Composable Core-sets for Diversity and Coverage Maximization Piotr - - PowerPoint PPT Presentation

Composable Core-sets for Diversity and Coverage Maximization

Piotr Indyk (MIT) Sepideh Mahabadi (MIT) Mohammad Mahdian (Google) Vahab S. Mirrokni (Google)

Core-Set Definition

• Setup

– Set of 𝑜 points 𝑸 in 𝑒-dimensional space – Optimize a function 𝑔

Core-Set Definition

• Setup

– Set of 𝑜 points 𝑸 in 𝑒-dimensional space – Optimize a function 𝑔

• 𝒅-Core-set: Small subset of points S ⊂ 𝑄

which suffices to 𝑑-approximate the optimal solution

• Maximization:

𝑔

𝑝𝑝𝑝 𝑄

𝑑

≤ 𝑔

𝑝𝑝𝑝 𝑇 ≤ 𝑔 𝑝𝑝𝑝(𝑄)

Core-Set Definition

• Setup

– Set of 𝑜 points 𝑸 in 𝑒-dimensional space – Optimize a function 𝑔

• 𝒅-Core-set: Small subset of points S ⊂ 𝑄

which suffices to 𝑑-approximate the optimal solution

• Maximization:

𝑔

𝑝𝑝𝑝 𝑄

𝑑

≤ 𝑔

𝑝𝑝𝑝 𝑇 ≤ 𝑔 𝑝𝑝𝑝(𝑄)

• Example

– Optimization Function: Distance of the two farthest points

Core-Set Definition

• Setup

– Set of 𝑜 points 𝑸 in 𝑒-dimensional space – Optimize a function 𝑔

• 𝒅-Core-set: Small subset of points S ⊂ 𝑄

which suffices to 𝑑-approximate the optimal solution

• Maximization:

𝑔

𝑝𝑝𝑝 𝑄

𝑑

≤ 𝑔

𝑝𝑝𝑝 𝑇 ≤ 𝑔 𝑝𝑝𝑝(𝑄)

• Example

– Optimization Function: Distance of the two farthest points – 1-Core-set: Points on the convex hull.

Composable Core-sets

• Setup

– 𝑸𝟐, 𝑸𝟑, … , 𝑸𝒏 are set of points in 𝑒-dimensional space – Optimize a function 𝑔 over their union 𝑸.

Composable Core-sets

• Setup

– 𝑸𝟐, 𝑸𝟑, … , 𝑸𝒏 are set of points in 𝑒-dimensional space – Optimize a function 𝑔 over their union 𝑸.

• 𝒅-Composable Core-sets: Subsets of

points S1 ⊂ 𝑄

1, S2 ⊂ 𝑄2, … , Sm ⊂ 𝑄 𝑛

points such that the solution of the union

• f the core-sets approximates the solution
• f the point sets.
• Maximization :

1 𝑑 𝑔

𝑝𝑝𝑝 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ≤ 𝑔

• pt S1 ∪ ⋯ ∪ 𝑇𝑛 ≤ 𝑔

𝑝𝑝𝑝(𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛)

Composable Core-sets

• Setup

– 𝑸𝟐, 𝑸𝟑, … , 𝑸𝒏 are set of points in 𝑒-dimensional space – Optimize a function 𝑔 over their union 𝑸.

• 𝒅-Composable Core-sets: Subsets of

points S1 ⊂ 𝑄

1, S2 ⊂ 𝑄2, … , Sm ⊂ 𝑄 𝑛

points such that the solution of the union

• f the core-sets approximates the solution
• f the point sets.
• Maximization :

1 𝑑 𝑔

𝑝𝑝𝑝 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ≤ 𝑔

• pt S1 ∪ ⋯ ∪ 𝑇𝑛 ≤ 𝑔

𝑝𝑝𝑝(𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛)

• Example: two farthest points

Composable Core-sets

• Setup

– 𝑸𝟐, 𝑸𝟑, … , 𝑸𝒏 are set of points in 𝑒-dimensional space – Optimize a function 𝑔 over their union 𝑸.

• 𝒅-Composable Core-sets: Subsets of

points S1 ⊂ 𝑄

1, S2 ⊂ 𝑄2, … , Sm ⊂ 𝑄 𝑛

points such that the solution of the union

• f the core-sets approximates the solution
• f the point sets.
• Maximization :

1 𝑑 𝑔

𝑝𝑝𝑝 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ≤ 𝑔

• pt S1 ∪ ⋯ ∪ 𝑇𝑛 ≤ 𝑔

𝑝𝑝𝑝(𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛)

• Example: two farthest points

Composable Core-sets

• Setup

– 𝑸𝟐, 𝑸𝟑, … , 𝑸𝒏 are set of points in 𝑒-dimensional space – Optimize a function 𝑔 over their union 𝑸.

• 𝒅-Composable Core-sets: Subsets of

points S1 ⊂ 𝑄

1, S2 ⊂ 𝑄2, … , Sm ⊂ 𝑄 𝑛

points such that the solution of the union

• f the core-sets approximates the solution
• f the point sets.
• Maximization :

1 𝑑 𝑔

𝑝𝑝𝑝 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ≤ 𝑔

• pt S1 ∪ ⋯ ∪ 𝑇𝑛 ≤ 𝑔

𝑝𝑝𝑝(𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛)

• Example: two farthest points

Applications – Streaming Computation

• Streaming Computation:

– Processing sequence of 𝑜 data elements “on the fly” – limited Storage

Applications – Streaming Computation

• Streaming Computation:

– Processing sequence of 𝑜 data elements “on the fly” – limited Storage

• 𝒅-Composable Core-set of size 𝒍

– Chunks of size 𝑜𝑜 , thus number of chunks = 𝑜/𝑜

Applications – Streaming Computation

• Streaming Computation:

– Processing sequence of 𝑜 data elements “on the fly” – limited Storage

• 𝒅-Composable Core-set of size 𝒍

– Chunks of size 𝑜𝑜 , thus number of chunks = 𝑜/𝑜 – Core-set for each chunk – Total Space: 𝑜 𝑜/𝑜 + 𝑜𝑜 = 𝑃( 𝑜𝑜) – Approximation Factor: 𝑑

Applications – Distributed Systems

• Streaming Computation
• Distributed System:

– Each machine holds a block of data. – A composable core-set is computed and sent to the server

Applications – Distributed Systems

• Streaming Computation
• Distributed System:

– Each machine holds a block of data. – A composable core-set is computed and sent to the server

• Map-Reduce Model:
• One round of Map-Reduce
• 𝑜/𝑜 mappers each getting 𝑜𝑜 points
• Mapper computes a composable core-set of size 𝑜
• Will be passed to a single reducer

Applications – Similarity Search

• Streaming Computation
• Distributed System
• Similarity Search: Small output size

Applications – Similarity Search

• Streaming Computation
• Distributed System
• Similarity Search: Small output size
• Good to have result from each

cluster: relevant and diverse

Applications – Similarity Search

• Streaming Computation
• Distributed System
• Similarity Search: Small output size
• Good to have result from each

cluster: relevant and diverse

• Diverse Near Neighbor Problem

[Abbar, Amer-Yahia, Indyk, Mahabadi

WWW’13] [Abbar, Amer-Yahia, Indyk, Mahabadi, Varadarajan, SoCG’13]

Applications – Similarity Search

• Streaming Computation
• Distributed System
• Similarity Search: Small output size
• Good to have result from each

cluster: relevant and diverse

• Diverse Near Neighbor Problem

[Abbar, Amer-Yahia, Indyk, Mahabadi

WWW’13] [Abbar, Amer-Yahia, Indyk, Mahabadi, Varadarajan, SoCG’13]

– uses Locality Sensitive Hashing (LSH) and Composable Core- sets techniques.

Diversity Maximization Problem

• A set of 𝑜 points 𝑄 in metric space

(Δ, 𝑒𝑒𝑒𝑒)

• Optimization Problem:

– Find a subset of 𝑜 points 𝑇 which maximizes Diversity

k=4 n = 6

Diversity Maximization Problem

• A set of 𝑜 points 𝑄 in metric space

(Δ, 𝑒𝑒𝑒𝑒)

• Optimization Problem:

– Find a subset of 𝑜 points 𝑇 which maximizes Diversity

• Diversity:

– Minimum pairwise distance (Remote Edge)

k=4 n = 6

Diversity Maximization Problem

• A set of 𝑜 points 𝑄 in metric space

(Δ, 𝑒𝑒𝑒𝑒)

• Optimization Problem:

– Find a subset of 𝑜 points 𝑇 which maximizes Diversity

• Diversity:

– Minimum pairwise distance (Remote Edge) – Sum of Pairwise distances (Remote Clique)

k=4 n = 6

Diversity Maximization Problem

• A set of 𝑜 points 𝑄 in metric space

(Δ, 𝑒𝑒𝑒𝑒)

• Optimization Problem:

– Find a subset of 𝑜 points 𝑇 which maximizes Diversity

• Diversity:

– Minimum pairwise distance (Remote Edge) – Sum of Pairwise distances (Remote Clique)

• Long list of variants [Chandra and

Halldorsson ‘01]

k=4 n = 6

Diversity Functions

Diversity function over a set 𝑇 of 𝑜 point Description Remote-edge Minimum Pairwise Distance: min

𝑝,𝑟∈𝑇 𝑒𝑒𝑒𝑒(𝑞, 𝑟)

Remote-clique Sum of Pairwise Distances : ∑ 𝑒𝑒𝑒𝑒(𝑞, 𝑟)

𝑝,𝑟∈𝑇

Remote-tree Weight of Minimum Spanning Tree (MST) of the set 𝑇 Remote-cycle Weight of minimum Traveling Salesman Tour (TSP) of the set 𝑇 Remote-star Weight of minimum star: min

𝑝∈𝑇 ∑

𝑒𝑒𝑒𝑒(𝑞, 𝑟)

𝑟∈𝑇

Remote-Pseudoforest Sum of the distance of each point to its nearest neighbor ∑ min

𝑟∈𝑇 𝑒𝑒𝑒𝑒(𝑞, 𝑟) 𝑝∈𝑇

Remote-Matching Weight of minimum perfect Matching of the set 𝑇 Max-Coverage How well the points cover each coordinate max

𝑝∈𝑇 𝑞𝑗 𝑒 𝑗=1

Our Results

Diversity function Offline ApproxFactor Composable Coreset Approx factor [Our Results] Remote-edge Minimum Pairwise Distance 𝑃(1) [Tmair 91][White 91] [Ravi et al 94] 𝑷(𝟐) Remote-clique Sum of Pairwise Distances 𝑃(1) [Hassin et al 97] 𝑷(𝟐) Remote-tree Weight of MST 𝑃(1) [Halldorsson et al 99] 𝑷(𝟐) Remote-cycle Weight of minimum TSP 𝑃(1) [Halldorsson et al 99] 𝑷(𝟐) Remote-star Weight of minimum star 𝑃(1) [Chandra&Halldorsson 01] 𝑷(𝟐) Remote-Pseudoforest Sum of the distance of each point to its nearest neighbor 𝑃(log 𝑜) [Chandra&Halldorsson 01] 𝑷(𝐦𝐦𝐦 𝒍) Remote-Matching Weight of minimum perfect Matching 𝑃(log 𝑜) [Chandra&Halldorsson 01] 𝑷(𝐦𝐦𝐦 𝒍) Max-Coverage How well the points cover each coordinate max

𝑝∈𝑇 𝑞𝑗 𝑒 𝑗=1

𝑃(1) [Feige 98] No Composable Coreset of Poly size in 𝒍 with app. factor

𝒍 𝒎𝒎𝒎 𝒍

Review of Offline Algorithms

• We have a set of 𝑜 point 𝑄
• Goal: find a subset 𝑇 of size 𝑜 which

maximizes the diversity

The Greedy Algorithm

• Used for minimum-pairwise distance

The Greedy Algorithm

• Used for minimum-pairwise distance
• Greedy Algorithm [Ravi, Rosenkrantz,

Tayi] [Gonzales]

– Choose an arbitrary point – Repeat k-1 times

• Add the point whose minimum distance to

the currently chosen points is maximized

The Greedy Algorithm

• Used for minimum-pairwise distance
• Greedy Algorithm [Ravi, Rosenkrantz,

Tayi] [Gonzales]

– Choose an arbitrary point – Repeat k-1 times

• Add the point whose minimum distance to

the currently chosen points is maximized

• Remote-edge: computes a 2-

approximate set

Local Search Algorithm

• Used for sum of pairwise distances

Local Search Algorithm

• Used for sum of pairwise distances
• Algorithm [Abbasi, Mirrokni, Thakur]

– Initialize 𝑇 with an arbitrary set of 𝑜 points which contains the two farthest points

Local Search Algorithm

• Used for sum of pairwise distances
• Algorithm [Abbasi, Mirrokni, Thakur]

– Initialize 𝑇 with an arbitrary set of 𝑜 points which contains the two farthest points – While there exists a swap that improves diversity by a factor of 1 +

𝜗 𝑜

Local Search Algorithm

• Used for sum of pairwise distances
• Algorithm [Abbasi, Mirrokni, Thakur]

– Initialize 𝑇 with an arbitrary set of 𝑜 points which contains the two farthest points – While there exists a swap that improves diversity by a factor of 1 +

𝜗 𝑜

» Perform the swap

Local Search Algorithm

• Used for sum of pairwise distances
• Algorithm [Abbasi, Mirrokni, Thakur]

– Initialize 𝑇 with an arbitrary set of 𝑜 points which contains the two farthest points – While there exists a swap that improves diversity by a factor of 1 +

𝜗 𝑜

» Perform the swap

Local Search Algorithm

• Used for sum of pairwise distances
• Algorithm [Abbasi, Mirrokni, Thakur]

– Initialize 𝑇 with an arbitrary set of 𝑜 points which contains the two farthest points – While there exists a swap that improves diversity by a factor of 1 +

𝜗 𝑜

» Perform the swap

• For Remote-Clique

– Number of rounds: log 1+𝜗

𝑜

𝑜2 = 𝑃(𝑜

𝜗 log 𝑜)

– Approximation factor is constant.

Composable Core-sets

• Greedy Algorithm Computes a 3-composable core-set for

minimum pairwise distance

• Local Search Algorithm Computes a constant factor

composable core-set for sum of pairwise distances.

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c Let 𝑃𝑄𝑃 = 𝑝1, ⋯ , 𝑝𝑙 be the optimal solution Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤(𝑃𝑄𝑃) / c

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c Let 𝑃𝑄𝑃 = 𝑝1, ⋯ , 𝑝𝑙 be the optimal solution Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤(𝑃𝑄𝑃) / c

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c Let 𝑃𝑄𝑃 = 𝑝1, ⋯ , 𝑝𝑙 be the optimal solution Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤(𝑃𝑄𝑃) / c Let 𝑠 be their maximum diversity , 𝑠 = max

𝑗 𝑒𝑒𝑤 𝑇𝑗 , Note: divk 𝑇 ≥ 𝑠

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c Let 𝑃𝑄𝑃 = 𝑝1, ⋯ , 𝑝𝑙 be the optimal solution Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤(𝑃𝑄𝑃) / c Let 𝑠 be their maximum diversity , 𝑠 = max

𝑗 𝑒𝑒𝑤 𝑇𝑗 , Note: divk 𝑇 ≥ 𝑠

Case 1: one of 𝑇𝑗 has diversity as good as the optimum: 𝑠 ≥ 𝑷 𝑒𝑒𝑤 𝑃𝑄𝑃

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c Let 𝑃𝑄𝑃 = 𝑝1, ⋯ , 𝑝𝑙 be the optimal solution Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤(𝑃𝑄𝑃) / c Let 𝑠 be their maximum diversity , 𝑠 = max

𝑗 𝑒𝑒𝑤 𝑇𝑗 , Note: divk 𝑇 ≥ 𝑠

Case 1: one of 𝑇𝑗 has diversity as good as the optimum: 𝑠 ≥ 𝑷 𝑒𝑒𝑤 𝑃𝑄𝑃 Case 2: : 𝑠 ≤ 𝑷(𝑒𝑒𝑤(𝑃𝑄𝑃))

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c Let 𝑃𝑄𝑃 = 𝑝1, ⋯ , 𝑝𝑙 be the optimal solution Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤(𝑃𝑄𝑃) / c Let 𝑠 be their maximum diversity , 𝑠 = max

𝑗 𝑒𝑒𝑤 𝑇𝑗 , Note: divk 𝑇 ≥ 𝑠

Case 1: one of 𝑇𝑗 has diversity as good as the optimum: 𝑠 ≥ 𝑷 𝑒𝑒𝑤 𝑃𝑄𝑃 Case 2: : 𝑠 ≤ 𝑷(𝑒𝑒𝑤(𝑃𝑄𝑃))

• find a one-to-one mapping 𝜈 from 𝑃𝑄𝑃 = {𝑝1, ⋯ , 𝑝𝑙} to 𝑇 = 𝑇1 ∪ ⋯ ∪ 𝑇𝑛 s.t.

𝑒𝑒𝑒𝑒 𝑝𝑗 , 𝜈 𝑝𝑗 ≤ 𝑷(𝑠)

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c Let 𝑃𝑄𝑃 = 𝑝1, ⋯ , 𝑝𝑙 be the optimal solution Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤(𝑃𝑄𝑃) / c Let 𝑠 be their maximum diversity , 𝑠 = max

𝑗 𝑒𝑒𝑤 𝑇𝑗 , Note: divk 𝑇 ≥ 𝑠

Case 1: one of 𝑇𝑗 has diversity as good as the optimum: 𝑠 ≥ 𝑷 𝑒𝑒𝑤 𝑃𝑄𝑃 Case 2: : 𝑠 ≤ 𝑷(𝑒𝑒𝑤(𝑃𝑄𝑃))

• find a one-to-one mapping 𝜈 from 𝑃𝑄𝑃 = {𝑝1, ⋯ , 𝑝𝑙} to 𝑇 = 𝑇1 ∪ ⋯ ∪ 𝑇𝑛 s.t.

𝑒𝑒𝑒𝑒 𝑝𝑗 , 𝜈 𝑝𝑗 ≤ 𝑷(𝑠)

• Replacing 𝑝𝑗 with 𝜈(𝑝𝑗) has still large diversity
• 𝑒𝑒𝑤

𝜈 𝑝𝑗 is approximately as good as 𝑒𝑒𝑤 𝑝𝑗

Proof Idea

Let 𝑄

1, ⋯ , 𝑄 𝑛 be the set of points , 𝑄 = ⋃𝑄 𝑗

𝑇1, ⋯ , 𝑇𝑛 be their core-sets, S = ⋃𝑇𝑗 Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤𝑙(𝑄) / c Let 𝑃𝑄𝑃 = 𝑝1, ⋯ , 𝑝𝑙 be the optimal solution Goal: 𝑒𝑒𝑤𝑙 𝑇 ≥ 𝑒𝑒𝑤(𝑃𝑄𝑃) / c Let 𝑠 be their maximum diversity , 𝑠 = max

𝑗 𝑒𝑒𝑤 𝑇𝑗 , Note: divk 𝑇 ≥ 𝑠

Case 1: one of 𝑇𝑗 has diversity as good as the optimum: 𝑠 ≥ 𝑷 𝑒𝑒𝑤 𝑃𝑄𝑃 Case 2: : 𝑠 ≤ 𝑷(𝑒𝑒𝑤(𝑃𝑄𝑃))

• find a one-to-one mapping 𝜈 from 𝑃𝑄𝑃 = {𝑝1, ⋯ , 𝑝𝑙} to 𝑇 = 𝑇1 ∪ ⋯ ∪ 𝑇𝑛 s.t.

𝑒𝑒𝑒𝑒 𝑝𝑗 , 𝜈 𝑝𝑗 ≤ 𝑷(𝑠)

• Replacing 𝑝𝑗 with 𝜈(𝑝𝑗) has still large diversity
• 𝑒𝑒𝑤

𝜈 𝑝𝑗 is approximately as good as 𝑒𝑒𝑤 𝑝𝑗

• The actual mapping 𝜈 depends on the specific diversity measure we are considering.

Maximum k-Coverage

• A set of 𝑜 points 𝑄 in 𝑒-dimensional space
• Each dimension corresponds to a feature.
• Goal: choose a set of 𝑜 points 𝑇 in 𝑄 which maximizes the total

coverage:

– cov S = ∑ max

𝑡∈𝑇 𝑒𝑗 𝑒 𝑗=1

Maximum k-Coverage

• A set of 𝑜 points 𝑄 in 𝑒-dimensional space
• Each dimension corresponds to a feature.
• Goal: choose a set of 𝑜 points 𝑇 in 𝑄 which maximizes the total

coverage:

– cov S = ∑ max

𝑡∈𝑇 𝑒𝑗 𝑒 𝑗=1

• Special Case hamming space:
• A collection of 𝑜 sets 𝑄
• Over the universe 𝑉 = 1, … , 𝑒
• Goal: choose 𝑜 sets 𝑇 = {𝑇1, … , 𝑇𝑙} in 𝑄 whose union is

maximized.

Maximum k-Coverage

• A set of 𝑜 points 𝑄 in 𝑒-dimensional space
• Each dimension corresponds to a feature.
• Goal: choose a set of 𝑜 points 𝑇 in 𝑄 which maximizes the total

coverage:

– cov S = ∑ max

𝑡∈𝑇 𝑒𝑗 𝑒 𝑗=1

• Special Case hamming space:
• A collection of 𝑜 sets 𝑄
• Over the universe 𝑉 = 1, … , 𝑒
• Goal: choose 𝑜 sets 𝑇 = {𝑇1, … , 𝑇𝑙} in 𝑄 whose union is

maximized.

• Theorem: for any 𝛽 <

𝑙 log 𝑙 and any constant 𝛾 > 1, there is

no 𝛽-composable core-set of size 𝑜𝛾

Proof Idea

Build a set of instances 𝑄

1, ⋯ , 𝑄𝑃 𝑙

let 𝑉 = 1, ⋯ , 𝑃 𝑜4

Proof Idea

Build a set of instances 𝑄

1, ⋯ , 𝑄𝑃 𝑙

let 𝑉 = {1, ⋯ , 𝑃 𝑜4 }

• Let 𝑊

𝑗 be subset of size 𝑜 of 𝑉

Proof Idea

Build a set of instances 𝑄

1, ⋯ , 𝑄𝑃 𝑙

let 𝑉 = {1, ⋯ , 𝑃 𝑜4 }

• Let 𝑊

𝑗 be subset of size 𝑜 of 𝑉

• 𝑄𝑗 is a collection of subsets of size

𝑜 from 𝑊

𝑗

Proof Idea

Build a set of instances 𝑄

1, ⋯ , 𝑄𝑃 𝑙

let 𝑉 = {1, ⋯ , 𝑃 𝑜4 }

• Let 𝑊

𝑗 be subset of size 𝑜 of 𝑉

• 𝑄𝑗 is a collection of subsets of size

𝑜 from 𝑊

𝑗

• 𝑄𝑗 has cardinality

𝑙 𝑙

Proof Idea

Build a set of instances 𝑄

1, ⋯ , 𝑄𝑃 𝑙

let 𝑉 = {1, ⋯ , 𝑃 𝑜4 }

• Let 𝑊

𝑗 be subset of size 𝑜 of 𝑉

• 𝑄𝑗 is a collection of subsets of size

𝑜 from 𝑊

𝑗

• 𝑄𝑗 has cardinality

𝑙 𝑙

We show there exists 𝑊

1, ⋯ , 𝑊 𝑃 𝑙 such that

– 𝑊

𝑗 ∖ 𝑊 1 has size 𝑜

– 𝑊

𝑗 ∖ 𝑊 1 and 𝑊 𝑘 ∖ 𝑊 1 are disjoint for 𝑒 ≠ 𝑘

Proof Idea

Build a set of instances 𝑄

1, ⋯ , 𝑄𝑃 𝑙

let 𝑉 = {1, ⋯ , 𝑃 𝑜4 }

• Let 𝑊

𝑗 be subset of size 𝑜 of 𝑉

• 𝑄𝑗 is a collection of subsets of size

𝑜 from 𝑊

𝑗

• 𝑄𝑗 has cardinality

𝑙 𝑙

We show there exists 𝑊

1, ⋯ , 𝑊 𝑃 𝑙 such that

– 𝑊

𝑗 ∖ 𝑊 1 has size 𝑜

– 𝑊

𝑗 ∖ 𝑊 1 and 𝑊 𝑘 ∖ 𝑊 1 are disjoint for 𝑒 ≠ 𝑘

• Using 𝑜 sets everything in ∪ 𝑊

𝑗 can be covered,

that is 𝑃(𝑜3/2) elements.

Proof Idea

Build a set of instances 𝑄

1, ⋯ , 𝑄𝑃 𝑙

let 𝑉 = {1, ⋯ , 𝑃 𝑜4 }

• Let 𝑊

𝑗 be subset of size 𝑜 of 𝑉

• 𝑄𝑗 is a collection of subsets of size

𝑜 from 𝑊

𝑗

• 𝑄𝑗 has cardinality

𝑙 𝑙

We show there exists 𝑊

1, ⋯ , 𝑊 𝑃 𝑙 such that

– 𝑊

𝑗 ∖ 𝑊 1 has size 𝑜

– 𝑊

𝑗 ∖ 𝑊 1 and 𝑊 𝑘 ∖ 𝑊 1 are disjoint for 𝑒 ≠ 𝑘

• Using 𝑜 sets everything in ∪ 𝑊

𝑗 can be covered,

that is 𝑃(𝑜3/2) elements.

• Using core-sets only 𝑊

1 + 𝑜 log 𝑜 = O(k log k )

can be covered

Conclusion

• Applications of composable core-sets

Conclusion

• Applications of composable core-sets
• We showed construction of composable core-sets for a

wide range of diversity measures

Conclusion

• Applications of composable core-sets
• We showed construction of composable core-sets for a

wide range of diversity measures

• We showed non existence of core-sets of polynomial

size in 𝑜 for maximum coverage

Conclusion

• Applications of composable core-sets
• We showed construction of composable core-sets for a

wide range of diversity measures

• We showed non existence of core-sets of polynomial

size in 𝑜 for maximum coverage

• Open Problems

– Are there any other applications of composable core-sets?

Conclusion

• Applications of composable core-sets
• We showed construction of composable core-sets for a

wide range of diversity measures

• We showed non existence of core-sets of polynomial

size in 𝑜 for maximum coverage

• Open Problems

– Are there any other applications of composable core-sets? – Is there a general characterization of measures for which composable core-sets exist?

Conclusion

• Applications of composable core-sets
• We showed construction of composable core-sets for a

wide range of diversity measures

• We showed non existence of core-sets of polynomial

size in 𝑜 for maximum coverage

• Open Problems

– Are there any other applications of composable core-sets? – Is there a general characterization of measures for which composable core-sets exist? – Better approximation factors?

Thank You!

Questions?