1. CSC 380: Design and Analysis of Algorithms
Dr. Curry Guinn

2. Quick Info Dr. Curry Guinn CIS 2015 guinnc@uncw.edu
Office Hours: MWF: 10:00am-11:00m and by appointment

3. Outline of today Satisfiability 3-SAT Brute force Hill climbing
Simulated annealing

4. The Satisfiability Problem
Given a set of propositional logic clauses, find an assignment of truth values to each symbol so that all of the clauses are true.
(P -> Q & S) & (~P & ~Q) & (S | ~Q)

5. 3SAT is a special case All clauses contain the disjunction of 3 terms
Disjunction means OR
Each term is of the form P or ~P.
(P | Q | ~S) & (~P | ~Q | ~S) & (P | ~Q | S)
It turns out that any SAT case, can be transformed into 3SAT

6. 3SAT is an NP-complete problem
Which means that it is map-able to other NP-complete problems such as
Traveling salesman
Clique
Map coloring
Knapsack,
Etc.
What do I mean by “map-able”?

7. Suppose we want to actually write code to solve this problem
How do we do it?
Analysis
Design
Implement
Test

8. Analysis
Can we write down the problem?

9. Write down the problem
Find a set of truth assignments for the propositional symbols such that are of the clauses are true. Each clause consist of a disjunction (OR) of three terms. Each term is either a proposition symbol or its negation (i.e., P or ~P).
More generally, for a particular problem, we will have n different propositional symbols and m different clauses.
Do you want to add anything?

10. Find the nouns. Why?

11. Find the nouns. Why?
Find a set of truth assignments for the propositional symbols such that all of the clauses are true. Each clause consist of a disjunction (OR) of three terms. Each term is either a proposition symbol or its negation (i.e., P or ~P).
More generally, for a particular problem, we will have n different propositional symbols and m different clauses.

12. Start making UML diagrams of the classes
Term
Clause
Symbol
TruthAssignment
Problem

13. See board and IDE

14. Iterative Improvement Algorithms
Essentially, guess at a solution and then make improvements until you find a solution.
Not important how you get there.
Hill climbing
Simulated annealing

15. Local search algorithms
In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution
State space = set of "complete" configurations
Find configuration satisfying constraints, e.g., n-queens
In such cases, we can use local search algorithms
keep a single "current" state, try to improve it

16. Example: n-queens
Put n queens on an n × n board with no two queens on the same row, column, or diagonal

17. Hill-climbing search
"Like climbing Everest in thick fog with amnesia"

18. Hill-climbing search
Problem: depending on initial state, can get stuck in local maxima

19. Hill-climbing search: 8-queens problem
h = number of pairs of queens that are attacking each other, either directly or indirectly
h = 17 for the above state

20. Hill-climbing search: 8-queens problem
A local minimum with h = 1

21. Hill climbing
From a guess, always choose a successor state that is closer to the solution.
Problems
Local maxima
Plateaux
Ridges
Solution: Random-restart ... or ...

22. Simulated annealing
Instead of always choosing the “best” successor, occasionally choose something that is worse at random.

23. Simulated annealing search
Idea: escape local maxima by allowing some "bad" moves but gradually decrease their frequency

24. Properties of simulated annealing search
One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1
Widely used in VLSI layout, airline scheduling, etc

25. For Next Class, Wednesday
Read about Hill Climbing