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1 Wroclaw University, Sept 18, 2007
Approximation via Doubling
Marek Chrobak
University of California, Riverside
Joint work with Claire Kenyon-Mathieu
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2 Wroclaw University, Sept 18, 2007
Doubling method:
(for a minimization problem)
Choose d1 < d2 < d3 … (typically powers of 2)
For j = 1, 2, 3, …
Assume that the optimum is ≤ dj
Use this bound to construct a solution of cost ≤ C·dj
• Simple and effective (works for many problems, offline and online)• Typically not best possible ratios
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3 Wroclaw University, Sept 18, 2007
Outline:
1. Online bidding2. Cow-path3. Incremental medians (size approximation)4. Incremental medians (cost approximation)5. List scheduling on related machines6. Minimum latency tours7. Incremental clustering
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4 Wroclaw University, Sept 18, 2007
Outline:
1. Online bidding2. Cow-path3. Incremental medians (size approximation)4. Incremental medians (cost approximation)5. List scheduling on related machines6. Minimum latency tours7. Incremental clustering
![Page 5: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/5.jpg)
5 Wroclaw University, Sept 18, 2007
Online Bidding
1
2
5
12
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6 Wroclaw University, Sept 18, 2007
Online Bidding
1
2
5
12
20 bags of gunpowder
but… 6 bags could have been enough
so ratio = 20/6
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7 Wroclaw University, Sept 18, 2007
Online Bidding
Item for sale of value u (unknown to bidder)
Buyer bids d1,d2,d3, … until some dj ≥ u
Cost: d1 + d2 + … + dj Optimum = u
Competitive ratio
€
maxu, jd1 +d2 + ...+ d j
u: d j−1 < u ≤ d j
⎧ ⎨ ⎩
⎫ ⎬ ⎭
≅ max j
d1 +d2 + ...+ d jd j−1
⎧ ⎨ ⎩
⎫ ⎬ ⎭
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8 Wroclaw University, Sept 18, 2007
Deterministic Bidding - Upper Bound
If 2j-1 < u ≤ 2j, the ratio is
Doubling strategy: bid 1, 2, 4, … , 2i, …
€
21 +22 + ...+ 2 j
2 j−1≤
2 j+1
2 j−1= 4
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11 Wroclaw University, Sept 18, 2007
Online Bidding
Theorem:
The optimal competitive ratio for online bidding is:
• 4 in the deterministic case
• e 2.72 in the randomized case
Randomized e-ing strategy: choose uniformly random x [0,1), and bid e x , e x+1, e x+2 , e x+3 , …
[folklore] [Chrobak, Kenyon, Noga, Young, ‘06]
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12 Wroclaw University, Sept 18, 2007
Outline:
1. Online bidding2. Cow-path3. Incremental medians (size approximation)4. Incremental medians (cost approximation)5. List scheduling on related machines6. Minimum latency tours7. Incremental clustering
![Page 11: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/11.jpg)
13 Wroclaw University, Sept 18, 2007
d1 d2d3 d40 u
Cow-Path
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14 Wroclaw University, Sept 18, 2007
Analysis:
d1 d2d3dj+10 udj-1dj
For dj-1 < u ≤ dj+1 (j odd)
2 bidding ratio extra ratio 1
So the ratio = 2 bidding ratio + 1 = 9 for dj = 2j
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16 Wroclaw University, Sept 18, 2007
Theorem:The optimal competitive ratio for the cow-path problem is
• 9 in the deterministic case
• 4.59 in the randomized case
Solution of (r-1)ln(r-1) = r 2e+1
Connection to online bidding does not work in randomized case -- why?
[Gal ‘80] [Baeza-Yates, Culberson, Rawlins ‘93]
[Papadimitriou, Yannakakis ‘91] [Kao, Reif, Tate ‘94] …
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17 Wroclaw University, Sept 18, 2007
Outline:
1. Online bidding2. Cow-path3. Incremental medians (size approximation)4. Incremental medians (cost approximation)5. List scheduling on related machines6. Minimum latency tours7. Incremental clustering
![Page 15: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/15.jpg)
18 Wroclaw University, Sept 18, 2007
The k-Median Problem
X = set of facilitiesY = set of customers X Y : metric space with distance function dxy
For F X let cost(F) = y Y dyF
where dyF = minf F dyf
The k-Median Problem: Find a facility set F of size k for which cost(F) is minimized.
optimal F = Qk (the k-median)
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19 Wroclaw University, Sept 18, 2007
customer
facility (potential)
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20 Wroclaw University, Sept 18, 2007
k = 2 facilities
3
1
2 2
3
1
4
1
cost = 17
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21 Wroclaw University, Sept 18, 2007
k = 4 facilities
1
2 21
1
1
1
3
cost = 12
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22 Wroclaw University, Sept 18, 2007
Offline Case
• k-Median is NP-hard
• Offline approximations: given k, find F such that
• |F | ≤ k and cost(F) ≤ C·optk
C-cost-approximation Upper bound C = 3+ [Arya, Garg, Khandekar, Munagala, Pandit ‘01] C ≥ 1+2/e for polynomial algorithms (unless P = NP) [Jain, Mahdian, Saberi ‘02] • cost(F) ≤ optk and |F| ≤ S·k S-size-approximation S = Ω(logn) for polynomial algorithms (unless P = NP)
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23 Wroclaw University, Sept 18, 2007
Size-Competitive Incremental Medians
• k not known, authorizations for additional facilities arrive over time
• Algorithm produces a sequence of facility sets: F1 F2 … Fn
An algorithm is S-size-competitive if |Fk| ≤ S·k and cost(Fk) ≤ optk for all k.
Goal: small competitive ratio
![Page 21: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/21.jpg)
24 Wroclaw University, Sept 18, 2007
k = 1
2
4
53
2
25
3
cost = 26opt = 26
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25 Wroclaw University, Sept 18, 2007
k = 2
2
1
22
2
24
3
cost = 18 !!! opt = 17
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26 Wroclaw University, Sept 18, 2007
k = 2
2
1
22
2
14
1
cost = 15 opt = 17
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27 Wroclaw University, Sept 18, 2007
Size-Competitive Incremental Medians
Algorithm:
1. choose d1 < d2 < d3 … 2. Compute Q1, Q2, … (optimal medians)
3. F1 = Qd(1) // d(j) = dj
for k = 2, 3, … if k = di+1
Fk = Fk-1 Qd(i+1)
… not a polynomial time algorithm …
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28 Wroclaw University, Sept 18, 2007
Qd(1)
Qk = optimal k-median
Qd(4)
1 2 3 4 5 … 11 12 13 … 19 20 21 … 31 32 … d1 d2 d3 d4
k k k k
Qd(2)
Qd(3)
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29 Wroclaw University, Sept 18, 2007
Qd(2)
1 2 3 4 5 … 11 12 13 … 19 20 21 … 31 32 … d1 d2 d3 d4
kkk
Qd(1)
Qd(3)
Qd(4)
Qk = optimal k-median
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30 Wroclaw University, Sept 18, 2007
Qd(3)
Qd(2)
1 2 3 4 5 … 11 12 13 … 19 20 21 … 31 32 … d1 d2 d3 d4
k
Qd(1)
k k
Qd(4)
Qk = optimal k-median
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31 Wroclaw University, Sept 18, 2007
Qd(4)
1 2 3 4 5 … 11 12 13 … 19 20 21 … 31 32 … d1 d2 d3 d4
k
Qd(3)
Qd(2)
Qd(1)
Qk = optimal k-median
![Page 29: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/29.jpg)
32 Wroclaw University, Sept 18, 2007
Analysis:At step k, for dj-1 < k ≤ dj
• cost(Fk) ≤ cost(Qd(j)) = opt(dj) ≤ optk
• |Fk| ≤ d1+d2+ … + dj
So the ratio is
€
maxk, jd1 +d2 + ...+ d j
k: d j−1 < k ≤ d j
⎧ ⎨ ⎩
⎫ ⎬ ⎭
Same as online bidding
So we get ratio = 4 for dj = 2j
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33 Wroclaw University, Sept 18, 2007
Theorem:
The optimal size-competitive ratio for incremental medians is:
• 4 in the deterministic case
• e ≈ 2.72 in the randomized case
(Lower bound: prove that online bidding reduces to incremental medians)
[Chrobak, Kenyon, Noga, Young, ‘06]
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34 Wroclaw University, Sept 18, 2007
Outline:
1. Online bidding2. Cow-path3. Incremental medians (size approximation)4. Incremental medians (cost approximation)5. List scheduling on related machines6. Minimum latency tours7. Incremental clustering
![Page 32: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/32.jpg)
35 Wroclaw University, Sept 18, 2007
Cost-Competitive Incremental Medians
• k not known, authorizations for additional facilities arrive over time
• Algorithm produces a sequence of facility sets: F1 F2 … Fn
An algorithm is C-cost-competitive if |Fk|≤ k and cost(Fk) ≤ C·optk for all k.
Goal: small competitive ratio (in polynomial time, if possible …)
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36 Wroclaw University, Sept 18, 2007
Example: Star with m arms, w farmers per cluster
1
0
1
1
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37 Wroclaw University, Sept 18, 2007
Example: Star with m arms, w farmers per cluster
1
0
1
1
k = 1
cost = 2(m-1)w ≈ 2 opt cost
So C 2
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38 Wroclaw University, Sept 18, 2007
Example: Star with m arms, w farmers per cluster
1
0
1
1
k = 1
cost = w
opt cost = 0
2 3 4 … m
So C ∞
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39 Wroclaw University, Sept 18, 2007
Cost-Competitive Incremental Medians
[Mettu, Plaxton ‘00]:
• Lower bound of 2
• Upper bound C ≈ 30 (in polynomial time)
use doubling to improve to 8
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40 Wroclaw University, Sept 18, 2007
Idea: construct sequence backwards, at each step extracting next set from previous one
for k’ < k we want to show that Fk contains a cheap subset Fk’
customers
facil
ities
Fk
Fk’
Fk”
![Page 38: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/38.jpg)
41 Wroclaw University, Sept 18, 2007
Lemma: F, Q facility sets.
|F| = k
H |Q| = k’ < k
H = H(Q,F) = k’ facilities in F closest to the points in Q
Then
cost(H) cost(F) + 2·cost(Q)
![Page 39: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/39.jpg)
42 Wroclaw University, Sept 18, 2007
Proof: Choose
H Q
F
customer x
f
h q
f F : closest to xq Q : closest to xh H : closest to q (in F)
dxH ≤ dxh
≤ dxq + dqh
≤ dxq + dqf
≤ dxq + (dxf + dxq) = 2dxq + dxf
= 2dxQ + dxFSo
cost(H) ≤ 2·cost(Q) + cost(F)
![Page 40: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/40.jpg)
43 Wroclaw University, Sept 18, 2007
Algorithm:
1. Choose d1 < d2 < d3 < … Wlog. optn = cost(X) = 12. Choose p(1) > … > p(m) = 1
s.t. cost(Qp(i)) = optp(i) = di (For simplicity assume they exist)
3. Construct sets Fk for k = n, p(1), p(2),… Fn X (all facilities) Fp(i+1) H (Fp(i) , Qp(i+1) ) for i= 2,…,m
4. For p(i+1) < k < p(i) set Fk Fp(i+1)
(So for these k we have |Fk| ≤ k)
5. Output F1, F2,…, Fn
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45 Wroclaw University, Sept 18, 2007
Fp(i-2)
cost(Fp(i)) ≤ cost(Fp(i-1)) + 2·di
≤ cost(Fp(i-2)) + 2·di-1 + 2·di
≤ …≤ 2 · (d1 + d2 + …. + di)
Analysis:
Fp(i-1)
Fp(i)
Qp(i-1)
optimal Qp(i)
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46 Wroclaw University, Sept 18, 2007
Suppose p(j) < k ≤ p(j-1)
Then
• optk ≥ optp(j-1) = dj-1
• cost(Fk) = cost(Fp(j)) ≤ 2 · (d1 + d2 + …. + dj)
€
ratio ≤ 2 ⋅d1 +d2 + ...+ d j
d j−1
This is 2 (bidding ratio)
So we get ratio = 8 for dj = 2j
![Page 43: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/43.jpg)
47 Wroclaw University, Sept 18, 2007
Theorem:
Upper bounds for cost-competitive incremental medians:
• Deterministic• 8• 24+ in polynomial time
• Randomized • 2e• 6e + ≈ 16.31 + in polynomial time
[Lin, Nagarajan, Rajamaran, Williamson ‘06][Chrobak, Kenyon, Noga, Young ‘06]
Use (3+ )-approximate medians instead of optimal ones
![Page 44: 1 Wroclaw University, Sept 18, 2007 Approximation via Doubling Marek Chrobak University of California, Riverside Joint work with Claire Kenyon-Mathieu](https://reader035.vdocument.in/reader035/viewer/2022062408/56649f335503460f94c50af1/html5/thumbnails/44.jpg)
48 Wroclaw University, Sept 18, 2007
Current world records:
• 16+, deterministic polynomial time• 4e +, randomized polynomial time[Lin, Nagarajan, Rajamaran, Williamson ‘06]
Deterministic (not polynomial-time)
• Lower bound of 2.0013• Upper bound of 7.65[Chrobak, Hurand ‘07]