The Shortest Path problem

I Given graph and a vertex s find shortest paths from s to all othervertices.

I Map routing, robot navigation, urban traffic planning

I Optimal pipelining of VLSI chip

I Routing of telecommunication messages

I Network routing protocols (OSPF, BGP, RIP)

I Seam carving, texture mapping, typesetting in TeX!

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Example with positive edge weights

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

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Example with negative edge weights

ν1

ν2

ν3

2

10

−15

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Unweighted shortest paths

I Given unweighted graph G

I Can assume all edge weights are 1

I Find shortest paths from s

I There is what is known as a shortest path tree!

I Can be found using Breadth First Search (BFS)

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Naive implementation: pseudo code

void Graph::unweighted( Vertex s ){

Vertex v,w;

s.dist = 0;

for(int currDist=0; currDist < NUM_VERTICES; currDist++)

for each vertex v

if( !v.known && v.dist == currDist ){

v.known = true;

for each w adjacent to v

if( w.dist == INFINITY ){

w.dist = currDist + 1;

w.path = v;

}

}

}

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Smarter implementation: pseudo code

void Graph::unweighted( Vertex s ){

Queue q( NUM_VERTICES );

Vertex v,w;

q.enqueue(s);

s.dist = 0;

while( !q.isEmpty() ){

v = q.dequeue();

v.known = true;

for each w adjacent to v

if( w.dist == INFINITY ){

w.dist = v.dist + 1;

w.path = v;

q.enqueue( w );

}

}

}

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Main structural properties of Shortest Paths

I Prefixes of shortest paths are themselves shortest paths

I Does a shortest path always exist?

I What about a shortest path tree?

I How can we compute such a tree

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Main concepts

I known vertices

I Relaxation of an edge (v, w) : d(w) = min(d(w), d(v) + cvw)

I Next: The Djikstra algorithm

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25

Edsger W. Dijkstra: select quotes

Edsger W. Dijkstra

Turing award 1972

“ Do only what only you can do. ”

“ In their capacity as a tool, computers will be but a ripple on the

surface of our culture. In their capacity as intellectual challenge,

they are without precedent in the cultural history of mankind. ”

“ The use of COBOL cripples the mind; its teaching should,

therefore, be regarded as a criminal offence. ”

“ It is practically impossible to teach good programming to

students that have had a prior exposure to BASIC: as potential

programmers they are mentally mutilated beyond hope of

regeneration. ”

“ APL is a mistake, carried through to perfection. It is the

language of the future for the programming techniques

of the past: it creates a new generation of coding bums. ”

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26

Edsger W. Dijkstra: select quotes

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Djikstra algorithm : arbitrary non-negative edge weights

I Store dv, known, pv

I Pick vertex with minimum dv (that is not known)

I Relax all edges outgoing from it

I Repeat until all vertices are known

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Example of Djikstra in action

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

12 / 27

Example of Djikstra in action

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

0 ∞

∞∞∞

∞∞

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Example of Djikstra in action

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

0 2

∞∞1

∞∞

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Example of Djikstra in action

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

0 2

331

95

12 / 27

Example of Djikstra in action

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

0 2

331

95

12 / 27

Example of Djikstra in action

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

0 2

331

95

12 / 27

Example of Djikstra in action

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

0 2

331

85

12 / 27

Example of Djikstra in action

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

0 2

331

65

12 / 27

Example of Djikstra in action

ν1 ν2

ν3 ν4 ν5

ν6 ν7

2

1 310

2

64

1

85

2

4

0 2

331

65

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Djikstra data-structuresstruct Vertex

{

List adj; // Adjacency list

bool known;

DistType dist;

Vertex path; // ref to parent in path

};

void Graph::createTable( vector<Vertex> & t){

readGraph( t ); //Read graph, fill in adj

for(int i=0; i < t.size(); i++){

t[i].known = false;

t[i].dist = INFINITY;

t[i].path = NOT_A_VERTEX;

}

NUM_VERTICES = t.size();

}

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Shortest Paths after Djikstra run

void Graph::printPath( Vertex v )

{

if(v.path != NOT_A_VERTEX)

{

printPath( v.path );

cout << " to ";

}

cout << v;

}

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The Djikstra algorithm: pseudo-code

void Graph::djikstra( Vertex s ){

Vertex v,w;

1. s.dist = 0;

2. for( ; ; ){

3. v = smallest unknown distance vertex;

4. if( v == NOT_A_VERTEX )

5. break;

6. v.known = true;

7. for each w adjacent to v;

8. if( !w.known )

9. if( v.dist + c(v,w) < w.dist ){

10. decrease w.dist to v.dist + c(v,w);

11. w.path = v;

}

}

}

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Implementing Djikstra

I Naive implementation (using array to find min dv) :O(|E|+ |V 2|) = O(|V |2)

I Could we be better for sparse graphs?

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Implementing Djikstra

I Naive implementation (using array to find min dv) :O(|E|+ |V 2|) = O(|V |2)

I Could we be better for sparse graphs?

PQ impl insert delete-min decrease-key

total

unorderedarray

1 V 1 V 2

binary heap log V log V log V E log V

d-way heap logd V d logd V logd V logEVV

Fibonacciheap

1 log V 1 E +V log V

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Negative edge weights!

ν1

ν2

ν3

2

10

−15

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Acyclic Graphs

I Important special case : Nonreversible chemcial reactions, criticalpath analysis

I Running time is O(|E|+ |V |)I Djikstra can be implemented along with Topological sort

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Example: Activity-node graph

start finish

A(3)

B(2)

C(3)

D(2)

E(1)

F (3)

G(2)

K(4)

H(1)

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Event-node graph

1

2

3

6′

4

6

5

7′

8′

7

89

10′ 10

A/3

B/2 0

0

C/3

D/2

E/1

0

0

0

0

F/3

G/2

K/4

0

0

0

H/1

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Earliest Completion times

1

2

3

6′

4

6

5

7′

8′

7

89

10′ 10

A/3

B/2 0

0

C/3

D/2

E/1

0

0

0

0

F/3

G/2

K/4

0

0

0

H/1

0

3

2

3

6

5

3

6

5

9

7

7

9 10

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Latest Completion times

1

2

3

6′

4

6

5

7′

8′

7

89

10′ 10

A/3

B/2 0

0

C/3

D/2

E/1

0

0

0

0

F/3

G/2

K/4

0

0

0

H/1

0

3

4

4

6

6

5

6

7

9

8

9

9 10

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EC, LC, Slack, Critical Path

Earliest and Latest Completion times

EC1 = 0

ECw = max(v,w)∈E

(ECv + cv,w)

LCn = ECn

LCv = min(v,w)∈E

(LCw − cv,w)

Slack of an edge (v, w)

Slack(v,w) = LCw − ECv − c(v,w)

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The Bellman Ford algorithm

Basic Pseudo code

d[s] = 0for i = 1 to |V |

Relax each edge

I Why does this work?

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Bellman Ford: Queue Based Implementation

void Graph::weightedNegative( Vertex s ){

Queue q(NUM_VERTICES);

Vertex v,w;

q.enqueue(s);

s.dist = 0;

while(! q.isEmpty()){

v=q.dequeue();

for each w adjacent to v

if(v.dist + c(v,w) < w.dist){

w.dist = v.dist + c(v,w);

w.path = v;

if(w is not already in q)

q.enqueue(w);

}

}

}

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Bellman Ford contd.

I Runtime is O(EV )

I Can be used to detect negative cycles

I Useful in finding arbitrage opportunities!

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All-Pairs Shortest Path

I Can run |V | Djikstra’s - O(|E||V | log |V |)I Floyd Warshall : Dynamic programming algorithm

I Works in O(|V |3)

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