# 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.](https://en.wikipedia.org/wiki/Online_algorithm) - [Competitive analysis of OA](https://en.wikipedia.org/wiki/Competitive_analysis_(online_algorithm)).

### 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*