528491

1. 528491 Not quite random

2. PotW Solution Scanner scan = new Scanner( System.in ); intx = scan.nextInt(); LinkedList<Integer> q = new LinkedList<Integer>(); for ( inti = 0; i < x; i++ ) q.add( scan.nextInt() ); intanswer = 0; inti = 1; while ( !q.isEmpty() ) { intcur = q.poll(); answer++; if ( cur != i ) q.add( cur ); else i++; } System.out.println( answer );

3. SIGN UP FOR HARKER • (if you haven’t already) • http://bit.ly/harkerproco12 • Deadline for registration is this Saturday (3/10) • Put N/A under chaperone info • Contest is Saturday, 3/17 • 9AM-4PM(actual contest is ~1.5 hours) • Worth 30 points PotW credit • If you need any more incentive

4. TAKE MARCH USACO • Today’s the last day! • 5 points of PotW credit (who would’ve guessed?)  Space filler

5. Some random facts about randomness

6. Randomness • Numbers in themselves are not actually random • Distributions of numbers can be random, however • The discrete uniform distribution: • A random integer between [a, b] is chosen • Two possible ways of obtaining “random” numbers • Artificially generated • Read from outside sources • Lava lamps • Atmospheric noise • Nuclear fission

7. Random number generation • For most purposes, pseudorandom works fine • Computers, without any additional input from the environment, cannot generate truly random numbers • Can use mathematical formulas or tables to generate “random” numbers • E.g. Linear Congruential Generator (LCG): • (c * x + a) (mod m) over x • This formula has a period of at most m, so it is fairly weak • Also suffers from points lying on common planes • For randomized algorithms • Even weak pseudorandom generators can work fine • If an evil person is able to manipulate the program in real-time, then the algorithm may suffer • E.g. if pivots are artificially chosen during quicksort

8. Randomized Algorithms • Certain classes of problems can be solved more easily using randomized algorithms • E.g. Sorting, Monte Carlo, Max-flow min-cut can all be approached with randomized algorithms • Quicksort: • Under each recursive step of the algorithm, a pivot must be chosen to partition the array into two halves • Optimally, the pivot should be the median of the values • Instead, choosing a random pivot offers good runtime, on average • Randomized algorithms can also assist in solving “unsolvable” NP-hard problems • Finding the optimal solution may take exponential time, but we can generate approximations with randomized algorithms • E.g. Traveling salesman, Simulated annealing, the PotW

9. PotW– Bovinekiin, Cowborn • Returning from your most recent adventures, you have noticed that there has been some pretty ridiculous inflation in the shop. Given N < 50 pieces of equipment available in the shop, maximize the total power of items you can buy with your X < 1012 hard-earned coins. Each piece of equipment costs C < 1010 and gives you P < 1010 power, and you can only buy each at most once. • Worth 40 points. If your answer is within 5% of the answer our awesome program gives, it will be deemed correct. • Time limit: 2 seconds

10. Sample Input/Output • Sample Input:6 9001(N, X)8980 3(C, P)8940 210 320 430 540 6 • Sample Output:2 3 4 5 (indices of the bought items, not necessarily in order)

11. Hints • The test data will not be especially tricky • Thus, 5% is fairly lenient • One possible solution: • Randomly shuffle the items • Buy as many items as possible left to right • Randomly swap an item that you have just bought with an item that you have not bought • Try step 2 and accept the swap if it produces better results • Repeat 90001 times, reshuffling if stalled for too long