Suppose that you are the general manager for a major-league baseball team. During
the off-season, you need to sign some free-agent players for your team. The team
owner has given you a budget of $X to spend on free agents. You are allowed to
spend less than $X altogether, but the owner will fire you if you spend any more
than $X.
You are considering N different positions, and for each position, P free-agent
players who play that position are available. Because you do not want to overload
your roster with too many players at any position, for each position you may sign
at most one free agent who plays that position. (If you do not sign any players at a
particular position, then you plan to stick with the players you already have at that
position.)
To determine how valuable a player is going to be, you decide to use a sabermetric
statistic (the application of statistical analysis to baseball records, providing several ways
to compare the relative values of individual players) known as “VORP,” or “value over replacement player.” A player with
a higher VORP is more valuable than a player with a lower VORP. A player with a
higher VORP is not necessarily more expensive to sign than a player with a lower
VORP, because factors other than a player’s value determine how much it costs to
sign him.
For each available free-agent player, you have three pieces of information:
� -the player’s position,
� -the amount of money it will cost to sign the player, and
� -the player’s VORP.
Devise an algorithm that maximizes the total VORP of the players you sign while
spending no more than $X altogether. You may assume that each player signs for a
multiple of $100,000. Your algorithm should output the total VORP of the players
you sign, the total amount of money you spend, and a list of which players you
sign. Analyze the running time and space requirement of your algorithm.
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To solve this problem, you can use a greedy algorithm. Here's how you can approach it:
1. Sort the available players for each position based on their VORP in descending order.
2. Initialize a variable named "total_VORP" to keep track of the total VORP of the players you sign. Set it to 0.
3. Initialize another variable named "total_cost" to keep track of the total amount of money you spend. Set it to 0.
4. Create an empty list called "signed_players" to store the players you sign.
5. Iterate through each position:
a. For each position, iterate through the available players sorted by VORP.
b. Check if signing the current player exceeds the budget limit. If it does, continue to the next player.
c. If signing the current player doesn't exceed the budget limit, add the player's VORP to "total_VORP", add the player's cost to "total_cost", and append the player to the "signed_players" list.
d. Continue this process until you have checked all available players for each position.
6. Output the "total_VORP", "total_cost", and "signed_players".
The running time of this algorithm depends on the sorting step, which has a time complexity of O(N log N), assuming there are N players in total. The subsequent steps have a time complexity of O(N), as we iterate through each available player. Therefore, the overall time complexity is O(N log N).
The space requirement of this algorithm is O(N) to store the signed players in the list "signed_players".