Combining Olympic Sports with Math and AI: 5 Key Questions
School
Simon Fraser University**We aren't endorsed by this school
Course
MATH 301
Subject
Computer Science
Date
Dec 11, 2024
Pages
7
Uploaded by LieutenantClover14697
Here are 5 questions that combine Olympic sports, mathematics, and computer science concepts, along with their answers:1. Question: In Olympic archery, an AI system is being developed to predict arrow trajectories. Design an algorithm that uses computer vision and machine learning to analyze an archer's form and predict the arrow's landing position. What key features would you extract, and how would you handle real-time processing?Answer: Key features to extract:- Bow angle and orientation- Archer's body posture and alignment- Release timing and smoothness- Wind speed and direction (from external sensors)Algorithm outline:1. Use computer vision to capture and track the archer's form in real-time.2. Extract the key features mentioned above using image processing techniques.3. Input these features into a pre-trained machine learning model (e.g., a neural network or random forest) that predicts the arrow's trajectory and landing position.4. Utilize parallel processing to handle multiple archers simultaneously.5. Implement a sliding window approach to continuously update predictions as the archer prepares to shoot.For real-time processing:- Use GPU acceleration for computer vision tasks- Optimize the machine learning model for inference speed (e.g., model pruning, quantization)
- Implement efficient data streaming and buffering to handle continuous input2. Question: In Olympic sprinting, develop a mathematical model and algorithm to determine the optimal strategy for a relay team to minimize the total race time. Consider factors such as individual runner speeds, acceleration profiles, and baton exchange zones. How would you use dynamic programming to solve this optimization problem?Answer:Mathematical model:Let t_i(s) be the time function for runner i, where s is the distance run.Let e_ij be the time taken for baton exchange between runners i and j.Total race time T = t_1(s_1) + e_12 + t_2(s_2) + e_23 + t_3(s_3) + e_34 + t_4(s_4)where s_1 + s_2 + s_3 + s_4 = total race distanceDynamic programming approach:1. Discretize the race distance into small segments.2. Define a state as (runner, distance_covered).3. For each state, calculate the optimal remaining time using the recurrence relation:OPT(i, d) = min{t_i(s) + e_i(i+1) + OPT(i+1, d+s)} for all valid swhere i is the current runner, d is the distance covered so far, and s is the distance run by the current runner.4. Use memoization to store and reuse computed results.5. Backtrack to determine the optimal distances for each runner.
The algorithm would consider the acceleration profiles of each runner to determine optimal exchange points within the allowed zones, balancing the strengths of each team member.3. Question: For the Olympic decathlon, create a data structure and algorithm to efficiently store and query athlete performances across all ten events. Your system should support fast updates as events conclude and quick calculation of overall standings. How would you optimize for both space and time complexity?Answer:Data structure:Use a combination of a hash table and a balanced binary search tree (e.g., Red-Black tree).- Hash table: Key = Athlete ID, Value = Pointer to athlete's node in the BST- BST: Ordered by total points, each node contains:- Athlete ID- Array of 10 event scores- Total points- Pointers to left and right childrenAlgorithm:1. For updates:- Use the hash table to find the athlete's node in O(1) time- Update the event score and recalculate total points- Rebalance the BST if necessary (O(log n) time)
2. For querying standings:- Perform an in-order traversal of the BST (O(n) time)- To get top k athletes, use a partial traversal (O(k) time)3. For calculating overall standings:- Maintain a running total of points in each node- Update this total whenever an event score changesSpace complexity: O(n), where n is the number of athletesTime complexity: - Updates: O(log n)- Querying full standings: O(n)- Querying top k: O(k)Optimizations:- Use a cache to store frequently accessed standings- Implement lazy propagation for updates to amortize the cost of rebalancing- Use parallel processing for batch updates and large queries4. Question: In Olympic swimming, design a system using computer vision and fluid dynamics to analyze a swimmer's technique. How would you use the Navier-Stokes equations to model water flow around the swimmer, and what algorithm would you employ to optimize the swimmer's body position for minimal drag?Answer:System design:1. Use underwater cameras to capture the swimmer's motion
2. Apply computer vision techniques to create a 3D model of the swimmer3. Use this model as input for computational fluid dynamics (CFD) simulationsNavier-Stokes application:Use the incompressible Navier-Stokes equations to model water flow:∇· u = 0ρ(∂u/∂t + u · ∇u) = -∇p + μ∇²u + FWhere u is velocity, ρ is density, p is pressure, μ is viscosity, and F represents external forces.Solve these equations numerically using finite element or finite volume methods, with the swimmer's body as a boundary condition.Optimization algorithm:1. Parameterize the swimmer's body position (e.g., joint angles, body curvature)2. Use a gradient-based optimization method like adjoint optimization:a. Compute the adjoint of the Navier-Stokes equationsb. Use this to efficiently calculate gradients of drag with respect to body position parametersc. Update body position using gradient descent3. Implement a multi-start approach to avoid local minima4. Use parallel processing to evaluate multiple body positions simultaneouslyThe system would iterate between CFD simulations and body position updates, converging on an optimal low-drag configuration.
5. Question: For Olympic weightlifting, develop a machine learning model to predict an athlete's likelihood of successfully completing a lift based on real-time sensor data. What features would you extract, what type of model would you use, and how would you handle the time-series nature of the data?Answer:Features to extract:1. Bar path (vertical and horizontal displacement)2. Bar velocity and acceleration3. Ground reaction forces4. Joint angles (ankle, knee, hip, shoulder)5. Center of mass trajectory6. Muscle activation patterns (if EMG data available)Model selection:Use a Recurrent Neural Network (RNN), specifically a Long Short-Term Memory (LSTM) network, to handle the time-series nature of the data.Data handling and model architecture:1. Preprocess sensor data:- Normalize and scale features- Use sliding windows to create fixed-length input sequences2. LSTM architecture:- Input layer: Dimensionality matches the number of features- 2-3 LSTM layers with dropout for regularization
- Dense output layer with sigmoid activation for binary classification (success/failure)3. Training:- Use historical lift data, balanced for successful and failed attempts- Implement early stopping and learning rate scheduling- Use cross-validation with a time-based split to prevent data leakage4. Real-time prediction:- Feed live sensor data into the model as the lift progresses- Output updated success probability at each time step5. Post-processing:- Implement a smoothing function to reduce prediction volatility- Set a threshold for alerting coaches or athletes about potential failed liftsThis model would provide real-time feedback on lift quality and could be used to prevent injuries by identifying potentially dangerous lift attempts early in the movement.