Problem 1: Knapsack Variant (Pseudocode)
Consider the following version of Knapsack. Given are two weight limits W1 and W2, where W1 ≤ W2. Given are also n items (w1, c1),(w2, c2), . . . ,(wn, cn), where wi is the weight and ci the cost of the i-th item. We want to find a subset of these items of the highest cost, where the subset weights at least W1 and at most W2. Give an O(nW2) algorithm for this problem. (Re Block Crusher (Kattis) call that the cost (respectively weight) of a subset is the sum of the costs (respectively weights) of the items in the subset.)
Problem 2: Minimum Spanning Tree with Repeated Costs (Proof)
Consider the Minimum Cost Spanning Tree problem on an undirected graph G = (V, E) with a cost ce ≥ 0 on each edge, where the costs may not all be different. If the costs are not all distinct, there can in general be many distinct minimum-cost solutions. Suppose we are given a spanning tree T ⊆ E with the guarantee that, for every e ∈ T, e belongs to some minimum-cost spanning tree in G. Can we conclude that T itself must be a minimum-cost spanning tree in G? Give a proof or counterexample with explanation.
Problem 3: Block Crusher (Kattis)
Solve the “Block Crusher”, which you can find at this web address: https://open.kattis. com/problems/blockcrusher This is a coding problem; be sure to follow the relevant instructions for testing and submitting.
Problem 4: Continuous Median (Kattis)
Solve the “Continuous Median” problem, which you can find at this web address: https: //open.kattis.com/problems/continuousmedian This is a coding problem; be sure to follow the relevant instructions for testing and submitting. An O(n 2 ) algorithm will not be fast enough to pass all the tests, and will receive at most 3 points out of 10. You need to get the running time down to o(n 2 ), and ideally, O(n log n).
Hint 1: Suppose “median” in the problem statement were replaced by “min” (or “max”). How would you solve that problem? Suppose we added the additional condition that, after even-numbered inputs, you must return the min value, and thereafter treat it as having been deleted from the data set. How would you solve it now?
Hint 2: Notice that the median element is (ignoring rounding for now) the maximum element of the bottom half of the input, and the minimum element of the top half of the input. How can you make use of that fact?
Hint 3: Choosing the right data structure is essential.
