Heuristic Search: Greedy, A* & IDA* for Optimal Solutions - Prof. Dana S. Nau, Study notes of Computer Science

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Last update: September 9, 2008

Informed search algorithms

CMSC 421: Chapter 4, Sections 1–

Motivation

♦ In Chapter 3 we were talked about trial-and-error search

♦ In the worst case, most searches take exponential time (unless P=NP)

♦ Can sometimes do much better on the average, using heuristic techniques

heuristic: A rule of thumb, simplification, or educated guess that reduces or limits the search for solutions in domains that are difficult and poorly understood.

Depending on what heuristic you use, you won’t necessarily find an optimal solution, or even a solution at all.

Heuristic search

function Tree-Search( problem) returns a solution, or failure fringe ← a list containing Make-Node(Initial-State[problem]) loop do if fringe is empty then return failure node ← Remove-Front(fringe) if Goal-Test[problem] applied to State(node) succeeds return node fringe ← InsertAll(Expand(node, problem), fringe)

Heuristic choice in search algorithms: what node to expand next Use an evaluation function for each node – estimate of “desirability” ⇒ Expand most desirable unexpanded node

This depends on InsertAll Want to keep fringe sorted in decreasing order of desirability

Heuristic search on graphs

function Graph-Search( problem, fringe) returns a solution, or failure closed ← an empty set fringe ← Insert(Make-Node(Initial-State[problem]), fringe) loop do if fringe is empty then return failure node ← Remove-Front(fringe) if Goal-Test(problem, State[node]) then return node if State[node] is not in closed then add State[node] to closed fringe ← InsertAll(Expand(node, problem), fringe) end

Same as for Tree-Search: Use InsertAll to keep fringe sorted in decreasing order of desirability

Greedy search

Evaluation function h(n) (heuristic) = estimate of cost from n to the closest goal

E.g., hSLD(n) = straight-line distance from n to Bucharest

Greedy search keeps fringe ordered in increasing value of h, hence always expands whatever node appears to be closest to a goal

Greedy search example

Arad 366 ↗ straight-line distance to Bucharest

Greedy search example

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara 329 374

366 176 380 193

Greedy search example

Rimnicu Vilcea

Zerind

Arad

Sibiu

Arad Fagaras Oradea

Timisoara

Sibiu Bucharest

329 374

366 380 193

253 0

Properties of greedy search

Complete? No. Can get stuck in loops: Iasi → Neamt → Iasi → Neamt → Complete in finite space with repeated-state checking

Time?

Properties of greedy search

Complete? No. Can get stuck in loops: Iasi → Neamt → Iasi → Neamt → Complete in finite space with repeated-state checking

Time? O(bm), but a good heuristic can give dramatic improvement

Space?

Properties of greedy search

Complete? No. Can get stuck in loops: Iasi → Neamt → Iasi → Neamt → Complete in finite space with repeated-state checking

Time? O(bm), but a good heuristic can give dramatic improvement

Space? O(bm)—keeps all nodes in memory

Optimal? No

Problem with terminology: Greedy search is not the same as an ordinary greedy algorithm.

An ordinary greedy algorithm doesn’t remember all of fringe. It remembers only the current path, and never backtracks. Hence: ♦ Repeated-state checking cannot make it complete ♦ It runs in time O(l) if it finds a solution of length l

A∗^ search

Idea: avoid expanding paths that are already expensive

Evaluation function f (n) = g(n) + h(n)

g(n) = cost so far to reach n h(n) = estimated cost to goal from n f (n) = estimated total cost of path through n to goal

First requirement for A∗^ search:

A∗^ needs an admissible heuristic i.e., 0 ≤ h(n) ≤ h∗(n) where h∗(n) is the true cost from n. (Thus h(G) = 0 for any goal G.)

E.g., hSLD(n) never overestimates the actual road distance

Romania with step costs in km

Recall that we want to get from Arad to Bucharest:

Bucharest

Giurgiu

Urziceni

Hirsova

Eforie

Neamt Oradea

Zerind

Arad

Timisoara

Lugoj Mehadia

Dobreta

Craiova

Sibiu

Fagaras

Pitesti Rimnicu Vilcea

Vaslui

Iasi

Straight−line distanceto Bucharest

0 160 242 161 77 151

241

366

193

178

253 329 80 199

244

380

226

234

374

98

Giurgiu

Urziceni Hirsova

Eforie

Neamt

Oradea

Zerind

Arad

Timisoara Lugoj

Mehadia

Dobreta Craiova

Sibiu (^) Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111 70

75 120

151 140 99 80

97 101

211

138

146 85

90

98

142

92

87

86

A∗^ search

Arad 366=0+