SINGLE-SOURCE SHORTEST PATHS

1
SINGLE-SOURCE SHORTEST PATHS
2
Shortest Path Problems

• Directed weighted graph.
• Path length is sum of weights of edges on path.
• The vertex at which the path begins is the source
  vertex.
• The vertex at which the path ends is the
  destination vertex.

3
Example

• Consider Source node 1 and destination node 7
• A direct path between Nodes 1 and 7 will cost 14

4
Example

• A shorter path will cost only 11

5
Shortest-Path Variants

• Single-source single-destination (1-1) Find the
  shortest path from source s to destination v.
• Single-source all-destination(1-Many) Find the
  shortest path from s to each vertex v.
• Single-destination shortest-paths (Many-1) Find
  a shortest path to a given destination vertex t
  from each vertex v.
• All-pairs shortest-paths problem (Many-Many)
  Find a shortest path from u to v for every pair
  of vertices u and v.

We talk about directed, weighted graphs
6
Shortest-Path Variants (Contd)

• For un-weighted graphs, the shortest path problem
  is mapped to BFS
• Since all weights are the same, we search for the
  smallest number of edges

BFS creates a tree called BFS-Tree
7
Shortest-Path Variants
No need to consider different solution or
algorithm for each variant (we reduce it into
two types)

• Single-source single-destination (1-1) Find the
  shortest path from source s to destination v.
• Single-source all-destination(1-Many) Find the
  shortest path from s to each vertex v.
• Single-destination shortest-paths (Many-1) Find
  a shortest path to a given destination vertex t
  from each vertex v.
• All-pairs shortest-paths problem (Many-Many)
  Find a shortest path from u to v for every pair
  of vertices u and v.

8
Shortest-Path Variants
Single-Source Single-Destination (1-1) No good solution that beats 1-M variant Thus, this problem is mapped to the 1-M variant Single-Source All-Destination (1-M) Need to be solved (several algorithms) We will study this one
All-Sources Single-Destination (M-1) Reverse all edges in the graph Thus, it is mapped to the (1-M) variant All-Sources All-Destinations (M-M) Need to be solved (several algorithms) We will study it (if time permits)
9
Shortest-Path Variants
Single-Source Single-Destination (1-1) No good solution that beats 1-M variant Thus, this problem is mapped to the 1-M variant Single-Source All-Destination (1-M) Need to be solved (several algorithms) We will study this one
All-Sources Single-Destination (M-1) Reverse all edges in the graph Thus, it is mapped to the (1-M) variant All-Sources All-Destinations (M-M) Need to be solved (several algorithms) We will study it (if time permits)
10
Introduction

• Generalization of BFS to handle weighted graphs
  
• Direct Graph G ( V, E ), edge weight fn w E
  ? R
• In BFS w(e)1 for all e Î E
• Weight of path p v1 v2 vk is

11
Shortest Path

• Shortest Path Path of minimum weight
• d(u,v)

p
min?(p) u v if there is a path from u
to v, ? otherwise.
12
Several Properties
13
1- Optimal Substructure Property

• Theorem Subpaths of shortest paths are also
  shortest paths
• Let d(1,k) ltv1, ..i, .. j.., .. ,vk gt be a
  shortest path from v1 to vk
• Let d(i,j) ltvi, ... ,vj gt be subpath of d(1,k)
  from vi to vj
• for any i, j
• Then d(I,j) is a shortest path from vi to vj

14
1- Optimal Substructure Property

• Proof By cut and paste
• If some subpath were not a shortest path
• We could substitute a shorter subpath to create a
  shorter total path
• Hence, the original path would not be shortest
  path

v5
v7
v1
v4
v6
v3

v0
v2
15
2- No Cycles in Shortest Path

• Definition
• d(u,v) weight of the shortest path(s) from u to
  v
• Negative-weight cycle The problem is undefined
• can always get a shorter path by going around the
  cycle again
• Positive-weight cycle can be taken out and
  reduces the cost

cycle lt 0
s
v
16
3- Negative-weight edges

• No problem, as long as no negative-weight cycles
  are reachable from the source (allowed)

17
4- Triangle Inequality

• Lemma 1 for a given vertex s in V and for every
  edge (u,v) in E,
• d(s,v) d(s,u) w(u,v)
• Proof shortest path s v is no longer than
  any other path.
• in particular the path that takes the shortest
  path s v and
• then takes cycle (u,v)

u
v
s
18
Algorithms to Cover

• Dijkstras Algorithm
• Shortest Path is DAGs

19
Initialization

• Maintain dv for each v in V
• dv is called shortest-path weight estimate
• and it is upper bound on d(s,v)

INIT(G, s) for each v ? V do dv ?
8 pv ? NIL ds ? 0
Upper bound
Parent node

20
Relaxation
RELAX(u, v) if dv gt duw(u,v)
then dv ? duw(u,v)
pv ? u
When you find an edge (u,v) then check this
condition and relax dv if possible
u
v
u
v
2
2
5
5
9
6
Change dv
No chnage

5
7
5
6
2
2
v
u
v
u
21
Algorithms to Cover

• Dijkstras Algorithm
• Shortest Path is DAGs

22
Dijkstras Algorithm For Shortest Paths

• Non-negative edge weight
• Like BFS If all edge weights are equal, then use
  BFS, otherwise use this algorithm
• Use Q min-priority queue keyed on dv values

23
Dijkstras Algorithm For Shortest Paths

• DIJKSTRA(G, s)
• INIT(G, s)
• S?Ø gt set of discovered nodes
• Q?VG
• while Q ?Ø do
• u?EXTRACT-MIN(Q)
• S?S U u
• for each v in Adju do
• RELAX(u, v) gt May cause
• gt DECREASE-KEY(Q, v, dv)

24
Example Initialization Step
25
Example
3
26
Example
2
27
Example
2
28
Example
10
29
Example
30
Example
u
5
31
Dijkstras Algorithm Analysis
O(V)
O(V Log V)
Total in the loop O(V Log V)
Total in the loop O(E Log V)
Time Complexity O (E Log V)
32
Algorithms to Cover

• Dijkstras Algorithm
• Shortest Path is DAGs

33
Key Property in DAGs

• If there are no cycles ? it is called a DAG
• In DAGs, nodes can be sorted in a linear order
  such that all edges are forward edges
• Topological sort

34
Single-Source Shortest Paths in DAGs

• Shortest paths are always well-defined in dags
• no cycles gt no negative-weight cycles even if
  there are negative-weight edges
• Idea If we were lucky
• To process vertices on each shortest path from
  left to right, we would be done in 1 pass

35
Single-Source Shortest Paths in DAGs

• DAG-SHORTEST PATHS(G, s)
• TOPOLOGICALLY-SORT the vertices of G
• INIT(G, s)
• for each vertex u taken in topologically
  sorted order do
• for each v in Adju do
• RELAX(u, v)

36
Example
6
1
u
r
t
s
v
w
5
2
7
2
1
?
0
?
?
?
?
4
3
2
37
Example
6
1
u
r
t
s
v
w
5
2
7
2
1
?
0
?
?
?
?
4
3
2
38
Example
6
1
u
r
t
s
v
w
5
2
7
2
1
?
0
2
6
?
?
4
3
2
39
Example
6
1
u
r
t
s
v
w
5
2
7
2
1
?
0
2
6
6
4
4
3
2
40
Example
6
1
u
r
t
s
v
w
5
2
7
2
1
?
0
2
6
5
4
4
3
2
41
Example
6
1
u
r
t
s
v
w
5
2
7
2
1
?
0
2
6
5
3
4
3
2
42
Example
6
1
u
r
t
s
v
w
5
2
7
2
1
?
0
2
6
5
3
4
3
2
43
Single-Source Shortest Paths in DAGs Analysis
O(VE)

• DAG-SHORTEST PATHS(G, s)
• TOPOLOGICALLY-SORT the vertices of G
• INIT(G, s)
• for each vertex u taken in topologically
  sorted order do
• for each v in Adju do
• RELAX(u, v)

O(V)
Total O(E)
Time Complexity O (V E)