Heuristic Functions - Interactive Basics

Learn how to calculate Euclidean and Manhattan distances on a grid

Click any cell to explore heuristic values • Foundation for A* Search

Select Heuristic Function

Euclidean Distance
Straight-line distance
√[(x₂-x₁)² + (y₂-y₁)²]
Manhattan Distance
City-block distance
|x₂-x₁| + |y₂-y₁|
Compare Both
Side-by-side view
Both calculations
Interactive Learning

Click any cell to see how heuristic distances are calculated from that position to the goal.

  • Red cell: Goal position (fixed at coordinates 8,2)
  • Yellow cell: Your selected current position
  • Color intensity: Shows distance values (light = close, dark = far)
Tip: Try clicking cells at different distances to see how the calculations work!

Euclidean Distance Heuristic

Calculates the straight-line distance between two points, like measuring with a ruler on a map. This is the shortest possible distance "as the crow flies".

Interactive Controls
Instructions: Click any white cell to calculate heuristic distance to the red goal cell.
Legend
Goal
Selected
Close
Far
Step-by-Step Calculation
Click any cell to see calculation
Current Selection

No cell selected

Distance Comparison
Position Euclidean Manhattan Difference
Click cells to compare distances
Quick Facts

Euclidean: Always ≤ Manhattan distance

Manhattan: Only horizontal/vertical moves

Equal when: Moving only horizontally OR vertically

Biggest difference: On pure diagonals

Understanding Heuristic Functions

What are Heuristics?

Heuristics are "educated guesses" about the cost to reach a goal from any position.

Key Properties:
  • Estimate: Not exact, but reasonable guess
  • Fast: Quick to calculate
  • Informative: Guide search toward goal

Why use them? Help AI algorithms like A* search more efficiently by focusing on promising directions.

Euclidean Distance
h = √[(x₂-x₁)² + (y₂-y₁)²]
When to use:
  • Movement allowed in any direction
  • Diagonal movement permitted
  • Flying, swimming, or direct movement
Characteristics:
  • More optimistic: Lower estimates
  • Geometric: Based on coordinate geometry
  • Continuous: Real-valued results
Manhattan Distance
h = |x₂-x₁| + |y₂-y₁|
When to use:
  • Movement in 4 directions only (↑↓←→)
  • Grid-based worlds
  • City street navigation
Characteristics:
  • More conservative: Higher estimates
  • Discrete: Based on grid steps
  • Admissible: Never overestimates
Real-World Applications
Euclidean Distance Examples:
  • GPS Navigation: Straight-line distance to destination
  • Video Games: Character movement in open worlds
  • Robotics: Flying drones pathfinding
  • Computer Graphics: 2D/3D animation and rendering
Manhattan Distance Examples:
  • Urban Planning: City block distances
  • Puzzle Games: Sliding puzzles, maze solving
  • Circuit Design: Wire routing on chips
  • Data Mining: Feature distance in machine learning
Key Takeaway

The choice of heuristic function depends on your problem constraints. Euclidean works for free movement, Manhattan for grid-constrained movement. Understanding these basics prepares you for advanced AI search algorithms like A*!