ADVANCED ALGORITHMS 2 LECTURE 10 ANNOUNCEMENTS Homework 2 due - - PowerPoint PPT Presentation

ADVANCED ALGORITHMS

LECTURE 10: SHORTEST PATH

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LECTURE 10

ANNOUNCEMENTS

▸ Homework 2 due next Friday ▸ Contacting the TAs: utah-algo-ta@googlegroups.com ▸ Notes for this class: http://jeffe.cs.illinois.edu/teaching/algorithms/

notes/21-sssp.pdf

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LECTURE 10

LAST CLASS

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▸ Local search and gradient descent ▸ Shortest path problem

if 2 approx formatching

LECTURE 10

GRAPH ALGORITHMS

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LECTURE 10

SHORTEST PATH PROBLEM

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Shortest path: given a (possibly directed) graph G = (V, E), and two vertices u, v, find the length of the shortest path from u to v

▸ What if the graph is not weighted? — breadth first search ▸ Negative weight edges?

all lengths 1

01mm

Problem is not

Fell if

u

grapmhtfsdes.ve

LECTURE 10

BFS + DISTANCE UPDATES?

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▸ Maintain array dist[w] for all vertices w ▸ Initialize using BFS ▸ For each edge ij, if dist[j] < dist[i] + length(i,j), update dist[j] ▸ Repeat until the dist[] values stop changing

OU

O W

List

Ew

i

• j

dhe w

w

LECTURE 10

OBSERVATIONS

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▸ At every point of the algorithm, dist[w] is an upper bound on true

shortest path length d(u, w)

▸ dist[w] is monotone

Claim: when the algorithm terminates, dist[w] = d(u, w) for all w

true because dist w

is the lengthof sp to w found

so far

LECTURE 10

PROOF

8 Suppose we did k iterations of the algorithm. Is there something nice to be said about dist[w] values?

Ow

• f

it

t

• Consider all

vertices w for a

which shortest path from

Ito

u

w

has lingulatek

a

ti

i

LECTURE 10

PROOF (CONTINUED)

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Prone claim byinduction

k

is

trivial

Suppose it holds for

k

now

connider htt

IrmasutEf for

all vertices

w

s't

hops in shortestpath Ek

O

we found the path

Consider any

w

s t

shortest pathfrom u has

htt

hops

go

ee

affirmedIttihad

w

THE

correctvalue ftp.estfwI

Optimum

path from

a

to w has

hops

Ek

Reach

Optimal sub structure

property of

shortest paths

LECTURE 10

RUNNING TIME BOUND (SHIMBEL’S ALGORITHM)

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Bellman

forT

algorithm

Previous

claim

aftern iterations

we havefoundthortest

paths

toll

w

iterations In

Time

for each iteration

0cm

Overall runtime

Ofm n It

also works when some edgewts are

we

LECTURE 10

BETTER THAN M*N?

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▸ Remember, this is an algorithm that also works when there are

negative edges

▸ Can we use positivity in a nicer way?

▸ Can we “fix” BFS? ▸ Recall: main idea of BFS — find all vertices at certain distance from u ▸ Problem: in BFS, vertex at distance 2 is nbr of vertex at distance 1

LECTURE 10

MAIN IDEA — DIJKSTRA’S ALGORITHM

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1958

Dijkstra

u

x

m

• 9K

LECTURE 10

EXTENDING BFS

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W

WI

Kes

win

Intuitiveidas

Grow a ball of a certainradiusaround

u

Add one vertex at

a time

Iqqjo

formalitalgorithmi

start

with

vertex

n

Add

vertex

that has

min distancetofu

Add

vertex

l

st

u

via

u E

it

i

E

tou

via E EE't

LECTURE 10

PRIORITY QUEUE

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Want

a data structure that

maintains 6h dist w

pair distfw

is

defined

quicklyobtain the

w thathas minimum

t.be dstme

distf

value

to

a

via the sitar

quickly

the

disowns

Oflog

n

Read

in

notes

Offmtn login

fm 7 nlognl

LECTURE 10

CORRECTNESS

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Claim

whenever

we add

a vertex to the set

ball

we

are

maintaining the distt

value is

indeed

the

shortest distance

to

n

i

LECTURE 10

RUNNING TIME

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LECTURE 10

RUNNING TIME

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LECTURE 10

SHORTEST PATHS — SUMMARY

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▸ Shimbel/Bellman-Ford algorithm — can handle negative weights, but

is slower

▸ Dijkstra’s algorithm — (m+n) log n ▸ Don’t know if log n is necessary!