Greedy Random Start Algorithms: From TSP to Daily Life
03/10/25 • 16 min
Greedy Random Start Algorithms: From TSP to Daily Life
Key Algorithm Concepts
Computational Complexity Classifications
- Constant Time O(1): Runtime independent of input size (hash table lookups)
- "The holy grail of algorithms" - execution time fixed regardless of problem size
- Examples: Dictionary lookups, array indexing operations
- Logarithmic Time O(log n): Runtime grows logarithmically
- Each doubling of input adds only constant time
- Divides problem space in half repeatedly
- Examples: Binary search, balanced tree operations
- Linear Time O(n): Runtime grows proportionally with input
- Most intuitive: One worker processes one item per hour → two items need two workers
- Examples: Array traversal, linear search
- Quadratic O(n2), Cubic O(n3), Exponential O(2n): Increasingly worse runtime
- Quadratic: Nested loops (bubble sort) - practical only for small datasets
- Cubic: Three nested loops - significant scaling problems
- Exponential: Runtime doubles with each input element - quickly intractable
- Factorial Time O(n!): "Pathological case" with astronomical growth
- Brute-force TSP solutions (all permutations)
- 4 cities = 24 operations; 10 cities = 3.6 million operations
- Fundamentally impractical beyond tiny inputs
Polynomial vs Non-Polynomial Time
- Polynomial Time (P): Algorithms with O(nk) runtime where k is constant
- O(n), O(n2), O(n3) are all polynomial
- Considered "tractable" in complexity theory
- Non-deterministic Polynomial Time (NP)
- Problems where solutions can be verified in polynomial time
- Example: "Is there a route shorter than length L?" can be quickly verified
- Encompasses both easy and hard problems
- NP-Complete: Hardest problems in NP
- All NP-complete problems are equivalent in difficulty
- If any NP-complete problem has polynomial solution, then P = NP
- NP-Hard: At least as hard as NP-complete problems
- Example: Finding shortest TSP tour vs. verifying if tour is shorter than L
The Traveling Salesman Problem (TSP)
Problem Definition and Intractability
- Formal Definition: Find shortest possible route visiting each city exactly once and returning to origin
- Computational Scaling: Solution space grows factorially (n!)
- 10 cities: 181,440 possible routes
- 20 cities: 2.43×1018 routes (years of computation)
- 50 cities: More possibilities than atoms in observable universe
- Real-World Challenges:
- Distance metric violations (triangle inequality)
- Multi-dimensional constraints beyond pure distance
- Dynamic environment changes during execution
Greedy Random Start Algorithm
Standard Greedy Approach
- Mechanism: Always select nearest unvisited city
- Time Complexity: O(n2) - dominated by nearest neighbor calculations
- Memory Requirements: O(n) - tracking visited cities and current path
- Key Weakness: Extreme sensitivity to starting conditions
- Gets trapped in local optima
- Produces tours 15-25% longer than optimal solution
- Visual metaphor: Getting stuck in a valley instead of reaching mountain bottom
Random Restart Enhancement
- Core Innovation: Multiple independent greedy searches from different random starting cities
- Implementation Strategy: Run algorithm multiple times from random starting points, keep best result
- Statistical Foundation: Each restart samples different region of solution space
- Performance Improvement: Logarithmic improvement with iteration count
- Implementation Advantages:
- Natural parallelization with minimal synchronization
- Deterministic runtime regardless of problem instance
- No parameter tuning required unlike metaheuristics
Real-World Applications
Urban Navigation
- Traffic Light Optimization: Avoiding getting stuck at red lights
- Greedy approach: When facing red light, turn right if that's green
- Local optimum trap: Always choosing "shortest next segment"
- Random restart equivalent: Testing multiple routes from different entry points
- Implementation example: Navigation apps calculating multiple route options
...
Greedy Random Start Algorithms: From TSP to Daily Life
Key Algorithm Concepts
Computational Complexity Classifications
- Constant Time O(1): Runtime independent of input size (hash table lookups)
- "The holy grail of algorithms" - execution time fixed regardless of problem size
- Examples: Dictionary lookups, array indexing operations
- Logarithmic Time O(log n): Runtime grows logarithmically
- Each doubling of input adds only constant time
- Divides problem space in half repeatedly
- Examples: Binary search, balanced tree operations
- Linear Time O(n): Runtime grows proportionally with input
- Most intuitive: One worker processes one item per hour → two items need two workers
- Examples: Array traversal, linear search
- Quadratic O(n2), Cubic O(n3), Exponential O(2n): Increasingly worse runtime
- Quadratic: Nested loops (bubble sort) - practical only for small datasets
- Cubic: Three nested loops - significant scaling problems
- Exponential: Runtime doubles with each input element - quickly intractable
- Factorial Time O(n!): "Pathological case" with astronomical growth
- Brute-force TSP solutions (all permutations)
- 4 cities = 24 operations; 10 cities = 3.6 million operations
- Fundamentally impractical beyond tiny inputs
Polynomial vs Non-Polynomial Time
- Polynomial Time (P): Algorithms with O(nk) runtime where k is constant
- O(n), O(n2), O(n3) are all polynomial
- Considered "tractable" in complexity theory
- Non-deterministic Polynomial Time (NP)
- Problems where solutions can be verified in polynomial time
- Example: "Is there a route shorter than length L?" can be quickly verified
- Encompasses both easy and hard problems
- NP-Complete: Hardest problems in NP
- All NP-complete problems are equivalent in difficulty
- If any NP-complete problem has polynomial solution, then P = NP
- NP-Hard: At least as hard as NP-complete problems
- Example: Finding shortest TSP tour vs. verifying if tour is shorter than L
The Traveling Salesman Problem (TSP)
Problem Definition and Intractability
- Formal Definition: Find shortest possible route visiting each city exactly once and returning to origin
- Computational Scaling: Solution space grows factorially (n!)
- 10 cities: 181,440 possible routes
- 20 cities: 2.43×1018 routes (years of computation)
- 50 cities: More possibilities than atoms in observable universe
- Real-World Challenges:
- Distance metric violations (triangle inequality)
- Multi-dimensional constraints beyond pure distance
- Dynamic environment changes during execution
Greedy Random Start Algorithm
Standard Greedy Approach
- Mechanism: Always select nearest unvisited city
- Time Complexity: O(n2) - dominated by nearest neighbor calculations
- Memory Requirements: O(n) - tracking visited cities and current path
- Key Weakness: Extreme sensitivity to starting conditions
- Gets trapped in local optima
- Produces tours 15-25% longer than optimal solution
- Visual metaphor: Getting stuck in a valley instead of reaching mountain bottom
Random Restart Enhancement
- Core Innovation: Multiple independent greedy searches from different random starting cities
- Implementation Strategy: Run algorithm multiple times from random starting points, keep best result
- Statistical Foundation: Each restart samples different region of solution space
- Performance Improvement: Logarithmic improvement with iteration count
- Implementation Advantages:
- Natural parallelization with minimal synchronization
- Deterministic runtime regardless of problem instance
- No parameter tuning required unlike metaheuristics
Real-World Applications
Urban Navigation
- Traffic Light Optimization: Avoiding getting stuck at red lights
- Greedy approach: When facing red light, turn right if that's green
- Local optimum trap: Always choosing "shortest next segment"
- Random restart equivalent: Testing multiple routes from different entry points
- Implementation example: Navigation apps calculating multiple route options
...
Previous Episode
Hidden Features of Rust Cargo
Hidden Features of Cargo: Podcast Episode Notes
Custom Profiles & Build Optimization
Custom Compilation Profiles: Create targeted build configurations beyond dev/release
- [profile.quick-debug] opt-level = 1 # Some optimization debug = true # Keep debug symbols
- Usage: cargo build --profile quick-debug
- Perfect for debugging performance issues without full release build wait times
- Eliminates need for repeatedly specifying compiler flags manually
Profile-Guided Optimization (PGO): Data-driven performance enhancement
- Three-phase optimization workflow:# 1. Build instrumented version cargo rustc --release -- -Cprofile-generate=./pgo-data # 2. Run with representative workloads to generate profile data ./target/release/my-program --typical-workload # 3. Rebuild with optimization informed by collected data cargo rustc --release -- -Cprofile-use=./pgo-data
- Empirical performance gains: 5-30% improvement for CPU-bound applications
- Trains compiler to prioritize optimization of actual hot paths in your code
- Critical for data engineering and ML workloads where compute costs scale linearly
Workspace Management & Organization
Dependency Standardization: Centralized version control
- # Root Cargo.toml [workspace] members = ["app", "library-a", "library-b"] [workspace.dependencies]
serde = "1.0"
tokio = { version = "1", features = ["full"] }Member Cargo.toml
[dependencies]
serde = { workspace = true }- Declare dependencies once, inherit everywhere (Rust 1.64+)
- Single-point updates eliminate version inconsistencies
- Drastically reduces maintenance overhead in multi-crate projects
Dependency Intelligence & Analysis
Dependency Visualization: Comprehensive dependency graph insights
- cargo tree: Display complete dependency hierarchy
- cargo tree -i regex: Invert tree to trace what pulls in specific packages
- Essential for diagnosing dependency bloat and tracking transitive dependencies
Automatic Feature Unification: Transparent feature resolution
- If crate A needs tokio with rt-multi-thread and crate B needs tokio with macros
- Cargo automatically builds tokio with both features enabled
- Silently prevents runtime errors from missing features
- No manual configuration required—this happens by default
Dependency Overrides: Direct intervention in dependency graph
- [patch.crates-io] serde = { git = "https://github.com/serde-rs/serde" }
- Replace any dependency with alternate version without forking dependents
- Useful for testing fixes or working around upstream bugs
Build System Insights & Performance
Build Analysis: Objective diagnosis of compilation bottlenecks
- cargo build --timings: Generates HTML report visualizing:
- Per-crate compilation duration
- Parallelization efficiency
- Critical path analysis
- Identify high-impact targets for compilation optimization
Cross-Compilation Configuration: Target different architectures seamlessly
- # .cargo/config.toml [target.aarch64-unknown-linux-gnu] linker = "aarch64-linux-gnu-gcc" rustflags = ["-C", "target-feature=+crt-static"]
- Eliminates need for environment variables or wrapper scripts
- Particularly valuable for AWS Lambda ARM64 deployments
- Zero-configuration alternative: cargo zigbuild (leverages Zig compiler)
Testing Workflows & Productivity
Targeted Test Execution: Optimize testing efficiency
- Run ignored tests only: cargo test -- --ignored
- Mark resource-intensive tests with #[ignore] attribute
- Run selectively when needed vs. during routine testing
- Module-specific testing: cargo test module::submodule
- Pinpoint tests in specific code areas
- Critical for large projects where full test suite takes minutes
- Sequential execution: cargo test -- --test-threads=1
- Forces tests to run one at a time
- Essential for tests with shared state dependencies
Continuous Testing Automation: Eliminate manual test cycles
- Install automation tool: cargo install cargo-watch
- Continuous validation: cargo watch -x check -x clippy -x test
- Automatically runs validation suite on file changes
- Enables immediate feedback without manual test triggering
Advanced Compilation Techniques
Link-Time Optimization Refinement: Beyond boolean LTO settings
- [profile.release] lto = "thin" # Faste...
Next Episode
K-means basic intuition
Finding Hidden Groups with K-means Clustering
What is Unsupervised Learning?
Imagine you're given a big box of different toys, but they're all mixed up. Without anyone telling you how to sort them, you might naturally put the cars together, stuffed animals together, and blocks together. This is what computers do with unsupervised learning - they find patterns without being told what to look for.
K-means Clustering Explained Simply
K-means helps us find groups in data. Let's think about students in your class:
- Each student has a height (x)
- Each student has a weight (y)
- Each student has an age (z)
K-means helps us see if there are natural groups of similar students.
The Four Main Steps of K-means
1. Picking Starting Points
First, we need to guess where our groups might be centered:
- We could randomly pick a few students as starting points
- Or use a smarter way called K-means++ that picks students who are different from each other
- This is like picking team captains before choosing teams
2. Making Teams
Next, each student joins the team of the "captain" they're most similar to:
- We measure how close each student is to each captain
- Students join the team of the closest captain
- This makes temporary groups
3. Finding New Centers
Now we find the middle of each team:
- Calculate the average height of everyone on team 1
- Calculate the average weight of everyone on team 1
- Calculate the average age of everyone on team 1
- This average student becomes the new center for team 1
- We do this for each team
4. Checking if We're Done
We keep repeating steps 2 and 3 until the teams stop changing:
- If no one switches teams, we're done
- If the centers barely move, we're done
- If we've tried enough times, we stop anyway
Why Starting Points Matter
Starting with different captains can give us different final teams. This is actually helpful:
- We can try different starting points
- See which grouping makes the most sense
- Find patterns we might miss with just one try
Seeing Groups in 3D
Imagine plotting each student in the classroom:
- Height is how far up they are (x)
- Weight is how far right they are (y)
- Age is how far forward they are (z)
- The team/group is shown by color (like red, blue, or green)
The color acts like a fourth piece of information, showing which group each student belongs to. The computer finds these groups by looking at who's clustered together in the 3D space.
Why We Need Experts to Name the Groups
The computer can find groups, but it doesn't know what they mean:
- It might find a group of tall, heavier, older students (maybe athletes?)
- It might find a group of shorter, lighter, younger students
- It might find a group of average height, weight students who vary in age
Only someone who understands students (like a teacher) can say:
- "Group 1 seems to be the basketball players"
- "Group 2 might be students who skipped a grade"
- "Group 3 looks like our regular students"
The computer finds the "what" (the groups), but experts explain the "why" and "so what" (what the groups mean and why they matter).
The Simple Math Behind K-means
K-means works by trying to make each student as close as possible to their team's center. The computer is trying to make this number as small as possible:
"The sum of how far each student is from their team's center"
It does this by going back and forth between:
- Assigning students to the closest team
- Moving the team center to the middle of the team
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