Dictionaries and their Implementation: Binary Trees, ...

(辞書とその実装: 二分木など)

Data Structures and Algorithms

8th lecture, November 5, 2015

http://www.sw.it.aoyama.ac.jp/2015/DA/lecture8.html

Martin J. Dürst

Today's Schedule

Summary of last lecture, homework
Sorting algorithms faster than O(n log n)
The dictionary ADT
Binary trees and their traversal methods
Binary search tree
Balanced tree

Summary of Last Lecture

Quicksort is a very efficiont algorithm for sorting, and a good example to learn about algorithms and their implementation
In the worst case, quicksort is O(n²); on average, O(n log n)
Quicksort is a good example for the use of average time complexity and randomized algorithms
Implementing quicksort requires attention to many details
Animation of many sorting algorithms: sort.svg
Sorting based on pairwise comparison is Θ(n log n)

Report: Manual Sorting

Deadline: November 4th, 2015 (Wednesday), 19:00.

Problem: Propose and describe an algorithm/algorithms for manual sorting, for the following two cases:

One person sorts 6000 pages
20 people together sort 60000 pages

Each page is a sheet of paper of size A4, where a 10-digit number is printed in big letters.

The goal is to sort the pages by increasing number. There is no knowledge about how the numbers are distributed.

You can use the same algorithm for both cases, or a different algorithm.

Details:

Describe the algorithm(s) in detail, so that e.g. your friends who don't understand computers can execute them.
Describe the equipment/space that you need.
Calculate the overall time needed for each case.
Analyse the time complexity (O()) of the algorithm(s).
Comment on the relationship to other algorithms you know, and on the special needs of manual (as opposed to computer) execution.
If you use any Web pages, books, ..., list them as references at the end of your report
Caution: Use IRIs (e.g. http://ja.wikipedia.org/wiki/情報), not URLs (e.g. http://ja.wikipedia.org/wiki/%E6%83%85%E5%A0%B1)

Problems Seen in Reports

218341.368 seconds (⇒about 61 hours)
6^10¹⁰·10^3·10¹⁰ (units? way too big)
O(60000) (how many seconds could this be)
Calulation of actual time backwards from big-O notation (1second/operation, n=6000, O(n²) ⇒ 3600000 seconds?)
A O(n) algorithm (example: "5 seconds per page")
For 20 people, having only one person work at the end of the algorithm
For humans, binary sorting is constraining (sorting into 3~10 parts is better)
Using bubble sort (868 days without including breaks or sleep)
Preapare 10¹⁰ boxes (problem: space, cost, distance for walking)
Forgetting time for preparation, cleanup, breaks,...
Submitting just a program
Report too short

`Sorting Faster than` `O`(n log n)

All sorting algorithms studied so far assume an arbitrary distribution of values
Decisions are made by comparisons of values;
the depth of the decision tree is at least O(n log n)
If there is some knowledge about the value distribution, improvements are possible
Extreme example: Integers from 1 to n
→Final place of data can be predicted exactly
→O(n)
Radix sort、
Bin sort

Bin Sort

(also called bucket sort)

Example: Sorting by student number

Separate data into 10 parts using most significant digit
Apply recursively to less significant digits
To manage memory, split separation into two phases
(one_digit_stable_sort in 8binradix.rb)
1. Counting number of items in each part
2. Moving data items
Complexity is O(n k), where k is the number of digits
Implementation in Ruby: bin_sort in 8binradix.rb

Radix Sort

Sort once for each digit, starting with the least significant digit
No need to partition data
A stable sorting method is necessary
Complexity is O(n k), where k is the number of digits
Implementation in Ruby: radix_sort 8binradix.rb

Parallelization of Sorting

These days, computers do not get faster, but smaller and larger in numbers
Parallelization becomes important!
For some tasks, parallelization can be very difficult
For some tasks, parallelization can be quite easy
Many sorting algorithms are eazy to parallelize:
- Bubble sort
- Merge sort
- Quick sort

The Dictionary ADT

(caution: Not exactly the same as a (book) dictionary)

For each data item, there is:
- A key: Used to identify the data item, e.g. during search
- A value: All the information besides the key (may be empty)
Operations
- Insert
- Delete
- Search/find

Simple Dictionary Implementations

Sorted array: Search is O(log n) (binary search), insertion/deletion is O(n)
Unordered array/linear list: Search is O(n)
Ideally, search/insertion/deletion should all be O(log n)screen
- Binary search tree
- Balanced tree
- Hashing

Binary Tree

Graph: Consisting of nodes and edges
Tree: The root does not have any parent; all other nodes have exactly one parent
Binary tree: Each node has ≦2 children

Traversal Methods for Binary Trees

Depth first
- Preorder
- Inorder
- Postorder
Breadth first

Binary Search Tree

(definition/invariants)

Each node contains one data item
For any node with key k:
- All the keys in the left subtree will be <k (or ≦k)
- All the keys in the right subtree will be >k (or ≧k)
What to do with multiple identical keys is implementation-dependent

Search in a Search Tree

Start searching from the root node
If the current node, compared to the search key, is
- The same: Return the data item at the current node
- Greater: Search in left subtree
- Smaller: Search the right subtree
- The empty node: Give up seaching

Insertion into a Search Tree

Start insertion from the root node
If the current node, compared to the inserted key, is
- Greater: Insert item into left subtree
- Smaller: Insert item into right subtree
- Empty: Insert item here as a new node (with two empty nodes as children)
- The same: Give up/insert into right subtree, ... (implementation dependent)

Deletion from a Search Tree

Find the node to delete (using search)
If the number of (real, non-empty) children is
- 0: Delete current node (and replace with empty node)
- 1: Replace current node with child
- 2: Replace current node with smallest child in the right subtree
  (or largest node in left subtree)

Implementation of Search Tree

Share a single special node (NilNode) everywhere there is no child node
Pseudocode/implementation in Ruby: 8bintree.rb

Evaluation of Simple Search Tree

The execution speed depends on the height of the tree
The best height is O(log n)
The worst height is O(n)
The average height is O(log n)
(assuming that all input orders have the same probability)

Balanced Trees

In the worst case, the shape of a general search tree is the same as the shape of a linear list
Because the order of insertions/deletions cannot be changed,
using randomization (as used for quicksort) to select the dividing item is impossible
In the case of a complete binary tree, insertion/deletion take too much time

Solution: A tree that is to some degree (but not perfectly) balanced

Top-down 2-3-4 Tree

The number of children for each node is 2, 3, or 4
If a node has k children, it stores k-1 keys and data items
(if the number of children is 2, then this is the same as for a binary search tree)
The keys stored in a node are the separators for the subtrees
The tree is of uniform height
In the lowest layer of the tree, the nodes have no children

Summary

Homework

(no need to submit)

Calculate the minimum and maximum height of a binary search tree with n data items
Using various examples, think about how to insert items into a 2-3-4 tree and propose an algorithm

Glossary

bin sort: ビンソート
most significant digit: 最上位の桁
radix sort: 基数整列
balanced tree: 平衡木
traversal method: 辿り方
binary search tree: 二分探索木
balanced tree: 平衡木
hashing: ハッシュ法
binary tree: 二分木
parallelization: 並列化
depth first: 深さ優先
preorder: 行きがけ順
inorder: 通りがけ順
postorder: 帰りがけ順
breadth first: 幅優先
key: キー、鍵
implementation-dependent: 実装依存
top-down 2-3-4 tree: トップダウン 2-3-4 木