Lecture 16. Shortest Path Algorithms
 The single-source shortest path problem is the following: given a
source vertex s, and a sink vertex v, we'd like to find the shortest
path from s to v. Here shortest path means a sequence of directed
edges from s to v with the smallest total weight.
 There are some subtleties here. Should we allow negative edges?
Of course, there are no negative distances; nevertheless, there are
actually some cases where negative edges make logical sense. But
then there may not be a shortest path, because if there is a cycle
with negative weight, we could simply go around that cycle as many
times as we want and reduce the cost of the path as much as we
like. To avoid this, we might want to detect negative cycles. This can
be done in the algorithm itself.
 Non-negative cycles aren't helpful, either. Suppose our shortest path
contains a cycle of non-negative weight. Then by cutting it out we
get a path with the same weight or less, so we might as well cut it
out.
Weighted Graphs
 In a weighted graph, each edge has an associated numerical value,
called the weight of the edge
 Edge weights may represent distances, costs, etc.
 Example:

In a flight route graph, the weight of an edge represents the distance in
miles between the endpoint airports
SFO
PVD
ORD
LGA
HNL
LAX
DFW
MIA
Shortest Path Problem
 Given a weighted graph and two vertices u and v, we want to find a path
of minimum total weight between u and v.

Length of a path is the sum of the weights of its edges.
 Example:

Shortest path between Providence and Honolulu
 Applications



Internet packet routing
Flight reservations
Driving directions
SFO
PVD
ORD
LGA
HNL
LAX
DFW
MIA
Shortest Path Properties
Property 1:
A subpath of a shortest path is itself a shortest path
Property 2:
There is a tree of shortest paths from a start vertex to all the other vertices
Example:
Tree of shortest paths from Providence
SFO
PVD
ORD
LGA
HNL
LAX
DFW
MIA
Dijkstra’s Algorithm
 The distance of a vertex v
 We grow a “cloud” of vertices,
from a vertex s is the length
of a shortest path between
s and v
 Dijkstra’s algorithm
computes the distances of
all the vertices from a given
start vertex s
 Assumptions:
 the graph is connected
 the edges are
undirected
 the edge weights are
beginning with s and eventually
covering all the vertices
 We store with each vertex v a label
d(v) representing the distance of v
from s in the subgraph consisting of
the cloud and its adjacent vertices
 At each step
 We add to the cloud the vertex
u outside the cloud with the
smallest distance label, d(u)
 We update the labels of the
vertices adjacent to u
nonnegative
Edge Relaxation
 Consider an edge e = (u,z) such
that


d(u) = 50
u is the vertex most recently
added to the cloud
z is not in the cloud
s
u
e
d(z) = 75
z
 The relaxation of edge e
updates distance d(z) as follows:
d(z)  min{d(z),d(u) + weight(e)}
d(u) = 50
s
u
d(z) = 60
e
z
Example
A
8
0
4
A
8
2
B
8
7
2
C
2
1
D
9
E

F
A
8
4
5
B
8
7
5
E
2
C
3
B
2
7
5
E
D
8
A
8
3
5
0
4
2
C
3
1
9
0
4
2
F
2
8
4
2
3

0
2
1
9
D
11
F
5
3
B
2
7
7
5
E
C
3
2
1
9
D
8
F
5
3
Example (cont.)
A
8
0
4
2
B
2
7
7
C
3
5
E
2
1
9
D
8
F
3
5
A
8
0
4
2
B
2
7
7
C
3
5
E
2
1
9
D
8
F
5
3
Dijkstra’s Algorithm
 A priority queue stores the
vertices outside the cloud


Key: distance
Element: vertex
 Locator-based methods


insert(k,e) returns a
locator
replaceKey(l,k) changes
the key (distance) of an
item
 We store two labels with
each vertex:


Distance (d(v) label)
locator in priority queue
Algorithm DijkstraDistances(G, s)
Q  new heap-based priority queue
for all v  G.vertices()
if v = s
setDistance(v, 0)
else
setDistance(v, )
l  Q.insert(getDistance(v), v)
setLocator(v,l)
while Q.isEmpty()
u  Q.removeMin()
for all e  G.incidentEdges(u)
{ relax edge e }
z  G.opposite(u,e)
r  getDistance(u) + weight(e)
if r < getDistance(z)
setDistance(z,r)
Q.replaceKey(getLocator(z),r)
Analysis
 Graph operations
Method incidentEdges is called once for each vertex
 Label operations
 We set/get the distance and locator labels of vertex z O(deg(z)) times
 Setting/getting a label takes O(1) time
 Priority queue operations
 Each vertex is inserted once into and removed once from the priority
queue, where each insertion or removal takes O(log n) time
 The key of a vertex in the priority queue is modified at most deg(w)
times, where each key change takes O(log n) time
 Dijkstra’s algorithm runs in O((n + m) log n) time provided the graph is
represented by the adjacency list structure

Recall that Sv deg(v) = 2m
 The running time can also be expressed as O(m log n) since the graph is
connected

Extension
 Using the template
method pattern, we can
extend Dijkstra’s
algorithm to return a tree
of shortest paths from the
start vertex to all other
vertices
 We store with each
vertex a third label:

parent edge in the
shortest path tree
 In the edge relaxation
step, we update the
parent label
Algorithm DijkstraShortestPathsTree(G, s)
…
for all v  G.vertices()
…
setParent(v, )
…
for all e  G.incidentEdges(u)
{ relax edge e }
z  G.opposite(u,e)
r  getDistance(u) + weight(e)
if r < getDistance(z)
setDistance(z,r)
setParent(z,e)
Q.replaceKey(getLocator(z),r)
Why Dijkstra’s Algorithm Works
 Dijkstra’s algorithm is based on the greedy method. It
adds vertices by increasing distance.




Suppose it didn’t find all shortest
distances. Let F be the first wrong
vertex the algorithm processed.
When the previous node, D, on the
true shortest path was considered,
its distance was correct.
But the edge (D,F) was relaxed at
that time!
Thus, so long as d(F)>d(D), F’s
distance cannot be wrong. That is,
there is no wrong vertex.
A
8
0
4
2
B
2
7
7
C
3
5
E
2
1
9
D
8
F
5
3
Why It Doesn’t Work for
Negative-Weight Edges
Dijkstra’s algorithm is based on the greedy
method. It adds vertices by increasing distance.

If a node with a negative
incident edge were to be added
late to the cloud, it could mess
up distances for vertices already
in the cloud.
A
8
0
4
6
B
2
7
7
C
0
5
E
5
1
-8
D
9
F
5
C’s true distance is 1, but
it is already in the cloud
with d(C)=5!
4
Bellman-Ford Algorithm
 Works even with negative



weight edges
Must assume directed edges
(for otherwise we would have
negative-weight cycles)
Iteration i finds all shortest
paths of length i .
Running time: O(nm).
Can be extended to detect a
negative-weight cycle if it
exists.
Algorithm BellmanFord(G, s)
for all v  G.vertices()
if v = s
setDistance(v, 0)
else
setDistance(v, )
for i  1 to n-1 do
(*)
for each directed edge e = u  z
{ relax edge e }
r  getDistance(u) + weight(e)
if r < getDistance(z)
setDistance(z,r)
Correctness
 Lemma. After i repetitions of the "for" loop in BELLMAN-FORD, if there
is a path from s to u with at most i edges, then d[u] is at most the length
of the shortest path from s to u with at most i edges.
 Proof. By induction on i. The base case is i=0. Trivial.
For the induction step, consider the shortest path from s to u with at
most i edges. Let v be the last vertex before u on this path. Then the
part of the path from s to v is the shortest path from s to v with at most
i-1 edges. By the inductive hypothesis, d[v] after i-1 executions of the
(*) "for" loop is at most the length of this path. Therefore, d[v] + w(u,v) is
at most the length of the path from s to u, via v (as i-1st node). In the i'th
iteration, d[u] gets compared with d[v] + w(u,v), and is set equal to it if
d[v] + w(u,v) is smaller.
Therefore, after i iterations of the (*) "for" loop, d[u] is at most the length
of the shortest path from s to u that uses at most i edges. So the lemma
holds.
Bellman-Ford Example
Nodes are labeled with their d(v) values
0
8
4
0
8
-2
7

-2
1

3
-2
8

7

9


3
5

0
8
-2
4
7
1
-2
6

1


5
9
0
8
4
-2
1
-2
3
-2

-2
5 8
4
9
9

4
-1
5
7
3
5
-2
1
1
-2
9
4
9
-1
5
4
DAG-based Algorithm
 Works even with negative



weight edges
Uses topological order
Doesn’t use any fancy
data structures
Is much faster than
Dijkstra’s algorithm
Running time: O(n+m).
Algorithm DagDistances(G, s)
for all v  G.vertices()
if v = s
setDistance(v, 0)
else
setDistance(v, )
Perform a topological sort of the vertices
for u  1 to n do {in topological order}
for each e  G.outEdges(u)
{ relax edge e }
z  G.opposite(u,e)
r  getDistance(u) + weight(e)
if r < getDistance(z)
setDistance(z,r)
1
DAG Example
Nodes are labeled with their d(v) values
1
1
0
8
4
0
8
-2
3
2
7

-2

4
1

3
-5
3


8
2
7
9

6
5
-5

5
0
8
-5
2
1

3
6
4
1

9

6
4
5
5
0
8
-2
7
-2
1
8
5

3
1
3
4
4
-2
4
1
-2
4
-1
3
5
2
7
9
7

5
5
-5
0
1
3
6
4
1
-2
9
4
7
(two steps)
-1
5
5
4
All-Pairs Shortest Paths: Dynamic
Programming
Algorithm AllPair(G) {assumes vertices 1,…,n}
every pair of vertices in a
for all vertex pairs (i,j)
weighted directed graph G.
if i = j
D0[i,i]  0
 We can make n calls to
Dijkstra’s algorithm (if no
else if (i,j) is an edge in G
negative edges), which
D0[i,j]  weight of edge (i,j)
takes O(nmlog n) time.
else
D0[i,j]  + 
 Likewise, n calls to Bellman2
Ford would take O(n m)
for k  1 to n do
time.
for i  1 to n do
for j  1 to n do
 We can achieve O(n3) time
Dk[i,j]  min{Dk-1[i,j], Dk-1[i,k]+Dk-1[k,j]}
using dynamic programming
(Floyd-Warshall algorithm).
return Dn
 Find the distance between
i
Uses only vertices numbered 1,…,k
(compute weight of this edge)
Uses only vertices
numbered 1,…,k-1
j
k
Uses only vertices
numbered 1,…,k-1