Lecture 6 Announcements Reading Start on Chapter 4 Graph Theory G V E V vertices V n E edges E m Undirected graphs Edges sets of two vertices u v Directed graphs Edges ordered pairs u v ID: 783534
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Slide1
CSE 421Algorithms
Autumn 2019
Lecture 6
Slide2Announcements
Reading
Start on Chapter 4
Slide3Graph Theory
G = (V, E)
V: vertices, |V|= n
E: edges, |E| = m
Undirected graphs
Edges sets of two vertices {u, v}Directed graphsEdges ordered pairs (u, v)Many other flavorsEdge / vertices weightsParallel edgesSelf loops
Path: v1, v2, …, vk, with (vi, vi+1) in ESimple PathCycleSimple CycleNeighborhoodN(v)DistanceConnectivityUndirectedDirected (strong connectivity)TreesRootedUnrooted
Explain that there will be some review from 326
By default |V| = n and |E| = m
Slide4Last Lecture
Bipartite Graphs : two-colorable graphs
Breadth First Search algorithm for testing two-
colorability
Two-colorable
iff no odd length cycleBFS has cross edge iff graph has odd cycle
Slide5Graph Search
Data structure for next vertex to visit determines search order
Slide6Graph search
Breadth First Search
S
= {s
}
while S is not empty
u = Dequeue(S) if u is unvisited visit u foreach v in N(u) Enqueue(S, v)
Depth First Search S = {s} while S is not empty u = Pop(S) if u is unvisited
visit u
foreach
v in N(u)
Push(S, v)
Slide7Breadth First Search
All edges go between vertices on the same layer or adjacent layers
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Slide8Depth First Search
Each edge goes between vertices on the same branch
No cross edges
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Slide9Connected Components
Undirected Graphs
Slide10Computing Connected Components in O(n+m) time
A search algorithm from a vertex v can find all vertices in v’s component
While there is an unvisited vertex v, search from v to find a new component
Slide11Directed Graphs
A Strongly Connected Component is a subset of the vertices with paths between every pair of vertices.
Slide12Identify the Strongly Connected Components
Slide13Strongly connected components can be found in O(n+m) time
But it’s tricky!
Simpler problem: given a vertex v, compute the vertices in v’s scc in O(n+m) time
Slide14Topological Sort
Given a set of tasks with precedence constraints, find a linear order of the tasks
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Slide15Find a topological order for the following graph
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Slide16If a graph has a cycle, there is no topological sort
Consider the first vertex on the cycle in the topological sort
It must have an incoming edge
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Definition: A graph is Acyclic if it has no cycles
Slide17Lemma: If a
(finite)
graph is acyclic, it has a vertex with in-degree 0
Proof:
Pick a vertex v
1, if it has in-degree 0 then doneIf not, let (v2, v1) be an edge, if v2 has in-degree 0 then doneIf not, let (v3, v2) be an edge . . .
If this process continues for more than n steps, we have a repeated vertex, so we have a cycle
Slide18Topological Sort Algorithm
While there exists a vertex v with in-degree 0
Output vertex v
Delete the vertex v and all out going edges
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Slide19Details for O(n+m) implementation
Maintain a list of vertices of in-degree 0
Each vertex keeps track of its in-degree
Update in-degrees and list when edges are removed
m edge removals at O(1) cost each