A Simple and General Problem and its Optimal Randomized Online Algorithm Design with Competitive Analysis

The online algorithm design was proposed to handle the caching problem when the future information is unknown [3]. And currently, it draws more and more attentions from the researchers from the areas of microgrid, where the production of renewables are unpredictable, [5], [4], etc. In this note, we present a framework of randomized online algorithm design for the simple and tractable problem. This framework hopes to provide a tractable design to design a randomized online algorithm, which can be proved to achieve the best competitive ratio by Yao’s Principle [6]. I. A SIMPLE BUT GENERAL PROBLEM REQUIRING ONLINE SOLUTION In this note, we consider a simple problem, which needs to be solved in the online manner. Suppose its input can be denoted by the parameter p ∈ P and its online algorithm can be denoted by s ∈ S. For example, in the ski rental problem [2], [3], p represents how many times the player goes to ski totally, and s represents how many days the player rents the ski before he buys the ski. In our consideration, p and s can be numbers, vectors or matrixes. 1 We use the probability distributions of p and s to denote the randomized input and the randomized online algorithm. Obviously the optimal offline cost is uniquely determined by the input p, which we denote as Costoff(p), while the online cost is jointly determined by the input p and the algorithm s, which we denote as Coston(s, p). The ratio of the online cost and offline cost R(s, p) = Coston(s,p) Costoff evaluates how well the online algorithm s performs on the input p: a smaller R(s, p) means better s, and R(s, p) ≥ 1. We assume we can obtain a closed form of R(s, p). 2 II. A LOWER BOUND FOR THE COMPETITIVE RATIO BY Yao’s Principle For a given randomized online algorithm Ar, we can obtain its competitive ratio by CR(Ar) = maxinput Coston Costoff . To show that this randomized online algorithm is the best in terms of competitive ratio, technically, we need to show that given any other randomized online algorithm, the competitive ratio is larger. This is nontrivial because it is difficult to enumerate all possible randomized online algorithms in the design space, or we can think that it’s difficult to enumerate all distributions. 1The problem should be simple enough such that we can characterize its input and its online algorithm by a limited number of parameters. 2Again, since the problem is so simple In the following analysis, we denote the randomized online algorithm and the randomized input by two randomized variables S with the distribution f(s) and P with the distribution g(p), which are supported by S and P respectively. For convenience, we define two functions Ug(s) and Vf (p) as follows, • Given the randomized input g(p), Ug(s) represents the expectation of the ratio when the online algorithm is deterministically s, i.e.