# Graphs

• A graph $G=(V,E)$ is a pair of sets: vertices V and edges E

• To give an adjacency list representation of a graph, for each vertex v list all the vertices adjacent to v

• To get an adjacency matrix representation of a graph create a square matrix A and label the rows and columns with the vertices: the entry in row i column j is 1 if vertex j is adjacent to vertex i and 0 if it is not

• Can also represent a graph by an array of its edges

## Representations

For each representation:

• How much space do we need to store it?

• How long does it take to initialize an empty graph?

• How long does it take to make a copy?

• How long does it take to insert an edge?

• How long does it take to list the vertices adjacent to a vertex u?

• How long does it take to find out if the edge (u,v) belongs to G?

• Input: a graph $G=(V,E)$ and a source vertex s

• Aim: to find the distance from s to each of the other vertices in the graph

• Idea: send out a wave from s

• The wave first hits vertices at distance 1

• Then the wave hits vertices at distance 2

• and so on

• BFS maintains a queue that contains vertices that have been discovered but are waiting to be processed

• BFS colours the vertices:

• White indicates that a vertex is undiscovered

• Grey indicates that a vertex is discovered but unprocessed

• Black indicates that a vertex has been processed

• The algorithm maintains an array d (distance)

• $d[s]=0$ where s is the source vertex

• if we discover a new vertex v while processing u, we set $d[v]=d[u]+1$

## Example

• Initialization: source vertex grey, others are while; distance to source is 0; add source to the queue

• While the queue is not empty

• Remove first vertex v from the queue

• add white neighbours of v to queue and colour them grey; distance is 1 greater than to v

• colour v black

## Pseudocode

BFS(G,s)

for each vertex u in V[G]-{s}
do colour[u]=White
d[u]=infinity
pi[u]=null
colour[s]=grey
d[s]=0
pi[s]=null
Q=[]
enqueue(Q,s)
while Q != []
do u=dequeue(Q)
do if color[v] = white
then colour[v]=grey
d[v]=d[u]+1
pi[v]=u
enqueue(Q,v)
colour[u]=black


## Analysis of running time

• We want an upper bound on the worst case running time

• Assume that it takes constant time for each operation such as to teat and update colours, to make changes to distance (and predecessor) and to enqueue and dequeue

• Initialization takes time $\mathcal{O}(V)$

• Each vertex enters (and leaves) the queue exactly one. So queueing operations take $\mathcal{O}(V)$

• In the loop the adjacency lists of each vertex are scanned. Each list is read once, and the combined lengths of the lists is $\mathcal{O}(E)$

• Thus the total running time is $\mathcal{O}(V+E)$

## More than distances

• What if as well as finding the distance to each vertex, we want to be able to find a shortest possible path from the source to each vertex?

• Recursively ask predecessors of nodes until you get back to the start node

• BFS used the predecessor of v and v for each vertex v. Note that the predecessor is denoted by $\Pi$

• The path from the source S in the Breadth First Tree is a shortest path from S to V

• What should we add to the algorithm to achieve this?

## Notes

• Note that the algorithm runs on both directed and undirected graphs

• Notice that the highlighted edges (the ones used to discover new vertices) form a tree: we call this a Breadth-first tree. A path from s to another vertex v through the tree is the shortest path between s and v

• The predecessor of a vertex is the one from which is was discovered (i.e. its parent in the Breadth-first tree). We can record predecessors in an array $\Pi$ when we run the algorithm and then use this array to construct the breadth-first tree

# Proofs

## Notation

Let’s call the graph we consider G and the source vertex s. The distance in G from s to a vertex v is denoted $\delta(s,v)$ and the distance found by BFS is $d[v]$

So BFS is correct if $d[v]=\delta(s,v)$ for every vertex $v$ in G

Let us first show that the d values found cannot be too small

## Lemma 1

When BFS terminates, for each vertex v we have $d[v]\geqslant \delta(s,v)$

### Proof

We use induction on the number of vertices added to the queue

The base case is when the first vertex s is added to the queue. Clearly $d[s]=0=\delta(s,s)$

Now suppose a vertex $v$ is being added to the queue. This means that $v$ has just been discovered from some other vertex $u$ and that as $v$ is added to the queue $d[v]$ is set to be $d[u]+1$

By induction, we know that $d[u]\geqslant \delta (s,u)$ so we have

$$d [ v ] = d [ u ] + 1 \geq \delta ( s , u ) + 1 \geq \delta ( s , v )$$

Where the last inequality follows from the fact that a shortest path to $u$ can be extended to a shortest path to $v$ by adding the edge from $u$ to $v$ (so the distance to $v$ is at most one greater than the distance to $u$ - it might, of course, be less if there is an alternative path to $v$ and this is why we cannot replace the inequality in the lemma by an equals sign and use the same proof). The lemma is proved

Now we know that the value of $d[v]$ cannot be smaller than it should be, we have to think about how to prove it cannot be too big. To do this, we first think about what we can say about the values in the array $d$ if we know the order in which vertices enter the queue

## Lemma 2

In $u$ is enqueued before $v$ then $d[u]\leqslant d[v]$

### Proof

In fact, we shall prove the following claim

If a is at the head of the queue and b is at the tail, then $d[b]$ is either $d[a]$ or $d[a]+1$, and $d$ values of the vertices as you go along the queue never decrease

Notice what this is saying: either $d[w]$ is the same for every vertex in the queue or the first so many vertices have one $d$ value and the ones behind have $d$ value one greater. Also not that the claim implies the lemma: if $v$ enters the queue later than $u$ then $d[v]$ must be at least $d[u]$

We prove this claim by using induction on the number of vertices added to the queue:

The base case - when the source is first added to the queue - is clearly true.

Now suppose a vertex $w$ is added to the queue. This happens when a vertex $x$ is removed from the front of the queue and $w$ is a neighbour of $x$. Let $y$ be the vertex at the tail of the queue at the moment $w$ is added.

By the inductive hypothesis, $d[y]$ is $d[x]$ or $d[x]+1$ and we set $d[w]$ to be $d[x]+1$ so the claim remains true (since $w$ is given a $d$ value at least as large as the vertex in front of it and no more than one more than the vertex at the front of the queue).

The lemma is proved

## Theorem 1

When BFS terminates, for each vertex $v$ we have $d[v]=\delta(s,v)$

### Proof

Let us assume the Theorem is not true. Then by lemma 1, we have $d[v]>\delta(s,v)$ for some vertex $v$. Let us say that $v$ is the vertex closest to the source $s$ for which this is true.

Consider the shortest path from $s$ to $v$. Let $u$ be the penultimate vertex on that shortest path. So $\delta(s,v)=\delta(s,u)+1$. Because $u$ is nearer to $s$ than $v$, we have $d[u]=\delta(s,u)$ (by the way that we chose $v$. So)

$$d [ v ] > \delta ( s , v ) = \delta ( s , u ) + 1 = d [ u ] + 1$$

That is $d[v]$ is at least $d[u]+2$. But think about what happens when $u$ is dequeued

• If $v$ was already dequeued, when $d[v]\leqslant d[u]$ (by Lemma 2); a contradiction

• If $v$ is in the queue, then it was added, when some vertex $w$ was dequeued and $d[v]$ was assigned the value $d[w]+1$. But $d[w]\leqslant d[u]$ (by Lemma 2 again, since $w$ must have been added ahead of $u$ in the queue) so $d[v]\leqslant d[u]+1$; again, a contradiction

• Finally if $v$ is not yet in the queue, then when $u$ is discovered and $d[v]$ is given the value $d[u]+1$. This last contradiction proved the Theorem