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Assignment 9 Solutions: Graph Algorithms (DFS, BFS, Prim's, Kruskal's, Dijkstra's), Assignments of Algorithms and Programming

The solutions to assignment 9 of a computer science course focusing on various graph algorithms, including depth-first, breadth-first, prim's, kruskal's, and dijkstra's algorithms. The solutions include recursive call stacks, priority queues, and sorted lists.

Typology: Assignments

Pre 2010

Uploaded on 08/18/2009

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CScD-320 Assignment 9 Solution Set, Page 1 Spring 2009
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Recursive depth-first
A(1) B(2) C(3) D(4) E(6) F(7) G(5)
From A recursive call stack shown
A to B AB
B to C AB BC
C to D AB BC CD
D to G AB BC CD DG
G to E AB BC CD DB GE
E to F AB BC CD DB GE EF
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Breadth-first
A(1) B(2) C(3) D(4) E(5) F(7) G(6)
From A (queue) AB AC AD AE AG
A to B AC AD AE AG BC BE . . .
A to C AD AE AG BC BE CD CF . . .
A to D AE AG BC BE CD CF . . .
A to E AG BC BE CD CF . . .
A to G BC BE CD CF . . .
Drop BC BE CD CF . . .
Drop BE CD CF . . .
Drop CD CF . . .
C to F . . . (discards)
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Prim's algorithm Note that using a sorted list may select a different set of edges.
A(1) B(4) C(6) D(7) E(2) F(5) G(3)
From A: AE(1) | AG(1) AD(2) | AC(2) AB(2)
A to E (1): AG(1) | EB(1) EF(1) | AC(2) AB(2) AD(2) EG(2) |
A to G (1): EB(1) | GF(1) EF(1) | EG(2) AB(2) AD(2) GD(1) | AC(2)
E to B (1): GF(1) | BC(1) EF(1) | AC(2) AB(2) AD(2) GD(1) | EG(2)
G to F (1): BC(1) | FC(1) EF(1) | EG(2) AB(2) AD(2) GD(1) | AC(2)
B to C (1): FC(1) | CD(1) EF(1) | AC(2) AB(2) AD(2) GD(1) | EG(2)
Discard FC: CD(1) | EG(2) EF(1) | AC(2) AB(2) AD(2) GD(1)
C to D (1): GD(1) | EG(2) EF(1) | AC(2) AB(2) AD(2)
Six edges in use; terminate run
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Kruskal's Algorithm for MST:
Accept AE1 AG1 BC1 BE1 CD1 CF1 --- this bring us to 6 edges
Printed 29-Nov-20 16:11
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Download Assignment 9 Solutions: Graph Algorithms (DFS, BFS, Prim's, Kruskal's, Dijkstra's) and more Assignments Algorithms and Programming in PDF only on Docsity!

CScD-320 Assignment 9 Solution Set, Page 1 Spring 2009

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Recursive depth-first

A(1) B(2) C(3) D(4) E(6) F(7) G(5)

From A recursive call stack shown

A to B AB

B to C AB BC

C to D AB BC CD

D to G AB BC CD DG

G to E AB BC CD DB GE

E to F AB BC CD DB GE EF

A

B C

D

E

F

G

Breadth-first

A(1) B(2) C(3) D(4) E(5) F(7) G(6)

From A (queue) AB AC AD AE AG

A to B AC AD AE AG BC BE...

A to C AD AE AG BC BE CD CF...

A to D AE AG BC BE CD CF...

A to E AG BC BE CD CF...

A to G BC BE CD CF...

Drop BC BE CD CF...

Drop BE CD CF...

Drop CD CF...

C to F_... (discards)_

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Prim's algorithm — Note that using a sorted list may select a different set of edges.

A(1) B(4) C(6) D(7) E(2) F(5) G(3)

From A: AE(1) | AG(1) AD(2) | AC(2) AB(2)

A to E (1): AG(1) | EB(1) EF(1) | AC(2) AB(2) AD(2) EG(2) |

A to G (1): EB(1) | GF(1) EF(1) | EG(2) AB(2) AD(2) GD(1) | AC(2)

E to B (1): GF(1) | BC(1) EF(1) | AC(2) AB(2) AD(2) GD(1) | EG(2)

G to F (1): BC(1) | FC(1) EF(1) | EG(2) AB(2) AD(2) GD(1) | AC(2)

B to C (1): FC(1) | CD(1) EF(1) | AC(2) AB(2) AD(2) GD(1) | EG(2)

Discard FC: CD(1) | EG(2) EF(1) | AC(2) AB(2) AD(2) GD(1)

C to D (1): GD(1) | EG(2) EF(1) | AC(2) AB(2) AD(2)

Six edges in use; terminate run

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B C

D

E

F

G

Kruskal's Algorithm for MST:

Accept AE1 AG1 BC1 BE1 CD1 CF1 --- this bring us to 6 edges

Printed 29-Nov-20 16:

CScD-320 Assignment 9 Solution Set, Page 2 Spring 2009

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Prim’s Algorithm --- sorted list priority queue, insertion from the

right, use of distance list.

Enter at A, initial priority queue: AE1 AG1 AB2 AC2 AD

Distance list: A: 0 ; B: -> 2 ; C: -> 2 ; D: -> 2 ; E: -> 1 ; F: ->  ;

G: -> 1

Edge AE1: enter EB1 after AG1, enter EF1 after EB1, discard EG2 (worse)

AG1 EB1 EF1 AB2 AC2 AD

Distance list: A: 0 ; B: -> 1 ; C: -> 2 ; D: -> 2 ; E: -> 1 ; F: -> 1 ;

G: -> 1

Edge AG1: enter GD1 after EF1, discard GF1 (same)

EB1 EF1 GD1 AB2 AC2 AD

Distance list: A: 0 ; B: -> 1 ; C: -> 2 ; D: -> 1 ; E: -> 1 ; F: -> 1 ;

G: -> 1

Edge EB1: enter BC1 after GD

EF1 GD1 BC1 AB2 AC2 AD

Distance list: A: 0 ; B: -> 1 ; C: -> 1 ; D: -> 1 ; E: -> 1 ; F: -> 1 ;

G: -> 1 — no more changes

Edge EF1: discard FC1 (same)

Edge GD1: discard DC1 (same)

Edge BC1: final edge

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Dijkstra's algorithm

Using a sorted list may select a different set of edges, depending on the list protocol.

A(1) B(7) C(5) D(6) E(2) F(4) G(3)

From A: AE(1) | AG(1) AD(2) | AC(2) AB(2)

A to E (1): AG(1) | AB(2) AD(2) | AC(2) EF(2) omit EB(2=2)

A to G (1): EF(2) | AB(2) AD(2) | AC(2) omit GD(2=2) GF(2=2)

E to F (2): AC(2) | AB(2) AD(2) |

A to C (2): AD(2) | AB(2) omit CD(3>2)

A to D (2): AB(2) |

A to B (2):

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Dijkstra’s Algorithm --- sorted list priority queue, insert from the right.

Enter at A, initial priority queue: AE1 AG1 AB2 AC2 AD

Distance list: A: 0 ; B: -> 2 ; C: -> 2 ; D: -> 2 ; E: -> 1 ; F: ->  ;

G: -> 1

Edge AE1: enter EF2 after AD2, discard EB2 (same), discard EG3 (worse)

AG1 AB2 AC2 AD2 EF

Distance list: A: 0 ; B: -> 2 ; C: -> 2 ; D: -> 2 ; E: -> 1 ; F: -> 2 ;

G: -> 1 — no more changes

Edge AG1: discard GD2 (same), discard GF2 (same)

AB2 AC2 AD2 EF

Edge AB2: discard BC3 (worse)

Edge AC2: discard CD3 (worse), discard CF3 (worse)

Edge AD2: all adjacent have been visited

Edge EF2: final edge

Printed 29-Nov-20 16: