# Define a function to implement the Minimum Violators Algorithm

minimum$\_$violators$\_$algorithm $<-$ function$($components$,$ roots, Ts$,$ K$,$ g$)\{$

# Step $1$: Create a singleton cluster for each component

clusters $<-$ lapply$($components$,$ function$($comp$)$ list$($components $=$ comp$))$

# Initialize set Q with roots of all clusters except the one containing the final product

Q $<-$ setdiff$($roots$,$ tail$($roots$,1))$

# Initialize set r with the root of the final product cluster

r $<-$ tail$($roots$,1)$

while $($length$($Q$)>0)\{$

# Step $2$: Find a cluster i in Q with minimal K$($Ci$)/$ g$($Ci$)$

min$\_$ratio $<-$ Inf

min$\_$cluster $<-$ NULL

for $($i in Q$)\{$

Ci $<-$ clusters$[[$i$]]$

ratio $<-$ K$($Ci$)/$ g$($Ci$)$

if $($ratio $<$ min$\_$ratio $||($ratio $==$ min$\_$ratio && length$($Ci$components$)<$ length$($min$\_$cluster$components$)))\{$

min$\_$ratio $<-$ ratio

min$\_$cluster $<-$ Ci

$\}$

$\}$

# Step $3$: Remove i from Q

Q $<-$ setdiff$($Q$,$ which$($roots $==$ min$\_$cluster$components$[[1]]))$

# Check the condition K$($Ci$\text{'})/$ g$($Ci$\text{'})>=$ K$($Ci$)/$ g$($Ci$)$

if $($K$($min$\_$cluster$)/$ g$($min$\_$cluster$)>=$ K$($clusters$[[$r$]])/$ g$($clusters$[[$r$]]))\{$

# Add i to r

r $<-$ c$($r$,$ min$\_$cluster$components$[[1]])$

$\}$ else $\{$

# Collapse cluster i into cluster r

clusters$[[$r$]]$$components $<-$ c$($clusters$[[$r$]]$$components, min$\_$cluster$components$)$

$\}$

$\}$

# Output the optimal clusters

optimal$\_$clusters $<-$ lapply$($r$,$ function$($root$)$ clusters$[[$root$]])$

return$($optimal$\_$clusters$)$

$\}$