1. Introduction to Artificial Intelligence (G51IAI)
Dr Rong Qu
Heuristic Searches

2. Blind Search vs. Heuristic Searches
Randomly choose where to search in the search tree
When problems get large, not practical any more
Heuristic search
Explore the node which is more likely to lead to the goal state
Quite often, using knowledge
Both based on tree search
blind, systematic, random choose which child node to go
Heuristic, use knowledge, go to the promising child node / branches
G51IAI - Heuristic

3. Heuristic Searches - Characteristics
Heuristic searches work by deciding which is the next best node to expand
Has some domain knowledge
Use a function to tell us how close the node is to the goal state
Usually more efficient than blind searches
Sometimes called an informed search
There is no guarantee that it is the best node
G51IAI - Heuristic

4. Heuristic Searches - Characteristics
Heuristic searches estimate the cost to the goal from its current position. It is usual to denote the heuristic evaluation function by h(n)
Compare this with something like Uniform Cost Search which chooses the lowest code node thus far ( g(n) )
UCS: only guarantee the lowest cost route from the root to the current
Heuristic: estimate the cost from the current node to the goal node – how?
G51IAI - Heuristic

5. Heuristic Searches - Characteristics
Heuristic searches vs. Uniform Cost Search
Uniform cost search
expand the path with the lowest path cost
chooses the lowest cost node thus far
Heuristic search
estimate how close the solution is to the goal
not how cheap the solution is thus far
Heuristic: better and precise evaluation of the current solution being built
G51IAI - Heuristic

6. Heuristic Searches - Characteristics
Heuristic searches vs. Uniform Cost Search
Heuristic searches evaluation function
h(n): how close is the current node to the solution
Uniform Cost Search path cost function
g(n): the cost of the path thus far
Draw tree example
G51IAI - Heuristic

7. Heuristic Searches - Definition
Heuristics are "rules of thumb", educated guesses, intuitive judgments or simply common sense.
A heuristic method is particularly used to rapidly come to a solution that is hoped to be close to the best possible answer, or 'optimal solution'.
- Wikipedia
G51IAI - Heuristic

8. Heuristic Searches - methods
Tree searches (G51IAI)
A way to reduce the search effort by pushing search in good directions
Not losing completeness
Search algorithms
Not complete
Find good solutions quickly
Genetic Algorithms, Tabu Search, Ant Algorithms
G51IAI - Heuristic

9. Heuristic Searches - Implementation 1
Implementation is achieved by sorting the nodes based on the evaluation function: h(n)
Search is based on the order of the nodes
G51IAI - Heuristic

10. Heuristic Searches - Implementation 2
Function BEST-FIRST-SEARCH(problem, EVAL- FN) returns a solution sequence
Inputs: a problem
Eval-Fn: an evaluation function
Queuing-Fn: a function that orders nodes by EVAL-FN
Return GENERAL-SEARCH (problem, Queuing-Fn)
Based on general search, now define the best first search, rather than picking the child randomly
What’s important is the evaluation function, other operations are the same, i.e. remove front node, insert child nodes
G51IAI - Heuristic

11. Heuristic Searches – Example
Go to the city which is nearest to the goal city
Available infor: SLD to goal from each city on the map
Evaluation function (heuristic): SLD, a good estimation of how good the solution is
Hsld(n) = straight line distance between n and the goal location

12. Heuristic Searches - Greedy Search
So named as it takes the biggest “bite” it can out of the problem.
That is, it seeks to minimise the estimated cost to the goal by expanding the node estimated to be closest to the goal state
Function GREEDY-SEARCH(problem) returns a solution of failure
Return BEST-FIRST-SEARCH(problem,h)
Important: easy to define
Simple, quick – still being used most of the time as initialisation approach
G51IAI - Heuristic

13. Heuristic Searches - Greedy Search
It is only concerned with short term aims
It is possible to get stuck in an infinite loop
Fsg: two child nodes (Siblu and Buchrest); Fsg  Bucharest (nearest to Iasi)
Buchsrest: four child nodes; Buchsrest  Fsg (nearest to Iasi)
Iasi - Buchsrest: 226
Even bigger problem: get stuck to local optimal
G51IAI - Heuristic

14. Heuristic Searches - Greedy Search
It is not optimal
It is not complete
Time and space complexity is O(bm); where m is the depth of the search tree
Big O: to learn in ADS
G51IAI - Heuristic

15. Performed well, but not optimal
Greedy Search
Luckily not get in a loop, but Sibiu  Bucharest is not optimal
Performed well, but not optimal

16. UCS Arad Nodes Expanded 1.Sibiu 2.Rimnicu 3.Faragas 4.Arad 5.Pitesti
6.Zerind
7.Craiova
8.Timisoara
9.Bucharest 278
GOAL!!
Neamt
286
215
Oradea 71
Zerind 87
Iasi
Fringe in RED with g Visited in BLUE 75
140 92
Arad
Optimal route is ( ) = 278 miles
140
140
Vaslui
118 99 99
Sibiu
Faragas
Timisoara
142
258 80
111 80
211
Lugoj
Rimnicu 98
Urziceni
369
Hirsova
177 70 86 97
Exam the sub-problem now: Sibiu  Bucharest
UCS: optimal and complete
Pitesti
Mehadia
146
101
Bucharest 86 75
138
310 (F)
Dobreta 90
278 (R,P)
226 (R)
120
Craiova
336
Eforie
Giurgui
[315 (R,P)]

17. Heuristic Searches vs. UCS
goal
outline of graph
increasing cost
start
goal
outline of graph
increasing cost
start
This region is basically wasted effort
UCS: give all search directions the same priority as long as the current cost is the least
Heuristic: use estimation (function) to guide search towards more promising direction to the goal
Estimation from the current to the goal! Make sue of knowledge, not blind search
G51IAI - Heuristic

18. Heuristic Searches vs. UCS
goal
outline of graph
increasing cost
start
Want to achieve this but stay
complete
optimal
If bias the search “too much” then could miss goals or miss shorter paths
However, heuristic search, guided by evaluation function not designed properly, not complete, nor optimal
Greedy search: bias the search too much
All background knowledge – now think how!
G51IAI - Heuristic

19. Heuristic Searches - A* Algorithm
Combines the cost so far and the estimated cost to the goal
That is fn = g(n) + h(n)
This gives us an estimated cost of the cheapest solution through n
g: cost so far (not losing the best from root to node so far)
h: estimate to the goal (good guidance to goal from what’s the best so far)
G51IAI - Heuristic

20. Heuristic Searches - A* Algorithm
We need to have a proper way to estimate h
goal
outline of graph
start A B gA gB hA hB
Best so far, corrected by good estimate
gB < gA, but (gB + hB) > (gA + hA)
G51IAI - Heuristic

21. Heuristic Searches - A* Algorithm
A search algorithm to find the shortest path through a search space to a goal state using a heuristic.
f = g + h
f - function that gives an evaluation of the state
g - the cost of getting from the initial state to the current state
h - the cost of getting from the current state to a goal state
G51IAI - Heuristic

22. Heuristic Searches - A* Algorithm
A search algorithm to find the shortest path through a search space to a goal state using a heuristic
h=0 A* becomes UCS
complete & optimal* but search pattern undirected
h too large
if h is large enough to dominate g then becomes like Greedy, lose optimality
*when cost along path never decrease
G51IAI - Heuristic

23. Heuristic Searches - A* Algorithm
It can be proved to be optimal and complete providing that the heuristic is admissible.
That is the heuristic must never over estimate the cost to reach the goal
h(n) must provide a valid lower bound on cost to the goal
But, the number of nodes that have to be searched still grows exponentially
Minimisation problem here
Intelligent: still exponentially
G51IAI - Heuristic

24. Heuristic Searches - A* Algorithm
Function A*-SEARCH(problem) returns a solution of failure
Return BEST-FIRST-SEARCH(problem, g + h)
G51IAI - Heuristic

25. Straight Line Distances to Bucharest
Town
SLD
Arad
366
Bucharest
Craiova
160
Dobreta
242
Eforie
161
Fagaras
178
Giurgiu 77
Hirsova
151
Iasi
226
Lugoj
244
Town
SLD
Mehadai
241
Neamt
234
Oradea
380
Pitesti 98
Rimnicu
193
Sibiu
253
Timisoara
329
Urziceni 80
Vaslui
199
Zerind
374
We can use straight line distances as an admissible heuristic as they will never overestimate the cost to the goal. This is because there is no shorter distance between two cities than the straight line distance.
G51IAI - Heuristic

26. ANIMATION OF A*. Arad Nodes Expanded 1.Sibiu 2.Rimnicu 3.Pitesti
4.Fagaras
5.Bucharest GOAL!!
Neamt
Oradea 71
Zerind 87
Iasi
Annotations: “g+h=f”
Fringe in RED Visited in BLUE 75
=506 92
Arad
Optimal route is ( ) = 278 miles
140
140
Vaslui
118
0+253=253
99+178=277 99
Sibiu
Fagaras
Timisoara
Why not 211?
142 80
111
80+193=273
211
Lugoj
Rimnicu 98
Urziceni
Hirsova
177+98=275 70 86 97
Give chance to other direction if it’s good enough, while the overall direction is towards the goal.
Pitesti
Mehadia
146
101
Bucharest 86 75
138
310+0=310 (F)
Dobreta 90
278+0=278 (R,P)
=386(R)
120
Craiova
Eforie
Giurgui
=475(P)

27. Arad
Oradea
Zerind
Faragas
Neamt
Iasi
Vaslui
Hirsova
Eforie
Urziceni
Giurgui
Pitesti
Sibiu
Dobreta
Craiova
Rimnicu
Mehadia
Timisoara
Lugoj 87 92
142 86 98
211
101 90 99 71 75
140
118
111 70
120
138
146 97 80
Bucharest
Optimal route is ( ) = 278 miles
Arad
Oradea
Zerind
Fagaras
Neamt
Iasi
Vaslui
Hirsova
Eforie
Urziceni
Giurgui
Pitesti
Sibiu
Dobreta
Craiova
Rimnicu
Mehadia
Timisoara
Lugoj 87 92
142 86 98
211
101 90 99 71 75
140
118
111 70
120
138
146 97 80
Bucharest
Optimal route is ( ) = 278 miles
UCS A*
A*: give chance to other direction if it’s good enough, while the overall direction is towards the goal.
UCS: give all directions the same chance.
Difference: heuristic in A*: admissible
Nodes expanded:
1.Sibiu; 2.Rimnicu; 3.Faragas; 4.Arad; 5.Pitesti; 6.Zerind; 7.Craiova; 8.Timisoara; 9.Bucharest 278
Nodes Expanded:
1.Sibiu; 2.Rimnicu; 3.Pitesti; 4.Fagaras; 5.Bucharest 278

28. In terms of a search tree we could represent this as follows ....
101 Bu
278+0 =278 5 97 Pi
=275 Ri
=273 3 2 80
138 Cr
=475
146 99 Cr
=386
Si.
0+253 =253 1 Fa
=277 4
Maths: “g + h = f”
211 Bu
310+0 =310
140 Ar
=506
The only problem in working through the algorithm was that when we found our goal state the first time we ignored it as we had to expand an even lower cost node to see if it led to an even better solution. This has shown to be a problem in previous examinations
The goal state is achieved. In relation to path cost, A* has found the optimal route with 5 expansions. Press space to end.
Press space to begin the search
A* SEARCH TREE

29. Heuristic Searches - A* Algorithm
Clearly the expansion of the fringe is much more directed towards the goal
The number of expansions is significantly reduced
G51IAI - Heuristic

30. Heuristic Searches - A* Algorithm
A* is optimal and complete, but it is not all good news
It can be shown that the number of nodes that are searched is still exponential to the size of most problems
This has implications not only for the time taken to perform the search but also the space required
Of these two problems the space complexity is more serious
A*: more like a BFS, space is bigger problem
G51IAI - Heuristic

31. Heuristic Searches - A* Algorithm
If you examine the animation on the previous slide you will notice an interesting phenomenon
Along any path from the root, the f-cost never decreases
This is no accident
It holds true for all admissible heuristics
Admissible: never over estimate
G51IAI - Heuristic

32. Heuristic Searches - A* Example
Initial State
Goal State
Map search: good example
8-puzzle: another widely studied example in AI
G51IAI - Heuristic

33. Heuristic Searches - A* Example
Typical solution is about twenty steps
Branching factor is approximately three. Therefore a complete search would need to search 320 states. But by keeping track of repeated states we would only need to search 9! (362,880) states
But even this is a lot (imagine having all these in memory)
Our aim is to develop a heuristic that does not over estimate (it is admissible) so that we can use A* to find the optimal solution
G51IAI - Heuristic

34. Heuristic Searches - A* Example
8 puzzle and 15 puzzle
Online demo of A* algorithm for 8 puzzle
Noyes Chapman’s 15 puzzle
A lot online game: mainly based on A*? Key issue: how to design admissible heuristic!
15 puzzle: with only 15 and 14 reverse - not solvable!
a solvable 15 puzzle: take the shortcut of solving the squares 1,2,3,4,5,9 and 13 - only have a 3x3 grid to solve. This speeds the process up quite considerably.
G51IAI - Heuristic

35. Heuristic Searches - Possible Heuristics in A* Algorithm
= the number of tiles that are in the wrong position H2
= the sum of the distances of the tiles from their goal positions using the Manhattan Distance
We need admissible heuristics (never over estimate)
Both are admissible but which one is better?
Lot of research attention: design efficient heuristic so better solutions can be found in shorter time
Question: do we need to consider the path cost so far?
G51IAI - Heuristic

36. Heuristic Searches - Possible Heuristics in A* Algorithm
= the number of tiles that are in the wrong position (=4) H2
= the sum of the distances of the tiles from their goal positions using the Manhattan Distance (=5)
What’s the effect of higher and lower h?
Why we don’t need to cost so far?
G51IAI - Heuristic

37. Heuristic Searches - Possible Heuristics in A* Algorithm1 3 4 8 6 2 7 5 1 3 4 8 6 2 7 5 5 4 1 3 4 8 2 7 6 5 1 3 4 8 6 2 7 5 6 1 3 4 8 6 2 7 5 1 3 4 8 6 2 7 5 1 3 4 8 6 2 7 5 4 6 5 1 3 4 8 2 7 6 5 6 3
H1 = the number of tiles that are in the wrong position (=4)
H2 = the sum of the distances of the tiles from their goal positions using the Manhattan Distance (=5)
G51IAI - Heuristic

38. Possible Heuristics in A* Algorithm1 3 4 8 6 2 7 5 5 1 3 4 8 2 7 6 5 1 3 4 8 6 2 7 5 6 1 3 4 8 6 2 7 5 4 6 1 4 8 3 2 7 6 5 1 3 4 8 2 7 6 5 5 1 3 4 8 2 7 6 5 5 3 1 3 8 2 4 7 6 5 1 3 4 8 2 5 7 6 2 4 1 3 8 2 4 7 6 5 1
What’s wrong with this search? is it A*?
H2 = the sum of the distances of the tiles from their goal positions using the Manhattan Distance (=5)

39. Possible Heuristics in A* Algorithm1 3 4 8 6 2 7 5 4 1 3 4 8 2 7 6 5 1 3 4 8 6 2 7 5 5 1 3 4 8 6 2 7 5 3 5 1 4 8 3 2 7 6 5 1 3 4 8 2 7 6 5 4 1 3 4 8 2 7 6 5 3 3 1 4 8 3 2 7 6 5 1 4 8 3 2 7 6 5 4 3 1 4 2 8 3 7 6 5 3
What’s wrong with this search? is it A*?
H1 = the number of tiles that are in the wrong position (=4)

40. Test from 100 runs with varying solution depths using h1 and h2
Solutions at different depths
the average number of nodes that were expanded
H2 looks better as fewer nodes are expanded. But why?
G51IAI - Heuristic

41. Effective Branching Factor
h2 is better estimation compared with h1, closer to real cost of solution, more precise.
Admissible, but as closer to the optimal solution as possible
Effective branching factor: average number of branches expanded
H2 has a lower branching factor and so fewer nodes are expanded
Therefore, one way to measure the quality of a heuristic is to find its average branching factor
H2 has a lower EBF and is therefore the better heuristic
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42. Domination For any node: h2(n) <= h1(n) h2 dominates h1
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43. G51IAI - Heuristic Previous final year project
G51IAI - Heuristic

44. Show at each step what fringe nodes are in the queue (5 marks).
Use Uniform Cost Search to find the shortest path from A to F in the map below (not drawn to scale). You should not re-visit a node that you have just come from.
Show at each step what fringe nodes are in the queue (5 marks).
Show the list of nodes that are expanded (3 marks).
State the shortest route you take and its cost (2 marks).
Can Uniform Cost Search guarantee to find the optimal solution for this problem? Explain the reason. (3 marks) A 15 25 B 20 C 15 12 25 D E 16 8 F