$57.$ We construct a program that obtains the solutions of the dual and primary fused Lasso problems. Fill in the blanks, and execute the procedure.

$```$

def fused$\_$dual $($y$,$ D$)$:

m $=$ D$.$shape $[0]$

lambda$\_$seq $=$ np$.$zeros$($m$)$

s $-$ np$.$zeros$($m$)$

alpha $=$ np$.$zeros$(($n$,$ n$))$

alpha$[0,$ : $]=$ np$.$l$1$nalg $.$ p$1$nv$($D D$.$T$)$ D @ y

for $\}1\backslash$mathrm$\{$ in range$($n$)$:

if np$.$abs$($alpha$[0,1])>$ lambda$\_$seq$[0]$:

lanbda$\_$seq$[0]=$ np$.$abs$($alpha$[0,1])$

$1$ndex $=[1]$

if alpha$[0,1]>0$:

## Blank$($l$)$ ##

I$\_$s $=$ list $($range$($n$))$

for k in range$(1,$ n$)$:

sub$\_$s $=$ list$($set$(1\_$s$)-$ set$(1$ndex$))$

U $=$ np$\mathrm{.}11$nalg.p$1$nv$($D$[$sub$\_$s$,$ : $]$ D $[$sub$\_$s$,$ :$].$T$)$

V $=$ D$[$sub$\_$s$,$ :$]0$ D$[1$ndex $,$ :$].$T

u $=$ U @ D$[$sub$\_$s$,$ : $]$ y

v $=$ U a v a s$[1$ndex$]$

t $=$ u $/($v$+1)$

for $1$ in range$(0,$ m$-$k$)$:

if t$[1]>$ lambda$\_$seq$[$k$]$:

lambda$\_$seq$[$k$]=$ t$[1]$

n $=1$

r $=1$

t $=$ u $/($v$-1)$

for $1$ in range$(0,$ m$-$k$)$:

if t$[1]>$ lambda$\_$seq$[$k$]$:

lambda$\_$seq$[$k$]-$ t$[1]$

h $=1$

r $=-1$

alpha$[$k$,1$ndex$]-$ ## Blank$(2)$ ##

alpha$[$k$,$ sub$\_$s$]=$ ## Blank$(3)$ ##

h $=$ sub$\_$s$[$h$]$

$1$ndex, append$($n$)$

if r $--1$:

s$[$n$]-1$

else:

s$[$h$]=-1$

return $[$alpha$,1$ambda$\_$seq$]$

$```$