1. 2012 = 2 2 * 503 Might be useful tomorrow. – MG

2. PotW Solution int dp(int pos, int k) { if (k > lim) return -(1 << 30); if (pos == n) return 0; int res = mem[pos][k]; if (res != -1) return res; res = 0; int cur = k % 2; int stay = (s[pos] == '0' + (1 - cur)); res = max(res, dp(pos + 1, k) + stay); int flip = (s[pos] == '0' + (cur)); res = max(res, dp(pos + 1, k + 1) + flip); return res; }

3. Gunn Programming Contest Feb. 25, 9AM – 3PM ProCo-like contest at Gunn High School o Teams of up to 3 o 2 hour round, 12 problems o A good number of the problems at APCS A level, with more advanced problems also included Will be worth PotW credit (exact credit TBA) Register at http://bit.ly/gunnproco12http://bit.ly/gunnproco12 Other dates: o March 17 – Harker Programming Contest o May 19 – Stanford ProCo

4. February USACO! Todays the last day to take the February USACO! o Take it if you havent already! (Even though AMC is tomorrow…) 5 points PotW credit for participating (as usual)

5. Compression Representing data with smaller amounts of data o Note that all data is essentially binary There are various file formats for compressed files o General files: "zip", "tgz", "tar" o In fact, "jar"s are really just augmented "zip" files o Images: "jpg", "png", "gif" o Sound: "mp3", "wav", "wma" o Movies: "so many we're not even going to try to list some of them here" o Executables, 3D Models, etc. Two distinct variations: lossless and lossy

6. Lossless Compress the data so that the process is reversible No scheme can guarantee decrease in file size o One-to-one mapping between possible file input/output makes this impossible Instead, schemes often take advantage of repetition in data o Text files often repeat certain letters more frequently o Images might feature certain colors o Neighboring pixels are often close in color Examples: ".png", most general formats

7. Simple Encoding Algorithms Run-Length Encoding (RLE) o Scan left to right on the data o Group identical elements that appear consecutively o Store the first element + a count Huffman Trees o Depending on how often a certain character or word appears, assign it a binary id o More common words should be assigned more bits o However, no id can be a prefix of another

8. Lossy Take advantage of "unnoticeable" imperfections in data o Audio, image, and movie files depend on this technique Often use differencing techniques o E.g.: In a movie, each successive frame can be differenced from the previous one to reduce data storage Often have ability to specify quality

9. Images Specialized optimizations are designed to trick the human eye o ".jpg" files are notorious for their blockiness and failure to preserve sharp edges o ".gif' files omit colors and then use a dithering technique to emulate shades of color

10. PotW – Silly Compression As part of a new compression algorithm, Bessie is trying to modify a sequence of data of length N so that it contains at most K distinct values. However, it costs |a - b| additional bits to change an integer a into an integer b. Tell her the optimal sequence she should modify the original sequence into. If there are multiple solutions, output any. For 30 points, solve for N < 20. For 50 points, solve for N < 100. For bragging rights, solve for N < 1000. Time Limit: 2 seconds. Note: The second and third tasks are very difficult

11. Silly Compression (cont.) Sample Input #1: 3 2 (N K) 7 6 3 (elements of the original sequence) Output: 1 (minimal cost is 1) 6 6 3 (7 7 3 is another option) Sample Input #2: 7 2 3 4 5 6 9 10 11 Output: 6 4 4 4 4 10 10 10 (5 5 5 5 10 10 10 is another option)

12. Hints Note that an optimal sequence always exists such that: o Each element of the original sequence should be shifted to its closest relative in the new sequence o Each element of the new sequence should appear in the original sequence o Thus, the first task can be solved with a exponential-time brute force Try all n choose k possibilities 20 choose 10 fits well within time limit Second task: o Note that the element of a sequence that minimizes the total distance to the other elements is the median o Use dynamic programming after sorting