Gridland

Time Limit: 2 Seconds
Memory Limit: 65536 KB

**Background**

For years, computer scientists have been trying to find efficient solutions
to different computing problems. For some of them efficient algorithms are already
available, these are the "easy" problems like sorting, evaluating a polynomial
or finding the shortest path in a graph. For the "hard" ones only exponential-time
algorithms are known. The traveling-salesman problem belongs to this latter
group. Given a set of N towns and roads between these towns, the problem is
to compute the shortest path allowing a salesman to visit each of the towns
once and only once and return to the starting point.

**Problem**

The president of Gridland has hired you to design a program that calculates
the length of the shortest traveling-salesman tour for the towns in the country.
In Gridland, there is one town at each of the points of a rectangular grid.
Roads run from every town in the directions North, Northwest, West, Southwest,
South, Southeast, East, and Northeast, provided that there is a neighbouring
town in that direction. The distance between neighbouring towns in directions
North-South or East-West is 1 unit. The length of the roads is measured by the
Euclidean distance. For example, Figure 7 shows 2 * 3-Gridland, i.e., a rectangular
grid of dimensions 2 by 3. In 2 * 3-Gridland, the shortest tour has length 6.

Figure 7: A traveling-salesman tour in 2 * 3-Gridland.

**Input**

The first line contains the number of scenarios.

For each scenario, the grid dimensions m and n will be given as two integer
numbers in a single line, separated by a single blank, satisfying 1 < m <
50 and 1 < n < 50.

**Output**

The output for each scenario begins with a line containing "Scenario #i:", where
i is the number of the scenario starting at 1. In the next line, print the length
of the shortest traveling-salesman tour rounded to two decimal digits. The output
for every scenario ends with a blank line.

**Sample Input**

2

2 2

2 3

**Sample Output**

Scenario #1:

4.00

Scenario #2:

6.00

Source:

**Northwestern Europe 2001**
Submit
Status