2012 = 2 2 * 503

1. 2012 = 22 * 503 “Might be useful tomorrow.” – MG

2. PotW Solution intdp(intpos, int k){ if(k >lim)return-(1<<30); if(pos== n)return0; int res =mem[pos][k]; if(res !=-1)return res; res =0; intcur = k %2; intstay =(s[pos]=='0'+(1- cur)); res = max(res, dp(pos+1, k)+ stay); intflip =(s[pos]=='0'+(cur)); res = max(res, dp(pos+1, k +1)+ flip); returnres; }

3. Gunn Programming Contest • Feb. 25, 9AM – 3PM • ProCo-like contest at Gunn High School • Teams of up to 3 • 2 hour round, 12 problems • “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/gunnproco12 • Other dates: • March 17 – Harker Programming Contest • May 19 – Stanford ProCo

4. February USACO! • Today’s the last day to take the February USACO! • Take it if you haven’t already! (Even though AMC is tomorrow…) • 5 points PotW credit for participating (as usual)

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

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

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

8. Lossy • Take advantage of "unnoticeable" imperfections in data • Audio, image, and movie files depend on this technique • Often use differencing techniques • 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 • ".jpg" files are notorious for their blockiness and failure to preserve sharp edges • ".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: • Each element of the original sequence should be shifted to its closest relative in the new sequence • Each element of the new sequence should appear in the original sequence • 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: • Note that the element of a sequence that minimizes the total distance to the other elements is the median • Use dynamic programming after sorting