[PPT] - Single-Source Shortest Paths Chapter 24 1 CPTR 430 Algorithms PowerPoint Presentation

SLIDE 1

Single-Source Shortest Paths

Chapter 24

CPTR 430 Algorithms Single-Source Shortest Paths

1

SLIDE 2

Motivation

■ We plan a trip from Collegedale to Silicon Valley ■ Our map has distances between interstate highway interchanges

marked

■ We want to layout the shortest route ■ Proposal: Try all possible routes and select the shortest one

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 3

Try-All-Routes Algorithm

The try-all-routes approach is combinatorially explosive!

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 4

Shortest-Path Problem

■ G

✁

V

✂

E

✄

is a weighted, directed graph

■ Weight function w : E

☎ ✆

■ Weight of a path p

✝

v0

✂

v1

✂✞ ✞ ✞ ✂

vk

✟

is the sum of the weights of the edges that make up the path:

w

✁

p

✄

k

∑

i

✠

1

w

✁

vi

✡

1

✂

vi

✄

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 5

Shortest-Path Weight

The shortest-path weight from u to v is

δ

✁

u

✂

v

✄

min
w

✁

p

✄ ✁

u p

✂

v

✄

if there is a path from u to v

∞

therwise

A shortest path from vertex u to vertex v is any path with weight

w

✁

p

✄

δ

✁

u

✂

v

✄

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 6

Model

■ In our trip planner problem ❚ Each interchange is a vertex in the graph ❚ Each road connecting interchanges is an edge in the graph ❚ The distance between two interchanges represents an edge weight ■ Edge weights may be interpreted as metrics other than distance: ❚ time ❚ cost ❚ penalties

Any quantity that accumulates linearly along a path that we wish to minimize

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 7

Breadth-first Search

BFS (Chapter 22) is one type of shortest path problem

■ BFS is (usually) applied to unweighted, undirected graphs ■ Viewed another way, unweighted graphs are just weighted graphs with

unit-weight edges

■ The concepts of BFS can be applied to shortest paths in weighted,

directed graphs

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 8

Shortest Path Variants

■ Single-source SP ❚ Find the shortest path from a source vertex s

V to every vertex

v

V

■ Single-destination SP ❚ Find the shortest path to a destination vertex t

V from every vertex

v

V

❚ For undirected and symmetric directed graphs, just use the single-

source SP algorithm and reverse all the edges

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 9

Shortest Path Variants

■ Single-pair SP ❚ Find the shortest path from a source vertex u

V to a destination

vertex v

V

❚ Apply the single-source SP algorithm with u as the source vertex; the

shortest path to v is found in the solution

❚ No known algorithms for this problem run faster asymptotically than

the single-source algorithm applied to this problem

■ All-pairs SP ❚ Find the SP between u and v,

u

✂

v

V

❚ Can run single-source SP algorithm on every vertex ❚ A more efficient algorithm exists

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 10

Optimal Substructure

■ The shortest path between two vertices contains other shortest paths ■ SP problems thus exhibit an optimal substructure; recall ❚ Dynamic programming

(Chapter 15)

❚ Greedy algorithms

(Chapters 16, 23) Dijkstra’s SP algorithm is a greedy algorithm

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 11

Lemma 24.1

Subpaths of shortest paths are shortest paths

■ Given a weighted digraph G

✁

V

✂

E

✄

with weight function w : E

☎ ✆

■ Let p

✝

v0

✂

v1

✂✞ ✞ ✞ ✂

vk

✟

be a shortest path from vertex v0 to vertex vk

■ For any i

✂

j such that 0

i
j
k, let pi j
✝

vi

✂

vi

✁

1

✂✞ ✞ ✞ ✂

v j

✟

be a subpath of p from vertex vi to vertex v j

✂

pi j is a shortest path from vi to v j

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 12

Proof of Lemma 24.1

We’ll use a proof by contradiction

■ Let p be a shortest path from v0 to vk ■ Decompose path p into three segments: v0

p0i

✂

vi vi pi j

✂

v j v j p jk

✂

vk

w

✁

p

✄

w

✁

p0i

✄✂✁

w

✁

pi j

✄✂✁

w

✁

p jk

✄

■ Assume there exists a path p

✄

i j from vi to v j such that w

✁

p

✄

i j

✄✆☎

w

✁

pi j

✄

v0

p0i

✂

vi vi p

✄

i j

✂

v j v j p jk

✂

vk is a path

from v0 to vk with weight w

✁

p

✄

w

✁

p0i

✄✂✁

w

✁

p

✄

i j

✄✂✁

w

✁

p jk

✄ ☎

w

✁

p

✄

■ This contradicts the assumption that p is a shortest path

from v0 to vk

✝

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 13

Negative-Weight Edges

■ Some SP algorithms can handle graphs with negative edges ❚ E.g., Bellman-Ford algorithm

while others cannot

❚ E.g., Dijkstra’s algorithm ■ We will focus on graphs with non-negative edges ■ Most practical problems involve non-negative edges exclusivley

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 14

Cycles

■ Any path that contains a positive-weight cycle can be made shorter if

the cycle is removed from the path

■ Negative-weight cycles are pathological for SP problems, so they are

not allowed (Why?)

■ 0-weight cycles do not affect the path lengths, so we generally remove

0-weight cycles from paths, if necessary

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 15

Representing Shortest Paths

■ Use a representation similar to BFS (Chapter 22) ■

v
V, π
v

✁

is the predecessor of v on a shortest path from s (start vertex) to v

❚ π

v

✁

is either another vertex or null

❚ The predecessors can be traced back from a given vertex v to s to

determine a shortest path from s to v

❚ Given a vertex v with π

v

✁ ✂

null, the printPath() algorithm

(Chapter 22) will print a shortest path from s to v

■ The predecessor subgraph Gπ

✁

Vπ

✂

Eπ

✄

is induced by the π values

❚ Vπ

v
V

✁

π

v

✁ ✂

null

✄ ✄

s

✄

❚ Eπ

✁

π

v

✁ ✂

v

✄

E

✁

v

Vπ

☎

s

✄ ✄

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 16

Shortest-paths Tree

■ When the shortest-path algorithm is finished, Gπ is a tree that contains

a shortest path from the source vertex s to every other vertex reachable from s

■ A shortest-path tree, G

✄

✁

V

✄ ✂

E

✄ ✄

, is a directed subgraph of G, where

❚ V

✄

V and E

✄

E

❚ V

✄

is the set of vertices reachable from s

❚ G

✄

is a tree rooted at s

❚

v
V

✄

, the unique simple path from s to v is a shortest path from s to v in G

■ Shortest-path trees for given graph G and start vertex s are not unique ■ Shortest-path trees are like breadth-first trees (Chapter 22)

But here shortest paths are defi ned based on edge weights, not simply the number of edges in the path

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 17

Shortest-path Estimate

■ Each vertex v

V maintains a shortest-path estimate, d
v

✁

, an upper bound on the weight of a shortest path from s to v

■ Initially, ❚ d

v

✁

∞

✂

v
V

☎

s

✄

❚ d

s

✁

❚ π
v

✁

null

✂

v
V

■ The SP algorithm traverses the graph and updates the d

v

✁

and π

v

✁

values as necessary

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 18

Relaxation

■ An edge

✁

u

✂

v

✄

is relaxed when the SP algorithm determines if the shortest path found so far to v can be improved if

✁

u

✂

v

✄

is used in the shortest path

■

✁

u

✂

v

✄

results a shorter path

d
v

✁

and π

v

✁

are updated

■ d

v

✁

d
u

✁ ✁

w

✁

u

✂

v

✄

d
v

✁ ✁

d

u

✁ ✁

w

✁

u

✂

v

✄

, and

π

v

✁ ✁

u

■ Relaxation is the only way shortest-path estimates and predecessors

are updated

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 19

Relaxation Example

u v u v π[ ] = v x π[ ] = v x d u = [ ] 5 d u = [ ] 5 6 d v = [ ] 6 d v = [ ] ( , ) u v Relax π[ ] = v x d u = [ ] 5 u v u v 9 d v = [ ] d u = [ ] 5 7 d v = [ ] π[ ] = v u ( , ) u v Relax 2 2 2 2

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 20

Properties of Shortest Paths and Relaxation

■ Triangle inequality ❚

✁

u

✂

v

✄

E, δ

✁

s

✂

v

✄

δ

✁

s

✂

u

✄ ✁

w

✁

u

✂

v

✄

❚ This is a standard metric property ■ Upper-bound property ❚ It is always true that

v
V

✂

d

v

✁✁

δ

✁

s

✂

v

✄

❚ Once d

v

✁

is assigned δ

✁

s

✂

v

✄

, it never changes

■ No-path property ❚ If no path exists from s to v, then d

v

✁

δ

✁

s

✂

v

✄

∞ at all times

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 21

Properties of Shortest Paths and Relaxation

■ Convergence property ❚ If s

✂

u

☎

v is a shortest path in G for some u

✂

v

V, and if

d

u

✁

δ

✁

s

✂

u

✄

at any time before relaxing edge

✁

u

✂

v

✄

, then d

v

✁

δ

✁

s

✂

v

✄

at all times afterward

■ Path-relaxation property ❚ If p

✝

v0

✂

v1

✂✞ ✞ ✞ ✂

vk

✟

is a shortest path from s

v0 to vk, and the

edges of p are relaxed in the order

✁

v0

✂

v1

✄

,

✁

v1

✂

v2

✄

, . . . ,

✁

vk

✡

1

✂

vk

✄

, then d

vk

✁

δ

✁

s

✂

vk

✄

❚ The path-relaxation property holds regardless of other intermixed

relaxations

■ Predecessor-subgraph property ❚ d

v

✁

δ

✁

s

✂

v

✄

for all v

V, the predecessor subgraph is a shortest-

paths tree rooted at s

CPTR 430 Algorithms Single-Source Shortest Paths

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SLIDE 22

Dijkstra’a Algorithm

■ Solves the single-source SP problem on graphs ❚ that are weighted ❚ directed ❚ contain only non-negative edge weights ■ S is a set of vertices whose final SP weights from source s have already

been determined

■

CPTR 430 Algorithms Single-Source Shortest Paths