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--- courses:cs211:winter2018:journals:mccaffreyk:home:6 [2018/03/26 01:45] – mccaffreyk
+++ courses:cs211:winter2018:journals:mccaffreyk:home:6 [2018/03/26 04:18] (current) – mccaffreyk
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 =====Chapter 6: Dynamic Programming =====
+Dynamic Programming is the opposite of Greedy and is similar to divide and conquer. It
+is a good tool for problems which do not have greedy aspects and can only produce exponential divide and
+conquer solutions. Dynamic Programming is the process of decomposing all information into a collection
+of sub-problems and then using their solutions to solve progressively larger sub-problems. Overall,
+it is more powerful than the other strategies we have tried so far.
+=== Section 6.1: Weighted Interval Scheduling: A Recursive Procedure ===
+First, we will apply DP to the weighted interval scheduling problem. This is like the old
+scheduling problem except that now intervals can have differing values, rendering the old
+greedy algorithm(minimizing overlaps) obsolete. No other greedy algorithm is currently known.
+We will take a recursive approach although we could take an iterative approach if we chose.
+First, the book derives a complex equation representing a recurrence relationship. It simply
+means that we can find a solution by checking the final interval points vs the overlap points,
+applying and then recursing back to the next interval. Unfortunately, this algorithm is still
+exponential as it requires Fibonacci like computations. Luckily, this poor efficiency is
+caused by redundancy: making the same calls over and over again to re-find various sub-optimums.
+The solution is memoization: storing and then referencing the results of each of these calls with
+computer memory. We store all of our memorized values in an array and reference from it when able
+rather than making calls again. Amazingly, this new memoized algorithm runs in only O(n) time.
+This however requires that intervals are sorted by finish times before the algorithm is run. We derive
+this running time from the fact that the algorithm may recursively fill in at most n-entries in the memoization array
+and each call fills in one. This section was often difficult to read given the high amounts of math and abstraction
+combined. Thus, I will rate its ease of reading only 5/10.
+=== Section 6.2: Principles of Dynamic Programming: Memoization or Iteration over Sub-problems ===
+Here, we focus on the iterative way of applying this algorithm. we originally used a recursive style
+but both achieve the same results for this type of dynamic programming. The iterative algorithm iterates
+n times, always adding one value to our memoized array m. The values added into m are the maximum number of
+points if x-i intervals are considered, moving from 0 to n intervals where n is the interval that starts last.
+We find the values by computing each optimum solution (overlap vs number of points). I really liked the diagram
+present in this section. It helped me grasp key concepts which I did not yet understand when reading section 6.1.
+This gets an 8/10.
+=== Section 6.3: Segmented Least Squares: Multi-way Choices ===
+Here we focus on the line of best fit problem. For a given line through a set of points
+on a 2d graph, we define error value as the sum of the squares of distances between the
+line and each point. The interesting thing here is that we allow more than one line.
+However, additional lines incur two penalties: segment number is multiplied by a constant
+and summed with the error of each segment. Clearly, we are attempting to minimize total
+error. We can define the constant C as we choose to encourage or discourage additional
+lines more. The exact algorithm is very abstract. We use iteration somewhat similar
+to that of section 6.2 to find and memoize the least square error sums in O(n^3) time. Next, we deal with
+how many line segments we will need, also with a recursive function. This part takes O(n^2) time. This
+section was hard for me to an extent similar to 6.1. This is because it gave complex mathematical formulas
+without explaining them adequately. Further, the algorithms were very abstract forcing me to make assumptions
+and constantly decipher them. Thus, my score for this section is 5/10.