1. Discrete Mathematics CS 2610
February 24, part 5

2. Algorithm- Insertion Sort
For each element:
The elements on its left are already sorted
Shift the element with the element on the left until it is in the correct place.
See animation

3. Algorithm- Insertion Sort
procedure insertSort (a1, a2, …, an: distinct integers) for j=2 to n
begin
i=j - 1
while i > 0 and ai > ai+1
swap ai and ai+1
i=i-1
end
Worst-Case: The sequence is in decreasing order
At step j,
the while loop condition is executed j times
the body of the loop is executed j-1 times

4. Greedy Algorithms Can you think of a better solution ?
Problem: Assign meeting to conference rooms
Policy: In decreasing order of room capacity, assign a meeting to the next largest available room
Room A 200
Room B 150
Room C 150
Room D 100
Room E 75
Room F 50
Meeting 1: 70
Meeting 2: 46
Meeting 3: 125
Meeting 4: 110
Meeting 5: 30
Meeting 6: 87
M 1
M 2
Can you think of a better solution ? M3 X M5 X

5. Greedy Algorithm
Policy: : In ascending order of room capacity, assign a meeting to the smallest room that can hold the meeting.
Room A 200
Room B 150
Room C 150
Room D 100
Room E 75
Room F 50
Meeting 1:70
Meeting 2:46
Meeting 3:125
Meeting 4:110
Meeting 5:30
Meeting 6:87
M 2
M 1 M5 M3 M4 M6

6. Order of Growth Terminology
O(1) Constant
O(log cn) Logarithmic (c  Z+)
O(logc n) Polylogarithmic (c  Z+)
O(n) Linear
O(nc) Polynomial (c  Z+)
O(cn) Exponential (c  Z+)
O(n!) Factorial
Best
Worst

7. Complexity of Problems
Tractable
A problem that can be solved with a deterministic polynomial (or better) worst-case time complexity.
Also denoted as P
Example:
Search Problem
Sorting problem
Find the maximum

8. Complexity of Problems
Intractable
Problems that are not tractable.
Example:
Traveling salesperson problem
Wide use of greedy algorithms to get an approximate solution.
For example under certain circumstances you can get an approximation that is at most double the optimal solution.

9. P vs. NP
NP: Solvable problems whose solution can be checked in polynomial time.
PNP
The most famous unproven conjecture in computer science is that this inclusion is proper.
PNP rather than P=NP

10. Complexity of Problems
Not Solvable
Proven to have no algorithm that computes it
Example:
Halting problem (Alan Turing)
Determine whether an arbitrary given algorithm, will eventually halt for any given finite input.
Corollary: The question of whether or not a program halts for a given input is unsolvable.