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--- courses:cs211:winter2012:journals:jeanpaul:chaptersixsectioniv [2012/03/28 03:02] – [Designing the Algorithm] mugabej
+++ courses:cs211:winter2012:journals:jeanpaul:chaptersixsectioniv [2012/03/28 03:13] (current) – [Designing the Algorithm] mugabej
@@ Line 28: / Line 28: @@
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> new weight limit = w – w<sub>i</sub> \\
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> OPT selects best of { 1, 2, …, i–1 } using new weight limit, w – w<sub>i</sub>\\
->>>>>>>>>>>>>>>>>>> SO, OPT(i,w) is:\\
+>>>>>>>>>>>>>>>>>>> So, OPT(i,w) is:\\
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 0 if i =0\\
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> OPT(i-1,w) if w<sub>i</sub> > w\\
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> max{OPT(i-1,w), v<sub>i</sub> + OPT(i-1,w-w<sub>i</sub>)} otherwise\\
+\\
+====== Algorithm ======
+>>>>>>>>>>>>>>>>>>> Input: N, w<sub>1M/sub>,…,w<sub>N</sub>, v<sub>1</sub>,…,v<sub>N</sub>\\
+>>>>>>>>>>>>>>>>>>> for w = 0 to W: O(W) time\\
+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> M[0, w] = 0\\
+>>>>>>>>>>>>>>>>>>> for i = 1 to N: --> For all items\\
+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> for w = 1 to W: --> For all possible weights( So O(NW) time\\
+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> if w<sub>i</sub> > w : --> item's weight is more than available\\
+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> M[i, w] = M[i-1, w]\\
+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> else:\\
+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> M[i, w] = max{ M[i-1, w], vi + M[i-1, w-wi] }\\
+>>>>>>>>>>>>>>>>>>> return M[n,W]
+** Algorithm Analysis**
+\\
+  * The algorithm takes Θ(nW) time to compute the optimal value of the problem.
+  * Given a table M of the optimal values of the subproblems, the optimal set S can be found in O(n) time.
+  * Thus the Knapsack Problem can be solved in O(nW) time.
+I give this section an 8/10.