{
"metadata": {
"name": "Algo HW3"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": "4-1"
},
{
"cell_type": "markdown",
"metadata": {},
"source": "a. $T(n)=2T(n/2)+n^4$
\nLet
Therefore,
\nLet
Therefore,
\nLet
\nLet
\nLet
Therefore,
\nLet
\nAssume T(n) is constant if
\n$= T(\alpha) + (\lfloor \frac{n}{2} \rfloor -1)[n^2 - 2n\lfloor \frac{n}{2} \rfloor + \frac{2}{3} \lfloor \frac{n}{2} \rfloor(2\lfloor \frac{n}{2} \rfloor -1)] \dots ()$
\n$\implies \frac{n^3}{48} = (\frac{n}{4})[ \frac{2}{3}(\frac{n}{4})(\frac{n}{2})] \leq (\frac{n}{2} -2)[n^2 - 2n \frac{n}{2} + \frac{2}{3}(\frac{n}{2}-1)(n-3)] \leq () \leq (\frac{n}{2})[n^2 - 2n \frac{n}{4} + \frac{2}{3} \frac{n}{2} (2 \cdot \frac{n}{2})] \leq n^3$
\n(Because
\nTherefore, we can conclude that
4-3"
},
{
"cell_type": "markdown",
"metadata": {},
"source": "a. $T(n)=4T(n/3)+n \lg{n}$
\nLet
Then we have
\nAssume that
\nPut
\n$T(n)=\sqrt{n} T(\sqrt{n}) + n$
\n$ = b^{2^{m-1}}T(b^{2^{m-1}}) + b^{2^{m}}$
\n$= b^{2^{m-1}} [ b^{2^{m-2}}T(b^{2^{m-2}}) + b^{2^{m-1}} ] + b^{2^{m}}$
\n$= \dots = T(b)\prod_{k=1}^{m}{b^{2^{m-k}}} + \sum_{k=1}^{m}{\prod_{h=1}^{k}(b^{2^{m-h}} b^{2^{m-k}})} + b^{2^{m}}$
\n$= T(b)b^{\sum_{k=1}^{m}{2^{m-k}}} + \sum_{k=1}^{m}{b^{2^{m-k}} b^{\sum_{h=1}^{k}{2^{m-h}}}} + b^{2^{m}}$
\n$= T(b)b^{2^{m}-1} + \sum_{k=1}^{m}{b^{2^{m-k}} b^{2^{m}-2^{m-k}}} + b^{2^{m}}$
\n$= T(b)b^{2^{m}-1} + m b^{2^{m}} + b^{2^{m}}$
\n$= T(b) \sqrt{n} + n \lg \log_{b}{n} + n$
\n$= \Theta(n \lg \lg n)$
\n"
},
{
"cell_type": "markdown",
"metadata": {},
"source": "
4-5"
},
{
"cell_type": "markdown",
"metadata": {},
"source": "<style type="text/css">\n.tg {border-collapse:collapse;border-spacing:0;}\n.tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;}\n.tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;}\n.tg .tg-s6z2{text-align:center}\n</style>\n<table class="tg">\n \n <th class="tg-s6z2">Result\n <th class="tg-s6z2">$\alpha$ says
"
},
{
"cell_type": "markdown",
"metadata": {},
"source": "\nFor each pair $(a_k,a_k')$:
\n If we get testing result 2, 3 or 4, then remove $a_k$ and $a_k'$ from $A$.
\n If we get testing result 1, then remove one of $a_k$, $a_k'$ from $A$ in arbitrariness.
\n\n
"
},
{
"cell_type": "markdown",
"metadata": {},
"source": "Proof of Correctness:
\nClearly, we do
"
}
],
"metadata": {}
}
]
}
Created
March 11, 2014 12:36
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