Algorithms

1
Algorithms

• Irena Pevac
• CCSU

2
Design Techniques

• A design technique or strategy is general
  approach to solving problems algorithmically that
  is applicable to a variety of problems from
  different aras
• Brute force
• Divide and conquer
• Dynamic
• Greedy

3
Methods to specify an Algorithm

• Pseudocode
• Mixture of natural language and programming
  language
• Flowchart
• Method of expressing an algorithm by a collection
  of connected geometric shapes containing
  descriptions of the algorithms steps.

4
Analyzing an Algorithm

• Time efficiency
• Space efficiency
• Simplicity
• Generality

5
Proving Correctness

• Prove that the algorithm produces required result
  for every single input in finite amount of time.
• Mathematical induction is often used to prove
  that certain assertion holds for every integer.

6
Problem Types

• Sorting
• Searching
• String processing
• Graph problems
• Combinatorial problems
• Geometric problems
• Numerical problems

7
Sorting

• Quadratic ?(n2) running time sorting
• Insertion sort
• Selection sort
• Bubble sort
• ?(n lg n) running time sorting
• Quick sort
• Merge sort
• Heap sort

8
Stable Sorting Algorithms

• Sorting algorithm is stable if it preserves the
  relative order of any two equal elements in the
  input
• Important in case when we sort students by gpa
  that were sorted alphabetically. All students
  with same gpa stay sorted.
• Another example sort students by departments
  (major or minor)

9
In Place Algorithm

• An algorithm is called in place if it does not
  require more that some constant number of extra
  spaces in memory.
• If the extra amount of space is proportional to
  the size of input it is not stable.

10
Searching

• Find a given value called search key in a given
  list, set or multiset
• In multiset several items have the same value.

11
Sequential search

• Check one by one item in the given collection of
  items.
• For each item check if it equals to the key that
  we are searching for.
• If the key is found stop further search.
• If the end of collection is reached without
  finding the key the search is stopped without
  success.

12
Binary Search

• Algorithm applicable to sorted list only.
• Check if item in the middle equals to the key.
• If not if key is smaller than item in the middle
  proceed with search in the sublist left of the
  item in the middle.
• if key is larger than item in the middle proceed
  with search in the sublist right of the item in
  the middle.
• Stop if key is found (successful search) or if
  sublist is empty (unsuccessful search).

13
String Processing

• String is a sequence of characters from some
  given alphabet.
• TEXT Strings strings on alphabet of letters,
  numbers, and spec chars.
• BIT strings strings on 0,1
• GENE SEQUENCES strings on the alphabet
  A,C,G,T

14
String Matching

• INPUT Given text and given pattern
• String matching algorithm has to find the
  location of the pattern in the text.
• Variations
• First occurrence
• N-th occurrence
• All occurrences

15
Graph Problems

• Set of vertices (nodes) some of which are
  connected by edges.
• Directed
• Undirected
• Labeled edges (weighted)

16
Graph algorithms

• Graph traversal
• Shortest path
• Topological sorting
• Traveling salesman
• Graph coloring- event scheduling

17
Combinatorial Problems

• Find a combinatorial object
• Find all permutations
• Find all combinations n-choose-m
• Find all subsets of a given set
• Find a combinatorial object with desired property
  ( to maximize or minimize cost)

18
Geometric Problems

• Greeks
• Construct a triangle with unmarked ruler and a
  compass
• Closest pair
• Find the closest pair of n points in a plane.
• Convex hull problem
• Find the smallest convex polygon that contains
  all n points in the plane

19
Numerical Algorithms

• Solving system of equations
• Computing definite integrals
• Evaluate functions
• Most algorithms are approximate
• Round off error
• Error may accumulate if large number of
  operations is performed on approximate numbers.