Course Review

15-211 Fundamental Structures of Computer Science

Margaret Reid-Miller

29 April 2004

What was this course about?

How to solve computing problems.

• Problem analysis, to abstract away details and divide into smaller subproblems.

• Mathematical foundations for precise formulations of problems and solutions.

• Data structures and algorithms to solve problems correctly and efficiently.

• Java programming and modular software construction for good implementations.

The science in CS

Not “hacking”, but:

• Thinking about problems abstractly.

• Selecting good data structures and obtaining correct and fast algorithms.

• Implementing programs that are understandable and correct.

Topics Covered

Simple list algorithms

Solving simple recurrences

Dictionaries: Binary trees, tries, hash tables

Priority queues and binary heaps

Data compression

Sorting

String matching

Topics Covered ctd..

Graphs

Greedy algorithms

Dynamic programming

Game trees

Client-server computing

Equivalence relations and union find

Computational geometry

Studying for the Final Exam

Review all material first

• Lectures, programs, quizzes

Do sample finals – under time restrictions

Old Finals are in blackboard/assignments

Types and difficulty of problems may be similar

List Algorithms

List Interface

Reversing a list

• rev(L) = L if |L|=1

• rev(L) = append(first(L),rev(tail(L)))

• What is the complexity?

Solving Recurrences, Asymptotics

Solve T(n) = 3 T(n/3) + 1

Prove the correctness

• By induction

• Or other methods

Big-Oh, little-oh, Big-Omega, Theta

Binary Trees

1. Traverse the tree inorder

2. Do exactly two rotations so that the tree is balanced

3. Consider a binary tree of height h

a. What are the max and min number of nodes?

b. What are the max and min number of leaves?

4. Perfect, full, complete trees

AVL trees

The single rotation

§ For the case of insertion into left subtree of left child:

Z

YX

ZYX

Deepest node of X has depth 2 greater than deepest node of Z.

Depth reduced by 1

Depth increased by 1

Double rotation

§ For the case of insertion into the right subtree of the left child.

Z

X

Y1 Y2

ZX Y1 Y2

Dictionaries

Keypad IM Trie4 5 9

4 6 6

5 8 8

3 3

I

like love

you

5

9

lovely

…

…

Separate chaining

§ Build chains when keys hash to the same table location

0123456789

101112

Aswithlongkeyint

hash

find

in

testinfo

foo

linkchar

onto

fail

foo

urlopen

type

list

queue

Hashing

Describe why hashing a string to sum of its characters is not a good idea.

Assume S = a1a2….an is a string

Define H(S) = ai2i-1 (i=1..n)

Describe a way to efficiently calculate H(S)

Priority Queues and Heaps

Suppose you implement a priority queue using following

• Unsorted array

• Linked list (smallest at front)

• Heap

What is the complexity in each case for

• deleteMin

• insert

Binary heaps

Representing complete binary trees

• Arrays (1-based)§ Parent at position i§ Children at 2i (and 2i+1).

2

1

98

4

10

5 76

3

1 2 3 4 5 6 7 8 9 10

Percolation down

• Bubble the transplanted leaf value down the tree until the heap order property is satisfied.

14

31

2665

24

32

21 6819

1614

--

2665

24

32

21 6819

16

31

1 2

Compression and Huffman’s

Compare two images

One image is 400K the other is 1100K. Which is which?

Tree representation

• Represent prefix free codes as full binary trees

• Full: every node§ Is a leaf, or§ Has exactly 2 children.

• The encoding is then a (unique) path from the root to a leaf.

c

a

b

d0

0

0

1

1

1

a=1, b=001, c=000, d=01

LZW compression

Binary LZW: Compression example

10010110011Input:^

0 1

Dictionary:

Output: 1034

1 2

0 1

0

0 1

3

0 1

4 5

0 1

Data Compression

Encode “I AM SAM SAM I AM SAM SAM” using

• Huffman compression

• LZW

• In each case calculate the compression ratio

Is it possible to apply Huffman after applying LZW?

If so apply Huffman to output generated by LZW above

Sorting and lower bounds

N2 vs Nlog N

N^2Nlog N

Quicksort idea

§ Choose a pivot.

§ Rearrange so that pivot is in the “right” spot.

§ Recurse on each half and conquer!

Upper and lower bounds

N

d× ( )g N( )T N

( ) = T N O( ( ))f N( ) = T N Ω( ( ))g N

c× ( )f N

Sorting

Insert the random set of numbers {10, 12, 5, 15, 8, 14,11,7} into a BST with AVL properties.

Fred has suggested that he can sort a list using hash table. The idea is the following. Assume that there are n numbers in the set. He would find the max and min and create a hash table of size (max-min). Then Fred would simply run an iterator I from min to max and use the hash map H(I) = I mod (max-min) if I is in the set. He claims that this is a sorting algorithm of O(n). What can you tell Fred? Politely please.

String matching

KMP B -M

16 comparisons

This is a string

-----------tring

This is a string

--

-gn

irt

tring

8 comparisons

Graphs

What is breadth first traversal?

What is depth first traversal?

What data structures can be used to implement each one?

Fred claims that if the degree of each node in an undirected graph of n nodes is (n-1), then graph has n(n-1) edges. Can this statement be true?

Graphs

Undirected

Graphs — an overview

PIT

BOS

JFK

DTW

LAX

SFO

Vertices (aka nodes)

Edges

1987

2273

3442145

2462

618

211

318

190

Weights

Dijkstra’s algorithm

Visiteds (D = 0)b (D = 2)d (D = 3)c (D = 4)

a e f g

4 6 6 ∞

s

f

a

b

d

e

c

g

4

2

5

1

11

4

2

2

33

How Can We Generate a MST?

a

ce

d

b2

45

9

6

4

5

5

a

ce

d

b2

45

9

6

4

5

5

Greedy Algorithms

100

5060

3050

60

20

50

20

6040

50

150

40

20

40

20

40 40

10

Find the Shortest Path from Node 1 to every other node

Dynamic Programming

Dependent subproblems, recursive solutions, memoizing, explicit tables.

Knapsack

Longest Common Subsequence

Matrix Multiplication

Floyd-Warshall-Kleene

Dynamic Programming

Consider a sequence of n numbers, A = {a1, a2, a3, ..., an}. A subsequence of a given sequence is just the given sequence with 0 or more of the elements removed.

Let's find an algorithm that finds the subsequence, A' = {ai1, ai2, ai3, ...aik} of A that maximizes ai1 - ai2 + ai3 - ... +- aik (i.e. the sign alternates).

Game trees

A Tic Tac Toe Game Tree

moves

moves

movesAlpha Beta Example

10 2 12 2 7

10 12 7

10 7

10Max

Max

Min

Min

α = 10

β =7

α > β!

Games

2-person, deterministic, complete information, ...

Backtracking

MiniMax

Alpha-beta pruning

Heuristics, iterative deepening, move order, tricks, ...

Client-Server Computing

14

Socket communication

process

writeread socket

process

read writesocket

network

Host Host

Sockets provide a bi-directional communicationchannel from one process to another.

Messages are queued• at sending socket until transmitted across the network and• at receiving socket until received by the receiving process.

18

Client-server interaction

listenSocket = ServerSocket(port)

connectionSocket = listenSocket.accept()

read request(s) from connectionSocket

write reply(s) to connectionSocket

connectionSocket .close()

clientSocket = Socket( hostid ,port)

send request(s) to clientSocket

read reply(s) from clientSocket

clientSocket .close()

Server

Client

TCPsession

Union-find

Union, v.0

13

0

42

5§ {1,2}{0,3}{4}{5}{1,2,0,3}{4}{5}

§ union(0,2)s: 3 -1 1 -1 -1 -1 before

s’: 3 -1 1 1 -1 -1 after0 1 2 3 4 5

public void union(int x, int y){s[find(x)] = find(y);

}

1 30

42

5

Trick 2 : Path compression

§ find flattens trees• Redirect nodes to point directly to the root

§ Example: find(0)

§ Do this whenever traversing a path from node to root.

13

0

42

5 10

42

53

Computational Geometry

Intersections

How do we check if two lines intersect?

A + λ ( - ) = + B A C κ ( D – )C

, Two linear equations two unknowns λ, κ.

Add constraints onλ, κ for , .intersection of rays line segments

A

D

CB

Gift Wrapping

Then find the other extremal points as follows:tie a string to p0 and then wrap it all around P.

extremalp0

p1p2p3

p4

p0p1

p2

p3

p4

Computational Geometry

Points, lines, rays, line segments

Intersection, turns

Membership in region

Jarvis' March

Graham Scan

Divide-and-Conquer (QuickHull, ...)

Lower Bound

Grades

1 “warmup”.

3 small homeworks.

3 large homeworks.

3 quizzes.

Midterm exam.

Final exam.

TA discretion.

39 points.

50 points each.

100 points each.

20 points each.

125 points.

275 points.

50 points.

SUBJECT TO CHANGE

Final Exam

Tuesday, May 4, 5:30 – 8:30 pm

McConomy (*not* in DH)

Sugary stuff will be provided ...

Make sure not to be late.

Final Exam

Closed book, no calculators, cell phones, pagers, IM, …

You can bring 1 (one) page of notes.

Conclusion

Review all material

Do all sample quizzes and finals

Good luck!!