1

CS 2710, ISSP 2610

Chapter 4, Part 1Heuristic Search

2

Heuristic Search

• Take advantage of information about the problem

3

Best-First-Search

• More general use of the term than in the 1st edition of the text

• An evaluation function f is used to determine the ordering of nodes on the fringe (there are variations, depending on the search algorithm)

4

Best-First-Search

• In our framework: – treesearch or graphsearch, with

nodes ordered on the fringe in increasing order by an evaluation function, f(n).

5

def treesearch (qfun,fringe): while len(fringe) > 0: cur = fringe[0] fringe = fringe[1:] if goalp(cur): return cur fringe = qfun(makeNodes(successors(cur)),fringe) return []

best-first search: qfun appends the liststogether and sorts them in increasing order by f-value

[In the more efficient version, a heap is used to maintain the queue in increasing order by f-value]

6

Heuristic Evaluation Function, h(n)

• There is a family of best-first search algorithms with different evaluation functions, f(n)

• A key component is the “heuristic evaluation function”, h(n)

7

h(n)

• Metric on states. Estimate of shortest distance to some goal.

• h : state estimate of distance to goal

• h (goal) = 0 for all goal nodes

8

Greedy Best-First Search

• f (n) = h (n)• Greedy best-first search may

switch its strategy mid-search. For example, it may go depth-first for awhile, but then return to the shallow parts of the tree.

9

Greedy Example

• In the map domain, h(n) could be the straight line distance from a city to Bucharest

• Greedy search expands the node that currently appears to be closest to the goal

10

Go from Arad to BucharestOradea

Zerind

AradSibiu

Timisoara

Lugoj

Mehadia

Dobreta

Rimnicu Vilcea

Fagaras

Craiova

Pitesti

Giurgiu

Bucharest

Urziceni

Vaslui

Iasi

Neamt

Hirsova

Eforie

71

75

151

140

118

99

80

97

146

138

120

75

70

111101

90

211

85

366

329

374

380

253 176

0

193

160

244

241

11

Greedy Example

Arad 366

Sibiu 253Zerind 374 Timisoara 329

Arad 366 Oradea 380 Fagaras 178 Rimniciu 193

Bucharest 0Sibiu 253

12

Greedy Search

• Complete?– Nope

• Optimal?– Nope

• Time and Space?– It depends

13

Best of Both

• In an A* search we combine the best parts of Uniform-Cost and Best-First.

• We want to use the cost so-far to allow optimality and completeness, while at the same time using a heuristic to draw us toward a goal.

14

A*: f(n) = g(n) + h(n)

g(n): actual cost from start to nh(n): estimated distance from n.state to a

goalEven if h continuously returns good

values for states along a path, if no goal is reached, g will eventually dominate h and force backtracking to more shallow nodes.

15

Arad 646 Oradea526 Fagaras 417 Rimniciu 413

Arad 366

Sibiu 393Zerind 449 Timisoara 447

Bucharest 450Sibiu 591 Sibiu 553Pitesti 415Craiova 526

Bucharest418

Craiova615

Rimniciu607

16

A*: f(n) = g(n) + h(n)

• If h(n) does not overestimate the real cost then the search is optimal.

• An h function that does not overestimate is called admissible

17

A* with an admissible heuristic is optimal

• Let: G2 be a suboptimal goal on fringe and GO be an optimal goal, g(GO) = C*

• C* < g(G2) (since G2 is suboptimal)

• h(G2) = 0 (since G2 is a goal)• So f(G2) = g(G2) and • C* < f(G2)

18

Proof continued

• Let n be a node on the fringe that is on an optimal solution path

• Since h is admissible: f(n) = g(n) + h(n) <= C*

• For G2 to be the first goal found, it would need to be first on the fringe

• But f(n) <= C* < f(G2)

19

Proof continued

• Is it possible that g2 is the first node on the fringe but there is no such node n on the fringe?

• No: by virtue of how it is generated, the search tree is a connected graph, and start is an ancestor of both n and g2. Let p be the first node on the path from start to g2 such gval(p) > C* (this could be g2). The ancestors of n all have f-vals <= C* (since h is admissible). So, it isn’t possible for p to be ordered before those ancestors on the fringe.

20

A* with an admissible heuristic is complete

• If it is guaranteed to find the optimal solution, it is guaranteed to find a solution

21

A* and Memory

• Does A* solve the memory problems with BFS and Uniform Cost?– A* has same or smaller memory

requirement than BFS or Uniform Cost – How is A* related to BFS And UC?– BFS = A* with edgecost(n) = 1, h(n) = 0– UC = A* with h(n) = 0– But it might not be sufficiently better to

make A* practically feasible

22

Note

• Placement of goalp test (and return if successful) in algorithm is critical.

• Optimality guarantee lost if nodes are tested when they are generated– The only specification successor

function must meet is that it return all legal successors of its input

23

Note for A*

• Assuming f-costs are nondecreasing along any path:– Can draw contours in the state space– Inside a contour labeled 300 are all nodes

with f(n) less than or equal to 300– A* fans out from start, expanding nodes in

bands of increasing f-cost.– h(n) = 0: contours are round– With better heuristics, the bands narrow

and stretch toward the goal node

24

EG Admissible Heuristics

The 8-puzzle (a small version of the 15 puzzle).

Sample heuristicsNumber of misplaced tilesManhattan distance

25

8 Puzzle Example

• H1(S) = 7• H2(S) = 2+3+3+2+4+2+0+2 = 18Which heuristic is better?

26

Informedness

• Let h1 and h2 be admissible heuristics. If h1(n) <= h2(n) for all n, then h2 is more informed than h1 and

• Fewer nodes will be expanded, on average, with h2 than with h1

• The larger values the better (without going over)

27

A* is often not feasible

• Still a memory hog• What can we do?• Use an iterative deepening style

strategy!

28

IDA*

• Like iterative deepening, but search to f-contours rather than fixed depths.

• Each iteration expands all nodes within a particular f-value range.

29

Def fLimSearch(fringe,fLim): nextF = INFINITY while fringe: cur = fringe[0] fringe = fringe[1:] curF = cur.gval + h(cur) if curF <= fLim: if goalp(cur): return(cur,curF) succNodes = makeNodes(cur,successors(cur)) for s in succNodes: fVal = s.gval + h(s) if fVal > fLim and fVal < nextF: nextF = fVal fringe = succNodes + fringe return ([],nextF)

30

def IDAstar(start): result = [] startNode = Node(start) fLim = h(startNode) while not result: result, FLim = fLimSearch([startNode],fLim)

31

IDA*

• Worst case, space is O(bd)• Optimal, if h is admissible • The number of iterations grows as

the number of possible f values grow. Let x = average # nodes with the same f-value. The lower x is, the fewer new nodes, on average, are expanded on each iteration.

32

General Notes before Continuing

33

Search strategies differ along many dimensions

• Basic strategy: depth-first, breadth-first, least-actual-cost (g(n)), best first (h(n)), or a mixture?

• Is the algorithm iterative, starting by looking at a small part of the state space and then successively looking at larger parts of it? (e.g., iterative deepening and IDA*)

34

Search strategies differ along many dimensions

• Does it pay attention to cycles? (i.e., our treesearch vs. graphsearch)

• Can it backtrack? Or are parts of the search tree/graph irrevocably pruned? (e.g., beam search)

• Does it only look ahead toward goal, or does it also consider how far it has come so far?

35

A note on optimality

• It might be desirable to be greedy (e.g., greedy best-first vs. A*)

• Simon: people are often “satisficers”: often, they stop as soon as they find a satisfactory solution

• Consider choosing a line at the grocery store, or finding a parking space

36

Another note on optimality

• Distinguish between correctness of h(n) and the optimality of the search.

• An optimal search may use an incorrect h(n)!

• In fact, entirely correct h(n) functions are rare (otherwise, why perform heuristic search?)

37

What do we hope to gain by using h(n)?

• Now that you have seen a few types of best-first search, we can ask: what do we hope to gain by using a heuristic evaluation function?

• Ans: reduce the number of nodes explored before finding a solution