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Searching in an unknown environment - An optimal randomized algorithm for the cow-path problem
Introduction
- Classical search problems: **cost of a search == number of queries **made to an oracle which knows the position of the goal.
- w-lane Cowpath Problem: position unknown (no oracle). Cost: proportional to the distance between queries. Example: time required to travel between two query points.
(problem description) (problem application - robotics/hybrid algorithms/AI examples etc.)
This problem has various common points (see later) with online algorithms and because of this we use the notion of competitive analysis of online algorithms to measure the efficiency of the w-lane Cowpath problem. Quick description of online algorithms. - Competitive analysis of OA.
Competitive ratio for the Cow-Path problem
Competitive analysis uses an optimal offline algorithm (read: one in which all data is avaiable from the start) and defines a competitive ratio by comparing the performance of the online and offline algorithms. An algorithm is competitive if its competitive ratio is bounded.
We define a competitive ratio for the w-lane Cowpath problem: The competitive ratio for an algorithm solving the cow-path problem is the worst-case ratio of the expected distance traveled by the algorithm to the shortest-path distance from origin to goal.
In particular, if the worst-case expected distance traveled by a randomized algorithm is at most cn+d, where n is the distance to the goal and d is a fixed constant, then c is the competitive ratio of this algorithm.
Deterministic algorithm
(baeza-yaetes paper)
Randomized algorithm
(quick abstract, see paper for detailed version / formulas)
Definitions
A deterministic algorithm for the cow-path problem has competitive ratio c if:
cost(goal) <= c*dist(goal) + d // c,d are constants independent from g
Considering a randomized algorithm, the distance traveled to find a given goal position is no longer fixed. This means that cost(goal) is a random variable, and the competitive ratio c is computed on the expected value of that random variable.
E[cost(goal)] <= c*dist(g) + d
This means that a randomized algorithm has competitive ratio c if the expected value of the distance it has to travel is at most c*n + /small constant/.
Algorithm
(see paper)
Theorems
3.1
For any fixed geometric ratio r (explain) the algorithm has competitive ratio R(r,w)
4.1
(see paper)
Lower Bound Analysis
(see paper)
Minimization of the competitive ratio
(see paper)
Growth w. number of paths
(see paper)
Note: here we could prove that the algorithm is optimal with w=2