1. Data Structures and Algorithms
Equivalence Classes
PLSD210(ii)

2. Lectures 17 - Key Points Graphs Collections of vertices and edges
Paths
Spanning Trees
Provide a path from any vertex to any other vertex

3. Lecture 17 - Key Points Minimum Spanning Tree
Spanning Tree of minimum cost
Cost measure may be anything!
Operation
All vertices into single element trees,
Sort edges,
Add next cheapest edge unless it forms a cycle
Finish when we’ve added |V|-1 edges
Greedy algorithms
Don’t look ahead ..
Make a “naive” choice at every step

4. Lecture 17 - Key Points Greedy algorithms
Don’t look ahead ..
Make a “naive” choice at every step
Sometimes difficult to prove
Usually proof by contradiction employed
Completes classes of efficient algorithms
Divide and Conquer
Dynamic
Greedy

5. Equivalence Classes Cycle Determination in MST Program verification
Relies on Union Find structure
Lists pointing to a representative
Vertices are partitioned into equivalence classes
Equivalence relation: a path connects them
Edge creates a cycle if both its end-points
have the same representative
are in the same equivalence class
Program verification
Partition input data into equivalence classes
Relation: processed in the same way

6. Equivalence Classes Program verification
Partition input data into equivalence classes
Relation: processed in the same way
Choose a representative of the class
Testing the representative
Testing all inputs in the same class
Experienced Software Engineeers
Include special values also
Boundaries
Identities - 0 and 1
Never choose boundaries, identities, etc as a representative

7. Equivalence Classes Program verification
Experienced Software Engineeers
Include special values also
Boundaries
Identities - 0 and 1
Never choose boundaries, identities, etc as a representative
Even though this might be formally OK!

8. Equivalence Classes Program verification Assignment 4 Don’t forget
Identify equivalence classes of function to be tested
Set up a program to apply a representative of each class to the function
Table or file driven input best
Don’t forget
Nulls
Specification should tell you what will happen on empty or degenerate data sets
Empty lists, single node graphs, etc.
Illegal input
Violates pre-conditions
Should raise assertions!

9. Equivalence Classes Program verification Don’t forget Nulls
Specification should tell you what will happen on empty or degenerate data sets
Empty lists, single node graphs, etc.
Illegal input
Violates pre-conditions
Should raise assertions!