CS1022 Computer Programming & Principles

CS1022Computer Programming & Principles Lecture 8.1 Digraphs (1)

Plan of lecture • Digraphs (definition and terminology) • Simple digraphs • Paths and cycles • PERT charts • Topological sort algorithm CS1022

Digraphs, again • Directed graphs = digraphs • We have used digraphs to represent relations • We did not define them formally • Model partial ordering • A before B, A before C, • B before C? C before B? • Networks of dependences useful for • Data flow analysis • Task scheduling • Edges are directed • Finding paths require following a direction 1 5 6 2 3 4 CS1022

Directed graphs • A digraph is a pair G (V, E) where • V is a finite set of vertices • E is a relation on V • Visually, a digraph is • A set of labelled vertices with • Directed edges linking pairs of vertices • Directed edges are elements of E • Pairs of vertices, where the order is important • Also called arcs • If u, v V are vertices and (u, v)  E is an arc • We write simply uv a b CS1022

Simple digraphs (1) • A simple digraph has no loops or multiple arcs • There is at most one arc uv from u to v and • There is at most one arc vu from v to u • If uv is an arc then we say u is an antecedent of v CS1022

Simple digraphs (2) Example: digraph G (V, E) where • Vertex set V a, b, c, d • Arc set E ab, bd, cb, db, dc Graphically: b a d c CS1022

Simple digraphs (3) Adjacency matrix (set E ab, bd, cb, db, dc): CS1022

Paths and cycles in digraphs • A path of lengthk is a • Sequence of vertices v0, v1, , vk • Such that vi – 1vi is an arc, 1  i  k • Example: a, b, d, c is a path • A cycle is a • Sequence of vertices v0, v1, , vk • Such that vi – 1vi is an arc, 1  i  k • v0 vk (first and last vertices are the same) • vi vj, 0  i, j  k, i  0 or j  k (no other repetition) • Example: b, d, c, bis a cycle; a, b, d, c, b, ais not a cycle • A graph with no cycles in it is an acyclic graph b a d c CS1022

PERT chart (1) • Acyclic graphs useful to model situations in which tasks have to be carried out in a certain order • A cycle means that a task had to precede itself! • In task-scheduling problems the corresponding acyclic digraph is known as PERT chart • Project Evaluation and Review Technique (PERT) CS1022

PERT chart (2) • Suppose (partial) degree programme below • Pre-requisites, so order is important CS1022

PERT chart (3) • PERT chart shows interdependence of modules A H G F B C D E CS1022

Topological sort algorithm (1) • We want to help students find an order of modules • Consistent with pre-requisites • Classic solution: topological sort algorithm • Consistent labelling for vertices of acyclic digraphs • Labelling 1, 2, 3, , n of vertices such that • If uv is an arc and • Vertex u has label i, and • Vertex v has label j, then • i  j CS1022

Topological sort algorithm (2) Gives consistent labelling of acyclic digraph G (V, E) • Antecedents of each vertex stored in A(v) CS1022

Topological sort algorithm (3) Find consistent labelling for digraph of modules Step 0 – Antecedent sets are: • A(A)  {B} • A(B)  {C} • A(C)  {H} • A(D)  {C} • A(E)  {D, G} • A(F)  {E} • A(G)  {C} • A(H)   A H G F B C D E CS1022

Topological sort algorithm (4) Step 1 – Enter while loop: • Assign label 1 to H • Delete H from remaining A(v) • A(A)  {B} • A(B)  {C} • A(C)   • A(D)  {C} • A(E)  {D, G} • A(F)  {E} • A(G)  {C} CS1022

Topological sort algorithm (5) Step 2 – second pass through while loop: • Assign label 2 to C • Delete C from remaining A(v) • A(A)  {B} • A(B)   • A(D)   • A(E)  {D, G} • A(F)  {E} • A(G)   CS1022

Topological sort algorithm (6) Step 3 – third pass through while loop: • There is a choice of labels to choose from • Each choice leads to distinct consistent labelling • Assign label 3 to B and delete B from remaining A(v) • A(A)   • A(D)   • A(E)  {D, G} • A(F)  {E} • A(G)   CS1022

Topological sort algorithm (7) Step 4 – fourth pass through while loop: • There is again a choice of labels to choose from • Assign label 4 to A and delete A from remaining A(v) • A(D)   • A(E)  {D, G} • A(F)  {E} • A(G)   CS1022

Topological sort algorithm (8) Step 5 – fifth pass through while loop: • Assign label 5 to D and delete D from remaining A(v) • A(E)  {G} • A(F)  {E} • A(G)   CS1022

Topological sort algorithm (9) Step 6 – sixth pass through while loop: • Assign label 6 to G and delete G from remaining A(v) • A(E)   • A(F)  {E} CS1022

Topological sort algorithm (10) Step 7 – seventh pass through while loop: • Assign label 7 to E and delete E from remaining A(v) • A(F)   CS1022

Topological sort algorithm (11) Step 7 – final pass through while loop: • Assign label 8 to F a • There are no remaining vs to delete E from CS1022

Topological sort algorithm (12) • Algorithm found one possible consistent labelling: H, C, B, A, D, G, E, F • This gives an order in which modules can be taken • Consistent with pre-requisites A H G F B C D E CS1022

Some remarks • Algorithm analysed a graph and ordered vertices • “Sort” vertices based on incidence of arcs • Approach was exhaustive... • However, it did not try all traversals of the digraph • It relied on visiting vertices (labelling them) in some order • Why should you care? • If you ever need to perform similar process you can (you should!) re-use the algorithm • Algorithm can be implemented in different languages CS1022

Further reading • R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd. 2002. (Chapter 8) • Wikipedia’s entry on directed graphs • Wikibooks entry on graph theory CS1022

CS1022 Computer Programming & Principles