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all-pairs shortest paths in undirected graphs

Liam Roditty Uri Zwick

Tel Aviv University

approximateDynamic

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A graph

Dynamic Graph Problems

Initialize

InsertDelete

Query

n - number of verticesm - number of edges

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Shortest Paths Problems

Single-Source Shortest Paths (SSSP)

All-Pairs Shortest Paths (APSP)

“Distance Oracles”

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Compact datastructure

APSP

algorith

m

mn 1/k time

n 1+1/k space

Approximate Distance Oracles (TZ’01)

O(1) query time stretch 2k-1

Stretch-Space tradeoff is

essentially optimal!

n by ndistancematrix

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Dynamic Shortest Paths Algorithms

ProblemTotal

update timeAmortized*

update timeQuerytimeAuthors

Decremental SSSPO(mn)O(n)O(1)Even-Shiloach ’81

ProblemAmortizedupdate time

QuerytimeAuthors

Fully-dynamic APSPO*(n2)O(1)Demetrescu-Italiano ’03

(Thorup ’04)

Decremental (1+)-approximate

APSPO*(mn)O*(n)O(1)This talk

Fully-dynamic (1+)- approximate

APSPO*(m1/2n)O(m1/2)This talk

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Compact datastructure

mn1/k timen1+1/k space

Decremental Approximate Distance Oracles

O(1) query time stretch 2k-1

Static

mn timem+n1+1/k space

Decremental

dmn1/k timem+n1+1/k space

Decremental(up to distance d)

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Three ingredients

Random Sampling Let P be a set of vertices of size k.

Let S be a random subset of vertices of size (cn ln n)/k.Then with high probability PS .

Even-Shiloach ’81A decremental algorithm for maintaining a single-

source shortest path tree of depth d with a total running time of O(dm).

Thorup-Zwick ’01Static approximate distance oracles.

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Decremental SSSP [Even-Shiloach ’81]

Every edge is only examined once

per level!

Total complexityis O(dm).

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Random Sampling

Select each elementindependently with probability

n

k

The probability that agiven set of k elements

is not hit is

ln1(1 )

kk cc n

nk

p

lnc np

k

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A sequence of SamplesLet Si be a subset obtained by selecting each

vertex with probability (c ln n)/(2i)

If v is contained in a connected component with at least 2i vertices, then:

2i

pi(v)

Let pi(v) be the vertex of Si closest to v

δ(v,pi(v)) ≤ 2i

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Maintaining the closest centers

To compute pi(v) for every v:

1. Add a new source vertex si.

2. Connect it to all vertices of Si.

3. Maintain a SSSP tree from si.

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Maintaining distances from centers

From each center of Si maintain a SSSP tree of depth 2i+2

222

ii

n mnm

Complexity:

2i+2

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Answering a distance query

Suppose:12 ( , ) 2i iu v

( , ( )) ( ( ), )i iu p u p u v

( ( ), ) ( , )ip u u u v

Return:

≤

12 ( , ) (1 2 ) ( , )i u v u v

≤

u v

pi(u)

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Answering a distance query

( , ( )) ( ( ), )i iu p u p u v Return:

u v

pi(u)

where i is smallest such that v is in the tree of pi(u), i.e.,

2( ( ), ) 2iip u v

This value of i can be found using binary search.

Query time: O(log log n)Stretch: 1+

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Reporting long distances in O(1) time

u v

pr(u) pr(v)

Suppose: 1/ 2( , )u v n

Let: 12 logr n

Return: ( , ( )) ( ( ), ( )) ( ( ), )r r r ru p u p u p v p v v

Maintain a table of all Sr to Sr distances.

Size of table is O(n).

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Reporting short distances in O(1) time

Suppose: 1/ 2( , )u v n

Develop a decremental version of the static approximate distance oracles of TZ.

The 3-approximation can be refined into a (1+)-approximation in O(1) time.

dmn1/k timem+n1+1/k space

stretch 2k-1

For distancesup to d

Taked=n1/2

k=2

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Approximate Distance Oracles [TZ’01]A hierarchy of centers

A0V ; Ak ; Ai sample(Ai-1,n-1/k) ;

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Clusters A0=A1=A2=

1 1( ) { | ( , ) ( , )} ,i i iC w v V w v A v w A A

w

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Bunches (inverse clusters)

1

1

1

1

( ) { | ( , ) ( , )} ,

( ) { | ( , ) (

( ) (

, }

)

)i i

i

ii

i i

C w v V w v A v

if w

w

B v w A A w v A

B

A A

v v C w

v

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Bunches

1 1( ) { | ( , ) ( , )}i i ii

B v w A A w v A v

A0=A1=A2=

v

p1(v)

p2(v)

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Approximate Distance Oracles [TZ’01]

The data structure

For every vertex vV:

• The centers p1(v), p2(v),…, pk-1(v)

• A hash table holding B(v)

For every wV, we can check, in

constant time, whether wB(v), and if so,

what is (v,w).

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Lemma: E[|B(v)|] ≤ kn1/k

Proof: |B(v)Ai| is stochastically

dominated by a geometric random

variable with parameter p=n-1/k.

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Query answering algorithm Algorithm

distk(u,v)

wu , i0

while wB(v)

{ i i+1

(u,v) (v,u)

w pi(u) }

return (w,u)+

(w,v)

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Query answering algorithm

u v

w1=p1(v)A

1

w2=p2(u)A

2

w3=p3(v)A3

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u v

wi-1=pi-1(v)Ai-1

wi=pi(u)Ai

Analysis

(i-1)ii

(i+1)

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Open problems

• Faster dynamic SSSP algorithm?

• Exact dynamic APSP algorithm?

• Weighted graphs?

• Directed graphs?