CS660: Intro to Trees

CS660 Combinatorial Algorithms
Fall Semester, 1996
Intro to Trees

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San Diego State University -- This page last updated Sep 26, 1996

Contents of Intro to Trees

Intro to Trees Slide # 1

References

Introduction to Algorithms by Cormen, Leiserson, Rivest. Chapter 13

Data Structures and Algorithms 1: Sorting and Searching, Mehlhorn, Kurt, pages 174-177

Intro to Trees Slide # 2

Basic Operations Access (k, t): Return the item in dictionary t with key k; if no item in t has key k, return null

Insert (j, t): Insert item j into t, not previously containing j

Delete (j, t): Delete item j from t
Operations based on order Minimum (t): Return the item in t with the smallest key

Maximum (t): Return the item in t with the largest key

Successor (k, t): Return the item in t with the smallest key larger than k

Predecessor (k, t): Return the item in t with the largest key smaller than k
Multiple-Structure Operations Join (a, i, b): Return dictionary formed by combining dictionary a, item i, and dictionary b. Assumes that every item in a has key less then key(i) and every item in b has key larger than key(i)

Split (i, s): Split dictionary s, containing item i, into three dictionarys: a, containing all items with key less than key(i); single dictionary i; and b, containing all items with key greater than key(i)

Intro to Trees Slide # 3

Types of Structures & Algorithms for Dictionaries
Lists in arrays, unordered
Lists in arrays, ordered
Linked Lists, unordered
Linked Lists, ordered
Hash tables
Binary trees
Skip Lists
B-Trees
Heaps

What are the advantages of each?

Intro to Trees Slide # 4

Types of Algorithms for Trees
Off-line

Totally balanced BST
Know all items in list Minimize worst case search

Optimum BST Know all items and probability of access Minimize total search cost over all access

On-line

Balanced BST (Weight, Height), Skip lists Add items as they show up Minimize worst case search
Self-Adjusting BST
Modify tree based on access pattern

Intro to Trees Slide # 5

Tree Operations on BST

Access, Minimum, Maximum, Successor, Predecessor, Insert

Intro to Trees Slide # 6

Tree Operations Delete Case 1 Delete leaf Delete(20)

Intro to Trees Slide # 7

Tree Operations Delete Case 2 Delete node with one child Delete(7)

Intro to Trees Slide # 8

Tree Operations Delete Case 3 Delete node with two children Delete(6)

Intro to Trees Slide # 9

Tree Operations - Performance

Theorem

The operations Access, Minimum, Maximum, Successor, Predecessor, Insert, and delete run in O(h) on a BST of height h

Randomly Built BST

Assume that we have n distinct keys in random order

Each n! permutations of the input keys is equally likely

Insert the keys in an empty tree, using random order

Theorem

The average height of a randomly built BST on n distinct keys is O( lgn )

Intro to Trees Slide # 10

General Terms

We have a list of n items: a₁, a₂, ..., a_n in BST

Probability of accessing item a_k is P(a_k) = Alpha_k

Let Beta_k be the probability of accessing a key that is between a_k and a_k+1

Intro to Trees Slide # 11

More Terms
Let b_k be the leaf between a_k and a_k+1

Beta_k is the probability of accessing leaf b_k

Ordered by keys we have , b₀< a₁<b₁< a₂ < ... < a_n< b_n

(Beta₀ ,Alpha₁ ,Beta₁ , Alpha₂, Beta₂,... , Alpha_n, Beta_n)¹ is called the access distribution

Let

Intro to Trees Slide # 12

Weighted path length of a tree
Let D(a_k) = depth of node a_k

Define the weighted path length of tree T as:

is the average number of comparisons in a search of T

Intro to Trees Slide # 13

Entropy
Let (Gamma₁ ,Gamma₂,... , Gamma_n) be a discrete probability distribution, i.e.

Gamma_k >= 0 and SigmaGamma_k =1

H(Gamma₁ ,Gamma₂,... , Gamma_n) =

is the entropy of the distribution.

Use the convention that 0*lg 0 = 0

Gamma₁	Gamma₂	Gamma₃	Gamma₄	Gamma₅	Gamma₆	Gamma₇	Gamma₈	H
.125	.125	.125	.125	.125	.125	.125	.125	3
.250	.250	.250	.250	.000	.000	.000	.000	2
.500	.500	.000	.000	.000	.000	.000	.000	1
1.000	.000	.000	.000	.000	.000	.000	.000	0
.900	.010	.010	.010	.010	.010	.010	.010	.602
.800	.160	.032	.0064	.00128	.00026	.00005	.00001	.902
.368	.184	.123	.092	.074	.061	.053	.046	.265

Intro to Trees Slide # 14

Lower bound on weighted Path Length
We have a list of n items: a₁, a₂, ..., a_n in BST

Probability of accessing item a_k is P(a_k) = Alpha_k

Let b_k be the leaf between a_k and a_k+1

Beta_k is the probability of accessing leaf b_k

Let

H = H(Beta₀ ,Alpha₁ ,Beta₁ , Alpha₂, Beta₂,... , Alpha_n, Beta_n)

P =

Theorem 5[2] For any BST we have:

a)

b)