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| courses:cs211:winter2012:journals:paul:home [2012/03/26 20:57] – [Chapter 6] nguyenp | courses:cs211:winter2012:journals:paul:home [2012/04/06 01:44] – [Chapter 7] nguyenp | ||
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| * Case 2: j is not in OPT(j) | * Case 2: j is not in OPT(j) | ||
| * Must include OPT(j-1) | * Must include OPT(j-1) | ||
| + | * This is pretty much what we do. It's a divide and conquer type of thing. It's recursive. The dynamic programming part comes in where we memoize the values for OPT(whatever) | ||
| + | * The concept is very similar to the idea of tail recursion on the Fib Seq. Pretty much the same thing. | ||
| + | * Non-Dynamic Algorithm: Exponential and Very Slow (Page 254) | ||
| + | * Sort jobs by finish times so that f_1 ≤ f_2 ≤ ... ≤ f_n #this is the labeling step from above | ||
| + | * Compute p(1), p(2), ..., p(n) #compute the compatible jobs. This has to be done. No way around it. | ||
| + | * Compute(n) | ||
| * | * | ||
| + | * Compute-Opt(j): | ||
| + | * if j = 0 | ||
| + | * return 0 | ||
| + | * else | ||
| + | * #Max determines which on the optimal solution would pick since the optimal solution is trying to get the max weight | ||
| + | * # picks j does not pick j | ||
| + | * return max(v_j + Compute-Opt(p(j)), | ||
| + | * Having to constantly recompute Compute-Opt(p(j)) is stupid. Let's not do that and just compute it once. | ||
| + | * Dynamic Algorithm: Fast (Page 256) | ||
| + | * Sort jobs by finish times so that f_1 ≤ f_2 ≤ ... ≤ f_n | ||
| + | * Compute p(1), p(2), ..., p(n) | ||
| + | * | ||
| + | * for j=1 to n | ||
| + | * M[j]=Empty #This is a global array we use to store all the memoized values for M-compute so we only compute if we have to | ||
| + | * M[0]=0 | ||
| + | * | ||
| + | * M-Compute(n) | ||
| + | * | ||
| + | * M-Compute-Opt(j): | ||
| + | * if j = 0 | ||
| + | * return 0 | ||
| + | * else # v_j is the value/ | ||
| + | * return max(v_j + M-Compute-Opt(p(j)), | ||
| + | * Efficiency: | ||
| + | * Sorting jobs by finish time is O(n log n) | ||
| + | * Computing p(whatever) is O(n log n) | ||
| + | * I don't really see how this is not O(n^2) | ||
| + | * I think it is O(n^2) because we have n elements, so we must compute p(_) n times. | ||
| + | * p(_) is linear right? don't we have to go through all n elements to find one that it compatible with it? | ||
| + | * M-Compute-Opt(j) is O(n) | ||
| + | * This is awesome. | ||
| + | * Everything we do is pretty much just constant accessing actions. | ||
| + | * The thing that is not is recursively calling M-Compute-Opt. | ||
| + | * The thing is, we only do it once for each value since we memoized everything. | ||
| + | * So, we only do it n times. | ||
| + | * When we actually compute something, all we really do is look up a weight/ | ||
| + | * So, this takes O(n log n) | ||
| + | * This only gives us the total value/ | ||
| + | * We can figure this out after the fact. | ||
| + | * We still have our M[] array, which is nice. | ||
| + | * Informal Algorithm Explanation: | ||
| + | * We essentially go through M and find the intervals that would have been chosen using this inequality | ||
| + | * v_j + M[p(j)] > M[j-1] | ||
| + | * This means if the value of this current interval + that of the one compatible with it that ends the latest is greater than the value of the interval right before this one, then this is one would have been chosen. | ||
| + | * Algorithm: | ||
| + | * M-Compute-Opt(n) # compute M[] | ||
| + | * Find-Solution(n) | ||
| + | * | ||
| + | * Find-Solution(j): | ||
| + | * if j = 0: | ||
| + | * output nothing | ||
| + | * else if v_j + M[p(j)] > M[j-1]: | ||
| + | * print j | ||
| + | * Find-Solution(p(j)) | ||
| + | * else: | ||
| + | * Find-Solution(j-1) | ||
| + | * Section 6.2 - Principles of Dynamic Programming: | ||
| + | * The book first goes over what makes the previous algorithm dynamic and efficient | ||
| + | * If I was the author, I would have put that in the last chapter. It has no business being here. | ||
| + | * It says nothing else useful. | ||
| + | * Section 6.3 - Segmented Least Squares: Multi-way Choices | ||
| + | * The problem we are focusing in this section is the one where we have a bunch of dots and want to know what line (if it even is a line) that is most similar to the data (you know what I mean) | ||
| + | * If it is not a line, then we do not try and form a curve, instead we form a sequence of lines. | ||
| + | * How do we do this? | ||
| + | * Consider the first and last points. | ||
| + | * For every point p_n, it can only belong to one line segment | ||
| + | * We define cost for each point as cost: c (segment cost) + error of segment | ||
| + | * Let OPT(j) = minimum cost for points p_1, p_i+1 , … , p_j (the min cost of all the points) | ||
| + | * Let e(i, j) = minimum sum of squares for points p_i, p_i+1 , … , p_j | ||
| + | * How to compute OPT(j)? | ||
| + | * Cost = e(i, j) + c + OPT(i-1). | ||
| + | * OPT(J): | ||
| + | * if j=0 | ||
| + | * return 0 | ||
| + | * else | ||
| + | * min {e(i, | ||
| + | * Algorithm: page 265 | ||
| + | * INPUT: n, p_1,…,p_N , c | ||
| + | * Segmented-Least-Squares() | ||
| + | * M[0] = 0 | ||
| + | * e[0][0] = 0 | ||
| + | * for j = 1 to n | ||
| + | * for i = 1 to j | ||
| + | * e[i][j] = least square error for the segment p_i, …, p_j | ||
| + | * | ||
| + | * for j = 1 to n | ||
| + | * M[j] = min 1 ≤ i ≤ j (e[i][j] + c + M[i-1]) | ||
| + | * | ||
| + | * return M[n] | ||
| + | * The magic really lies in the math behind the least square error. That's how we determine which segment something belongs to. | ||
| + | * Efficiency: | ||
| + | * The first group of nested for loops is O(n^3). | ||
| + | * This can be reduced to quadratic if we pre-compute and write down the values for p_whatever | ||
| + | * YAY MEMOIZING | ||
| + | * the last for loop is actually quadratic even though it initially looks linear because it's just a single for loop, but remember that min is a linear time function | ||
| + | * For some reason the book likes to do things in this very not straightforward way of finding all the data we need using one algorithm and then later using that data to find the information we really need to get our results. | ||
| + | * So, since we really need to know how to split up the points into a sequence of segments, we will go through all the segments we just assigned | ||
| + | * Algorithm: page 266 | ||
| + | * FindSegments(j): | ||
| + | * if j = 0: | ||
| + | * output nothing | ||
| + | * else: | ||
| + | * Find an i that minimizes ei,j + c + M[i-1] | ||
| + | * Output the segment {pi, …, pj} | ||
| + | * FindSegments(i-1) | ||
| + | * Section 6.4 - Subset Sums and Knapsacks: Adding a variable | ||
| + | * We're focusing on the knapsack problem | ||
| + | * We have n things and a knapsack | ||
| + | * item i has weight w_i (always positive) and also has a value v_i | ||
| + | * We can only carry so much weight | ||
| + | * We want to maximize the value (more bang for the buck type of deal) | ||
| + | * So, for each item, we either pick it or we don't pick it | ||
| + | * OPT(i) = max profit subset of items 1,…, i | ||
| + | * How to compute OPT(i)? | ||
| + | * if it does not select object i | ||
| + | * OPT selects best of { 1, 2, …, i-1 } | ||
| + | * if it does | ||
| + | * Does not mean we must not pick everything lower than i | ||
| + | * So, we don't know what to do. | ||
| + | * MORE SUBPROBLEMS!!! w0000ooooooo... | ||
| + | * So, we set a new weight limit, our current available weight minus what we jsut added, i.e. w – w_i | ||
| + | * We let OPT select the best of these | ||
| + | * That's it! That's the algorithm | ||
| + | * Algoriddim be on Page 269 mahn | ||
| + | * Input: N, w_1, | ||
| + | * for w = 0 to W | ||
| + | * M[0, w] = 0 | ||
| + | * for i = 1 to N | ||
| + | * for w = 1 to W | ||
| + | * if wi > w : | ||
| + | * 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] | ||
| + | * The reason for why this works is suprisingly simple once we realize the important of just changing the amount of weight left over and over again | ||
| + | * Efficiency: | ||
| + | * This is a strange one. | ||
| + | * It's Θ(n W) | ||
| + | * So it both depends on the knap sack capacity and the number of objects | ||
| + | * they call it Pseudo-polynomial. | ||
| + | * Weird stuff. Never thought algorithms like this existed. | ||
| + | * | ||
| + | |||
| + | ====== Chapter 7 ====== | ||
| + | * Section 7.1 - The Maximum-Flow Problem and the Ford-Fulkerson Algorithm | ||
| + | * Problem | ||
| + | * You have a directed graph | ||
| + | * Each edge has a capacity | ||
| + | * Have a start node s and end node t | ||
| + | * Greedy | ||
| + | * Find an s-t path with highest capacity | ||
| + | * do it! | ||
| + | * keep doing this until you cant do it any more | ||
| + | * Locally optimal, but not globally optimal (it's very easy to think of a counter example, counter example is on page 339 in book) | ||
| + | * Awesome Solution: Residual Graphs | ||
| + | * Edges have a capacity, but also a flow | ||
| + | * We don't have to use all of it at once | ||
| + | * Residual edge is one where you have some going one way and some going the other | ||
| + | * The Ford-Fulkerson Algorithm uses residual edges and works awesomely (optimal) | ||
| + | * page 342 - 344 | ||
| + | * Informal explanation | ||
| + | * graphs | ||
| + | * keep updating the residual edges using the following formula: | ||
| + | * If edge is the same as it was originally, set the forward rate to the bottle neck + original rate (which is the edge with the smallest flow rate) | ||
| + | * Otherwise, set the backwards residual rate to what it originally was minus the bottle neck | ||
| + | * Section 7.2 - Maximum Flows and Minimmum Cuts in a Network | ||
| + | * Cut: s-t cut is a partition (A,B) where s is in A and t is in B | ||
| + | * Problem: Find an s-t cut of minimum capacity | ||
| + | * The capacity of a cut is the sum of all the possible stuff that can go through the edge at once | ||
| + | * Flow Value Lemma: Let f be any flow, and let (A, B) be any s-t cut. Then, the value of the flow is = f_out(A) – f_in(A). | ||
| + | * Weak Duality: let f be any flow and let (A, B) be any s-t cut. Then the value of the flow is at most the cut’s capacity. | ||
| + | * Corollary: Let f be any flow, and let (A, B) be any cut. If v(f) = cap(A, B), then f is a max flow and (A, B) is a min cut | ||
| + | * Augmenting path theorem: Flow f is a max flow iff there are no augmenting paths | ||
| + | * Max-flow min-cut theorem: The value of the max flow is equal to the value of the min cut. | ||
| + | * More explanation on page 350 | ||
| + | * Boom and we are done. Just use the same stuff as in the past section and we have all we need. | ||
| + | * Section 7.5 - A First Application: | ||
| + | * Given an undirected bipartite graph | ||
| + | * We must match so each node appears at most once on each edge | ||
| + | * How do we find one with the max number of nodes? This is our problem. | ||
| + | * Here's how we do it. We got a bunch of red and blue nodes. Add nodes s and t, which will be the start and end nodes. Have the start node s have a unit edge (edge of size 1) connect it to all the red nodes. Have all the blue nodes have a unit edge connect them to the end node t. Still have same edges in the middle. | ||
| + | * Now, use same algorithm as in the first section. BOOOM!!! Right? Yep. It's pretty intuitive, these later sections are more like practical uses rather than actual algorithms. I guess this is kind of good since pretty much all computer science problems go back to graph theory. | ||
| + | * | ||
