Chapter 9

Graph Theory

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Graphs

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✤ A graph consists of nodes and edges

✤ The set of all nodes (or vertices) in a graph is V

✤ The set of all edges in a graph is E

✤ Each edge connects one or two vertices (called its’ endpoints)

✤ A graph is formally represented (V, E)

✤ We will (usually) stick to simple graphs

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Simple Graphs

✤ No loops

✤ Endpoints of an edge are distinct

✤ No multiple edges

✤ between the same two vertices

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Directed Graphs

✤ A directed edge, or arc connects two vertices, but has a start and end

✤ Formally an arc (u, v) starts at u and ends at v

✤ A graph with directed edges is a directed graph or digraph

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5

Seven Bridges of Königsberg

Traveling Salesman Problem

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Terminology

✤ Two vertices u and v in an undirected graph G are neighbors if there exists an edge e in G which connects u, and v

✤ Equivalently, u and v are said to be adjacent if there exists an edge which connects them

✤ The set of all neighbors of u (u’s neighborhood) is N(u)

✤ This is defined similarly for sets of vertices

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Degree

✤ If an edge e has vertex u as one of its’ endpoints, e is incident with u

✤ The degree of a node u, deg(u), is the count of the number of edges incident with u

✤ A node with degree 0 is isolated

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2 3 1 0

4 3 1

In/Out -Degree

✤ In directed graphs, separate degrees

✤ In-Degree: # of edges ending at udeg-(u)

✤ Out-Degree: # of edges starting at udeg+(u)

✤ Vertices with in-degree 0 are sources

✤ Vertices with out-degree 0 are sinks

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1 2 2 1

1 2 1

2 1 0 1

4 1 0

deg−(v)v∈V∑ = deg+(v)

v∈V∑ = E

Representations

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Graph ADT

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✤ V = set of verticesE = set of edges

✤ Three operations

✤ getDegree(u)

✤ getAdjacent(u)

✤ isAdjacentTo(u, v)

Graph Representations

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✤ Adjacency list

✤ For each vertex, list the adjacent vertices

✤ Good for sparse graphs with few edges

✤ Adjacency matrix

✤ 2D matrix of 1s and 0s

✤ 1 iff there is an edge from i to j

✤ Good for dense graphs with many edges

Graph Representations

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V Adj.1 1,2,52 1,3,53 2,44 3,5,65 1,2,46 4

V 1 2 3 4 5 61 1 1 0 0 1 02 1 0 1 0 1 03 0 1 0 1 0 04 0 0 1 0 1 15 1 1 0 1 0 06 0 0 0 1 0 0

Graph Adjacency List Adjacency Matrix

Directed Graphs

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V Term1 6,72 5,63 1,2,44 5,75 16 47

V 1 2 3 4 5 6 71 0 0 0 0 0 1 12 0 0 0 0 1 1 03 1 1 0 1 0 0 04 0 0 0 0 1 0 15 1 0 0 0 0 0 06 0 0 0 1 0 0 07 0 0 0 0 0 0 0

Graph Adjacency List Adjacency Matrix

Practice

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Practice

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Performance of List vs Matrix

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Space getDegree(u) isAdjacentTo(u,v) getAdjacent(u)

AdjacencyList O(V+E) O(D(u)) O(D(u)) O(D(u))

AdjacencyMatrix O(V2) O(V) O(1) O(V)

Paths

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✤ A path (sometimes a walk) is a series of edges which starts at a vertex, travels from vertex to vertex along edges and ends at some vertex

✤ Formally a path p is a set of edges s.t. p.start = e0.start p.end = en.end en.end = en+1.start

✤ If p.start = p.end, then p is a circuit (cycle)

✤ If no edge is repeated, the path is simple

Cyclic vs Acyclic

✤ Directed graphs can be cyclic or acyclic

✤ Cyclic: contains a cycle

✤ Acyclic: does not contain any cycles

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Cycle Practice

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Cycle Practice

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How many cycles are there?

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Graph Search

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Depth-First Search (DFS)

✤ Visit some start vertex

✤ Follow an edge to a vertex which hasn’t been explored

✤ Visit that vertex

✤ Follow an edge from that vertex to another unexplored vertex

✤ If there are no edges to choose from, backtrack to the previous vertex

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DFS

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Order visited: a, b, c, g, h, d, k, j, f, e, i, m, l

A Recursive DFS Algorithm

procedure DFS(G: connected graph with vertices v1, v2, ..., vn)T := tree consisting only of the vertex v1fvisit(v1)

procedure visit(v: vertex of G)for each vertex w adjacent to v and not yet in T add vertex w and edge {v, w} to T visit(w)

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Practice

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Breadth-First Search

✤ Visit some start vertex

✤ Visit all neighbors of the start vertex

✤ Visit all those neighbors’ neighbors

✤ Repeat until all vertices visited

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BFS

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Order visited: a, b, e, c, f, i, g, j, m, h, k, l, d

A (non-recursive) BFS Algorithm

procedure BFS (G: connected graph with vertices v1, v2, ..., vn)T := tree consisting only of vertex v1L := empty listput v1 in the list L of unprocessed verticeswhile L is not empty remove the first vertex, v, from L for each neighbor w of v if w is not in L and not in T then add w to the end of the list L add w and edge {v, w} to T

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Shortest-Path Problems

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Shortest Path

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✤ The length of a path is the number of edges in it

✤ Many problems try to find the shortest path between two vertices

✤ Flights with the fewest stopovers

✤ Driving directions with few instructions

✤ Can use DFS or BFS

Practice

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Weighted Graphs

✤ Add a weight to every edge

✤ weights are normally “costs”

✤ Length of a path is the sum of the weights of each edge

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LA

NYC

Atlanta

Miami

Chicago

Vegas

Boston

SF 2:20

2:00

2:10

3:50

1:40

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2:104:05 0:

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1:55

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1:301:15

2:55

LA

NYC

Atlanta

Miami

Chicago

Vegas

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SF $89

$89

$69

$129

$99

$59

$79$129 $3

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$79

$99

$69$39

$99

LA

NYC

Atlanta

Miami

Chicago

Vegas

Boston

SF 957

834

908

2451

606

722

8602534 19

1

760

1090

595349

1855

Dijkstra’s Algorithm

✤ Finds shortest path between X and Z in O(n2) time

✤ Greedy algorithm:

✤ Start with C = {X}

✤ Search the neighbors of C to find next closest node to X not in C

✤ Add it to C

✤ Repeat until Z is the next closest node to X35

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Dijkstra’s Algorithm

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∞ ∞

∞

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{X}, 0

Dijkstra’s Algorithm

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Dijkstra’s Algorithm

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∞

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Dijkstra’s Algorithm

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Dijkstra’s Algorithm

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Dijkstra’s Algorithm

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{X}, 0

Dijkstra’s Algorithm

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{X,c,b,d,e}, 10{X,c}, 2

{X}, 0

Practice

✤ Find the shortest path from g to fusing Djikstra’s Algorithm

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Connectivity

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Network Reliability

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✤ What would happen if my router in NY went offline? If CA got knocked out?

✤ I often want there to always be a path available between all the nodes in my graph

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Connectedness

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✤ Two nodes are connected if there exists a path between them

✤ Otherwise they are disconnected

✤ If every pair of nodes in a graph is connected, then the whole graph is connected

✤ Otherwise it is disconnected

✤ If a graph is connected, then there exists a simple path between every pair of vertices in the graph

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Connected vs Disconnected

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Connected Components✤ A part of a graph which is

disconnected from all other parts is called a connected component

✤ Formally, a connected subgraph of G which is not a proper subgraph of any other connected subgraph of G is a connected component

✤ A connected graph has 1 connected component

✤ A disconnected graph has 2+48

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Directed Graphs

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✤ The underlying undirected graph of a digraph is the same graph, minus directions

✤ A digraph is weakly connected if the underlying undirected graph is connected

✤ i.e. if the digraph is “in one piece”

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Strongly Connected

✤ A digraph G is strongly connected if for every pair of vertices a,b ∈ G, there exists a path from a to b and from b to a

✤ Note: this includes the pair a,a

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Practice

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Strongly Connected Components

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✤ Analogous to connected components

✤ Maximal subgraphs of G which are strongly connectedi.e. they are not contained within any other such strongly connected subgraphs

✤ We can compress a digraph into a directed acyclic graph by reducing it to a graph of its strongly connected components

✤ Treat all nodes in a SCC as interchangeable

✤ Model how to hop from one SCC to another

Strongly Connected Components

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Euler and Hamilton Paths/Cycles

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Euler Paths and Circuits

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✤ Euler Path: path in G which uses every edge exactly once

✤ can visit a node more than once

✤ Euler Circuit: Euler path which is also a circuit

Practice

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Practice

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Hamilton Paths and Circuits

✤ Hamilton Path: path visiting every node exactly once

✤ each edge can only be used once

✤ Hamilton Circuit: hamilton path which is also a circuit

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Practice

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Traveling Salesperson Problem

✤ Traveling Salesperson Problem (TSP)or Traveling Salesman Problem

✤ Find the shortest hamilton cycle

✤ i.e. visiting every node once and returning to the start

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Saginaw

Detroit

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Toledo

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Traveling Salesperson Problem

✤ This is an NP-hard problem

✤ There are (n - 1)!/2 Hamilton cycles

✤ O(n!) to compute exhaustively

✤ FedEx pays lots and lots of money for improvements on this problem

✤ Often okay to find an approximate solution

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Route Total Distanc

eD - T - GR - S - K - D 610D - T - GR - K - S - D 516D - T - K - S - GR - D 588D - T - K - GR - S - D 458D - T - S - K - GR - D 540D - T - S - GR - K - D 504D - S - T - GR - K - D 598D - S - T - K - GR - D 576D - S - K - GR - T - D 682D - S - GR - T - K - D 646D - GR - S - T - K - D 670D - GR - T - S - K - D 728

Section 9.5

Spanning Trees

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Spanning Tree

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Etna Old Town

Herman

Hampden

Bangor

Orono Etna Old Town

Herman

Hampden

Bangor

Orono

Spanning Tree Tree w/ every

vertex of G

Finding Spanning Trees

✤ Trees are acyclic, graphs aren’t

✤ Remove edges to break up cycles

✤ Keep graph connected

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Minimum Spanning Trees

✤ Weighted graphs

✤ Find spanning tree with the smallest possible sum of edge weights

✤ Applications:

✤ Making connected graphs

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Prim’s Algorithm

procedure Prim (G)T := a minimum weight edgefor i := 1 to n - 2 e := an edge of minimum weight, incident to a vertex in T, not forming a cycle in T T := T with e addedreturn T

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Kruskal’s Algorithm

procedure Kruskal (G)T := an empty graphfor i := 1 to n - 1 e := any edge of minimum weight in G, not forming a cycle if added to T T := T with e addedreturn T

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