Question

Suppose that there are \(n=2^{k}\) teams in an elimination tournament, in which there are \(n / 2\) games in the first round, with the \(n / 2=2^{k-1}\) winners playing in the second round, and so on. a. Develop a recurrence equation for the number of rounds in the tournament. b. (b) How many rounds are there in the tournament when there are 64 teams? c. Solve the recurrence equation of part (a).

Short answer

a. The recurrence relation for the number of rounds in the tournament is \( t(k) = t(k-1) + 1 \) with \( t(0) = 0 \). b. There are 6 rounds in the tournament when there are 64 teams. c. The solution to the recurrence relation is \( t(k) = k \).

Step-by-step solution

Formulate the Recurrence Relation

In an elimination tournament with \( n = 2^k \) teams, half of the teams get eliminated in each round. So in the first round, there are \( n/2 = 2^{k-1} \) games. Since the winner of each game moves to the next round, the number of games becomes half in the next round. This keeps repeating till one team is left and it is declared the winner. Hence, the number of games is halved at each step till we reach 1. This leads to the recurrence relation: \[ t(k) = t(k-1) + 1, \] with initial condition \( t(0) = 0 \), where \( t(k) \) represents the number of rounds needed for \( 2^k \) teams.

Calculate the Total Rounds for 64 teams

Given \( n = 64 \) teams, convert it to the form of \( 2^k \), where \( k \) is the value to be found in the recurrence relation. We have \( 64 = 2^6 \), so \( k = 6 \). Using the recurrence relation established in Step 1, we get the total rounds equal to \( k \), so there are 6 rounds when 64 teams are participating.

Solve the Recurrence Relation

To solve the recurrence relation \( t(k) = t(k-1) + 1 \), we replace \( t(k-1) \) with its equivalent from the recurrence relation until we ultimately express everything in terms of \( t(0) \). Plugging \( k-1 \) into our recurrence relation, we obtain: \[ t(k-1) = t(k-2) + 1. \] Replacing \( t(k-1) \) in the initial recurrence relation, we get: \[ t(k) = t(k-2) + 1 + 1. \] Repeating this process \( k \) times, we get \( t(k) = t (0) + k = k \). This shows that the number of rounds in the tournament is given by \( k \), where \( k \) is such that \( n = 2^k \), the number of teams.

Key concepts

Elimination Tournament

An elimination tournament is a competitive format where participants are eliminated after a loss, ensuring only one winner emerges. In this specific type of tournament, each round consists of games matching up remaining competitors. Half of the competitors are eliminated each round until one remains.
- For example, consider having \( n = 64 \) teams. Immediately, you recognize this is an elimination tournament because each match ends in a knock-out.
- Such a format is straightforward because each team must win every game to proceed, and matches decrease as teams are eliminated.
- This simplicity makes it easy to predict the total number of rounds because for \( n \) competitors, there will be \( \log_2 (n) \) rounds. This formula arises because it always takes a certain number of rounds, effectively halving the pool until only one team remains.Thus, the elimination tournament ensures a clear and decisive resolution to a competitive event, making it popular in many sports and competitive scenarios.

Mathematical Induction

Mathematical induction is a critical principle in proving statements across mathematics, especially regarding sequences or propositions indexed over the natural numbers. It's akin to dominoes falling—in that it lays a foundation and repeatedly builds upon it.
- Consider you're tasked with proving that for an elimination tournament with \( n = 2^k \) teams, there are indeed \( k \) rounds. Induction is a tool tailored for such proofs.
- It initiates with a base case that's usually unmistakably simple—here observing a viable base is \( k = 0 \), i.e., one team and zero rounds.
- Moving to the induction step requires showing if a statement holds for \( k \), it must also hold for \( k + 1 \). This continuation ensures no gaps through logical attempts that lead towards your generalized conclusion.Each step in induction is carefully constructed to form a robust mechanism to prove solutions reliably, vital in tackling any recurrence based strategies.

Algorithm Analysis

Algorithm analysis is invaluable for understanding how well a process or set of operations performs, especially in terms of time and space efficiency. Whether determining strategies in a tournament or evaluating software performance, clarity on underlying algorithms helps assess efficiency.
- Take, for instance, a recurrence relation such as \( t(k) = t(k-1) + 1 \). This simple relationship hints toward its efficiency, translating to logarithmic growth \( \log_2(n) \).
- Analyzing algorithms also involves identifying Big O notation to describe how performance or costs grow as inputs expand. For our tournament example, the rounds grow linearly with \( k \), implying \( O(k) \) complexity.
- A good analysis identifies potential bottlenecks or inefficiencies within algorithms, thus ensuring effective execution even at scale, which is crucial for making informed decisions about which algorithm suits specific tasks.Through careful examination, algorithm analysis shines a spotlight on making competitive systems like tournaments both effective and fair.

Exponential Growth

Exponential growth is a phenomenon where quantities double over regular intervals, leading to rapid increases. In many domains, exponential progression signifies intense change, often starting unnoticeable and crescendoing rapidly.
- In our context of a tournament, initial stages see large numbers of competitors, but each round sees these halved—a form of exponential decrease.
- Conversely, if considering the growth of rounds as teams increase—since each doubling of teams demands an additional round to eliminate them, this clarifies how increases in \( n \) quickly escalate the rounds required.
- Mathematicians traditionally express exponential growth through equations like \( n = 2^k \), indicating such as in our tournament, \( k \) represents rounds growing alongside team numbers.Understanding exponential patterns is pivotal in recognizing rapid sequences and aptly predicting developments in competitive structures or even real-world scenarios beyond mathematics.