Math1800 Lec8 Bb

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MATH1800Mathematical Modelling

School of Mathematical and Physical Sciences,Faculty of Science and Information Technology,

University of Newcastle, Australia

Semester 2

Lec 8 Recap

What is the difference between a graph and digraph (i.e. network)?

What are three different ways of characterizing a tree?

What is a subgraph?

What is a spanning tree?

What is a minimum spanning tree (MST)?

How does Prims algorithm work?

What is a cut?

What is a cut induced by an edge of a spanning tree?What does the interchangeable property (i.e. Property 2) tell us aboutspanning trees?

What is the cut optimality condition for MST?

How can the cut optimality condition be used to prove that Prims

algorithm constructs an MST?

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Lec 8 Recap

What is the difference between a graph and digraph (i.e. network)?

What are three different ways of characterizing a tree?

What is a subgraph?

What is a spanning tree?

What is a minimum spanning tree (MST)?

How does Prims algorithm work?

What is a cut?

What is a cut induced by an edge of a spanning tree?

What does the interchangeable property (i.e. Property 2) tell us aboutspanning trees?

What is the cut optimality condition for MST?

How can the cut optimality condition be used to prove that Prims

algorithm constructs an MST?

Lec 8 Network Models: Shortest Paths in Directed Graphs (Networks)

The Shortest Path Problem

Given a directed graph with nodes N, and arcs A, find theshortest path from a node sto a node twith respect to arclength/cost cij.

Example: Find the shortest path from node 1to node 5(i.e. s=1, t=5)

1

2 3

4 5

1

1

2 2

5

9

8

Shortest path is (1, 2, 3, 5)with cost 8. (why?)

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1

2 3

4 5

1

1

2 2

5

9

8


Enumeration:

Path 1: (1,2,3,5)with cost 8

Path 2: (1,2,5)with cost 9







1

2 3

4 5

1

1

2 2

5

9

8


Enumeration:


Path 2: (1,2,5)with cost 9 Path 3: (1,2,4,5)with cost 12


Logical deduction:

The shortest path must use one of the arcs (4,5), (2,5),and (3,5). The cost of any path using arcs (2,5), or (4,5)is at least 9. The only path that does not use arcs (4,5)and (2,5)is the path (1, 2, 3, 5)with cost 8andtherefore optimal.

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Q. What about shortest paths in larger networks?




Q. What about shortest paths in larger networks?

No. of nodes

(n)

Max no. of arcs(n2)

Max no. of paths between a pair of nodes(n!)

10 100 3628800

20 400 2432902008176640000

50 2,500 30414093201713378043612608166065(32 more digits)

100 10,000 93326215443944152681699238856267(126 more digits)

: : :

1000 1,000,000 402387260077093773543702433923(2,538 more digits)

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Lec 8 Network Models: Applications of Shortest Paths

Transportat ion/Communicat ion networks (Logist ics, GPS, Google maps,internet routing , etc.)

Find the shortest (either time or distance) path from origin to destination


Capital Replacement Strategy

Each year a company car costs more to run and maintain, and depreciates invalue (see below). Projected new car prices are also given below.

If the company requires a car for 5 years, what is the optimal strategy?

Age of car Annual cost Trade-in Value

1 $1,000 $12,000

2 $2,000 $10,000

3 $5,000 $6,000

4 $9,000 $5,000

5 $12,000 $3,000

Start of Year Price

1 $19,000

2 $18,500

3 $18,000

4 $18,000

5 $17,000

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Each year a company car costs more to run and maintain, and depreciates in

value (see below). Projected new car prices are also given below.



1 $1,000 $12,000

2 $2,000 $10,000

3 $5,000 $6,000

4 $9,000 $5,000

5 $12,000 $3,000

Start of Year Price

1 $19,000

2 $18,500

3 $18,000

4 $18,000

5 $17,000

A network model:

inode Start of year i

iarc Buy a car at the start of year i and drive until start of year j (i < j)j

ilength Cost of buying a car at the start of year i+cost ofj-iyears of maintenancetrade-in value of (j-i)year-old car

jcij



Each year a company car costs more to run and maintain, and depreciates invalue (see below). Projected new car prices are also given below.



1 $1,000 $12,000

2 $2,000 $10,000

3 $5,000 $6,000

4 $9,000 $5,000

5 $12,000 $3,000

Start of Year Price

1 $19,000

2 $18,500

3 $18,000

4 $18,000

5 $17,000

1 2 3 4 5

19+1+2+5-6 =21

6

19+1+2-10 =12

19+1-12 =8 18.5+1-12 =7.5 18+1-12 =7

18.5+1+2-10 =11.5

Optimal strategy: shortestpath from node 1 to node 6.

Exercise: complete network and find shortest path.

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Very L arge Scale Integration (VLSI) circuit desig n

When designing VLSI circuits, need to isolate negative feedback loops. These

negative feedback loops correspond to negative cost cycles in the amplifier-gainnetwork of the circuit.


Robot ic Surgery

Planning shortest paths through patientsanatomy and avoid obstacles

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Image Segmentation

Find a line in an image that separates distinguishable objects of interest, i.e. line

that cuts through the fewest number of black pixels. This grid of pixels can bemodelled as a graph, with any edge across a black pixel given a high cost. Theshortest path defines the best separation.

Lec 8 Network Models: Shortest Path Algorithms

DijkstrasAlgori thm fo r Short est Paths (1956)

Given network with nodes Nand arcs A, find shortestpath from node sto node t.

1. Assign a distance label di, and a predecessor labelpito each node. Set dito infinity for all nodesexcept s. Set dsto 0. Mark all nodes as unlabelled.

2. Pick an unlabellednode iwith the minimumdistance label dito explore.

3. Update distance and predecessor labels forneighbouring nodes of i: for all nodesjs.t. (i,j)is anarc if dj> di + cijthen set djto di + cijand set pjto i.

4. Mark ias labelled.

5. Repeat second step until all nodes are labelled.

ditracks the distance of the shortest path found so far from sto node i.

pitracks the node that precedes ion the shortest path found so far from sto i.

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3. Update distance and predecessor labels forneighbouring nodes of i: for all nodesjs.t. (i,j)is anarc if dj> di + cijthen set djtodi + cijand set pjto i.










3. Update distance and predecessor labels forneighbouring nodes of i: for all nodesjs.t. (i,j)is anarc if dj> di + cijthen set djtodi + cijand set pjto i.





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1

2 3

4 5

1

1

2 2

5

9

8

Find the shortest path from node 1 to node 5.


1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)


Step 1.Initialise distance and predecessor labels

Distance and predecessor label (di,pi) for node i

Unlabelled nodes

Labelled nodes

Node we are currently exploring

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1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)

(0,-) (1,1) (,-) (1,1) (,-)



Step 2.Pick unlabelled node with minimum distance label, i.e, node 1.

Step 3.Update distance and predecessor labels for nodes 2 and 4.

Updatedistance and predecessor labels forneighbouring nodes of i: for all nodesjs.t. (i,j)is anarc if dj> di + cijthen set djtodi + cijand set pjto i.

Unlabelled nodes

Labelled nodes



1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)

(0,-) (1,1) (,-) (1,1) (,-)





Step 4.Mark node 1 as labelled.


Unlabelled nodes

Labelled nodes


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1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)

(0,-) (1,1) (,-) (1,1) (,-)

(0,-) (1,1) (6,2) (1,1) (9,2)






Step 2.Pick unlabelled node with minimum distance label, i.e, node 2(node 4 is also possible).

Step 3.Update distance and predecessor labels for nodes 3,4 and 5.


Unlabelled nodes

Labelled nodes



1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)

(0,-) (1,1) (,-) (1,1) (,-)

(0,-) (1,1) (6,2) (1,1) (9,2)








Step 4.Mark node 2 as labelled.Unlabelled nodes

Labelled nodes


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1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)

(0,-) (1,1) (,-) (1,1) (,-)

(0,-) (1,1) (6,2) (1,1) (9,2)

(0,-) (1,1) (6,2) (1,1) (9,2)










Step 3.Update distance and predecessor label for node 5.


Unlabelled nodes

Labelled nodes



1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)

(0,-) (1,1) (,-) (1,1) (,-)

(0,-) (1,1) (6,2) (1,1) (9,2)

(0,-) (1,1) (6,2) (1,1) (9,2)










Step 3.Update distance and predecessor label for node 5.Step 4.Mark node 4 as labelled.

Unlabelled nodes

Labelled nodes


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1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)

(0,-) (1,1) (,-) (1,1) (,-)

(0,-) (1,1) (6,2) (1,1) (9,2)

(0,-) (1,1) (6,2) (1,1) (9,2)

(0,-) (1,1) (6,2) (1,1) (8,3)













Step 3.Update distance and predecessor labels for node 5.


Unlabelled nodes

Labelled nodes



1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)

(0,-) (1,1) (,-) (1,1) (,-)

(0,-) (1,1) (6,2) (1,1) (9,2)

(0,-) (1,1) (6,2) (1,1) (9,2)

(0,-) (1,1) (6,2) (1,1) (8,3)














Unlabelled nodes

Labelled nodes


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1

2 3

4 5

1

1

2 2

5

9

8

Node

1 2 3 4 5

(0,-) (,-) (,-) (,-) (,-)

(0,-) (1,1) (,-) (1,1) (,-)

(0,-) (1,1) (6,2) (1,1) (9,2)

(0,-) (1,1) (6,2) (1,1) (9,2)

(0,-) (1,1) (6,2) (1,1) (8,3)
















Step 3.No distance and predecessor labels need updating.

Step 4.Mark 5 as labelled. No unlabelled labels remaining. STOP

Unlabelled nodes

Labelled nodes



1

2 3

4 5

1

1

2 2

5

9

8















Step 3.No distance and predecessor labels need updating.

Step 4.Mark 5 as labelled. No unlabelled labels remaining. STOP

Iteration1

Iteration2

Iteration3

Iteration4

Iteration5

Iterat

ionNode

1 2 3 4 5

0 (0,-) (,-) (,-) (,-) (,-)

1 (0,-) (1,1) (,-) (1,1) (,-)

2 (0,-) (1,1) (6,2) (1,1) (9,2)

3 (0,-) (1,1) (6,2) (1,1) (9,2)

4 (0,-) (1,1) (6,2) (1,1) (8,3)

Unlabelled nodes

Labelled nodes


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1

2 3

4 5

1

1

2 2

5

9

8


The shortest path from 1to 5is: (1,2,3,5) with cost 8

Property of Distance and Predecessor Labels

The distance and predecessor labels inform us about notonly the optimal path from sto t, but from sto every othernode.

Q. What is the shortest path from 1 to 2?

A. (1,2) with cost 1


A. (1,2,3) with cost 6




A. (1,2,3,5) with cost 8

Iterat

ionNode

1 2 3 4 5

0 (0,-) (,-) (,-) (,-) (,-)1 (0,-) (1,1) (,-) (1,1) (,-)

2 (0,-) (1,1) (6,2) (1,1) (9,2)

3 (0,-) (1,1) (6,2) (1,1) (9,2)

4 (0,-) (1,1) (6,2) (1,1) (8,3)

Unlabelled nodes

Labelled nodes



1

2 3

4 5

1

1

2 2

5

9

8


The shortest path from 1to 5is: (1,2,3,5) with cost 8

Property of Distance and Predecessor Labels

The distance and predecessor labels inform us about notonly the optimal path from sto t, but from sto every othernode.

Q. What is the shortest path from 1 to 2?A. (1,2) with cost 1


A. (1,2,3) with cost 6




A. (1,2,3,5) with cost 8

Iterat

ionNode

1 2 3 4 5

0 (0,-) (,-) (,-) (,-) (,-)

1 (0,-) (1,1) (,-) (1,1) (,-)

2 (0,-) (1,1) (6,2) (1,1) (9,2)

3 (0,-) (1,1) (6,2) (1,1) (9,2)

4 (0,-) (1,1) (6,2) (1,1) (8,3)

Unlabelled nodes

Labelled nodes


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1

2 3

4 5

50

200

6

40

150

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

Unlabelled nodes

Labelled nodes




1

2 3

4 5

50

200

6

40

150

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (100,1) (,-) (200,1) (,-) (,-)

Unlabelled nodes

Labelled nodes


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1

2 3

4 5

50

200

6

40

150

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (100,1) (,-) (200,1) (,-) (,-)

2 (0,-) (100,1) (300,2) (150,2) (200,2) (,-)

Unlabelled nodes

Labelled nodes




1

2 3

4 5

50

200

6

40

150

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (100,1) (,-) (200,1) (,-) (,-)

2 (0,-) (100,1) (300,2) (150,2) (200,2) (,-)

3 (0,-) (100,1) (300,2) (150,2) (190,4) (,-)Unlabelled nodes

Labelled nodes


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1

2 3

4 5

50

200

6

40

150

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (100,1) (,-) (200,1) (,-) (,-)

2 (0,-) (100,1) (300,2) (150,2) (200,2) (,-)

3 (0,-) (100,1) (300,2) (150,2) (190,4) (,-)

4 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

Unlabelled nodes

Labelled nodes




1

2 3

4 5

50

200

6

40

150

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (100,1) (,-) (200,1) (,-) (,-)

2 (0,-) (100,1) (300,2) (150,2) (200,2) (,-)

3 (0,-) (100,1) (300,2) (150,2) (190,4) (,-)

4 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

5 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

Unlabelled nodes

Labelled nodes


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1

2 3

4 5

50

200

6

40

150

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (100,1) (,-) (200,1) (,-) (,-)

2 (0,-) (100,1) (300,2) (150,2) (200,2) (,-)

3 (0,-) (100,1) (300,2) (150,2) (190,4) (,-)

4 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

5 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

6 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

Unlabelled nodes

Labelled nodes



Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (100,1) (,-) (200,1) (,-) (,-)

2 (0,-) (100,1) (300,2) (150,2) (200,2) (,-)

3 (0,-) (100,1) (300,2) (150,2) (190,4) (,-)

4 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

5 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

6 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)


1

2 3

4 5

50

200

6

40

150

The shortest path from 1to 6is: (1,2,4,5,6) with cost 290

Note: once a node is labelled, its distance and predecessor label is neverchanged. Hence, once a node is labelled, its distance label gives the shortestdistance from sto that node. We could therefore have stopped at iteration 5.

Unlabelled nodes

Labelled nodes


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Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (100,1) (,-) (200,1) (,-) (,-)

2 (0,-) (100,1) (300,2) (150,2) (200,2) (,-)

3 (0,-) (100,1) (300,2) (150,2) (190,4) (,-)

4 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

5 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

6 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)


1

2 3

4 5

50

200

6

40

150



Q. Does Dikstrasalgorithm always find the shortest path?

Unlabelled nodes

Labelled nodes



Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (100,1) (,-) (200,1) (,-) (,-)

2 (0,-) (100,1) (300,2) (150,2) (200,2) (,-)

3 (0,-) (100,1) (300,2) (150,2) (190,4) (,-)

4 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

5 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)

6 (0,-) (100,1) (300,2) (150,2) (190,4) (290,5)


1

2 3

4 5

50

200

6

40

150




A. No.

Unlabelled nodes

Labelled nodes


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1

2 4

3

1

6

5-100

10 1

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1

2

3

4

5

6

Unlabelled nodes

Labelled nodes




1

2 4

3

1

6

5-100

10 1

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (1,1) (,-) (,-) (,-) (,-)

2

3

4

5

6

Unlabelled nodes

Labelled nodes


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1

2 4

3

1

6

5-100

10 1

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (1,1) (,-) (,-) (,-) (,-)

2 (0,-) (1,1) (11,2) (2,2) (,-) (,-)

3

4

5

6

Unlabelled nodes

Labelled nodes




1

2 4

3

1

6

5-100

10 1

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (1,1) (,-) (,-) (,-) (,-)

2 (0,-) (1,1) (11,2) (2,2) (,-) (,-)

3 (0,-) (1,1) (11,2) (2,2) (,-) (3,4)

4

5

6

Unlabelled nodes

Labelled nodes


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1

2 4

3

1

6

5-100

10 1

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (1,1) (,-) (,-) (,-) (,-)

2 (0,-) (1,1) (11,2) (2,2) (,-) (,-)

3 (0,-) (1,1) (11,2) (2,2) (,-) (3,4)

4 (0,-) (1,1) (11,2) (2,2) (,-) (3,4)

5

6

Unlabelled nodes

Labelled nodes




1

2 4

3

1

6

5-100

10 1

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (1,1) (,-) (,-) (,-) (,-)

2 (0,-) (1,1) (11,2) (2,2) (,-) (,-)

3 (0,-) (1,1) (11,2) (2,2) (,-) (3,4)

4 (0,-) (1,1) (11,2) (2,2) (,-) (3,4)

5 (0,-) (1,1) (11,2) (2,2) (-89,3) (3,4)

6

Unlabelled nodes

Labelled nodes


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1

2 4

3

1

6

5-100

10 1

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (1,1) (,-) (,-) (,-) (,-)

2 (0,-) (1,1) (11,2) (2,2) (,-) (,-)

3 (0,-) (1,1) (11,2) (2,2) (,-) (3,4)

4 (0,-) (1,1) (11,2) (2,2) (,-) (3,4)

5 (0,-) (1,1) (11,2) (2,2) (-89,3) (3,4)

6 (0,-) (1,1) (11,2) (-88,5) (-89,3) (3,4)

Unlabelled nodes

Labelled nodes





A. No. Not if there are negat ive cost arcs.

Q. Are there other algorithms that can find the shortest path when there arenegative cost arcs?

1

2 4

3

1

6

5-100

10 1

Iteration Node

1 2 3 4 5 6

0 (0,-) (,-) (,-) (,-) (,-) (,-)

1 (0,-) (1,1) (,-) (,-) (,-) (,-)

2 (0,-) (1,1) (11,2) (2,2) (,-) (,-)

3 (0,-) (1,1) (11,2) (2,2) (,-) (3,4)

4 (0,-) (1,1) (11,2) (2,2) (,-) (3,4)

5 (0,-) (1,1) (11,2) (2,2) (-89,3) (3,4)

6 (0,-) (1,1) (11,2) (-88,5) (-89,3) (3,4)

Unlabelled nodes

Labelled nodes


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1

2 3

4 5

50

200

6

40

150


Path 1: (1,2,3,6) with cost 400

Path 2: (1,2,3,5,2,3,6) with cost 250

Path 3: (1,2,3,5,2,3,5,2,3,6) with cost 100

Path 3: (1,2,3,5,2,3,5,2,3,5,2,3,6) with cost -50

Path 4: (1,2,3,5,2,3,5,2,3,5,2,3,5,2,3,6) with cost -200

:



1

2 3

4 5

50

200

6

40

150

Q. What is the shortest path from 1 to 6?A. There is none.

Q. What is the shortest path from 1 to 6 that does not contain any cycles?

Path 1: (1,2,3,6) with cost 300

Path 2: (1,2,3,5,2,3,6) with cost 50

Path 3: (1,2,3,5,2,3,5,2,3,6) with cost -100

Path 3: (1,2,3,5,2,3,5,2,3,5,2,3,6) with cost -250

Path 4: (1,2,3,5,2,3,5,2,3,5,2,3,5,2,3,6) with cost -400

:

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1

2 3

4 5

50

200

6

40

150



Path 1: (1,2,3,6) with cost 300

Path 2: (1,2,3,5,2,3,6) with cost 50

Path 3: (1,2,3,5,2,3,5,2,3,6) with cost -100

Path 3: (1,2,3,5,2,3,5,2,3,5,2,3,6) with cost -250

Path 4: (1,2,3,5,2,3,5,2,3,5,2,3,5,2,3,6) with cost -400

:

Q. Are there algorithms that can find the shortest path when there are negativecost arcs?

A. Yes, there are efficient algorithms when there are no negat ive cost cyc les.

Q. Are there algorithms that can find the shortest path that does not contain anycycles?



1

2 3

4 5

50

200

6

40

150



Path 1: (1,2,3,6) with cost 300

Path 2: (1,2,3,5,2,3,6) with cost 50

Path 3: (1,2,3,5,2,3,5,2,3,6) with cost -100

Path 3: (1,2,3,5,2,3,5,2,3,5,2,3,6) with cost -250

Path 4: (1,2,3,5,2,3,5,2,3,5,2,3,5,2,3,6) with cost -400

:

Q. Are there algorithms that can find the shortest path when there are negativecost arcs?

A. Yes, there are efficient algorithms when there are no negat ive cost cyc les.

Q. Are there algorithms that can find the shortest path that does not contain anycycles?

A. Yesbut they are not ef fic ient. This can be a very difficult problem.

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Practical variants of Dijkstrasalgorithm

Dijkstrasalgorithm on a road network



Bi-directional Dijkstrasalgorithm on a road network

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Dijkstrasalgorithm with A* search on a road network



Dijkstra Bi-directional Dijkstra

Bi-directional Dijkstra + A*

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Hierarchy in a road network



Bi-directional Dijkstrasalgorithm + A* + Hierarchy on a road network

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Lec 8 Network Models: Flows in Directed Graphs

Think of arcs in network as conduits carrying flow. Given a network with nodes Nand arcs A, for each arc e = (i,j)in Awe associate an amount xe(or xij)corresponding to the amount of flow through arc e.

A = {(1,2), (1,4), (2,3), (2,4), (4,3)}

X = (x1,2, x1,4, x2,3, x2,4, x4,3) = (4,3,2,1,6)

Flow

For a graph with nodes Nand arcs A, the vector Xis calleda f low.

Flow in Directed Graph

1 2

34

4

31

6

2

Flow vector



1 2

34

4

31

6

2

Net flow in/out of a node

The net flow in/out of a node is the total amount of flowentering/leaving the node.

Q. What is the net flow in tonode 3? 8

Q. What is the net flow in tonode 4? 4

Q. What is the net flow ou tof node 3? 0

Q. What is the net flow ou tof node 4? 6

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Bounds on f low

For each arc e, we may be given a lowerand/or upperbound leand uerespectively on the allowable flow along e.

A = {(1,2), (1,4), (2,3), (2,4), (4,3)}L = (l1,2, l1,4, l2,3, l2,4, l4,3) = (0,1,0,2,0)

U = (u1,2, u1,4, u2,3, u2,3, u4,3) = (5,3,4,3,5)

Bounds on flow

1 2

34

[0,5]

[1,3] [0,4]

[0,5]

Note: If a lower bound is not given then we can assume it is 0, i.e., we do not haveve flow.


Bounds on flow

1 2

34

[0,5]

[1,3] [0,4]

[0,5]

Feasib le flow from source to sink

A flow from a source node sto a sink node tis feasible if:

1. for all nodes except sand t, the

net flow into the node = net flow out of the node, and

2. the flow along each arc is within the stated bounds.

Flow balancerequirement

No leaks

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Bounds on flow

1 2

34

[0,5]

[1,3] [0,4]

[0,5]


1 2

34

4

31

6

2

Q. Is the following a feasible flow from node 1 to node 3 (i.e. s=1, t=3)?

A. No. Flow balance requirements are not satisfied at nodes 2 and 4, and the boundsare not satisfied on arcs (2,4) and (4,3).

Feasib le flow from source to sink

A flow from a source node sto a sink node tis feasible if:

1. for all nodes except sand t, the

net flow into the node = net flow out of the node, and

2. the flow along each arc is within the stated bounds.


Find threefeasible flows from node 1 to node 3:

Bounds on flow

1 2

34

[0,5]

[1,3] [0,4]

[0,5]

Flow 1

1 2

34

2

12

3

0

Flow 2

1 2

34

3

22

4

1

Flow 3

1 2

34

5

32

5

3

Flow balance Bounds

Flow balance Bounds

Flow balance Bounds

Note: if flow balance is satisfied, then the net flow out of s= net flow into t. Thisnet flow out of s(or into t) is the amount of flow that is pushed through from sto t.

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Find threefeasible flows from node 1 to node 3:

Bounds on flow

1 2

34

[0,5]

[1,3] [0,4]

[0,5]

Flow 1

1 2

34

2

12

3

0

Flow 2

1 2

34

3

22

4

1

Flow 3

1 2

34

5

32

5

3

Net flow out of s=net flow intot= 3



Flow from sto t= 3 Flow from sto t= 5 Flow from sto t= 8


The Maximum Flow Problem

The maximum f low prob lemis one of finding a feasibleflow from a node sto a node twhere the net flow leaving s(or entering t) is the maximum possible, i.e. finding themaximum amount of flow that can be pushed through from

sto t.

Flow 1

1 2

34

2

12

3

0

Flow 2

1 2

34

3

22

4

1

Flow 3

1 2

34

5

32

5

3




Flow from sto t= 3 Flow from sto t= 5 Flow from sto t= 8

Q. Is 8the maximum possible flow from sto t?Yes(why?)

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Lec 8 Network Models: Applications of Maximum Flow

Transporting Natural Gas through Pipelines

Western Australia Natural Gas Co. has developed a pipeline network to transportLiquefied Natural Gas (LNG) from exploration fields to refineries and other

locations. There are 10 pipelines in the network (see below). The maximum flowrate along each pipeline is given in hundreds of litres per hour.

What is the maximum possible flow rate of LNG from node 1 to node 8?

2 4

10

3

6

1

5 7

8

8

12 10

6 10

8 5

12

10

Maximum flow rate inL/ hour.

Math1800 Lec8 Bb

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