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COMM7370 
AI Theories and Applications 
Lecture 3 
Advanced Uninformed Search 
Informed Search 
Heuristics 
Dr. Paolo Mengoni 
 
Visiting Scholar @HKBU School of Communication 
Agenda 
▪Search strategies 
▪Tree Search recap 
▪Advanced Uninformed Search 
▪Informed Search 
▪Greedy Best First 
▪A* 
▪Heuristics 
COMM7370 AI Theories and Applications 
Tree Search 
COMM7370 AI Theories and Applications 
Tree-based Search 
▪State: a possible configuration for the problem 
▪State space: a set of the possible configurations 
▪Basic idea: 
▪Exploration of state space by generating successors of 
already-explored states (i.e. expanding states). 
▪Use a tree-like representation 
▪Every state is evaluated: is it a goal state? 
▪Example 8 puzzle game 
▪State configuration to tree node 
COMM7370 AI Theories and Applications 
State Spaces versus Search Trees 
▪State Space 
▪Set of valid states for a problem 
▪ Linked by operators 
▪e.g., 20 valid states (cities) in the Romanian travel problem 
▪Search Tree 
▪Root node = initial state 
▪Child nodes = states that can be visited from parent 
▪Note that the depth of the tree can be infinite 
▪ E.g., via repeated states 
▪Partial search tree 
▪ Portion of tree that has been expanded so far 
▪ Fringe 
▪ Leaves of partial search tree, candidates for expansion 
▪Search trees = data structure to search state-space 
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Search Tree for the 8 puzzle problem 
COMM7370 AI Theories and Applications 
Advanced Uninformed 
Search 
COMM7370 AI Theories and Applications 
Depth-First Search (DFS) 
▪Expand deepest unexpanded node 
▪Fringe 
▪ last-in-first-out (LIFO) queue 
▪ E.g. the elevator with only one door 
▪new successors go at beginning of the queue 
▪Repeated nodes? 
▪Simple strategy: 
▪To avoid loops do not add a state as a leaf if that state is 
on the path from the root to the current node 
COMM7370 AI Theories and Applications 
COMM7370 AI Theories and Applications 
Depth-limited search 
▪To overcome the problem of DFS search, set a depth limit 
and perform tree search 
▪DLS = Depth Limited Search 
▪ Set limit l 
▪ Use DFS 
▪When the limit l is reached the nodes have no successors 
▪ IDS = Iterative Deepening Search 
▪ Perform depth limited search 
▪ Increase the depth limit until the solution is found 
COMM7370 AI Theories and Applications 
Iterative deepening search l =0 
COMM7370 AI Theories and Applications 
Iterative deepening search l =1 
COMM7370 AI Theories and Applications 
Iterative deepening search l =2 
COMM7370 AI Theories and Applications 
Iterative deepening search l =3 
COMM7370 AI Theories and Applications 
Iterative deepening search 
▪Number of nodes generated in a depth-limited search to 
depth d with branching factor b: 
NDLS = b 
0 + b1 + b2 + … + bd-2 + bd-1 + bd 
▪Number of nodes generated in an iterative deepening 
search to depth d with branching factor b: 
NIDS = (d+1)b 
0 + d b^1 + (d-1)b^2 + … + 3bd-2 +2bd-1 + 1bd 
▪For b = 10, d = 5, 
▪NDLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111 
▪NIDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456 
▪Overhead = (123,456 - 111,111)/111,111 = 11% 
COMM7370 AI Theories and Applications 
Properties of iterative deepening search 
▪Complete? 
▪Yes 
▪Time? 
▪ (d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd) 
▪Space? 
▪O(bd) 
▪Optimal? 
▪Yes, if step cost = 1 
Informed search strategies recap 
COMM7370 AI Theories and Applications 
▪Comparison between DFS and BFS 
▪DFS use less space, in average may be faster 
▪BFS is complete and optimal 
Measure DFS BFS IDS 
Completeness No Yes Yes 
Time Complexity bm bm bd 
Space Complexity b x m bm b x d 
Optimality No Yes Yes 
Informed search 
COMM7370 AI Theories and Applications 
Informed Search 
▪Motivations: Limitations of uninformed search 
methods 
▪Too many nodes to expand 
▪Informed (or heuristic) search uses problem- 
specific heuristics to improve efficiency 
▪This leads to different strategies 
▪Best-first 
▪A* 
▪Can provide significant speed-ups in practice 
▪e.g., on 8-puzzle 
COMM7370 AI Theories and Applications 
Best-first search 
▪Idea: use an evaluation function (also called 
heuristic ℎ ) for each node estimate of "desirability“ 
▪Expand most desirable unexpanded node 
▪Implementation: 
▪Order the nodes in fringe by their heuristics value 
▪Most desirable nodes have higher priority 
▪Expands the node that appears to be closest to goal 
▪ Not always true 
▪Special cases: 
▪greedy best-first search 
▪A* search 
▪Note 
▪Evaluation function is an estimate of node quality 
▪More on heuristics later 
COMM7370 AI Theories and Applications 
Example heuristic functions for 8-puzzle 
▪8-puzzle 
▪A good heuristic function can reduce the search process 
▪Commonly used heuristics 
▪Number of misplaced tiles 
▪ Example: ℎ = 8 
COMM7370 AI Theories and Applications 
Example Romania with step costs in km 
▪Map navigation 
▪Commonly used heuristics 
▪Straight line distance to destination 
▪ Example: ℎ = 366 
COMM7370 AI Theories and Applications 
Greedy best-first search example 
COMM7370 AI Theories and Applications 
Greedy best-first search example 
COMM7370 AI Theories and Applications 
Greedy best-first search example 
COMM7370 AI Theories and Applications 
Greedy best-first search example 
COMM7370 AI Theories and Applications 
Optimal Path 
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Optimal vs Greedy BestFirst 
COMM7370 AI Theories and Applications 
h(Fagaras)=176 
h(RimnicuVilcea)=193 
Properties of greedy best-first search 
▪Complete? No 
▪can get stuck in loops 
▪e.g., Iasi → Neamt → Iasi → Neamt → … 
▪Time? O(bm) 
▪Worst case 
▪A good heuristic can give dramatic improvement 
▪Space? O(bm) 
▪Keeps all nodes in memory 
▪Optimal? No 
COMM7370 AI Theories and Applications 
A* Search 
▪Expand node based on estimate of total path cost 
through node 
▪Idea: avoid expanding paths that are already 
expensive 
▪Evaluation function = + ℎ considers 
▪cost so far 
▪estimated cost to goal ℎ 
▪Efficiency of search will depend on quality of 
heuristic 
COMM7370 AI Theories and Applications 
A* search example 
COMM7370 AI Theories and Applications 
A* search example 
COMM7370 AI Theories and Applications 
A* search example 
COMM7370 AI Theories and Applications 
A* search example 
COMM7370 AI Theories and Applications 
A* search example 
COMM7370 AI Theories and Applications 
A* search example 
COMM7370 AI Theories and Applications 
Heuristics 
COMM7370 AI Theories and Applications 
Admissible heuristics 
▪A heuristic h(n) is admissible if for every node n, 
h(n) ≤ h*(n) 
▪where h*(n) is the true cost to reach the goal state from n 
▪An admissible heuristic never overestimates the 
cost to reach the goal, i.e. it is optimistic 
▪E.g.: in the map navigation problem, the heuristic 
straight-line distance never overestimates the actual 
road distance) 
▪Theorem: If h(n) is admissible, A* is optimal 
COMM7370 AI Theories and Applications 
Optimality of A* 
▪A* expands nodes in order of increasing f value 
▪Gradually adds "f-contours" of nodes 
▪Contour i has all nodes with f=fi, where fi < fi+1 
COMM7370 AI Theories and Applications 
Properties of A* 
▪Complete? 
▪Yes (unless there are infinitely many nodes with f ≤ f(G) ) 
▪Time? 
▪Exponential 
▪Space? 
▪Keeps all nodes in memory 
▪Optimal? 
▪Yes 
COMM7370 AI Theories and Applications 
Admissible heuristics 
▪E.g., for the 8-puzzle: 
▪h1(n) = number of misplaced tiles 
▪h2(n) = total Manhattan distance 
▪ i.e. n. of squares from desired location of each tile 
▪h1(S) = ? 
▪h2(S) = ? 
COMM7370 AI Theories and Applications 
Admissible heuristics 
▪E.g., for the 8-puzzle: 
▪h1(n) = number of misplaced tiles 
▪h2(n) = total Manhattan distance 
▪ i.e. n. of squares from desired location of each tile 
▪h1(S) = 8 
▪h2(S) = 3+1+2+2+2+3+3+2 = 18 
COMM7370 AI Theories and Applications 
Dominance 
▪ If h2(n) ≥ h1(n) for all n (both admissible) 
▪ then h2 dominates h1 
▪ i.e. h2 is better for search 
▪Typical search costs (average number of nodes expanded): 
▪d=12 IDS = 3,644,035 nodes 
A*(h1) = 227 nodes 
A*(h2) = 73 nodes 
▪d=24 IDS = too many nodes 
A*(h1) = 39,135 nodes 
A*(h2) = 1,641 nodes 
▪ IDS = Iterative Deepening Search 
COMM7370 AI Theories and Applications 
Relaxed problems 
▪A problem with fewer restrictions on the actions is 
called a relaxed problem 
▪The cost of an optimal solution to a relaxed 
problem is an admissible heuristic for the original 
problem 
▪If the rules of the 8-puzzle are relaxed so that a tile 
can move anywhere, then h1(n) gives the shortest 
solution 
▪If the rules are relaxed so that a tile can move to 
any adjacent square, then h2(n) gives the shortest 
solution 
COMM7370 AI Theories and Applications 
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