User Bin Crash problem

Problem: This is a problem from facebook buzz. The problem is that a plane is gong to crash,  you need to throw out W lbs of freight to survive the crash. Each type of freight fi is associated with a weight wi and price pi. If we throw one unit of fi out, we lost pi but the plane becomes lighter by wi. Our mission is to throw out minmum freights measured by their aggregate price and avoid the crash, which means the total weight of those freights should be larger than W.

Solution: The problem is very similar to the 0-1 knapsack. We can also use similar strategy to solve it. The detail is as follow:

• Basically, we need first sort different types of freight by their weight. 
• Then we will need to fill an array dp[n][W+1] where n is the number of different types of freights. dp[i][j] represents the minmum cost to throw out at least j lbs freight and we can throw up to ith type of freights (ranked by their unit weight and from light to heavy). 
• To fill dp[i][j], we need first select the most cost/efficient type of freight among the i types of freights. Cost/efficient is measured by the unit weight of the freight divided by its unit price. Say the most cost/efficient type of freight among the i types of freights is kth type of freights, we have if j <= kth type's unit weight, dp[i][j] = min(dp[i-1][j], kth type's unit price); Otherwise,  dp[i][j] = min(dp[i-1][j], dp[i][j-kth type's unit weight] + kth type's unit price). We don't need to use greedy method, just do regular dp. dp[i][j] = min(dp[i-1][j], dp[i][j-ith type's unit weight] + ith type's unit price).
• The time complexity of this approach is O(nW).