes time proportional to the height of the tree.

● Since the trees is perfect or close to perfect, this is in O(log n) time.

Priority Queue- Code for adding an element into a priority queue

Priority Queue

Removing an Element from a Priority Queue

● For example, remove the root node from a priority queue (Fig 14-6)

– Remove the current root node.– Move the element in the last

position as the root and filtered it down until it is in a legitimate position.

● This takes O(log n) time

Priority Queue

- Code for removing an element from a priority queue

Priority Queue

Heapsort (Fig. 14-8)

● A very useful sorting algorithm.● Call the Heap constructor to use the input elements

to create a heap. ● Note, the constructor on lines 1-13 in Fig. 14-8 is

an overloaded constructor for the Heap class. The other was in Figure 14-3.

● Call the heapsort method to remove the root (smallest) element from the heap, one at a time.

● Call FliterDown() to make the heap valid.● The order to remove elements from the heap is in

the ascending order (i.e. the output elements are in ascending order).

● Heapsort algorithm is O(n log n)

Heapsort

Heaps● Java.util package contains a PriorityQueue

class.

Disjoint Set Clusters

● Disjoint sets– Sets are disjoint if they have no elements in

common.– Clusters of disjoint sets include the sets of players

on different baseball teams, the set of cities in different countries, and the sets of newspapers owned by different media companies.

– We want to allow more efficient performance by:● Determining whether two elements belong to the same

set.● Merge two sets.

Disjoint Set Clusters

● up-tree– Cluster represents nodes in these trees keep

track of their parents.

Disjoint Set Clusters

● To determine if two elements are in the same set, they lead to the same root, the elements are in the same tree.

Disjoint Set Clusters

● Represent the up-trees at each position we store the parent of the corresponding element.

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

● All of the methods take constant time except for findRoot()– Worst case Θ(n)

Disjoint Set Clusters

● We can keep the trees shorter by making the root of the shorter tree a child of the root of the taller tree.

Disjoint Set Clusters

● We need to keep track of the height of each tree.– We need to keep track of heights only for the roots.– Store the height of each root in the array parents.– To avoid confusion between a root with height 3

and a node whose parent is 3, we store the heights as negative numbers.

Disjoint Set Clusters

Disjoint Set Clusters

● A tree might have height 0– The entry for the root of a tree of height h is -h-1

Disjoint Set Clusters

● Path Compression– A second improvement

● Suppose we determine that the root of the tree containing 4 is 7.

● We can make this operation faster next time by making 7 the parent of 4.

● In fact, we might as well make 7 the parent of every node we visit on the way to the root.

● We can't alter these parents until after we've found the root.

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

● O(log* n), log* n is the number of times we have to take the logarithm to get down to 1.

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

● The entire word set can be represented as a digital search tree.– Words are represented as paths through the tree.– Each child of node is associated with a letter.

Disjoint Set Clusters

Disjoint Set Clusters

● Whenever the user enters a letter, we descend to the appropriate child.– If there is no such child, the user loses.– When we need to pick a letter, we randomly

choose a child and the corresponding letter.

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

Disjoint Set Clusters

● getChild() is nothing more than a has table lookup.– Makes a digital search tree an excellent choice

when searching for Strings containing a prefix that grows one character at a time.

Red-Black Trees

● A plain binary search tree performs poorly if the tree is not balanced.– Worst case occurs if the elements are inserted in

order, the tree is linear, so search, insertion, and deletion take linear time.

– Red-black tree● Ensures that the tree cannot become significantly

imbalanced.– In the Java collections framework, the classes

TreeSet and TreeMap use red-black trees.

Red-Black Trees

● Red-black tree is a binary search tree and the node has a color, either red or black.– The root is black.– No red node has a red child.– All paths must contain the same number of black

nodes.

Red-Black Trees

● These properties ensure that the tree cannot be significatnly imbalanced.– The shortest path to a null child contains d nodes.– The longest path can contain at most 2d nodes.– Height of a red-black tree containing n nodes is in

O(log n).– Search in red-black trees is identical to search in

binary search trees.

Red-Black Trees

● Start at the root and descend until we either find the target or try to descend from a leaf.– In the latter case, the target is not present in the

tree, so we attach a new, red leaf.– The new node may be a child of another red node.– We fix this by working our way back up the tree,

changing colors and rearranging nodes.– Repair operation, complicated, time for insertion is

still in O(log n).

Red-Black Trees

● A key step in tree repair is rotation.● When we rotate, we replace a node with one of its children.

Red-Black Trees

● Work back up the tree in several steps, performing color changes and rotations.– Each step either fixes the tree or moves the

problem closer to the root.– If we get the root and still have a red node, we can

simply color it black.– How do we fix this?

● Three cases to consider, depending on the color of the node's parent's sibling and on whether node is a left or right child.

Red-Black Trees

● Node has a red aunt– Since the great-grandparent may also be red, we

may have to do some repair there, too, but we're getting closer to the root.

Red-Black Trees

● Node has a black aunt.– Outer-child.

● No further work is necessary at this point.

Red-Black Trees

● Node has a black aunt and is an inner child– New outer child is red and has a red parent, so we

can repair it as before.

Red-Black Trees● Splicing out a red node can never cause any

problems.– Let node be the child of the node that was spliced

out.– If node is red, we can simply color it black to cancel

out the problem.– If node is black, the subtree rooted at node's parent

is short a black node.

Red-Black Trees

● Node's sibling is black and has two black children

● Repair closer to the root.

Red-Black Trees

● Node's sibling is black and the outer child is red– No further work is necessary at this point.

Red-Black Trees

● Node's sibling is black, has a black outer child and has a red inner child.– Leads to the previous case.

Red-Black Trees

● Fourth case node's sibling is red.– Lead to one of the other 3 cases.

Red-Black Trees

● We use references to a special black node called a sentinel.– Sentinel indicates that we can't go any farther.

– We use a single sentinel instance to represent all nonexistent children.

– We keep track of the parent of each node.– The root of a parent is the sentinel.

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Red-Black Trees

Summary

● A heap is a binary tree data structure used either to represent a priority queue or in the heapsort algorithm.– A heap is a perfect binary tree or close to it.– The value at each node is less than or equal to the

values at the node's children.– When changes are made to a heap, it is repaired by

filtering the offending element up or down until it is in the right place.

Summary

● A cluster of disjoint sets may be represented as a forest of up-trees.– Two elements in the same set are in the same tree,

which can be detected by following parents up to the root.

– This data structure supports efficient algorithm for determining compression, these operations take amortized time in O(log*n)

● A set of strings may be represented as a digital search tree.– Each string corresponds to a path through the tree.

Summary

● Java's TreeSet and TreeMap classes use red-black trees, which are similar to binary search trees.– The colors in the tree must have certain properties,

which guarantee that the tree cannot be badly out of balance.

– Worst-case running time in O(log n) for search, insertion, and deletion.

● A special sentinel node is used in place of absent parents and children, where there would normally be null references.

Summary

● Search in red-black tree works just as it does in a binary search tree.– After the basic insertion and deletion operations, it

may be necessary to repair the tree to satisfy the properties.

Chapter 14 Self-Study Homework

● Pages: 377● A. Do Exercise 14.1 ● B. Use heapsort to sort the following integer set

{5, 2, 1, 4, 9, 8, 10, 7, 6, 3} in ascending order.1. Use the input data to create a heap

2. Then output the root node one by one to sort the data set in ascending order.

3. Show the heap in step 1 (the heap with 10 integers).

4. Show the heap after output each integer (9 heaps totally)