CS5031w- Advance Algorithms Analysis
Quiz 2
Student ID: __________
The 0-1 knapsack problem is the following. A thief robbing a store finds items.
The th
item is worth dollars and weighs pounds, where and are integers. The thief wants to take as valuable a load as possible, but he can carry at most pounds in his knapsack, for some integer . Which items should he take?
(We call this the 0-1 knapsack problem because for each item, the thief must either take it or leave it behind; he cannot take a fractional amount of an item or take an item more than once.)
Give a bottom-up dynamic programming solution to the 0-1 knapsack problem.
Hint:
For each item, there are two possibilities:
•
We include current item in KnapSack and recurse for remaining items with decreased capacity of Knapsack. (If the capacity becomes negative, do not recurse or return – ∞ ).
• We exclude current item from KnapSack and recurse for remaining items.
Finally, we return maximum value we get by including or excluding current item.
The base case of the recursion would be when no items are left or capacity becomes 0.

CS5031w- Advance Algorithms Analysis
Quiz 2
Student ID: __________
Given an × binary matrix (i.e. matrix containing 0’s and 1’s), find the size of largest square sub-matrix of 1's present in it.
Hint:
The size of largest square sub-matrix ending at a cell [ , ] will be 1 plus minimum among largest square sub-matrix ending at [ , − 1] ,
[ − 1, ] and [ − 1, − 1 .
The result will be the maximum of all square sub-matrix ending at [ , ] for all possible values of and .