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--- courses:cs211:winter2011:journals:chen:chapter_4 [2011/02/26 23:50] – [4.5 The Minimum Spanning Tree Proble] zhongc
+++ courses:cs211:winter2011:journals:chen:chapter_4 [2011/03/02 03:13] (current) – zhongc
@@ Line 129: / Line 129: @@
 **Two Properties:**
+When is it safe to invclude an edge in the MST?
+Cut property. Let S be any subset of nodes, and let e
+be the min cost edge with exactly one endpoint in S.
+Then MST contains e.
+proof on P170
-Cut
+When is it safe to remove an edge from the MST?
+Cycle property. Let C be any cycle, and let f be the
+max cost edge belonging to C. Then MST does not
+contain f.
+proof on P170
-Cycle
+cycle-cut intersection proof
 **Three algorithms**
 Prim:
+Start with an arbitrary node s and add it to the tree T. Grow T outward by choosing the cheapest edge e with on endpoint in T and the other out of T.
+Proof on P172
+Cut property
 Kruskal:
+Sort the edges in G in ascending orders. pick the smallest edge that does not creat a cycle.
 Reverse-delete:
+It is the rever version of Kruskal
+**Implementations of the two algorithms**
@@ Line 146: / Line 165: @@
 I had some trouble understanding the proof for Cayley's therom...
+===== 4.6 Implementing Kruskal’s Algorithm: The Union-Find Data Structure =====
+Section Summary:
+Problem Motivation:
+Try to implement the Kruskal's algorithm
+pointer based implementation
+Find Operations
+Log(n)
+Union Operations
+Log(n)
+The Kruskal Algorithm in detail
+Intereting/readible : 5/5
+===== 4.7 Clustering =====
+Section Summary
+definition:
+Divide objects into clusters so that points in
+different clusters are far apart.
+Motivation/Application
+Identify patterns in gene expression
+K-Clustering: try to minimize the distance functions so that the
+farthest two elements in one cluster are not farther apart than any
+elements outside of the cluster.
+It is basically Kruskal's algo except we stop when there are k connected components we seek for.
+proof on P184
+Interesting/readable: 7/7
+===== 4.8 Huffman Codes and Data Compression =====
+summary
+Motivation:
+How to encode data so that transmission could be more efficient?
+Answer: use less bits on the data without much differentiation!(Low entropy data?)
+We use Huffman codes.
+If we use all letters in the same frequency, then there is nothing to encode or compress, but when we do not, which is often the case,
+we can always represent the more frequently used words with less bits.
+Watch out for prefix ambiguity!
+**variable-length encoding**
+Goal: minimize Average number of Bits per Letter (ABL):
+calculate abl using the expected value.
+Algorithm
+implemenation
+  * Binary tree for the prefix codes
+  * Priority queue (with heap) choosing the node with lowest frequency
+Cost (nlogn) Why? logn inserting and dequeuing. do it n times.
+Interesting/readable: 5/8