$1.$List the following asymptotic complexities from best $($top$)$ to worst. Which functions are the same rank? $\backslash$theta $($ n lg n $)\backslash$theta $($ n$2)\backslash$theta $($ lg n $)\backslash$theta $($ nn $)\backslash$theta $($ log n $)\backslash$theta $($n $)\backslash$theta $($ n $$n $)\backslash$theta $($ lg lg n $)\backslash$theta $($ n$!)\backslash$theta $(1)\backslash$theta $($ n lg n$2)\backslash$theta $($ n lg$2$ n $)\backslash$theta $($ n $)\backslash$theta $($ n$2$ lg n $)\backslash$theta $($ nlg n $)\backslash$theta $($ n$3$ lg n $)\backslash$theta $($ n$3)2.$Rank these $1-5($Fastest $=1,$ Slowest $=5)$: $\_\_\_\_\backslash$theta $($ nlg n $)\_\_\_\_\backslash$theta $($ n$2$ lg n $)\_\_\_\_\backslash$theta $($ n lg$2$ n $)\_\_\_\_\backslash$theta $($ n lg n$2)\_\_\_\_\backslash$theta $($ lgn n $)\_\_\_\_\backslash$theta $($ lg nn $)\_\_\_\_\backslash$theta $($ n lg n $)3.$Rank these $1-7($Fastest $=1,$ Slowest $=7)$: $\_\_\_\_\backslash$theta $($ nlg n $)\_\_\_\_\backslash$theta $($ lgn n $)\_\_\_\_\backslash$theta $($ lg nn $)\_\_\_\_\backslash$theta $($ n lg n $)\_\_\_\_\backslash$theta $($ n$2$ lg n $)\_\_\_\_\backslash$theta $($ n lg$2$ n $)\_\_\_\_\backslash$theta $($ n lg n$2)4.$ Reduce these and list them smallest to largest. Note any equalities by listing them together as the same rank. $\backslash$theta $($ n lg$2$ n $+$ n lg n$2)\backslash$theta $($ nlg n $+$ n$2$ lg n $)\backslash$theta $($ lgn n $+$ lg nn $)\backslash$theta $($ n lg n $+$ nlg n$)\backslash$theta $($ lg$2$ n $+3$n$)$