Graph Algorithms

Peter Lammich

27. Januar 2020

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Outline

1 Directed GraphsFormal DefinitionImplementation

2 Graph Traversal AlgorithmsGeneric Graph TraversalDFS and BFSTopological SortingShortest Paths

3 Shortest Path in Weighted GraphsSingle-Source Shortest Path

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Directed Graphs

• A directed graph is a set of nodes V and edges E ⊆ V × V

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V = {1, 2, . . . , 9}E = {(1, 2), (1, 3), (2, 4), (2, 5), (3, 6),

(3, 7), (4, 8), (4, 9), (6, 5), (9, 2)}

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Directed Graphs

• A directed graph is a set of nodes V and edges E ⊆ V × V

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Path: Sequence of nodes u1 . . . un,with ∀i ∈ {1 . . . n − 1}. (ui , ui+1) ∈ Ee.g. 1, 2, 4, 8If not noted otherwise: paths are cycle-free, e.g.,no repeated nodes!

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Directed Graphs

• A directed graph is a set of nodes V and edges E ⊆ V × V

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Cycle: Path with same start and end nodee.g. 2, 4, 9, 2

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Operations

• Here: nodes implicitly given by edges:V := {u | ∃v . (u, v) ∈ E ∨ (v , u) ∈ E}• empty returns empty graph. E ← ∅• addEdge(u,v) adds an edge. E ← E ∪ {(u, v)}• removeEdge(u,v) removes an edge. E ← E \ {(u, v)}• isEdge(u,v) checks for edge. (u, v) ∈ E• succs(u) returns successors of node. {v | (u, v) ∈ E}

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Implementation

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Implementation

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Adjacency listStore list/set of successors for each nodesuccs[1] = {2, 3}succs[2] = {4, 5}succs[3] = {6, 7}succs[4] = {8, 9}succs[5] = {}succs[6] = {5}succs[7] = {}succs[8] = {}succs[9] = {2}

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Implementation

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Adjacency Matrixm(u, v) = T iff edge from u to v

1 2 3 4 5 6 7 8 91 F T T F F F F F F2 F F F T T F F F F3 F F F F F T T F F4 F F F F F F F T T5 F F F F F F F F F6 F F F F T F F F F7 F F F F F F F F F8 F F F F F F F F F9 F T F F F F F F F

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Adjacency Lists

• Map each node to list of successors• E = {(u, v) | v ∈ succs(u)}• directly implements succs(u)• for integer nodes: use array of (dynamic) arrays

• Memory O(|V |+ |E |)

Operation Implementation ComplexityaddEdge(u, v) append v to succs(u) O(1) (amortized)removeEdge(u, v) remove v from succs(u) O(|V |)isEdge(u, v) search for v in succs(u) O(|V |)succs(u) return succs(u) O(1)

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Adjacency Matrix

• Store |V | × |V | map to Booleans• E = {(u, v) | m(u, v) = true}• for integer nodes: use 2 dimensional array

• Memory O(|V |2).• Bad for sparse graphs (|E | << |V |2).

Operation Implementation ComplexityaddEdge(u, v) m(u, v)← true O(1)removeEdge(u, v) m(u, v)← false O(1)isEdge(u, v) return m(u, v) O(1)succs(u) return {v | m(u, v) = true} O(|V |)

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Adjacency Matrix + Adjacency List

• Store graph simultaneously as adjacancy list and matrix• Memory: O(|V |2)• addEdge, isEdge, succs — O(1)• removeEdge — O(|V |)

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Outline

1 Directed GraphsFormal DefinitionImplementation

2 Graph Traversal AlgorithmsGeneric Graph TraversalDFS and BFSTopological SortingShortest Paths

3 Shortest Path in Weighted GraphsSingle-Source Shortest Path

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Generic Graph Traversal

• Explore edges to new nodes, until all nodes discovered

procedure explore(s)D ← {s}, F ← ∅while D 6= ∅ do

u ← Some u ∈ DD ← D \ {u}, F ← F ∪ {u}D ← D ∪ {v | (u, v) ∈ E ∧ v /∈ F}

return F

• F finished, outgoing edges have been explored• D discovered, but outgoing edges yet to be explored• explore(s) returns exactly the nodes reachable from s

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Example

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D = { 1 }11 / 31

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D = { 2 3 }11 / 31

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Example

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Generic Graph Traversal (Correctness)

explore(s) returns exactly the nodes reachable from s• Any node in D ∪ F is reachable:

• Initially, only s ∈ D ∪ F• only successors of nodes in D ∪ F added

• If a node is reachable, it will be included in F :• During the loop, successors of finished nodes are finished or discovered• Finally, D = ∅, thus successors of finished nodes are finished

=⇒ every reachable node is finished (follow path from s)

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DFS and BFSprocedure explore(s)

D ← {s}, F ← ∅while D 6= ∅ do

u ← Some u ∈ DD ← D \ {u}, F ← F ∪ {u}D ← D ∪ {v | (u, v) ∈ E ∧ v /∈ F}

return F

• In which order do we process nodes from D• Last in / first out (stack): Depth First Search (DFS)• First in / first out (queue): Breadth First Search (BFS)

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Example: DFS

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Example: DFS

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Example: DFS

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Example: DFS

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Example: DFS

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Example: DFS

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Example: DFS

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Example: DFS

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Example: DFS

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Example: DFS

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Example: BFS

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Example: BFS

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Example: BFS

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Example: BFS

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Example: BFS

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Example: BFS

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Example: BFS

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Example: BFS

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Example: BFS

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Example: BFS

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Recursive DFS

• DFS has nice recursive implementation

procedure dfsRec(F , u)if u /∈ F then

F ← F ∪ {u}for all v with (u, v) ∈ E do

F ← dfsRec(F,v)return F

procedure dfs(s) return dfsRec(∅, s)

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Topological Sorting

• Set of tasks (e.g. build jobs in Makefile)• Dependencies, i.e. tasks that needs to be completed before a task can

be started• Model as directed graph. Edge (u, v): v depends on u.• Find a build sequence

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Examplemain: main.o hashtable.o dynarray.o graph.o

gcc ...

main.o: main.chashtable.o: hashtable.h hashtable.c dynarray.hdynarray.o: dynarray.h dynarray.cgraph.o: graph.h graph.c

main

main.o

hashtable.o

dynarray.o

graph.o

main.c

hashtable.h

hashtable.c

dynarray.h

dynarray.c

graph.h

graph.c

Possible build sequence:graph.o, dynarray.o, hashtable.o, main.o,main

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Topological Sorting

• Arrange nodes sequentially, such that all edges point forwards• Intuition: All prerequisites come earlier in sequence

• Topological sorting possible iff graph has no cycles• Directed Acyclic Graph (DAG)

• Now: DFS based algorithm for topological sorting and cycle detection

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Cycle detection with DFS

• When first encountering a node, mark it as open.• Only when finished exploring its children, mark it as done• Whenever we encounter an open node again, it’s a cycle

procedure dfsRec(F , u)if F (u) = N then

F (u)← O // Start processing nodefor all v with (u, v) ∈ E do

F ← dfsRec(F,v)F (u)← D // Done processing node

else if F (u) = O thenError: found cycle

return Fprocedure dfs(s)

F (u)← N for all nodes u return dfsRec(F , s)

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Topological Sorting

procedure dfsRec(F , u)if F (u) = N then

F (u)← O // Start processing nodefor all v with (u, v) ∈ E do

F ← dfsRec(F,v)F (u)← D // Done processing node

else if F (u) = O thenError: found cycle

return Fprocedure dfs(s)

F (u)← N for all nodes u return dfsRec(F , s)

• When node is done, all its successors are already done=⇒ Nodes done in reverse topological order• to get topological sort: prepend nodes to list when marked as done

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Example

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Error: found cycle

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Error: found cycle

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Error: found cycle

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Shortest paths with BFS

• Shortest path from s to some node?• shortest = minimal number of edges

• BFS visits nodes in order of their distance from s!• Obtain path via predecessor map:

• For each node, store node from which it was discovered• Follow these nodes backwards, until s is reached• Convention: predecessor of s is s itself.

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BFS Shortest Path

procedure shortestPaths(s)D ← [s], F ← ∅, π(s)← swhile D 6= [] do

(u, D)← dequeue(D)D ← D \ {u}, F ← F ∪ {u}for all v with (u, v) ∈ E ∧ v /∈ F do

D ← enqueue(v), π(v)← ureturn π

procedure getPath(π, u)p ← [u]while π(u) 6= u do

u ← π(u), p ← up

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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Example: BFS shortest paths

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predecessor map

getPath(8): p = [

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Example: BFS shortest paths

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getPath(8): p = [

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Example: BFS shortest paths

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Example: BFS shortest paths

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Outline

1 Directed GraphsFormal DefinitionImplementation

2 Graph Traversal AlgorithmsGeneric Graph TraversalDFS and BFSTopological SortingShortest Paths

3 Shortest Path in Weighted GraphsSingle-Source Shortest Path

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