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<pre align="left"><font face="Arial, Helvetica, sans-serif" size="4" color="#000000"><b>
<font color="#FF0000">brute force algorithm</font>
<i>O(n) -- the cost of each of the binary search tree
N(n) -- the total number of distinct binary search trees</i>
<font color="#FF0000">complexity: </font>
O( nN(n) ) N(n) = O( <img src="image/4n.gif" width="76" height="20" align="middle"> )
<font color="#FF0000">efficient algorithm</font>
Let <i> a1 < a2 < … < an</i>
<i><font color="#0033CC">T ij </font></i><font color="#0033CC"> </font> ---- an optimal binary search tree for <i>ai+1, …, aj ,</i> i < j.
<i><font color="#0033CC">T ii </font> </i> ---- an empty tree for <i>0 <= i <= n </i>
<i> <font color="#0033CC">cij </font></i> ---- the cost of the search tree <i><font color="#0033CC">Tij </font></i> ( cii = 0 )
<i><font color="#0033CC"> r ij </font></i>---- the root of <i><font color="#0033CC">T ij </font></i> ( <i><font color="#0033CC">rii</font></i> = 0 )
<i><font color="#0033CC"> wij</font></i> ---- the weight of <i><font color="#0033CC">T ij</font></i> , <i><font color="#0033CC">wij </font></i> = <img src="image/qi.gif" width="175" height="52" align="middle"> , ( <i>wii</i> = 0 )
<i><font color="#0033CC">T 0n</font></i> an optimal binary search tree for <i> a1, a2,…, an</i>
<i><font color="#0033CC">T ij </font></i><font color="#0033CC"> </font> --- an optimal binary search tree for <i>ai+1, …, aj </i>, <img src="image/text10-m.gif" width="144" height="63" align="middle">
<i><font color="#0033CC">r ij </font></i> = k (<i> i < k <= j </i>)
L --- the left subtree <i> ( ai+1, …, ak-1 )</i>
R --- the right subtree <i> ( ak, …, ak-1 )</i>
weight (L ) = weight <i>(<font color="#0033CC">T i, k-1 </font>) = <font color="#0033CC">w i,k-1</font></i>
weight (R ) = weight<i> (<font color="#0033CC">T kj </font>) = <font color="#0033CC">w kj</font></i>
<i><font color="#0033CC">cij </font></i> = pk + cost(L ) + cost (R ) + weight (L ) + weight (R )
if cost(L ) =<i><font color="#0033CC"> ci,k-1</font></i> and cost (R ) = <i><font color="#0033CC"> c kj</font></i> , then cij is minimal
<i><font color="#0033CC">cij </font></i> = <i>pk + ci,k-1 + c kj + w i,k-1 + w kj = w ij + ci,k-1 + c kj
</i><font color="#FF0000"><i>w ij + ci,k-1 + c kj</i> = <img src="image/min.gif" width="43" height="35" align="middle"> { <i>w ij + ci,l-1 + c lj </i>}
<font color="#000000">or </font> <i>ci,k-1 + c kj </i> = </font><font face="Arial, Helvetica, sans-serif" size="4" color="#000000"><b><font color="#FF0000"><img src="image/min.gif" width="43" height="35" align="middle"></font></b></font><font color="#FF0000"> {<i> ci,l-1 + c lj</i> }</font></b></font></pre>
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