Approximation Schemes for Euclidean k-Medians and Related Problems - - PowerPoint PPT Presentation

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Approximation Schemes for Euclidean k-Medians and Related Problems S. Arora, P. Raghavan, S. Rao STOC '98 A Nearly Linear-Time App. Scheme for the Euclidean k-median Problem ESA ' 99 S. Kolliopoulos, S. Rao k-medians Problem Given S={x i

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Approximation Schemes for Euclidean k-Medians and Related Problems

S. Arora, P. Raghavan, S. Rao

STOC '98

A Nearly Linear-Time App. Scheme for the Euclidean k-median Problem

S. Kolliopoulos, S. Rao

ESA '99

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k-medians Problem

Given

S={xi} be n points in metric space Rd
Positive integer k

Goal

Find M={mi} ∈ Rd which minimizes

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k-medians Problem

Paper 1 Paper 2

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k-medians Problem

Paper 1 Paper 2

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k-medians Problem

Paper 1 Paper 2

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Facility Location Problem

Given

S={xi} be n points in metric space Rd
Positive cost function c()

Goal

Find M={mi} ∈ S which minimizes

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Facility Location Problem

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Facility Location Problem

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Previous Results: Facility Location

O(logn) approx Hochbaum '82 3.16 approx Shmoys et al. '97 2.41 approx Guha and Khuller '98 1.74 approx Chudak '98 but no (1+ε) approx before this paper

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Previous Results: k-medians

(1+ε) approx using (1+1/ε)(1+lnn)k medians Lin and Vitter '92 2(1+ε) approx using (1+1/ε)k medians Lin and Vitter '92

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Results: Paper 1

In 2D, for the k-median problem, given any constant positive ε with probability 1-o(1) achieve solution at most (1+ε)OPT in time O(nO(1/ε)nk logn)

Proof of existence
Dynamic programming: bound table size

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Results: Paper 1

In 2D, for the facility problem, given any constant positive ε with probability 1-o(1) achieve solution at most (1+ε)OPT in time O(n1+O(1/ε)logn)

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Results: Paper 2

In 2D, for the k-median problem, given any constant positive ε with probability 1-o(1) achieve solution at most (1+ε)OPT in time O(2O(1+log(1/ε)/ε)nklogn)

Proof of existence
Dynamic programming: bound table size

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Approximation Schemes for Euclidean k-Medians and Related Problems

S. Arora, P. Raghavan, S. Rao

STOC '98

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Integer Point Coordinates

Given n points, minmax solution D is

2OPT for facility assignment

Optimal facility cost in [D/2,Dn]
Arora-TSP '98: Mapping each point to jD/n2

increases cost by maximum O(D/n)

Length of BBox(S)

– L=O(n4)

(<ε?)

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Terminology

Dissection vs quadtree
L = O(n4)
Number of nodes O(L2)
Minimum size of box = 1
Depth of tree log(L)
Dissection with (a,b) shift
Quadtree derived from (a,b)-shifted dissection

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A Result from Arora-TSP-'98

S={li} is a collection of line segments t(S,l) number of lines in S crossing l

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More Notations

BBox at level 0

its 4 children at level 1, …

Level of an edge

level of the corresponding box

Edge is in i level ⇒ in (i+1), (i+2), .., logL
maximal level

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Charging Scheme

R-charging For a maximal edge which is crossed g times by S, charge

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Charging Lemma

Expected total cost for top i-level edges is
Maximal level of grid line l be j

– Length L/2j – Charge t(S,l)/R L/2j – Probability 2j/L

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Charging Lemma

i ≤ logL
Choose R = logL/ε
Cost ≤ ε t(S,l)
Total ≤ O(ε cost(S))

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m-portal

m-regular portals for a shifted dissection
Total number of points ⇒ 4m

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m-portal

m-regular portals for a shifted dissection
Total number of points ⇒ 4m

3-portal

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m-light Solution

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m-light Solution

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Existence

m>1 and (a,b)∈U[0,L]
OPT ⇒ deflect to form m-portal

– for l sized square, cost of deflection O(l/m) – m-charging scheme

w.p. ½ or in expectation or w.h.p.

m-light solution exists for (a,b) shifted dissection with cost at most

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Existence

Solution is O( (1+logL/m)OPT )

L=O(n4)
Choose m to get O( (1+ε)OPT )

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Dynamic Programming: Sketch

Finds solution

within (1+1/4m) of m-light OPT

O((1+ε/4logn)(1+ε)OPT) = O((1+ε)OPT) Dynamic Programming

1. Nearest facility within (1+1/4m)
2. Nearest facilities similar for neighbors

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Dynamic Programming: Sketch

Given f and sub-boxes (children boxes Si-s) solve for current level

Table build for all choices f(≤k) and S
Table size O(nc)

⇒ worst case time for algorithm

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Summary

In 2D, for the k-median problem, given any constant positive ε with probability 1-o(1) achieve solution at most (1+ε)OPT in time O(nO(1/ε)nk logn)

Proof of existence
Dynamic programming: bound table size

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A Nearly Linear-Time App. Scheme for the Euclidean k-median Problem

S. Kolliopoulos, S. Rao

ESA '99

SLIDE 32

k-medians

– Given

S={xi} be n points in metric space Rd
Positive integer k

– Goal

Find M={mi} ∈ S in metric space Rd which minimizes

SLIDE 33

Paper 2

O(nO(1/ε)nk logn) reduced to O(O(1/ε)nk logn)

Reduce the number of portals from

O(logn/ε) ⇒ O(1+log(1/ε)/ε)

Different construction (no (a,b) shifting)

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Adaptive Dissection

Sub-rectangle

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Adaptive Dissection

Sub-rectangle

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Adaptive Dissection

Sub-rectangle

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Adaptive Dissection

Cut-rectangle (randomization)

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Adaptive Dissection

Cut-rectangle (randomization)

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Adaptive Dissection

Cut-rectangle (randomization)

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Lemma (just one of many)

Given two parallel cut-lines (due to cut-

rectangle) are L apart, the line segments has side length less than 3L.

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Structure Theorem

Error due to assignment using m-portal respecting paths is bounded by

Choose

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Extensions

d-dimension ⇒ md-1-portal
Facility location

– Same structure

Capacitated k-median

– Tweak dynamic programming

Few medians (small k)

– Guess position of facilities – Number of choices

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k-medians Problem

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