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Sadna in AlgorithmsSadna in Algorithms

Haim Kaplan and Haim Kaplan and

Svetlana OlonetskySvetlana Olonetsky

Tel Aviv University, fall 07/08Tel Aviv University, fall 07/08

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Suggested topics

• Shortest path

• Maximum flow

• Min cost flow

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What do you have to do:

• Choose algorithms to implement

• Understand them well

• Choose the graphs you’d run on

• The structure/language of your program

What is the purpose ? Why do you do it ?

There should be a goal, something that you do not know in advance, interesting graph classes

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Inspiration

There has been a lot of work recently on so called “algorithm engineering”:

Three yearly conferences:

1) WAE

2) ALENEX

3) ESA (track B)

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Some examples

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Distances and Shortest Distances and Shortest PathsPaths

u

v

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There are many There are many variationsvariations

graph is undirected/directed

weights: negative/nonnegative real/integer

given pair(s)/single source/all pairs

one shot/build a data structure or spaner

exact/approximate

randomized/deterministic

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Dijkstra’s shortest path algorithm

Let G = (V,E) be a weighted (weights are non-negative) undirected/directed graph, let s V. Want to find the distance (length of the shortest path), d(s,v) from s to every other vertex.

s

3

3 2

3

2

1

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Dijkstra: Maintain an upper bound d(v) on d(s,v).

Every vertex is either scanned, labeled, or unlabeled.

Initially: d(s) = 0 and d(v) = for every v s.s is labeled and all others are unlabeled.

• Pick a labeled vertex with d(v) minimum. Make v scanned.

For every edge (v,w) if d(v) + w(v,w) < d(w) then

1) d(w) := d(v) + w(v,w)

2) label w if it is not labeled already

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Dijkstra’s shortest path algorithm (implementation)

Maintain the labeled vertices in a heap, using d(v) as the key of v.

We perform n delete-min operations and n insert operations on the heap. O(n log(n))

For each edge we may perform a decrease-key.

With regular heaps O(m log (n)).

But if you can do decrease-key in O(1) time then you can implement Dijkstra’s algorithm to run in O(n log(n) + m) time !

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Dial’s implementation

• Assume that weights are small integers

• Maintain vertices in an array according to their distance label

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An Example

1

2

3

4

5

6

2

4

2 1

3

4

2

3

2

Initialize distance labels

0

Initialize array

1

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Scan vertex 1

2

3

4

5

6

24

2 1

3

4

2

3

2

2

0

1

1 2

4

3

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Scan vertex 2

2

3

4

5

6

24

2 1

3

4

2

3

2

2

0

1

1 2

4

3

2

3

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Scan vertex 2

2

3

4

5

6

24

2 1

3

4

2

3

2

2

0

1

1 2

4

3

2

34

5

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Analysis

• Let C-1 be the maximum length of an arc

• What should be the length of the array ?

• What is the running time ?

• How can you do better ?

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2-level buckets0 C-1

0, C

.........

,C C C , 2C C

.........

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2-level buckets0 C-1

0, C

.........

,C C C , 2C C

.........

0 1C

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2-level buckets0 C-1

0, C

.........

,C C C , 2C C

.........

0 1C

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Analysis

• What is the running time ?

• How can you do better ?

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Goldberg et al

• Has done extensive experimental work on Dijkstra’s algorithm

• It may be interesting to repeat some of them ? Run on different graph classes, check new ideas, etc

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k simple shortest paths

a

b

e

f

t

s

c

d

g

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Maintain a trie that described the paths found so far

a

b

e

f

t

s

c

d

g t

s

sgt

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The heap

a

b

e

f

t

s

c

d

g t

s

sgt

sabeft sgft

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a

b

e

f

t

s

c

d

g t

ssg

sabeft

g

t

gt gft

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a

b

e

f

t

s

c

d

g t

ssg

g

t

gt gftt

sabeft

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a

b

e

f

t

s

c

d

g t

ssg

g

t

gt gftt

sabeft

saceft

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a

b

e

f

t

s

c

d

g t

ssg

g

t

gt gft

tt

sa

abeft

aaceft

sabdft sacdft

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a

b

e

f

t

s

c

d

g t

ssg

g

t

gt gft

t

t

saa

aceft

sacdft

t

b

ab

bdftbeft

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a

b

e

f

t

s

c

d

g t

ssg

g

t

gt gft

t

t

saa

aceft

sacdft

t

b

ab

bdftbeft

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Point to point shortest paths

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Running Dijkstra...

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Bidirectional Dijkstra

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• Define potential function π(v) and modify lengths:– ℓ(v,w) = c(v,w) − π(v) + π(w)– ℓ(v,w): reduced cost of arc (v,w).

• All s-t paths change by same amount: π(t) − π(s).

• A* search: – Equivalent to Dijkstra on the modified graph: correct if ℓ(v,w) ≥ 0 (π is feasible). • Take π(v) to be lower bounds on dist(v, t)

A* search

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Bidirectional A*

• Could be made to work

• Need to be careful: lower bounds must be consistent

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Where do we get lower bounds ?

• Use landmarks

dist(v,w) ≥ dist(A,w) − dist(A,v)

dist(v,w) ≥ dist(v,A) − dist(w,A)

A

v w

A

v w

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Query with landmarks

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Landmark selection

• There has been research on it:– selection at preprocessing – selection for the query

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Reach

• Reach of v with respect to P:reach(v, P) = min{ dist(s, v) , dist(v, t) }

• Reach of v with respect to the whole graph: reach(v) = maxP {reach(v, P)}, over all shortest paths P that contain v [Gutman’04].

• Intuition:– vertices on highways have high reach;– vertices on local roads have low reach.

v ts

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How do you use reaches

• While scanning an edge (v,w):– If reach(w) < min{d(s, v) + ℓ(v,w), LB(w, t)}, then w can be pruned.

ts

w

• The lower bound is natural if the search is bidirectional

v

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Shortcuts• Consider a sequence of vertices of degree two on the path below:

– they all have high reach;

ss t

1000 1000

1000 1010 1020 1030 1040 1030 1020 1010 1000

10 1010 10 1010 10 10

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Shortcuts

ss t

1000 1000

1000 1010 1020 1030 1040 1030 1020 1010 1000

10 1010 10 1010 10 10

80

Add a shortcut:– single edge bypassing a path (with same length).– assume ties are broken by taking path with fewer nodes.

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Shortcuts

ss t

1000 1000

1000 60 50 40 30 40 50 60 1000

10 1010 10 1010 10 10

80

Decrease reaches

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Shortcuts

ss t

1000 1000

1000 60 50 40 30 40 50 60 1000

10 1010 10 1010 10 10

80

Can add more nested shortcuts

4040

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Shortcuts

ss t

1000 1000

1000 20 10 20 30 20 10 20 1000

10 1010 10 1010 10 10

80

Can add more nested shortcuts

4040

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Reaches + shortcuts

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Combine reaches+shortcuts and landmarks

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Spanners

Let G be a weighted undirected graph.

A subgraph H of G is a t-spanner of G

iff u,vG,

H(u,v) t G(u,v) .

Awerbuch ’85Peleg-Schäffer ‘89

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Example

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t-spanner

• Computing a t-spanner of smallest possible size is NP-complete.

• For a graph on n vertices, how large can a t-spanner be ?

u v 2-spanner may require Ω(n2) edges

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Theorem

For every k>1, every weighted

undirected graph on n vertices has

a (2k-1)-spanner with at most

n1+1/k edges.

Tight for k=1,2,3,5. Conjectured to be tight for

any k equivalent to a girth conjecture of

Erdös.

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Proof/Algorithm: Consider the edges in non-decreasing

order of weight.

If an edge closes a cycle of length ≤ 2k

discard it, otherwise add it.

The resulting graph is a (2k-1)-spanner

and it does not contain a cycle of length

≤ 2k.

Such graph has at most n1+1/k edges.[Althöfer, Das, Dobkin, Joseph, Soares ‘93]

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If |cycle|2k, then red edge can be removed.

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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k=2, Baswana & Sen

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Analysis

• What is the average # of edges in the spanner ?

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Correctness

• Why is this a 3-spanner ?

• Show:

Where the weight of the red edges is smaller than the weight of the green one

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Correctness

• This clearly holds if the edge became green at the first stage:

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Correctness

• Otherwise, an edge (u,v’) must exist

u

v

x

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Correctness

• Otherwise, an edge (u,v’) must exist

u

v

x

v’

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Implementation

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(a,b)-Spanners

Let G be an unweighted undirected graph.

A subgraph H of G is an (a,b)-spanner of G

iff u,vG,

H(u,v) a G(u,v) + b .

(Dor-Halperin-Zwick ’96, a=1)

Peleg-Elkin ’01Baswana, Kavitha, Mehlhorn, Pettie ‘05Thurup Zwick ‘06

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• Find interesting graph classes to test spanners on ?

• Should be relatively dense…

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Other SP problems

• Distance oracles• All pairs shortest paths

• geometric shortest paths/spanners

• time dependent shortest paths