Problem Description

In the game of tag described, we're given an array representing two types of players: those who are "it" (represented as 1) and those who are not "it" (represented as 0). Each player who is "it" has a specific reach (dist) which defines the range (inclusive) from their index position within which they can tag a player that is not "it". The goal is to calculate the maximum number of people who are not "it" that can be tagged by those who are "it".

The main challenge is to maximize the number of tags while considering the positions and the reach of the players who are "it". It's essentially a problem of finding the maximum pairing between the players who are "it" and those who are not, within the distance constraints set by dist.

Intuition

To arrive at the solution, consider that we must optimize the pairing between players who are "it" with those who are not. We can iterate through the team array, keeping track of two pointers: one for the current player who is "it" (i) and one for the potential player to catch (j).

As soon as we encounter a player who is "it", we look to find the nearest player who is not "it" within the legal range of dist. If such a player exists, we increment our answer ans, as this represents a successful tag, and move the j pointer forward to look for the next available player.

This approach naturally leads to a greedy solution, where we prioritize tagging the closest player who is not "it" for each "it" player, because this leaves the possibility open for more distant "it" players to tag those further away. We keep iterating until we either run out of "it" players or we exhaust checking all players who are not "it".

The solution code uses a single loop passing through the whole team, which ensures that the time complexity is linear with respect to the size of the team array.

Learn more about Greedy patterns.

Solution Approach

The implementation of the solution employs a single pass algorithm with a two-pointer technique, where one pointer (denoted as i) goes through the team array, and the other pointer (j) is used to find the nearest person to be caught.

Here's a step-by-step breakdown of the algorithm:

1. Initialize two variables: ans to accumulate the number of successful tags, and j to serve as the pointer for finding someone to catch.
2. Loop through the team array with the index i, representing the current person checking for potential tags.
3. When a person who is "it" (team[i] == 1) is encountered, start or continue the inner search with pointer j.
4. The inner loop increases the j pointer until it finds a person who is not "it" (team[j] == 0) and is within the catching range (i - j <= dist).
5. Once a person to catch is found, increase the answer counter ans, and increment j to avoid catching the same person more than once.
6. Repeat this process for each person who is "it" in the team array.
7. If j reaches the end of the array (j >= n), break out of the inner loop since there are no more people left to catch.
8. Continue until all "it" persons have been processed or no one left to catch.

In terms of data structures, only the original input array team is used, and no additional space is needed, making this an in-place algorithm with O(1) space complexity.

The time complexity is O(n), where n is the length of the team array, as each element in the team is visited up to twice – once by the i pointer and once by the j pointer. However, j does not reset after each iteration; it moves continuously forward.

The given implementation successfully captures the greedy nature of the problem, where each "it" person attempts to catch the nearest possible "not it" person in order to maximize the overall number of tags.

Example Walkthrough

Let's consider a small example to illustrate the solution approach. Suppose we have the following setup for the game of tag:

• Team array: team = [0, 1, 0, 0, 1, 0]
• Reach distance: dist = 2

Our goal here is to maximize the number of people who are not "it" (represented by 0) that can be tagged by the players who are "it" (represented by 1).

Let's walk through the algorithm:

1. We start with ans = 0 and j = 0, where ans will hold the number of successful tags and j is our pointer to look for the next player to catch.
2. Begin iterating through the team array (i = 0). Since team[0] is not "it", we do nothing and continue.
3. Move to team[1] (i = 1). Since team[1] is "it", we begin looking for the nearest player to catch using j.
4. Increase j until j reaches 2, satisfying the condition team[j] == 0 and i - j <= dist. We successfully tag the player at j = 2 and increment ans to 1.
5. We continue to iterate with i now at 2. But since team[2] is already tagged, we move forward.
6. At team[3] (i = 3), nothing happens since it's not "it".
7. At team[4] (i = 4), we find another "it" player. We start from the current position of j = 3.
8. Since the player at j = 3 is within reach (i - j <= dist), we tag them as well, increment j to 4, and increment ans to 2.
9. Moving to team[5] (i = 5), we notice j is at the end of the array, so no more players can be tagged.

At the end of this walkthrough, ans = 2, indicating two players were successfully tagged by players who were "it". This reflects a maximized number of tags based on the "greedy" method of tagging the nearest untagged player within reach for each "it" player.

Solution Implementation

Time and Space Complexity

The given Python code is designed to count the maximum number of people a team can catch within a certain distance dist. The algorithm uses a greedy two-pointer approach.

Time Complexity

The time complexity of the code is O(n), where n is the length of the team list. This is because:

• The for-loop iterates through the list once, which contributes O(n).
• Inside the loop, the while-loop moves the second pointer j forward until it finds a valid person to catch or reaches the end of the list. Each element is visited by the j pointer at most once throughout the entire execution of the algorithm. Therefore, the total number of operations contributed by the inner while-loop across all iterations of the for-loop is also O(n).

Hence, the combined time complexity remains O(n) since both the outer for-loop and the cumulative operations of the inner while-loop are linear with respect to the size of the input list.

Space Complexity

The space complexity of the code is O(1).

This is because the algorithm uses a constant amount of extra space for variables ans, j, n, and i. The space used does not depend on the input size and remains constant even as the input list team grows in length.

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