What happens if we insert keys $8,31,19,17,21,41,12,15,10$ into a hash table where colliam

keys are stored in linked lists $($as we did during the lecture $-$ this is called "chaining"$)$ with the

hash function

$h(k)=$kmod$11$

Let #$U=N=2^{?}$

$u,m=2^{}.$

Represent xinU as binary number of length

$u$

bin$(x)in{B}^{?}$

$u.$

Pick subset of $$ indices in $0$:

$u-1$

bin$(x)[j_y]z=$bin$(x)[i_{-1}],$dots,bin$(x)[0]h(x)=($:$z$:$)hxbin(x)$

$uzbin(x)p\{h|EE(xy),h(x)=h(y)\}{\displaystyle \underset{\{x)}{\overset{?}{}}},y,xyp\{h|h(x)=h(y)\}.S$#$S=np(S)=([N],[n]{)}^{-1}h$:$U[0$:$m-1]0j_0$

Concatenate bits bin$(x)[j_y]$

$z=$bin$(x)[i_{-1}],$dots,bin$(x)[0]$

and interpret $as$ number

$h(x)=($:$z$:$)$

Show that $h$ distributes keys evenly.

Figure $1.To$ hash $x$ convert $it$ into a binary number bin$(x)of$ length

$u,$ obtain $z$

$by$ concatenating a prescribed subsequence $of$ bits $of$ bin$(x)$ and convert back.

Explain why $(the$ proof $of$ Lemma $9,$ lecture $on$ Universal and Perfect Hashing$)$

$p\{h|EE(xy),h(x)=h(y)\}{\displaystyle \underset{\{x)}{\overset{?}{}}},y,xyp\{h|h(x)=h(y)\}.$

Suppose each set $S$ with #$S=nis$ chosen randomly with probability

$p(S)=([N],[n]{)}^{-1}$

and $h$:$U[0$:$m-1]$ distributes keys evenly.

$(1)$ Give asymptotic bounds for the expected times $to$ perform, insert, delete and find $ifwe$

resolve conflicts $by$ chaining.

$(2)$ How $do$ these bounds change $ifwe$ use$\frac{2}{3}-$trees instead $of$ linked lists? please solve second task