CS1022 Computer Programming &

Principles

Lecture 1Digraphs

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

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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

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• 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

Directed graphs

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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

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Simple digraphs (2)Example: digraph G (V, E) where• Vertex set V a, b, c, d• Arc set E ab, bd, cb, db, dcGraphically:

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a b c dabcd

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

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a b c da Fbcd

a b c da F T F Fbcd

a b c da F T F Fb F F F Tcd

a b c da F T F Fb F F F Tc F T F Fd

a b c da F T F Fb F F F Tc F T F Fd F T T F

Paths and cycles in digraphs• A path of length k 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, b is a cycle; a, b, d, c, b, a is not a cycle

• A graph with no cycles in it is an acyclic graph8CS1022

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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)

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

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Module Pre-requisitesAdvanced biotechnology BBiotechnology CCell biology HDNA structures CEnzyme activities D, GFood science EGenetic engineering CHuman biology None

PERT chart (3)• PERT chart shows interdependence of modules

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Module Pre-requisitesAdvanced biotechnology BBiotechnology CCell biology HDNA structures CEnzyme activities D, GFood science EGenetic engineering CHuman biology None

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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

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

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beginfor v V do

calculate A(v); label := 0; while unlabelled vertices v remain for which A(v) dobegin

label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V do

A(v) := A(v) {u}; % delete u from remaining vsend

end

Find consistent labelling for digraph of modulesStep 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)

beginfor v V do

calculate A(v); label := 0; while unlabelled vertices v remain for which A(v) dobegin

label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V do

A(v) := A(v) {u}; end

end

Topological sort algorithm (3)

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for v V docalculate A(v);

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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}

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...label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V doA(v) := A(v) {u};

...

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)

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...label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V doA(v) := A(v) {u};

...

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)

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...label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V doA(v) := A(v) {u};

...

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)

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...label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V doA(v) := A(v) {u};

...

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)

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...label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V doA(v) := A(v) {u};

...

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}

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...label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V doA(v) := A(v) {u};

...

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)

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...label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V doA(v) := A(v) {u};

...

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

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...label := label + 1;u := a vertex with A(u) ; assign label to u;for each unlabelled vertex v V doA(v) := A(v) {u};

...

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

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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

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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

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