Variations of Knapsack Problem Relevant to Interactive Rendering

We explain how the Knapsack problem and interactive rendering are related. We also discuss several additional concerns regarding interactive rendering and how these affect the solution to the original Knapsack problem. Lastly, we present two approximate algorithms and compare how they meet the needs of interactive rendering. 1 Background Knapsack Problem We are given a set of n items, S = x1, x2, ..., xn. Each item has a weight, w(x1), w(x2), ..., w(xn) and value, v(x1), v(x2), ..., v(xn). The Knapsack problem is to find S , a subset of S, such that: ∑ y∈S v(y) is maximized over S and ∑ y∈S w(y) ≤ c where c is the capacity of the knapsack. This problem is NP-Complete [8] because a polynomial time checker can verify a solution but the best algorithm for determining the solution is O(n2), where k is the number of bits used to represent the capacity of the knapsack and the weights of each item [4]. Interactive Rendering Rendering is the process of creating a 2-dimensional image from a 3-dimensional scene consisting of objects, lights, cameras and other entities. Interactive rendering is doing this conversion fast enough that a user can provide input and get feedback almost instantaneously. Interactive rendering is useful in different contexts such as architectural evaluation of buildings, CAD applications, game playing and simulations. Interactive rendering is made possible by advances in graphics card hardware. However, the time required to render a scene is input-dependent. That is, as more objects are added to the scene, the render time increases even if the final 2D image is of the same size. This makes interactive rendering of large scenes challenging.