Fair Max–Min Diversity Maximization in Streaming and Sliding-Window Models

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. 2023 Jul 14;25(7):1066. doi: 10.3390/e25071066

Algorithm 3 SFDM2

Input:
Stream $X = ⋃_{i = 1}^{m} X_{i}$ , distance metric d, parameter $ε \in (0, 1)$ , size constraints $k_{1}, \dots, k_{m} \in Z^{+}$ ( $k = \sum_{i = 1}^{m} k_{i}$ )
Output:
A set $S \subseteq X$ s.t. $| S \cap X_{i} | = k_{i}$ , $\forall i \in [m]$
▹
Stream processing
1:
$U = {\frac{d_{m i n}}{{(1 - ε)}^{j}} : j \in Z_{0}^{+} \land {(1 - ε)}^{j} \geq \frac{d_{m i n}}{d_{m a x}}}$
2:
Initialize $S_{μ}, S_{μ, i} = \emptyset$ for every $μ \in U$ and $i \in [m]$
3:
for all $x \in X$ do
4:
for all $μ \in U$ and $i \in [m]$ do
5:
Run Lines 3–6 of Algorithm 1 to update $S_{μ}$ w.r.t. x
6:
if $c (x) = i$ then
7:
Run Lines 3–6 of Algorithm 1 to update $S_{μ, i}$ w.r.t. x
▹
Post-processing
8:
$U^{'} = {μ \in U : | S_{μ} | = k \land | S_{μ, i} | \geq k_{i}, \forall i \in [m]}$
9:
for all $μ \in U^{'}$ do
10:
For each group $i \in [m]$ , pick $min (k_{i}, | S_{μ} \cap X_{i} |)$ elements arbitrarily from $S_{μ}$ as $S_{μ}^{'}$
11:
Let $S_{a l l} = (⋃_{i = 1}^{m} S_{μ, i}) \cup S_{μ}$ and $l = | S_{a l l} |$
12:
Create l clusters $C = {C_{1}, \dots, C_{l}}$ , each of which contains one element in $S_{a l l}$
13:
while there exist $C_{a}, C_{b} \in C$ s.t. $d (x, y) < \frac{μ}{m + 1}$ for some $x \in C_{a}$ and $y \in C_{b}$ do
14:
Merge $C_{a}, C_{b}$ into a new cluster $C^{'} = C_{a} \cup C_{b}$
15:
$C \leftarrow C ∖ {C_{a}, C_{b}} \cup {C^{'}}$
16:
Let $M_{1} = (S_{a l l}, I_{1})$ and $M_{2} = (S_{a l l}, I_{2})$ be two matroids, where $S \in I_{1}$ iff $| S \cap X_{i} | \leq k_{i}$ , $\forall i \in [m]$ and $S \in I_{2}$ iff $| S \cap C | \leq 1$ , $\forall C \in C$
17:
Run Algorithm 4 to augment $S_{μ}^{'}$ such that $S_{μ}^{'}$ is a maximum cardinality set in $I_{1} \cap I_{2}$
18:
return $S \leftarrow {arg max}_{μ \in U^{'} : | S_{μ}^{'} | = k} d i v (S_{μ}^{'})$