Faster than Weighted A*: An Optimistic Approach to Bounded - PowerPoint PPT Presentation

Faster than Weighted A*: An Optimistic Approach to Bounded Suboptimal Search Jordan Thayer and Wheeler Ruml { jtd7, ruml } at cs.unh.edu Jordan Thayer (UNH) Optimistic Search – 1 / 45

Motivation Finding optimal solutions is prohibitively expensive. ■ Introduction ■ Motivation Algorithm Overview 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Grid Pathfinding Grid Pathfinding A* A* Further Observations 200,000 Conclusion 1.6 Solution Cost (relative to A*) Nodes generated 1.4 100,000 1.2 0 1.0 200 400 600 800 1,000 200 400 600 800 1,000 Problem Size Problem Size Jordan Thayer (UNH) Optimistic Search – 2 / 45

Motivation Finding optimal solutions is prohibitively expensive. ■ Introduction Greedy solutions can be arbitrarily bad. ■ Motivation ■ Algorithm Overview 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Grid Pathfinding Four-way Grid Pathfinding (Unit cost) A* A* Further Observations 200,000 Conclusion Greedy Greedy 1.6 Solution Cost (relative to A*) Nodes generated 1.4 100,000 1.2 0 1.0 200 400 600 800 1,000 200 400 600 800 1,000 Problem Size Problem Size Jordan Thayer (UNH) Optimistic Search – 3 / 45

Motivation Finding optimal solutions is prohibitively expensive. ■ Introduction Greedy solutions can be arbitrarily bad. ■ Motivation ■ Algorithm Overview Weighted A* bounds suboptimality. ■ 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Grid Pathfinding Grid Pathfinding A* A* Further Observations 200,000 wA* wA* Conclusion Greedy Greedy 1.6 Solution Cost (relative to A*) Nodes generated 1.4 100,000 1.2 0 1.0 200 400 600 800 1,000 200 400 600 800 1,000 Problem Size Problem Size Jordan Thayer (UNH) Optimistic Search – 4 / 45

Motivation Finding optimal solutions is prohibitively expensive. ■ Introduction Greedy solutions can be arbitrarily bad. ■ Motivation ■ Algorithm Overview Weighted A* bounds suboptimality. ■ 1: Greedy Phase Optimistic Search: faster search within the same bound. ■ 2: Cleanup Phase Empirical Evaluation Grid Pathfinding Grid Pathfinding A* A* Further Observations 200,000 wA* wA* Optimistic Optimistic Conclusion Greedy Greedy 1.6 Solution Cost (relative to A*) Nodes generated 1.4 100,000 1.2 0 1.0 200 400 600 800 1,000 200 400 600 800 1,000 Problem Size Problem Size Jordan Thayer (UNH) Optimistic Search – 5 / 45

Introduction Algorithm Overview ■ Predecessors ■ Basic Idea 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Algorithm Overview Further Observations Conclusion Jordan Thayer (UNH) Optimistic Search – 6 / 45

Talk Outline Algorithm Overview ■ Introduction Run weighted A ∗ with a weight higher than the bound. Algorithm Overview ■ Predecessors Expand additional nodes to prove solution quality. ■ Basic Idea The Greedy Search Phase ■ 1: Greedy Phase The Cleanup Phase ■ 2: Cleanup Phase Empirical Evaluation ■ Empirical Evaluation Further Observations ■ Further Observations Conclusion Jordan Thayer (UNH) Optimistic Search – 7 / 45

Previous Algorithms: A ∗ A best first search expanding nodes in f order. ■ Introduction f ( n ) = g ( n ) + h ( n ) ■ Algorithm Overview ■ Predecessors If h ( n ) is admissible, returns optimal solution. ■ Basic Idea 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Further Observations Conclusion Jordan Thayer (UNH) Optimistic Search – 8 / 45

Previous Algorithms: Weighted A ∗ A best first search expanding nodes in f ′ order. ■ Introduction f ′ ( n ) = g ( n ) + w · h ( n ) ■ Algorithm Overview ■ Predecessors Solution quality bounded by w for admissible h ( n ) . ■ Basic Idea 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Further Observations Conclusion Jordan Thayer (UNH) Optimistic Search – 9 / 45

Optimistic Search: The Basic Idea Run weighted A ∗ with a high weight. 1. Introduction 2. Expand node with lowest f value after a solution is found. Algorithm Overview ■ Predecessors Continue until w · f min > f ( sol ) ■ Basic Idea This ’clean up’ guarantees solution quality. 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Further Observations Conclusion Jordan Thayer (UNH) Optimistic Search – 10 / 45

Optimistic Search: The Basic Idea Run weighted A ∗ with a high weight. 1. Introduction 2. Expand node with lowest f value after a solution is found. Algorithm Overview ■ Predecessors Continue until w · f min > f ( sol ) ■ Basic Idea This ’clean up’ guarantees solution quality. 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Further Observations Conclusion Jordan Thayer (UNH) Optimistic Search – 11 / 45

Optimistic Search: The Basic Idea Run weighted A ∗ with a high weight. 1. Introduction 2. Expand node with lowest f value after a solution is found. Algorithm Overview ■ Predecessors Continue until w · f min > f ( sol ) ■ Basic Idea This ’clean up’ guarantees solution quality. 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Further Observations Conclusion Jordan Thayer (UNH) Optimistic Search – 12 / 45

Optimistic Search: The Basic Idea Run weighted A ∗ with a high weight. 1. Introduction 2. Expand node with lowest f value after a solution is found. Algorithm Overview ■ Predecessors Continue until w · f min > f ( sol ) ■ Basic Idea This ’clean up’ guarantees solution quality. 1: Greedy Phase 2: Cleanup Phase Empirical Evaluation Further Observations Conclusion Jordan Thayer (UNH) Optimistic Search – 13 / 45

Introduction Algorithm Overview 1: Greedy Phase ■ Weighted A ∗ 2: Cleanup Phase Empirical Evaluation Further Observations 1: Greedy Phase Conclusion Jordan Thayer (UNH) Optimistic Search – 14 / 45

Talk Outline Algorithm Overview ■ Introduction The Greedy Search Phase ■ Algorithm Overview Weighted A ∗ becomes faster as the bound grows. 1: Greedy Phase Weighted A ∗ is often better than the bound. ■ Weighted A ∗ 2: Cleanup Phase The Cleanup Phase ■ Empirical Evaluation Empirical Evaluation ■ Further Observations Further Observations ■ Conclusion Jordan Thayer (UNH) Optimistic Search – 15 / 45

Large Bounds, Faster Solution wA ∗ returns solutions faster as the bound increases. ■ Introduction Algorithm Overview 1: Greedy Phase ■ Weighted A ∗ 2: Cleanup Phase Pearl and Kim Hard Empirical Evaluation Further Observations 0.9 Conclusion wA* Node Generations Relative to A* 0.6 0.3 0.0 1.0 1.1 1.2 Sub-optimality bound Jordan Thayer (UNH) Optimistic Search – 16 / 45

Weighted A ∗ is often better than the bound wA ∗ returns solutions better than the bound. ■ Introduction Algorithm Overview 1: Greedy Phase ■ Weighted A ∗ 2: Cleanup Phase Four-way Grid Pathfinding (Unit cost) Empirical Evaluation 3 y=x Further Observations wA* Conclusion Solution Cost (relative to A*) 2 1 1 2 3 Sub-optimality Bound Jordan Thayer (UNH) Optimistic Search – 17 / 45

Introduction Algorithm Overview 1: Greedy Phase 2: Cleanup Phase ■ w -Admissibility Empirical Evaluation Further Observations 2: Cleanup Phase Conclusion Jordan Thayer (UNH) Optimistic Search – 18 / 45

Talk Outline Algorithm Overview ■ Introduction The Greedy Search Phase ■ Algorithm Overview The Cleanup Phase ■ 1: Greedy Phase Expand additional nodes in f order. 2: Cleanup Phase ■ w -Admissibility Quit when the solution is provably within the bound. Empirical Evaluation Empirical Evaluation ■ Further Observations Further Observations ■ Conclusion Jordan Thayer (UNH) Optimistic Search – 19 / 45

Proving w -Admissibility p is the deepest node on an ■ Introduction optimal path to opt. Algorithm Overview f min is the node with the ■ 1: Greedy Phase smallest f value. 2: Cleanup Phase ■ w -Admissibility Empirical Evaluation Further Observations Conclusion Jordan Thayer (UNH) Optimistic Search – 20 / 45

Proving w -Admissibility p is the deepest node on an ■ Introduction optimal path to opt. Algorithm Overview f min is the node with the ■ 1: Greedy Phase smallest f value. 2: Cleanup Phase ■ w -Admissibility Empirical Evaluation Further Observations Conclusion f ( p ) ≤ f ( opt ) f ( f min ) ≤ f ( p ) f min provides a lower bound on solution cost. Determine f min by priority queue sorted on f Jordan Thayer (UNH) Optimistic Search – 20 / 45

Proving w -Admissibility p is the deepest node on an ■ Introduction optimal path to opt. Algorithm Overview f min is the node with the ■ 1: Greedy Phase smallest f value. 2: Cleanup Phase ■ w -Admissibility Empirical Evaluation Further Observations Conclusion f ( p ) ≤ f ( opt ) f ( f min ) ≤ f ( p ) f min provides a lower bound on solution cost. Determine f min by priority queue sorted on f Optimistic Search: Run a greedy search Expand f min until w · f min ≥ f ( sol ) Jordan Thayer (UNH) Optimistic Search – 20 / 45