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Algorithm Fundamentals

- Definition
- Steps in Construction
- Properties

Definition and role of Algorithms

- Algorithms are central to computer science
- An algorithm to solve problem P is
- a sequence of instructions
- sequence is finite
- each instruction is unambiguous
- initial instructions acquire valid input
- intermediate instructions perform processing
- final instructions produce valid output for P
- algorithm terminates after finite number of steps

Algorithm Issues

- Algorithm must be specified
- Pseudocode is combination of English and

programming language constructs - Algorithm is either exact or approximate
- An exact algorithm produces an output which is

valid for input while an approximate algorithm

produces an output close to valid output

Algorithm Issues

- Algorithm should be efficient
- If input is size n, time complexity of algorithm

may depend only on n (every case) or may depend

on particular input (worst case / best case)

will use Big Oh notation - Algorithm should be tested
- Before implementation by hand example
- After implementation by run time scenarios
- Algorithm is implemented in programming language

P - Each module and major control statement of P

should be commented

Algorithm Examples

- Euclidean Algorithm
- Problem Find greatest integer d which divides m

and n - Fact If m n then m d x n r , 0 ? r lt n
- Fact If p is a common divisor of m and n then
- m p x k1 and n p x k2 and
- r m d x n p x k1 d x p x k2 so p is a

common divisor of r - Exchange problem of finding greatest integer

which divides m and n for smaller problem of

finding greatest integer which divides n and m

mod n - Example Find gcd(96,36)
- gcd(96,36) gcd(36, 96 mod 36 24)

Euclidean Algorithm Pseudocode

- Euclid(m,n)
- Computes gcd(m,n)
- Input Two integers m, n with m gt n 0
- While n ? 0
- r ? m mod n
- m ? n
- n ? r
- Return m
- How do we know Euclids algorithm terminates?

Euclidean Algorithm Example

- Euclid(96,36)

m n r m mod n gcd

96 36 24

36 24 12

24 12 0

12 0 12

Euclidean Algorithm Implementation

Sieve of Erastothanes

Programming Assignment 1due Sept 16

- Find gcd(m,n)
- Using Euclidean Algorithm
- Measure time complexity by number of divisions
- Using
- sieve of Erastothanes
- Finding prime factors of m and of n
- Measure time complexity by number of divisions
- Finding lowest exponent on each prime factor
- Measure time complexity by number of comparisons

Fundamental Data Structures

- Arrays sequence of n items of same data type

with length. Each item is accessed by starting

location index. - Strings sequence of characters. Implemented as

array. Dynamic strings end with special

character. - Linked List sequence items implemented as item

value with link to next item. - Stacks list in which insertions and deletions

are made from same end of list (the top) - Queues list in which insertions are made one

end (tail) and deletions are made from other end

(head). - Graphs a set of vertices and a set of vertex

pairs called edges.

Graph Representation

- List of vertices and list of edges.
- Edge List (1,3), (2,1), (2,3), (3,2)
- Two dimensional array (adjacency matrix) with

boolean entries or edge length entries. - _ 1 2 3
- 1 - - 1
- 2 1 - 1
- 3 - 1 -
- Directory plus neighbor list directory has

start and stop into neighbor list - Directory Neigbor List
- Start Stop
- 1 1 1 1 3
- 2 2 3 2 1
- 3 4 4 3 3
- 4 2

Weighted Graphs and Paths

- A weighted graph is a graph in which each edge e

has an associated weight we. The graph is

symmetric if the weight of (u,v) weight of

(v,u). - A path is a sequence of vertices v1, v2 .. Vn for

which (vi , vi1) is an edge for each i. - The length of a path with edge weights w1, w2 ..

Wn is the sum of the edge weights - ?wi

Graph Properties

- Connected a graph is connected if there is a

path between every pair of vertices. - A cycle (circuit) is a path for which start and

end vertices coincide. - A tree is a graph which is connected and has no

cycles.

Tree Properties

- A tree with n vertices has n-1 edges.
- Proof by induction on the number of vertices
- If n 1, there is 1 vertex and 0 edges since a

tree by definition has no cycles and thus no

loops. - If a tree with n vertices has n 1 edges, let T

be a tree with n 1 vertices. Pick a vertex v

in T and if its degree is greater than one, move

to a neighbor v along edge e. If the degree of

v is greater than one, move to a neighbor v

along an edge e different from e. Eventually

must arrive at a vertex z with degree 1. Why?

Remove z and the edge by which we reached z. The

result is a tree T. Why? T has n vertices and

by inductive hypothesis, must have n-1 edges.

Therefor T has n (n1) 1 edges.

Sets and Dictionaries

- A set is a collection of objects. Objects belong

to a set at most once. The empty set is a set

with no elements or objects - A dictionary is a structure which has key objects

and a unique value associated with each key.

Graph theory The city of Königsberg

- The city is divided by a river. There are two

islands at the river. The first island is

connected by two bridges to both riverbanks and

is also connected by a bridge to the other

island. The second island has three bridges

connecting to a riverbank or the other island. - Question Is there a walk around the city that

crosses each bridge exactly once? - Swiss mathematician Leonhard Euler invented graph

theory to solve this problem.

The city of Königsberg

Graph of the Königsberg bridge system

Euler Circuits

- A graph has an Euler circuit iff it is connected

and every vertex is of even degree. - Necessity Euler circuit enters a vertex each

time on a new edge and leaves the vertex on a new

edge. So vertex has degree 2 number of times

on circuit - Sufficiency Pick starting vertex and traverse

graph, each time picking new edge. Can only be

blocked at start. Either have Euler circuit or

can pick vertex with unused edge on circuit and

build subtour starting with it. Splice subtour

in. Eventually will have used all edges once.

Service Delivery mail delivery, meter reading

etc. require visiting each street of a city.

Example of Garbage collection

Programming Assignment 2

- Find minimal Euler tour of an undirected graph
- Read number of vertices and edge list (u,v)
- Determine odd vertices (even number of them)
- Determine all pairs of vertex distances
- Pair odd vertices to minimize the sum of paired

distances - Double shortest paths connecting paired odd

vertices - Find Euler tour of resulting Euler graph

An undirected weighted graph

1

2

5

3

4

6

Dynamic ProgrammingAll Pairs Shortest Path

Dk(i,j) min Dk-1(i,j) , Dk-1(i,k) Dk-1(k,j)

This is decomposition of problem to shortest path

through vertices 1..k is min of going through

vertex k and not going through vertex k