Undirected Graphs

COS 265 - Data Structures & Algorithms

Undirected Graphs

introduction

undirected graphs

Graph: Set of vertices connected pairwise by edges

Why study graph algorithms?

Thousands of practical applications
Hundreds of graph algorithms known
Interesting and broadly useful abstraction
Challenging branch of computer science and discrete math

Protein-protein interaction network

[ Reference: Jeong et al Nature Review | Genetics ]

framingham heart study

[ "The Spread of Obesity in a Large Social Network over 32 Years" by Christakis and Fowler in New England Journal of Medicine, 2007 ]

the evolution of FCC lobbying coalitions

[ "The Evolution of FCC Lobbying Coalitions" by Pierre de Vries in JoSS Visualization Symposium 2010 ]

map of science clickstreams

[ http://www.plosions.org/article/info.doi/10.1371/journal.pone.0004803 ]

10 million facebook friends

[ "Visualizing Friendships" by Paul Butler ]

The Internet as mapped by the Opte Project

[ http://en.wikipedia.org/wiki/Internet ]

graph applications

graph	vertex	edge
communication	telephone, computer	fiber optic cable
circuit	gate, register, processor	wire
mechanical	joint	rod, beam, spring
financial	stock, currency	transactions
transportation	intersection	street
internet	class C network	connection
game	board position	legal move
social relationship	person	friendship
neural network	neuron	synapse
protein network	protein	protein-protein interaction
molecule	atom	bond
polygonal mesh	geometry	adjacency

graph terminology

Graph is made up of vertices (nodes, points) connected with edges (links, lines, arcs)

Path: sequence of vertices connected by edges
Cycle: path whose first and last vertices are the same
Vertex Degree: number of edges that connect to the vertex

Two vertices are connected if there is a path between them

graph terminology

vertex: black dot; edge: line; path: green lines (len=4); cycle: blue lines (len=5); 3 connected components; red vertex has degree 4

some graph-processing problems

problem	description
s-t path	Is there a path between \(s\) and \(t\)?
shortest s-t path	What is the shortest path between \(s\) and \(t\)?
cycle	Is there a cycle in the graph?
Euler cycle	Is there a cycle that uses each edge exactly once?
Hamilton cycle	Is there a cycle that uses each vertex exactly once?
connectivity	Is there a path between every pair of vertices?
biconnectivity	Is there a vertex whose removal disconnects the graph?
planarity	Can the graph be drawn in the plane with no crossing edges?
graph isomorphism	Are two graphs isomorphic?

Challenge: Which graph problems are easy? difficult? intractable?

Undirected Graphs

graph api

graph representation

Graph drawing provides intuition about the structure of the graph

Caveat: Intuition can be misleading

graph representation

Vertex representation

This lecture: use integers between \(0\) and \(V-1\)
Applications: convert between names and integers with symbol table

Anomalies: Not all graph representations can handle these

Graph API

// degree of vertex v in Graph G
public static int degree(Graph G, int V) {
    int degree = 0;
    for(int w : G.adj(v)) degree++;
    return degree;
}

graph representation: edge list

Maintain a list of edges as pairs of vertices.

edges = [(0,1), (0,2), (0,5), (0,6),
         (3,4), (3,5), (4,5), (4,6),
         (7,8), (9,10), (9,11),
         (9,12), (11,12)]

Note: (0,1) implies (1,0) is also an edge

undirected graphs: quiz 0

[ live view ]

Which is order of growth of running time of the following code fragment if the graph uses the edge-list representation, where \(V\) is the number of vertices and \(E\) is the number of edges?

// prints each edge exactly once
for(int v = 0; v < G.V(); v++)
    for(int w : G.adj(v))
        if(v < w)
            StdOut.println(v + "-" + w);

\(V\)
\(E+V\)
\(V^2\)
\(VE\)

graph representation: adjacency matrix

Maintain a two-dimensional \(V\)-by-\(V\) boolean array; for each edge \(v\)-\(w\) in graph: adj[v][w] = adj[w][v] = true.

    0 1 2 3 4 5 6 7 8 9 0 1 2
 0: . X X . . X X . . . . . .
 1: X . . . . . . . . . . . .
 2: X . . . . . . . . . . . .
 3: . . . . X X . . . . . . .
 4: . . . X . X X . . . . . .
 5: X . . X X . . . . . . . .
 6: X . . . X . . . . . . . .
 7: . . . . . . . . X . . . .
 8: . . . . . . . X . . . . .
 9: . . . . . . . . . . X X X
10: . . . . . . . . . X . . .
11: . . . . . . . . . X . . X
12: . . . . . . . . . X . X .

Two entries for each edge

undirected graphs: quiz 1

[ live view ]

Which is order of growth of running time of the following code fragment if the graph uses the adjacency-matrix representation, where \(V\) is the number of vertices and \(E\) is the number of edges?

// prints each edge exactly once
for(int v = 0; v < G.V(); v++)
    for(int w : G.adj(v))
        if(v < w)
            StdOut.println(v + "-" + w);

\(V\)
\(E+V\)
\(V^2\)
\(VE\)

graph representation: adjacency lists

Maintain vertex-indexed array of lists

undirected graphs: quiz 2

[ live view ]

Which is order of growth of running time of the following code fragment if the graph uses the adjacency-lists representation, where \(V\) is the number of vertices and \(E\) is the number of edges?

// prints each edge exactly once
for(int v = 0; v < G.V(); v++)
    for(int w : G.adj(v))
        if(v < w)
            StdOut.println(v + "-" + w);

\(V\)
\(E+V\)
\(V^2\)
\(VE\)

graph representations

In practice, use adjacency-lists representation

Algorithms based on iterating over vertices adjacenct to \(v\)
Real-world graphs tend to be sparse (huge number of vertices, small average vertex degree)

graph representations

representation	space	add edge	edge between \(v\) and \(w\)	iterate over verts adj to \(v\)
list of edges	\(E\)	\(1\)	\(E\)	\(E\)
adjacency matrix	\(V^2\)	\(1^\dagger\)	\(1\)	\(V\)
adjacency lists	\(E+V\)	\(1\)	\(\degree(v)\)	\(\degree(v)\)

\(^\dagger\)disallows parallel edges

adj-list graph repr: java implementation

public class Graph {
    private final int V;
    private Bag<Integer>[] adj;     // adj lists

    public Graph(int V) {
        // create empty graph with V vertices
        this.V = V;
        adj = (Bag<Integer>[]) new Bag[V];
        for(int v = 0; v < V; v++)
            adj[v] = new Bag<Integer>();
    }

    public void addEdge(int v, int w) {
        // add edge v-w (parallel edges and self-loops allowed)
        adj[v].add(w);
        adj[w].add(v);
    }

    // iterator for vertices adjacent to v
    public Iterable<Integer> adj(int v) {
        return adj[v];
    }
}

Undirected Graphs

depth-first search

Maze exploration

Maze graph:

Vertex = intersection
Edge = passage

Graph: vertex for every intersection, edge for every passage

Goal: Explore every intersection in the maze

maze exploration: national building museum

[ http://www.smithsonianmag.com/travel/winding-history-maze-180951998/?no-ist ]

trémaux maze exploration

Algorithm:

Unroll a ball of string behind you
Mark each newly discovered intersection and passage
Retrace steps when no unmarked options

maze exploration: easy

maze exploration: medium

maze exploration: challenge for the bored

depth-first search

Goal: systematically traverse a graph

Idea: mimic maze exploration (function-call stack acts as ball of string)

DFS (to visit a vertex v)
    Mark vertex v
    Recursively visit all unmarked verts w adj to v

Typical applications:

Find all vertices connected to a given source vertex
Find a path between two vertices

Design challenge: how to implement?

Depth-first search demo

To visit a vertex \(v\):

Mark vertex \(v\)
Recursively visit all unmarked vertices adjacent to \(v\)

design pattern for graph processing

Design pattern: decouple graph data type from graph processing

Create a Graph object
Pass the Graph to a graph-processing routine
Query the graph-processing routine for information

// print all verts connected to s
Paths paths = new Paths(G, s);
for(int v = 0; v < G.V(); v++)
    if(paths.hasPathTo(v))
        StdOut.println(v);

depth-first search: data structures

To visit a vertex \(v\)

Mark vertex \(v\)
Recursively visit all unmarked vertices adjacent to \(v\)

Data structures:

Boolean array marked[] to mark vertices
Integer array edgeTo[] to keep track of paths (edgeTo[w] == v) means that edge \(v\)-\(w\) taken to discover vertex \(w\)
Function-call stack for recursion

depth-first search: java implementation

public class DepthFirstPaths {
    private int s;
    private boolean[] marked;  // true if v connected to s
    private int[] edgeTo;      // prev vertex on path from s to v

    public DepthFirstPaths(Graph G, int s) {
        //  initialize data structures
        /* ...  */

        // find vertices connected to s
        dfs(G, s);
    }

    // recursive DFS does the work
    private void dfs(Graph G, int v) {
        marked[v] = true;
        for(int w : G.adj(v)) {
            if(!marked[w]) {
                edgeTo[w] = v;
                dfs(G, w);
            }
        }
    }
}

depth-first search: properties

Proposition: DFS marks all vertices connected to \(s\) in time proportional to the sum of their degrees (plus time to initialize the marked[] array).

Pf (correctness):

If \(w\) marked, then \(w\) connected to \(s\) (why?)
If \(w\) connected to \(s\), then \(w\) marked. (if \(w\) unmarked, then consider last edge on a path from \(s\) to \(w\) that goes from a marked vertex to an unmarked one)

Pf (running time): Each vertex connected to \(s\) is visited once

depth-first search: properties

Proposition: After DFS, can check if vertex \(v\) is connected to \(s\) in constant time and can find \(v\)-\(s\) path (if one exists) in time proportional to its length.

Pf. edgeTo[] is parent-link representation of a tree rooted at vert \(s\).

public boolean hasPathTo(int v) { return marked[v]; }

public Iterable<Integer> pathTo(int v) {
    if(!hasPathTo(v)) return null;
    Stack<Integer> path = new Stack<Integer>();
    for(int x = v; x != s; x = edgeTo[x])
        path.push(x);
    path.push(s);
    return path;
}

depth-first search: properties

Proposition: After DFS, can check if vertex \(v\) is connected to \(s\) in constant time and can find \(v\)-\(s\) path (if one exists) in time proportional to its length.

Pf. edgeTo[] is parent-link representation of a tree rooted at vert \(s\).

Exercise: Flood Fill

Problem: Implement flood fill (Photoshop magic wand)

Exercise: Nonrecursive DFS

Challenge: Implement DFS without recursion

One solution (see link)

Maintain a stack of vertices, initialized with \(s\)
For every vertex, maintain a pointer to current vertex in adjacency list
Pop next vertex \(v\) off the stack:
- let \(w\) be next unmarked vertex in adjacency list of \(v\)
- push \(w\) onto stack and mark it

Exercise: Nonrecursive DFS

Challenge: Implement DFS without recursion

Alternative solution (space proportional to \(E+V\), vertex can appear on stack more than once)

Maintain a stack of vertices, initialized with \(s\)
Pop next vertex \(v\) off the stack:
- if vertex \(v\) is marked, continue
- mark \(v\) and add to stack each of its unmarked neighbors

Undirected Graphs

breadth-first search

graph search

Tree traversal: Many ways to explore every vertex in binary tree

Inorder: ACEHMRSX
Preorder: SEACRHMX
Postorder: CAMHREXS
Level-order: SEXARCHM

graph search

Graph search: Many ways to explore every vertex in a graph

Preorder: vertices in order DFS calls dfs(G,v)
Postorder: vertices in order DFS returns from dfs(G,v)
Level-order: vertices in increasing order of distance from s

breadth-first search demo

Repeat until queue is empty:

Remove vertex \(v\) from queue
Add to queue all unmarked vertices adjacent to \(v\) and mark them

breadth-first search

Repeat until queue is empty:

Remove vertex \(v\) from queue
Add to queue all unmarked vertices adjacent to \(v\) and mark them

BFS (from source vertex s)
    Put s onto a FIFO queue, and mark s as visited
    Repeat until the queue is empty:
        Remove the least recently added vertex v
        For each unmarked neighbor of v:
            Add neighbor to the queue
            Mark neighbor

breadth-first search: java implementation

public class BreadthFirstPaths {
    private boolean[] marked;
    private int[] edgeTo;
    private int[] distTo;

    /* ... */

    private void BFS(Graph G, int s) {
        Queue<Integer> q = new Queue<Integer>();
        q.enqueue(s);   // init FIFO queue of vertices to explore
        marked[s] = true;
        distTo[s] = 0;

        while(!q.isEmpty()) {
            int v = q.dequeue();
            for(int w : G.adj(v)) {
                if(!marked[w]) {  // found new vertex w via edge v-w
                    q.enqueue(w);
                    marked[w] = true;
                    edgeTo[w] = v;
                    distTo[w] = distTo[v] + 1;
                }
            }
        }
    }
}

breadth-first search properties

Q. In which order does BFS examine vertices?

Increasing distance (number of edges) from s. Queue always consists of \(\geq 0\) vertices of distance \(k\) from \(s\), followed by \(\geq 0\) vertices of distance \(k+1\).

Proposition: In any connected graph \(G\), BFS computes shortest paths from \(s\) to all other vertices in time proportional to \(E+V\).

bfs application: routing

Fewest number of hops in a communication network

BFS application: kevin bacon numbers

Kevin Bacon graph

Include one vertex for each performer
Connect a movie to all performers that appear in the same movie
Compute shortest path from \(s = \text{Kevin Bacon}\)

BFS applications: Erdös numbers

Erdös-Bacon-Sabbath

[ erdosbaconsabbath.com via wayback ]

Undirected Graphs

challenges

Graph-processing challenge 1

Problem: Identify connected components

How difficult?

Any programmer could do it
Typical diligent algorithms student could do it
Hire an expert
Intractible
No one knows

graph-processing challenge 2

Problem: Is a graph bipartite?

How difficult?

Any programmer could do it
Typical diligent algorithms student could do it
Hire an expert
Intractible
No one knows

graph-processing challenge 3

Problem: Find a cycle in a graph (if one exists).

How difficult?

Any programmer could do it
Typical diligent algorithms student could do it
Hire an expert
Intractible
No one knows

graph-processing challenge 4

Problem: Is there a (general) cycle that uses every edge exactly once?

How difficult?

Any programmer could do it
Typical diligent algorithms student could do it
Hire an expert
Intractible
No one knows

graph-processing challenge 5

Problem: Is there a cycle that contains every vertex exactly once?

How difficult?

Any programmer could do it
Typical diligent algorithms student could do it
Hire an expert
Intractible
No one knows

graph-processing challenge 6

Problem: Are two graphs identical except for vertex names?

How difficult?

Any programmer could do it
Typical diligent algorithms student could do it
Hire an expert
Intractible
No one knows

graph-processing challenge 7

Problem: Can you draw a graph in the plane with no crossing edges?

How difficult?

Any programmer could do it
Typical diligent algorithms student could do it
Hire an expert
Intractible
No one knows

[ Try it yourself at link ]

graph traversal summary

BFS and DFS enables efficient solution of many (but not all) graph problems

graph problem	BFS	DF S	time
s-t path	✓	✓	\(E+V\)
shortest s-t path	✓		\(E+V\)
cycle	✓	✓	\(E+V\)
Euler cycle		✓	\(E+V\)
Hamilton cycle			\(\pow(2,1.657V)\)
bipartiteness (odd cycle)	✓	✓	\(E + V\)
connected components	✓	✓	\(E + V\)
biconnected components		✓	\(E + V\)
planarity		✓	\(E + V\)
graph isomorphism			\(\pow(2,c\cdot\sqroot(V \log V))\)