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--- courses:cs211:winter2018:journals:bairdc:chapter4 [2018/02/27 01:16] – bairdc
+++ courses:cs211:winter2018:journals:bairdc:chapter4 [2018/03/13 03:55] (current) – [4.8 – Huffman Codes and Data Compression] bairdc
@@ Line 10: / Line 10: @@
 <code>
 Sort jobs by finish times so that f1 ≤ f2 ≤ ... ≤ fn
 G = {}
 for j = 1 to n
@@ Line 34: / Line 33: @@
 </code>
 This algorithm runs in O(nlogn) time because for each classroom k, we maintain the finish time of the last job added, and we can keep the classrooms in a priority queue by the last job's finish time.
+I'd give this section an 8/10 on both readability and interestingness.
+===== 4.2 – Scheduling to Minimize Lateness: An Exchange Argument =====
+In this problem, we have a single resource and a set of n requests to use this resource for a given time interval. However, instead of a start and finish time, the request has a deadline, and it requires a time interval length, //t//, and it can be scheduled at any point before the deadline. The goal is to minimize the lateness of the request (finish time minus deadline). The correct greedy approach for this problem is to sort the jobs in increasing order of their deadlines, and then schedule them in this order. This way, we make sure that jobs with earlier deadlines get completed earlier.
+Minimize Lateness:
+<code>
+Sort n jobs by deadline so that d1 ≤ d2 ≤ … ≤ dn t = 0
+for j = 1 to n
+    Assign job j to interval [t, t + tj]
+    sj = t
+    fj = t + tj
+    t = t + tj
+output intervals [sj, fj]
+</code>
+I'd give this section a 10/10 on readability and a 7/10 on interestingness.
+===== 4.4 – Shortest Paths in a Graph =====
+This section examines a greedy algorithm that calculates the shortest paths with a designated starting node, //s//, and the assumption that //s// has a path to every other node in the graph. Such an algorithm was created by Edsger Dijkstra in 1959, and the algorithm became known as Dijkstra's Algorithm.
+Dijkstra's Algorithm:
+<code>
+Dijkstra's Algorithm (G, l):
+Let S be the set of explored nodes
+    For each u in S, we store a distance d(u)
+Initially S = {s} and d(s) = 0
+While S != V
+    Select a node v in S with at least one edge from S for which
+        d'(v) = min(e=(u,v):u in S) d(u) + le is as small as possible
+    Add v to S and define d(v) = d'(v)
+EndWhile
+</code>
+Dijkstra's Algorithm is greedy because "we always form the shortest new s-v path we can make from a path in //S// followed by a single edge". Its correctness can be proved using a Greedy Stays Ahead proof. Each time it selects a path to a node v, such path is shorter than any other possible path there. Its runtime is **O(mlogn)** where m is the number of edges and n is the number of nodes. Seemingly, the runtime seems like it would be O(mn), but using the right data structures drastically improves efficiency. We use a priority queue to keep the nodes V - S with d'(v) as their key. So, the algorithm can run in O(m) time, "plus the time for n ExtractMin and m ChangeKey operations".
+This section was very readable, and I'd give it a 8/10 on readability and a 7/10 on interestingness.
+===== 4.5 – The Minimum Spanning Tree Problem =====
+The goal of the minimum spanning tree problem is to find a tree from a graph that is fully connected, but with the minimum cost. This can be done with 3 different greedy algorithms. One, called Kruskal's algorithm, starts without any edges and builds a spanning tree by successively inserting edges in order of their increasing costs. You keep inserting edges unless a cycle is created. Another greedy algorithm is called Prim's Algorithm, which is essentially a modified version of Dijkstra's Algorithm. You start at a root node and greedily grow a tree outward based on the node that can be attached as cheaply as possible. Using a priority queue, Prim's Algorithm we can achieve a runtime of O(mlogn) because on a graph with n nodes and m edges, the algorithm runs in O(m) time, plus the n ExtractMin and m ChangeKey operations. The third greedy algorithm that we can run to get a minimum spanning tree is a backward version of Kruskal's Algorithm. We start with the full graph and delete nodes in order of decreasing cost; you delete edges as long as they don't disconnect the graph.
+Overall this section was very readable and pretty interesting. 9/10 on readability and 8/10 on interestingness.
+===== 4.6 – Implementing Kruskal's Algorithm: The Union-Find Data Structure =====
+A Union-Find data structure is a special one created for Kruskal's Algorithm, which stores a representation of the components in a way that supports rapid searching and updating. The data structure can test is two nodes are in the same set by comparing the Find operation on both. It can also merge two sets into a single set with the Union operation. The goal time for Find and Union both a are O(logn) time. We can use an array implementation so that Find takes O(1), MakeUnionFind takes O(n), and any sequence of k Union operations takes O(klogk) time. However, a better representation would be to use a pointer-based representation. This way, we get Union operations in O(1), MakeUnionFind in O(n), and Find operations in O(logn).
+Overall this section was pretty readable and interesting. 7/10 for both.
+===== 4.7 – Clustering =====
+Clustering happens when you're trying to classify a collection of objects into distinct groups. One way to divide them up into clusters is to calculate the distances between the distinct groups. However, there are infinitely many clusterings in a set; the issue is finding the ones with the max spacing. This procedure is the same as Kruskal's minimum spanning tree algorithm, except we stop the procedure when we obtain //k// connected components. This is the same as deleting the k-1 most expensive edges from the minimum spanning tree and returning the resulting components.
+Overall this section was very interesting and very readable to me: 10/10.
+===== 4.8 – Huffman Codes and Data Compression =====
+The goal of Huffman Codes is lossless data compression. The original top-down approach created by Shannon and Fano isn't guaranteed to always produce an optimal code. So, Huffman came up with a bottom-up approach that did. The idea in this is to have the most frequently used characters to have the shortest codes in order to optimize the overall compression. Furthermore, you don't want to have one character to be the prefix of another character, or you will get multiple ways to interpret data. To do this, we must minimize the average number of bits per letter ABL(y) = Sigma x in S fx|y(x)|.
+Using a Priority Queue, we can get Huffman's algorithm to run in O(klogk):
+<code>
+if S has two letters:
+  Encode one letter as 0 and the other letter as 1
+else:
+  Let y* and z* be the two lowest-frequency letters
+  Form a new alphabet S’ by deleted y* and z* and replacing
+    them with a new letter w of freq fy* + fz* Recursively construct a prefix code y’ for S’ with tree T’
+  Define a prefix code for S as follows:
+    Start with T’
+    Take the leaf labeled w and add two children below it
+      labeled y* and z*
+</code>
+I'd give this section a 10/10 on readability and interestingness. I was very interested in the topic, and it was explained well.