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Classified e-Material ©Copyrights Charotar Institute of Technology, Changa 1

CE806: AAD U. & P. U. Patel Dept. OF Computer Engineering

Introduction To Algorithms

CS 445





This Lecture

Single-source shortest paths in weightedgraphs

Shortest-Path Problems

Properties of Shortest Paths, Relaxation

Dijkstra¶s Algorithm

Bellman-Ford Algorithm

2





Shortest Path

Generalize distance to weighted setting Digraph G = (V ,E ) with weight functionW : E p R (assigning real values toedges)

Weight of path p = v 1 p v 2 p « p v k is

Shortest path = a path of the minimumweight

Applications static/dynamic network routing

robot motion planning

3

1

1

1

( ) ( , )k

i i

i

w p w v v

!

!§





Shortest-Path Problems

Shortest-Path problems Single-source (single-destination).

Find a shortest path from a given source(vertex s) to each of the vertices.

Single-pair. Given two vertices, find ashortest path between them. Solution tosingle-source problem solves this problemefficiently, too.

All-pairs. Find shortest-paths for everypair of vertices. Dynamic programmingalgorithm.

4





Negative Weights and Cycles?

Negative edges are OK, as long as thereare no negati v e w eight cycl es (otherwisepaths with arbitrary small ³lengths´ would be possible)

Shortest-paths can have no cycles(otherwise we could improve them byremoving cycles)

Any shortest-path in graph G can be nolonger than n ± 1 edges, where n is thenumber of vertices

5





Relaxation

For each vertex v in the graph, wemaintain v .d(), the estimate of theshortest path from s, initialized to gatthe start

Relaxing an edge (u,v ) means testingwhether we can improve the shortestpath to v found so far by going

through u

6

u v

vu

2

2

Relax(u,v)

u v

vu

2

2

Relax(u,v)

Relax (u,v,G)if v.d () > u.d ()+G. w(u,v) then

v.setd (u.d ()+G. w(u,v))

v.setparent(u)





Dijkstra's Algorithm

Non-negative edge weights Greedy, similar to Prim's algorithm for

MST

Like breadth-first search (if all weights = 1, one can simply use BFS)

Use Q, a priority queue ADT keyed byv .d() (BFS used FIFO queue, here weuse a PQ, which is re-organizedwhenever some d decreases)

Basic idea maintain a set S of solved vertices

at each step select "closest" vertex u, add itto S and relax all ed es from u7





Dijkstra¶s Pseudo Code

Input: Graph G , start vertex s

8

r elaxing

edges

Dijkstra(G,s)

01 for each vertex u � G. V ()

02 u.setd (g03 u.setparent(NIL)

04 s.setd (0)

05 S n� // Set S is used to explain the algorithm

06 Q.init(G. V ()) // Q is a priority queue ADT

07 while not Q.isEmpty()

08 u n Q.extractMin()09 S n S � {u}

10 for each v � u.adjacent() do

11 Relax(u, v, G)

12 Q. modifyKey(v)





Dijkstra¶s Example

9

g g

g g

s

u v

yx

10

5

1

2 39

4 67

2

g

g

s

u v

yx

10

5

1

2 39

4 67

2

Dijkstra(G,s)


02 u.setd (g

03 u.setparent(NIL)

04 s.setd (0)

05 S n�

06 Q.init(G. V ())


08 u n Q.extractMin()

09 S n S � {u}


11 Relax(u, v, G)

12 Q. modifyKey(v)





Dijkstra¶s Example (2)

10

u v

s

yx

10

5

1

2 39

4 67

2

s

u v

yx

10

5

1

2 39

4 67

2

Dijkstra(G,s)


02 u.setd (g

03 u.setparent(NIL)

04 s.setd (0)

05 S n�

06 Q.init(G. V ())



09 S n S � {u}


11 Relax(u, v, G)

12 Q. modifyKey(v)





Dijkstra¶s Example (3)

11

u v

yx

10

5

1

2 39

4 67

2

u v

yx

10

5

1

2 39

4 67

2

Dijkstra(G,s)


02 u.setd (g

03 u.setparent(NIL)

04 s.setd (0)

05 S n�

06 Q.init(G. V ())



09 S n S � {u}


11 Relax(u, v, G)

12 Q. modifyKey(v)





Dijkstra¶s Running Time

Extract-Min executed |V | time

Decrease-Key executed |E | time

Time = |V | T Extract-Min + |E | T Decrease-Key

T depends on different Q implementations

12

Q T(Extract-

Min)

T(Decrease-Key) Total

array 3 (V ) 3 (1) 3 (V 2)binary heap 3 (lg V ) 3 (lg V ) 3 (E lg V )

Fibonacci heap 3 (lg V ) 3 (1) (amort.) 3 (V lgV + E )





Bellman-Ford Algorithm

Dijkstra¶s doesn¶t work when there arenegative edges:

Intuition ± we can not be greedy any moreon the assumption that the lengths of paths

will only increase in the future

Bellman-Ford algorithm detects negativecycles (returns f al se) or returns the

shortest path-tree

13





Bellman-Ford Example

15

5

g g

g g

s

zy

6

7

8-3

72

9

-2xt

-4

g

g

s

zy

6

7

8-3

72

9

-2xt

-4

5

s

zy

6

7

8-3

72

9

-2xt

-4

5

s

zy

6

7

8-3

72

9

-2xt

-4

5





Bellman-Ford Example

Bellman-Ford running time:

(|V|-1)|E| + |E| = 5(VE)

16

s

zy

6

7

8-3

7

2

9

-2xt

-4

5

shortest path 1

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