649. Dota2 Senate

Problem Description

In the given problem, we are simulating a political power struggle within a senate of a fantasy world known as Dota2. The senate is made up of two parties, Radiant and Dire. Each senator, in turn, has the opportunity to exercise one of two rights—either banning another senator, effectively removing them from the game, or declaring victory for their party if all remaining senators are from their own party.

The outcome we seek to predict is which party will ultimately succeed in passing a change in the Dota2 game by eliminating the opposition's ability to vote. The senators are presented in a string where each character represents a senator from either the Radiant (as 'R') or the Dire (as 'D'). Senators act in the order they appear in the string, and once a senator is banned, they are skipped in subsequent rounds of the process.

Since each senator is smart and employs the best strategy for their party, we need to simulate the rounds of banning to determine which party wins.

Intuition

The intuition behind the solution involves simulating the process using a queue data structure. For simplicity and efficiency, we use a separate queue for each party, recording the positions (indices) of their senators in the senate list. Our goal is to simulate each round where senators ban their immediate opponent senator (if available).

The key insights to arrive at the solution are:

1. Senators will always ban the next senator of the opposite party to maximize their party's chance of victory.
2. Once a senator has made a move, they go to the end of the queue but with an increased index representing their new position in the "virtual" order. This is done to maintain the cyclic nature of the senate arrangement.
3. The process continues until one party's queue is empty, meaning no more senators from that party are left to cast votes or make bans.

By dequeuing the first senator of each party and having them ban the opponent's first senator, we simulate the banning process while keeping track of the new positions. If a Radiant senator acts before a Dire senator, they add to the end of their queue by considering the size of the senate (senate), effectively banning the first Dire senator. The same logic applies when a Dire senator acts before a Radiant senator.

The simulation continues until one party has no remaining senators, at which point the surviving party is declared the winner. This approach ensures that we correctly identify which party would win in an ideal scenario where each senator acts optimally for their party's interest.

Learn more about Greedy and Queue patterns.

Solution Approach

The solution uses a greedy algorithm to simulate the senate's round-based decision-making process. A greedy algorithm makes the locally optimal choice at each stage with the hope of finding a global optimum. In this context, the local optimum is for each senator to ban an opposing party senator as early as possible.

We utilize two queues represented by the deque data structure from Python's collections module:

• qr queue stores the indices of the Radiant senators.
• qd queue stores the indices of the Dire senators.

The indices allow us to keep track of the order of the senators and their relative positions in the simulated rounds.

Here's the approach step by step, in alignment with the code provided:

1. Iterate through the senate string and fill the queues qr and qd with indices of the senators belonging to the Radiant and Dire parties, respectively.

2. Enter a loop that will run until one of the queues is empty. The condition while qr and qd: ensures that the loop continues as long as there are senators from both parties available to take action.

3. In each iteration of the loop:

   • Compare the indices at the front of both qr and qd which represents the order in which the senators will take action. The senator with the lower index is able to act first.
   • If the first Radiant senator (qr[0]) is before the first Dire senator (qd[0]), the Radiant senator will ban the Dire senator. The Radiant senator's index is then added back to the qr queue, incremented by n, which is the length of the senate string. This effectively places them at the end of the order for the next round.
   • Similarly, if the Dire senator acts first, they will ban the Radiant senator, and their index (incremented by n) will be added back to the qd queue.
   • After a senator has acted (either banning or being banned), we remove them from the front of the queue using popleft().

4. After exiting the loop, we check which queue still has senators left, which determines the victorious party. If qr is not empty, Radiant wins; otherwise, Dire wins.

5. Finally, return "Radiant" if the qr queue has senators left, or "Dire" if the qd queue has senators left.

This approach ensures that each senator acts in the best interest of their party by banning the first available opposition senator and then waiting for their next turn at the end of the senate order.

Example Walkthrough

Let's walk through an example with the senate string RDD. We'll simulate the process to see which party comes out victorious.

1. We initialize two queues: qr for Radiant and qd for Dire. Given the senate string RDD, qr will initially contain [0] because the first senator (index 0) is from the Radiant party, and qd will contain [1, 2] because the second and third senators (indices 1 and 2) are from the Dire party.

2. Now, we enter the main loop where we process each senator's actions:

   • Both qr and qd are not empty, so we continue.
   • We compare the front of both queues: qr[0] is 0, and qd[0] is 1. Since the Radiant senator (R) at index 0 is the first in line, they act by banning the first Dire senator (D) at index 1.
   • We remove the banned Dire senator from queue qd by dequeuing it, and then we add the Radiant senator's index incremented by n (the length of the senate string) to represent their new virtual position at the end of the next round. Since n is 3 here, we add 0 + 3 to qr, which becomes [3].
   • The qd queue now looks like [2], as the first Dire senator was banned.

3. Continuing with the main loop:

   • Comparing the indices again, we see qr[0] is 3 (which is virtually 0 in the next round), and qd[0] is 2. The Dire senator at index 2 acts next, since they are the only one left and thus have the lowest current index.
   • The Dire senator bans the first Radiant senator positioned at virtual index 3 (original index 0 returned to the queue). Since nobody is left in the Radiant queue now, qd is decremented by dequeuing the acting senator, and their new indexed position 2 + 3 = 5 is added back to the qd queue.

4. We check the queues after the loop iteration:

   • The qr queue is empty, which indicates that there are no more Radiant senators to take action.
   • The qd queue has one senator left at index 5 (virtually 2).

5. Since qr is empty and qd still has a senator, the victorious party is Dire.

In this example, the final output is "Dire", and the process demonstrates the greedy approach of banning the opponent's next available senator to ensure the best outcome for one's own party.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code is O(N), where N is the length of the senate string.

Reasoning:

• We loop through each character of the string to build the initial qr and qd queues - this is an O(N) operation.
• In the while loop, in each iteration, one senator from each party (R and D) gets 'compared', and one is turned 'inactive' for the current round. Each senator will be dequeued and potentially re-queued once per round. The number of rounds is proportional to the number of senators, because in each round at least one senator is banned from further participation until all senators from the opposing party have been banned.
• Since qr.append(qr[0] + n) and qd.append(qd[0] + n) just change the index for the next round, the O(N) operations within the loop are repeated N times in the worst case (every senator goes once per round). However, due to the nature of the problem, the loop will terminate when one party has no more active senators, so it does not strictly go N rounds.

Space Complexity

The space complexity of the code is O(N), where N is the length of the senate string.

Reasoning:

• Two queues, qr for Radiant senators and qd for Dire senators, each can hold at most N elements if all the senators are from one party.
• No other data structures are used that grow with the input size - thus, the dominant term is the space used by the two queues.

Learn more about how to find time and space complexity quickly using problem constraints.