Dijkstra's Algorithm: A Greedy Approach to Finding Shortest Paths, Study notes of Algorithms and Programming

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Greedy Strategies:Dijkstra’s Algorithm

COMP157Nov^ 19,

Greedy

Technique

Constructs

a^ solution

to^ an^ optimization

problem

piece^ by

piece^ through

a^ sequence

of^ choices

that^ are:

-^ feasible –^ locally

optimal

-^ irrevocable For^ some^

problems,

this^ yields

an^ optimal

solution

for^ every

instance.

For^ most,^ this^ yields

non‐optimal,

but^ good,

fast^ approximations.

Single

Source

Shortest

Paths

Single^ Source

Shortest

Paths^

Problem

For^ a^ given

vertex

(the^ source)

in^ a^ weighted

connected

graph,^

find^ shortest

to^ all^ other

vertices

C DB 2 (^6) A^2

Example: find shortest paths 1 from source node A. G

AÆB: 2AÆBÆ

C: 4

AÆBÆ

D: 4

note: Floyd’s algorithm solved All Pairs Shortest Path

SSSP:

Dijkstra’s

Algorithm

Dijkstra’s

algorithm:

-^ Builds

a^ tree^ through

the^ graph, where^ each

node^ in

the^ tree

is^ a^ node

for^ which

the^ shortest

path^ is^

already^

determined.

-^ Each^

pass^ through

the^ algorithm

adds^ one

node^ to

the^ tree,

by^ finding

vertex^ u

with^ the

smallest

sum^ dv^

+^ w ( v , u

where^ v^ is^ a^ vertex

for^ which

shortest

path^ has

been^ already

found

on^ preceding

iterations

(such^ vertices

form^ a^ tree)

d^ is^ the v^

length^ of

the^ shortest

path^ form

source^

to^ v

w ( v , u )^ is

the^ length

(weight)

of^ edge

from^ v^ to

u

Dijkstra’s

Algorithm:

AnalogyA B 2 CD

6 2 2 1 C DB 2 (^6) A^2

(^1) G

A

C

B

E

4 D

3

2

5

6 7

4

ExampleA 3

C

B

E

4 D

3

2

5

6 7

(^47) (^57) A

C

B

E

4 D

3

2

5

6 7

4 5 3

7 10

9 7 3

Dijkstra:

Outline

of^ Algorithm

-^ Identify

u*^ =^ fringe

vertex

with^ min

cost

-^ move

u*^ from

fringe

to^ tree

-^ for^ each

fringe

vertex

u^ connected

to^ u*

if^ d^ +w(u,u)u^

<^ d^ :u^ p^ =^ uu^ d^ =^ d^ u^ u

+w(u*,u)

Example

Tree^

Remaining vertices a(-,0)^

b(a,3) c(-,

∞) d(a,7) e(-,

b(a,3)^

c(b,3+4) d(b,3+2) e(-,

d(b,5)^

c(b,7)^

e(d,5+4) c(b,7)^

e(d,9) e(d,9)

b a 4 c d

e 3 7

(^62 ) b a 4 c d

e 3 7

(^652 ) b a 4 c d

e 3 7

(^4) d 62 5 4 b^ a 4

e 3 7

6 c 2 5 d (^4) bc ad e 3 7

(^652 )

Notes

on^ Dijkstra’s

algorithm

-^ Doesn’t

work^

for^ graphs

with^

negative

weights • Applicable

to^ both

undirected

and^ directed

graphs • Efficiency^ –^ O(|V|

2 )^ for^ graphs

represented

by^ weight

matrix

and^ array

implementation

of^ priority

queue

-^ O(|E|log|V|)

for^ graphs

represented

by^ adj.^ lists

and^ min

‐heap^ implementation

of^ priority

queue

Graph

Algorithms

•^ DFS,^

BFS :^ visit

all^ nodes

-^ adj^ matrix:

2 Θ(|V |),

adj^ list:^

Θ(|V|+|E|)

-^ TSP^ (Brute

Force):^

shortest

tour

-^ O(n!) • Warshall

(DP):^ transitive

closure

Floyd^ (DP):

all^ pairs

shortest

path (^3) – Θ(|V

-^ Prim^

(greedy):

minimum

spanning

tree

Dijkstra

(greedy):

single^ source

shortest

path

-^ matrix/array:

2 Θ(|V |),

list/heap:

O(|E|log|V|)

-^ Kruskal

(greedy):

minimum

spanning

tree

-^ O(|E|log|E|)