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The document discusses informed search strategies, particularly focusing on best-first search, greedy best-first search, and A* search, which utilize heuristic functions to guide the search process more efficiently than uninformed methods. It explains the function of heuristics and provides illustrative examples related to pathfinding in Romania. Additionally, it details properties of these search algorithms regarding time, space, completeness, and optimality.

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Informed Search Bùi Tiến Lên 2022
Contents 1. Best-first search 2. Greedy best-first search 3. A* search 4. Memory-bounded heuristic search 5. Heuristic functions
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Informed search strategies Informed (heuristic) search strategies • Use problem-specific knowledge beyond the definition of the problem itself • Can find solutions more efficiently than can an uninformed strategy What is heuristic • Additional knowledge of the problem is imparted to the search algorithm. • Heuristic estimates how close a state is to a goal. 3
Best-first search
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Best-first search • An instance of the general tree/graph search algorithm • A node is selected for expansion based on an evaluation function, f (n), which is construed as a cost estimate. • The node with the lowest evaluation f (n) is expanded first • The implementation of best-first graph search is identical to that for uniform-cost search, except for the use of f instead of g to order the priority queue • The choice of f determines the search strategy 5
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Best-first search (cont.) • Most best-first algorithms include as a component of f a heuristic function, denoted h(n): h(n) estimated cost of the cheapest path from the state at node n to a goal state • h(n) depends only on the state at node n • Assumption of h • Arbitrary, nonnegative, problem-specific functions • Constraint: if n is a goal node, then h(n) = 0 6
Greedy best-first search
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Greedy best-first search • Try to expand the node that is closest to the goal, on the grounds that this is likely to lead to a solution quickly. • Thus, it evaluates nodes by using just the heuristic function f (n) = h(n) 8
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania • Use the straight-line distance heuristic, which we will call hSLD Table 1: Values of hSLD – straight-line distances to Bucharest. Urziceni Neamt Oradea Zerind Timisoara Mehadia Sibiu Pitesti Rimnicu Vilcea Vaslui Bucharest Giurgiu Hirsova Eforie Arad Lugoj Drobeta Craiova Fagaras Iasi 0 160 242 161 77 151 366 244 226 176 241 253 329 80 199 380 234 374 100 193 9
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania (cont.) • The initial state Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Arad 366 10
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania (cont.) • After expanding Arad Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Timisoara 253 329 374 11
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania (cont.) • After expanding Sibu Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Rimnicu Vilcea Zerind Arad Sibiu Arad Fagaras Oradea Timisoara 329 374 366 176 380 193 12
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania (cont.) • After expanding Fagaras Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Rimnicu Vilcea Zerind Arad Sibiu Arad Fagaras Oradea Timisoara Sibiu Bucharest 329 374 366 380 193 253 0 13
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Properties • Time: O(bm), but a good heuristic can give dramatic improvement • Space: O(bm), keeps all nodes in memory • Complete: No, can get stuck in loops, e.g., find a path from Iasi to Fagaras, Iasi → Neamt → Iasi → Neamt → ... • Yes in finite space with repeated-state checking • Optimal: No Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Drobeta 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 14
A* search
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions A* search • A* search (pronounced A-star search) is the most widely known form of best-first search • Ideas • Use heuristic to guide search, but not only • 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 of the cheapest path to goal from n • f (n): estimated cost of the cheapest solution through n • The algorithm is identical to Uniform-Cost-Search except that A* uses g + h instead of g 16
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Arad 366=0+366 17
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Timisoara 447=118+329 449=75+374 393=140+253 18
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Arad Timisoara Rimnicu Vilcea Fagaras Oradea 447=118+329 449=75+374 646=280+366 413=220+193 415=239+176 671=291+380 19
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Arad Timisoara Fagaras Oradea 447=118+329 449=75+374 646=280+366 415=239+176 Rimnicu Vilcea Craiova Pitesti Sibiu 526=366+160 553=300+253 417=317+100 671=291+380 20
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Arad Timisoara Sibiu Bucharest Rimnicu Vilcea Fagaras Oradea Craiova Pitesti Sibiu 447=118+329 449=75+374 646=280+366 591=338+253 450=450+0 526=366+160 553=300+253 417=317+100 671=291+380 21
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Arad Timisoara Sibiu Bucharest Rimnicu Vilcea Fagaras Oradea Craiova Pitesti Sibiu Bucharest Craiova Rimnicu Vilcea 418=418+0 447=118+329 449=75+374 646=280+366 591=338+253 450=450+0 526=366+160 553=300+253 615=455+160 607=414+193 671=291+380 22
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Properties • Time Exponential in [relative error in h × length of solution.] • Space Keeps all nodes in memory • Complete Yes, unless there are infinitely many nodes with f ≤ f (G) • Optimal Yes, cannot expand fi+1 until fi is finished • A* expands all nodes with f (n) < C∗ • A* expands some nodes with f (n) = C∗ • A* expands no nodes with f (n) > C∗ 23
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Admissibility and Consistency Concept 1 h(n) is an admissible heuristic • if it never overestimates the cost to reach the goal • or if for every node n, h(n) ≤ h∗(n) (h∗(n) is a the true optimal cost to reach the goal state from n) • An example of an admissible heuristic is the straight-line distance hSLD that we used in getting to Bucharest 24
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Admissibility and Consistency (cont.) Concept 2 h(n) is an consistent heuristic if for every node n and every successor n0 of n generated by any action a, the estimated cost of reaching the goal from n is no greater than the step cost of getting to n0 plus the estimated cost of reaching the goal from n0 h(n) ≤ c(n, a, n0 ) + h(n0 ) n c(n,a,n’) h(n’) h(n) G n’ • Consistent heuristic is also admissible 25
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Optimal for A* • Using tree search algorithm to find a path from S to G 1 1 3 S h = 5 a h = 4 G h = 0 26
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Optimal for A* (cont.) Theorem 1 The tree-search version of A* is optimal if h(n) is admissible Proof Suppose some suboptimal goal G2 has been generated and is in the frontier. Let n be an unexpanded node on a shortest path to an optimal goal G (graph separation) • f (G) = g(G) since h(G) = 0 • f (G2) = g(G2) since h(G2) = 0 • g(G2) > g(G) since G2 is suboptimal • f (G) > f (n) since h is admissible • Since f (G2) > f (n), A* will never select G2 for expansion G n G2 Start 27
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Optimal for A* (cont.) • Using graph search algorithm to find a path from S to G 1 1 1 2 3 S h = 2 a h =4 b h = 1 c h = 1 G h = 0 28
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Optimal for A* (cont.) Theorem 2 The graph-search version of A* is optimal if h(n) is consistent Proof • if h(n) is consistent, then the values of f (n) along any path are nondecreasing. Suppose n0 is a successor of n; then g(n0) = g(n) + c(n, a, n0) for some action a, and we have f (n0 ) = g(n0 ) + h(n0 ) = g(n) + c(n, a, n) + h(n0 ) ≥ g(n) + h(n) = f (n) • Whenever A* selects a node n for expansion, the optimal path to that node has been found. Were this not the case, there would have to be another frontier node n0 on the optimal path from the start node to n; f (n0) < f (n) ⇒ n0 would have been selected first. 29
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Contours • Search algorithm expands nodes in order of increasing f value • We can draw contours in the state space, just like the contours in a topographic map • Gradually adds “f -contours” of nodes, contour i has all nodes with f < fi, where fi < fi+1 O Z A T L M D C R F P G B U H E V I N 380 400 420 S 30
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Contours (cont.) • The bands of uniform-cost search will be “circular” around the start state. • The bands of A* with more accurate heuristics will stretch toward the goal state and become more narrowly focused around the optimal path 31
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions The 8-puzzle • The average solution cost for a randomly generated 8-puzzle instance is about 22 steps • The branching factor b is about 3 • 8-puzzle: • 322 ≈ 3.1 × 1010 states (using an exhaustive tree search) • 9!/2 = 181, 440 reachable states (using graph search) • 15-puzzle: around 1013 states 2 Start State Goal State 1 3 4 6 7 5 1 2 3 4 6 7 8 5 8 Figure 1: A typical instance of the 8-puzzle. The optimal solution is 26 steps long. 32
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Admissible heuristics for 8-puzzle • h1 = the number of misplaced tiles. h1 is an admissible heuristic because it is clear that any tile that is out of place must be moved at least once • h2 = the sum of the distances of the tiles from their goal positions. h2 is also admissible because all any move can do is move one tile one step closer to the goal 2 Start State Goal State 1 3 4 6 7 5 1 2 3 4 6 7 8 5 8 • h1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8 and h2 = 3 + 1 + 2 + 2 + 2 + 3 + 3 + 2 = 18 33
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Solve 8 puzzle • Using h2 2 3 1 4 5 8 7 6 2 3 1 4 5 8 7 6 1 2 3 4 5 8 7 6 2 3 1 4 5 8 7 6 1 2 3 4 5 8 7 6 1 2 3 8 4 5 7 6 1 2 3 8 4 7 6 5 Start Goal 6=0+6 8=1+7 6=1+5 6=2+4 8=2+6 8=2+6 34
Memory-bounded heuristic search
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Memory-bound heuristic search • In practice, A* usually runs out of space long before it runs out of time • Idea: try something like DFS, but not forget everything about the branches we have partially explored 36
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Iterative-deepening A* (IDA*) • The main difference with IDS • Cut-off use the f -value (g + h) rather than the depth • At each iteration, the cutoff value is the smallest f -value of any node that exceeded the cutoff on the previous iteration • Avoid the substantial overhead associated with keeping a sorted queue of nodes. • Practical for many problems with unit step costs, yet difficult with real valued costs 37
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Algorithm function IDA*(problem) returns a solution, or failure local variables: f _limit: the current f-COST limit root: a node root ← Make-Node(problem.Initial-State) f _limit ← root.f loop do solution, f _limit ← DFS-Contour(problem, root, f _limit) if solution is non-null then return solution if f _limit = ∞ then return failure 38
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Algorithm (cont.) function DFS-Contour(problem,node,f _limit) returns a solution, or failure and a new f-cost limit if node.f > f _limit then return null, node.f if problem.Goal-Test(node.State) then return Solution(node) successors ← ∅ for each action in problem.Actions(node.State) do add Child-Node(problem,node,action) into successors f _next ← ∞ for each s in successors do solution,f _new ← DFS-Contour(problem, s, f _limit) if solution is non-null then return solution, f _limit f _next ← Min(f _next, f _new) return null, f _next 39
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration • IDA* search for the shortest route from v0 (start node) to v25 (goal node); each node has a f value. 1 3 4 4 5 5 5 6 7 7 6 5 7 6 8 11 13 15 14 11 8 12 7 12 6 14 15 7 13 15 9 40
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Recursive best-first search (RBFS) • Keep track of the f -value of the best alternative path available from any ancestor of the current node • backtrack when the current node exceeds f _limit • As it backtracks, replace f -value of each along the path with the best value f (n) of its children 41
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Algorithm function Recursive-Best-First-Search(problem) returns a solution, or failure return Rbfs(problem,Make-Node(problem.Initial-State),∞) function Rbfs(problem,node,f _limit) returns a solution, or failure and a new f-cost limit if problem.Goal-Test(node.State) then return Solution(node) successors ← ∅ for each action in problem.Actions(node.State) do add Child-Node(problem,node,action) into successors if successors is empty then return failure,∞ for each s in successors do # update f with value from previous search, if any s.f ← Max(s.g + s.h, node.f ) loop do best ← lowest f-value node in successors if best.f > f _limit then return failure,best.f alternative ← the second-lowest f -value among successors result,best.f ← Rbfs(problem, best, Min(f _limit, alternative)) if result 6= failure then return result 42
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration • Stages in an RBFS search for the shortest route to Bucharest. The f -limit value for each recursive call is shown on top of each current node, and every node is labeled with its f -cost. Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu V. Sibiu Bucharest Craiova Pitesti Sibiu Bucharest Craiova Rimnicu V. 366 393 447 449 646 415 671 413 591 450 526 417 553 418 615 607 43
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration (cont.) (a) The path via Rimnicu Vilcea is followed until the current best leaf (Pitesti) has a value that is worse than the best alternative path (Fagaras). Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu V. Sibiu Bucharest Craiova Pitesti Sibiu Bucharest Craiova Rimnicu V. 366 393 447 449 646 415 671 413 591 450 526 417 553 418 615 607 44
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration (cont.) (b) The recursion unwinds and the best leaf value of the forgotten subtree (417) is backed up to Rimnicu Vilcea; then Fagaras is expanded, revealing a best leaf value of 450. Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu V. Sibiu Bucharest Craiova Pitesti Sibiu Bucharest Craiova Rimnicu V. 366 393 447 449 646 415 671 413 591 450 526 417 553 418 615 607 45
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration (cont.) (c) The recursion unwinds and the best leaf value of the forgotten subtree (450) is backed up to Fagaras; then Rimnicu Vilcea is expanded. This time, because the best alternative path (through Timisoara) costs at least 447, the expansion continues to Bucharest. Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu V. Sibiu Bucharest Craiova Pitesti Sibiu Bucharest Craiova Rimnicu V. 366 393 447 449 646 415 671 413 591 450 526 417 553 418 615 607 46
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Properties • Optimality • Like A*, optimal if h(n) is admissible • Time complexity • Difficult to characterize • Depends on accuracy of h(n) and how often best path changes • Can end up “switching” back and forth • Space complexity • Linear time: O(bd) • Other extreme to A* - uses too little memory even if more memory were available 47
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Simplified Memory-bound A* – SMA* • Similar to A*, but delete the worst node (largest f -value) when memory is full • SMA* expands the (newest) best leaf and deletes the (oldest) worst leaf • SMA* also backs up the value of the forgotten node to its parent. • If there is a tie (equal f -values), delete the oldest nodes first • Simplified-MA* finds the optimal reachable solution given the memory constraint. • The depth of the shallowest goal node, is less than the memory size (expressed in nodes) • Time can still be exponential. 48
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Algorithm function SMA*(problem) returns a solution, or failure frontier ← Make-Node(problem.Initial-State) loop do if frontier = ∅ then return failure node ← Pop(frontier) if problem.Goal-Test(node.State) then return Solution(node) s ← Next-Successor(node) if s is not a goal and is at maximum depth then s.f ← ∞ else s.f ← Max(s.g + s.h, node.f ) if all of node's successors have been generated then update node's f-cost and those of its ancestors if necessary if Successors(node) all in memory then remove node from frontier if memory is full then delete shallowest, highest f-cost node in frontier remove it from its parent's successor list insert its parent on frontier if necessary frontier ← Insert(s, frontier) 49
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration • Find the lowest-cost goalnode with enough memory • Max Nodes = 3 • A: start node and D, F, I, J: goal nodes • Each edge/node is labelled with its step cost/heuristic value A B 10 G 8 C 10 D 10 H 8 I 16 E 10 F 10 J 8 K 8 12 5 5 5 2 5 0 0 5 0 0 50
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Illustration (const.) • Consider only current f-cost A B G C D H I E F J K 12 15 13 25 18 35 30 24 29 20 24 51
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Learning to search better • Could an agent learn how to search better? Yes • Metalevel state space: in which each state captures the internal (computational) state of a program that is searching in an object-level state space. • A metalevel learning algorithm learn from these experiences to avoid exploring unpromising subtrees • For example, the map of Romania problem • The internal state of the A* algorithm is the current search tree. • Each action in the metalevel state space is a computation step that alters the internal state; for example, each computation step in A* expands a leaf node and adds its successors to the tree 52
Heuristic functions
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions The effect of heuristic accuracy on performance • Effective branching factor b∗ characterize the quality of a heuristic. • If the total number of nodes generated by A* for a particular problem is N and the solution depth is d, then b∗ is the branching factor that a uniform tree of depth d would have to have in order to contain N + 1 nodes N + 1 = 1 + (b∗ ) + (b∗ )2 + ... + (b∗ )d • b∗ varies across problem instances, but fairly constant for sufficiently hard problems • A well-designed heuristic would have a value of b∗ close to 1 ⇒ fairly large problems solved at reasonable cost 54
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions The effect of heuristic accuracy on performance (cont.) Search Cost (nodes generated) Effective Branching Factor d IDS A*(h1) A*(h2) IDS A*(h1) A*(h2) 2 10 6 6 2.45 1.79 1.79 4 112 13 12 2.87 1.48 1.45 6 680 20 18 2.73 1.34 1.30 8 6384 39 25 2.80 1.33 1.24 10 47127 93 39 2.79 1.38 1.22 12 3644035 227 73 2.78 1.42 1.24 14 – 539 113 – 1.44 1.23 16 – 1301 211 – 1.45 1.25 18 – 3056 363 – 1.46 1.26 20 – 7276 676 – 1.47 1.27 22 – 18094 1219 – 1.48 1.28 24 – 39135 1641 – 1.48 1.26 Table 2: Comparison of the search costs and effective branching factors for the IDS and A* algorithms with h1, h2. Data are averaged over 100 instances of the 8-puzzle for each of various solution lengths d. 55
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Dominance and Composite • If h2(n) ≥ h1(n) for all n (both admissible) then h2 dominates h1 and is better for search • A* using h2 will never expand more nodes than A* using h1 • Hence, it is generally better to use a heuristic function with higher values, provided it is consistent and that the computation time for the heuristic is not too long. • Given a collection of admissible heuristics {h1, ..., hm} is available for a problem and none of them dominates any of the others. • The composite heuristic function is defined as h(n) = max{h1(n), . . . , hm(n)} is consistent and dominates all component heuristics 56
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Relaxed problems Concept 3 A problem with fewer restrictions on the actions is called a relaxed problem. • Prove that the angle in a semicircle is a right angle. A O B C 57
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Generating admissible heuristics from relaxed problems • The state-space graph of the relaxed problem is a supergraph of the original state space because the removal of restrictions creates added edges in the graph. S a b c d G Figure 2: Original problem state-space S a b c d G Figure 3: Relaxed problem state-space 58
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Generating admissible heuristics from relaxed problems (cont.) • Hence, the cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem. • Furthermore, because the derived heuristic is an exact cost for the relaxed problem, it must obey the triangle inequality and is therefore consistent. 59
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Generating admissible heuristics from relaxed problems (cont.) Original problem of 8 puzzle • A tile can move from square A to square B if A is horizontally or vertically adjacent to B and B is blank We can generate three relaxed problems by removing one or both of the conditions Relaxed problems 1. A tile can move from square A to square B if A is adjacent to B → h2 2. A tile can move from square A to square B if B is blank → exercise 3. A tile can move from square A to square B → h1 60
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Generating admissible heuristics from subproblems • Admissible heuristics can also be derived from the solution cost of a subproblem of a given problem • The cost of the optimal solution of this subproblem ≤ the cost of the original problem (lower bound). • More accurate than Manhattan distance in some cases Start State Goal State 1 2 3 4 6 8 5 2 1 3 6 7 8 5 4 Figure 4: A subproblem of the 8-puzzle instance. The task is to get tiles 1, 2, 3, and 4 into their correct positions, without worrying about what happens to the other tiles. 61
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Goal fixed? • Assume that the goal state G is fixed • Using Uniform-Cost-Search to find the shortest path from G to the other states • The final g values can be considered as optimal h∗ values 62
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Goal fixed? (cont.) 9 3 6 8 5 4 1 8 17 21 4 4 G b d h c e f 9 3 6 8 5 4 1 8 17 21 4 4 G g = 0 b g = 7 d g = 3 h g = 6 c g = 4 e g = 9 f g = 26 9 3 6 8 5 4 1 8 17 21 4 4 G h = 0 b h = 7 d h = 3 h h = 6 c h = 4 e h = 9 f h = 26 63
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Pattern databases • Pattern databases (PDB) where store the exact solution costs for every possible subproblem instance • E.g., every possible configuration of the four tiles and the blank • The database itself is constructed by searching back from the goal and recording the cost of each new pattern encountered; the expense of this search is amortized over many subsequent problem instances • The complete heuristic is constructed using the patterns in the databases 64
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Computing the Heuristic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 10 14 7 8 3 6 1 15 12 2 11 4 9 13 • 31 moves needed to solve red tiles • 22 moves need to solve blue tiles • Overall heuristic is maximum of 31 = max(31, 22) moves 65
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Disjoint pattern databases • Limitation of traditional PDB: Take max → diminish returns on additional PDBs • Disjoint pattern databases: Count only moves of the pattern tiles, ignoring non-pattern moves. • If no tile belongs to more than one pattern, add their heuristic values. 66
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Computing the Heuristic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 10 14 7 8 3 6 1 15 12 2 11 4 9 13 • 20 moves needed to solve red tiles • 25 moves needed to solve blue tiles • Overall heuristic is sum, or 45 = 20 + 25 moves 67
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Performance • 15 Puzzle • 1000 × speedup vs. Manhattan distance • IDA* with the two DBs (the 7-tile database contains 58 million entries and the 8-tile database contains 519 million entries) solves 15-puzzles optimally in 30 milliseconds • 24 Puzzle • 12 million × speedup vs. Manhattan distance • IDA* can solve random instances in 2 days. • Requires 4 DBs (each DB has 128 million entries) • Without PDBs: 65,000 years 68
Best-first search Greedy best-first search A* search Memory- bounded heuristic search Heuristic functions Learning heuristics from experience • Experience means solving a lot of instances of a problem. • E.g., solving lots of 8-puzzles • Each optimal solution to a problem instance provides examples from which h(n) can be learned • Learning algorithms • Neural nets • Decision trees • Inductive learning 69
References Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep learning. MIT press. Lê, B. and Tô, V. (2014). Cở sở trí tuệ nhân tạo. Nhà xuất bản Khoa học và Kỹ thuật. Nguyen, T. (2018). Artificial intelligence slides. Technical report, HCMC University of Sciences. Russell, S. and Norvig, P. (2016). Artificial intelligence: a modern approach. Pearson Education Limited.

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lect03-informed-searchhhhhhhhhhhhhhhhh.pdf

  • 1.
  • 2.
    Contents 1. Best-first search 2.Greedy best-first search 3. A* search 4. Memory-bounded heuristic search 5. Heuristic functions
  • 3.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Informed search strategies Informed (heuristic) search strategies • Use problem-specific knowledge beyond the definition of the problem itself • Can find solutions more efficiently than can an uninformed strategy What is heuristic • Additional knowledge of the problem is imparted to the search algorithm. • Heuristic estimates how close a state is to a goal. 3
  • 4.
  • 5.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Best-first search • An instance of the general tree/graph search algorithm • A node is selected for expansion based on an evaluation function, f (n), which is construed as a cost estimate. • The node with the lowest evaluation f (n) is expanded first • The implementation of best-first graph search is identical to that for uniform-cost search, except for the use of f instead of g to order the priority queue • The choice of f determines the search strategy 5
  • 6.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Best-first search (cont.) • Most best-first algorithms include as a component of f a heuristic function, denoted h(n): h(n) estimated cost of the cheapest path from the state at node n to a goal state • h(n) depends only on the state at node n • Assumption of h • Arbitrary, nonnegative, problem-specific functions • Constraint: if n is a goal node, then h(n) = 0 6
  • 7.
  • 8.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Greedy best-first search • Try to expand the node that is closest to the goal, on the grounds that this is likely to lead to a solution quickly. • Thus, it evaluates nodes by using just the heuristic function f (n) = h(n) 8
  • 9.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania • Use the straight-line distance heuristic, which we will call hSLD Table 1: Values of hSLD – straight-line distances to Bucharest. Urziceni Neamt Oradea Zerind Timisoara Mehadia Sibiu Pitesti Rimnicu Vilcea Vaslui Bucharest Giurgiu Hirsova Eforie Arad Lugoj Drobeta Craiova Fagaras Iasi 0 160 242 161 77 151 366 244 226 176 241 253 329 80 199 380 234 374 100 193 9
  • 10.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania (cont.) • The initial state Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Arad 366 10
  • 11.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania (cont.) • After expanding Arad Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Timisoara 253 329 374 11
  • 12.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania (cont.) • After expanding Sibu Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Rimnicu Vilcea Zerind Arad Sibiu Arad Fagaras Oradea Timisoara 329 374 366 176 380 193 12
  • 13.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania (cont.) • After expanding Fagaras Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Rimnicu Vilcea Zerind Arad Sibiu Arad Fagaras Oradea Timisoara Sibiu Bucharest 329 374 366 380 193 253 0 13
  • 14.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Properties • Time: O(bm), but a good heuristic can give dramatic improvement • Space: O(bm), keeps all nodes in memory • Complete: No, can get stuck in loops, e.g., find a path from Iasi to Fagaras, Iasi → Neamt → Iasi → Neamt → ... • Yes in finite space with repeated-state checking • Optimal: No Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Drobeta 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 14
  • 15.
  • 16.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions A* search • A* search (pronounced A-star search) is the most widely known form of best-first search • Ideas • Use heuristic to guide search, but not only • 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 of the cheapest path to goal from n • f (n): estimated cost of the cheapest solution through n • The algorithm is identical to Uniform-Cost-Search except that A* uses g + h instead of g 16
  • 17.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Arad 366=0+366 17
  • 18.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Timisoara 447=118+329 449=75+374 393=140+253 18
  • 19.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Arad Timisoara Rimnicu Vilcea Fagaras Oradea 447=118+329 449=75+374 646=280+366 413=220+193 415=239+176 671=291+380 19
  • 20.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Arad Timisoara Fagaras Oradea 447=118+329 449=75+374 646=280+366 415=239+176 Rimnicu Vilcea Craiova Pitesti Sibiu 526=366+160 553=300+253 417=317+100 671=291+380 20
  • 21.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Arad Timisoara Sibiu Bucharest Rimnicu Vilcea Fagaras Oradea Craiova Pitesti Sibiu 447=118+329 449=75+374 646=280+366 591=338+253 450=450+0 526=366+160 553=300+253 417=317+100 671=291+380 21
  • 22.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration: Romania (cont.) Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to 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 Zerind Arad Sibiu Arad Timisoara Sibiu Bucharest Rimnicu Vilcea Fagaras Oradea Craiova Pitesti Sibiu Bucharest Craiova Rimnicu Vilcea 418=418+0 447=118+329 449=75+374 646=280+366 591=338+253 450=450+0 526=366+160 553=300+253 615=455+160 607=414+193 671=291+380 22
  • 23.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Properties • Time Exponential in [relative error in h × length of solution.] • Space Keeps all nodes in memory • Complete Yes, unless there are infinitely many nodes with f ≤ f (G) • Optimal Yes, cannot expand fi+1 until fi is finished • A* expands all nodes with f (n) < C∗ • A* expands some nodes with f (n) = C∗ • A* expands no nodes with f (n) > C∗ 23
  • 24.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Admissibility and Consistency Concept 1 h(n) is an admissible heuristic • if it never overestimates the cost to reach the goal • or if for every node n, h(n) ≤ h∗(n) (h∗(n) is a the true optimal cost to reach the goal state from n) • An example of an admissible heuristic is the straight-line distance hSLD that we used in getting to Bucharest 24
  • 25.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Admissibility and Consistency (cont.) Concept 2 h(n) is an consistent heuristic if for every node n and every successor n0 of n generated by any action a, the estimated cost of reaching the goal from n is no greater than the step cost of getting to n0 plus the estimated cost of reaching the goal from n0 h(n) ≤ c(n, a, n0 ) + h(n0 ) n c(n,a,n’) h(n’) h(n) G n’ • Consistent heuristic is also admissible 25
  • 26.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Optimal for A* • Using tree search algorithm to find a path from S to G 1 1 3 S h = 5 a h = 4 G h = 0 26
  • 27.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Optimal for A* (cont.) Theorem 1 The tree-search version of A* is optimal if h(n) is admissible Proof Suppose some suboptimal goal G2 has been generated and is in the frontier. Let n be an unexpanded node on a shortest path to an optimal goal G (graph separation) • f (G) = g(G) since h(G) = 0 • f (G2) = g(G2) since h(G2) = 0 • g(G2) > g(G) since G2 is suboptimal • f (G) > f (n) since h is admissible • Since f (G2) > f (n), A* will never select G2 for expansion G n G2 Start 27
  • 28.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Optimal for A* (cont.) • Using graph search algorithm to find a path from S to G 1 1 1 2 3 S h = 2 a h =4 b h = 1 c h = 1 G h = 0 28
  • 29.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Optimal for A* (cont.) Theorem 2 The graph-search version of A* is optimal if h(n) is consistent Proof • if h(n) is consistent, then the values of f (n) along any path are nondecreasing. Suppose n0 is a successor of n; then g(n0) = g(n) + c(n, a, n0) for some action a, and we have f (n0 ) = g(n0 ) + h(n0 ) = g(n) + c(n, a, n) + h(n0 ) ≥ g(n) + h(n) = f (n) • Whenever A* selects a node n for expansion, the optimal path to that node has been found. Were this not the case, there would have to be another frontier node n0 on the optimal path from the start node to n; f (n0) < f (n) ⇒ n0 would have been selected first. 29
  • 30.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Contours • Search algorithm expands nodes in order of increasing f value • We can draw contours in the state space, just like the contours in a topographic map • Gradually adds “f -contours” of nodes, contour i has all nodes with f < fi, where fi < fi+1 O Z A T L M D C R F P G B U H E V I N 380 400 420 S 30
  • 31.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Contours (cont.) • The bands of uniform-cost search will be “circular” around the start state. • The bands of A* with more accurate heuristics will stretch toward the goal state and become more narrowly focused around the optimal path 31
  • 32.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions The 8-puzzle • The average solution cost for a randomly generated 8-puzzle instance is about 22 steps • The branching factor b is about 3 • 8-puzzle: • 322 ≈ 3.1 × 1010 states (using an exhaustive tree search) • 9!/2 = 181, 440 reachable states (using graph search) • 15-puzzle: around 1013 states 2 Start State Goal State 1 3 4 6 7 5 1 2 3 4 6 7 8 5 8 Figure 1: A typical instance of the 8-puzzle. The optimal solution is 26 steps long. 32
  • 33.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Admissible heuristics for 8-puzzle • h1 = the number of misplaced tiles. h1 is an admissible heuristic because it is clear that any tile that is out of place must be moved at least once • h2 = the sum of the distances of the tiles from their goal positions. h2 is also admissible because all any move can do is move one tile one step closer to the goal 2 Start State Goal State 1 3 4 6 7 5 1 2 3 4 6 7 8 5 8 • h1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8 and h2 = 3 + 1 + 2 + 2 + 2 + 3 + 3 + 2 = 18 33
  • 34.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Solve 8 puzzle • Using h2 2 3 1 4 5 8 7 6 2 3 1 4 5 8 7 6 1 2 3 4 5 8 7 6 2 3 1 4 5 8 7 6 1 2 3 4 5 8 7 6 1 2 3 8 4 5 7 6 1 2 3 8 4 7 6 5 Start Goal 6=0+6 8=1+7 6=1+5 6=2+4 8=2+6 8=2+6 34
  • 35.
  • 36.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Memory-bound heuristic search • In practice, A* usually runs out of space long before it runs out of time • Idea: try something like DFS, but not forget everything about the branches we have partially explored 36
  • 37.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Iterative-deepening A* (IDA*) • The main difference with IDS • Cut-off use the f -value (g + h) rather than the depth • At each iteration, the cutoff value is the smallest f -value of any node that exceeded the cutoff on the previous iteration • Avoid the substantial overhead associated with keeping a sorted queue of nodes. • Practical for many problems with unit step costs, yet difficult with real valued costs 37
  • 38.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Algorithm function IDA*(problem) returns a solution, or failure local variables: f _limit: the current f-COST limit root: a node root ← Make-Node(problem.Initial-State) f _limit ← root.f loop do solution, f _limit ← DFS-Contour(problem, root, f _limit) if solution is non-null then return solution if f _limit = ∞ then return failure 38
  • 39.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Algorithm (cont.) function DFS-Contour(problem,node,f _limit) returns a solution, or failure and a new f-cost limit if node.f > f _limit then return null, node.f if problem.Goal-Test(node.State) then return Solution(node) successors ← ∅ for each action in problem.Actions(node.State) do add Child-Node(problem,node,action) into successors f _next ← ∞ for each s in successors do solution,f _new ← DFS-Contour(problem, s, f _limit) if solution is non-null then return solution, f _limit f _next ← Min(f _next, f _new) return null, f _next 39
  • 40.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration • IDA* search for the shortest route from v0 (start node) to v25 (goal node); each node has a f value. 1 3 4 4 5 5 5 6 7 7 6 5 7 6 8 11 13 15 14 11 8 12 7 12 6 14 15 7 13 15 9 40
  • 41.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Recursive best-first search (RBFS) • Keep track of the f -value of the best alternative path available from any ancestor of the current node • backtrack when the current node exceeds f _limit • As it backtracks, replace f -value of each along the path with the best value f (n) of its children 41
  • 42.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Algorithm function Recursive-Best-First-Search(problem) returns a solution, or failure return Rbfs(problem,Make-Node(problem.Initial-State),∞) function Rbfs(problem,node,f _limit) returns a solution, or failure and a new f-cost limit if problem.Goal-Test(node.State) then return Solution(node) successors ← ∅ for each action in problem.Actions(node.State) do add Child-Node(problem,node,action) into successors if successors is empty then return failure,∞ for each s in successors do # update f with value from previous search, if any s.f ← Max(s.g + s.h, node.f ) loop do best ← lowest f-value node in successors if best.f > f _limit then return failure,best.f alternative ← the second-lowest f -value among successors result,best.f ← Rbfs(problem, best, Min(f _limit, alternative)) if result 6= failure then return result 42
  • 43.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration • Stages in an RBFS search for the shortest route to Bucharest. The f -limit value for each recursive call is shown on top of each current node, and every node is labeled with its f -cost. Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu V. Sibiu Bucharest Craiova Pitesti Sibiu Bucharest Craiova Rimnicu V. 366 393 447 449 646 415 671 413 591 450 526 417 553 418 615 607 43
  • 44.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration (cont.) (a) The path via Rimnicu Vilcea is followed until the current best leaf (Pitesti) has a value that is worse than the best alternative path (Fagaras). Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu V. Sibiu Bucharest Craiova Pitesti Sibiu Bucharest Craiova Rimnicu V. 366 393 447 449 646 415 671 413 591 450 526 417 553 418 615 607 44
  • 45.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration (cont.) (b) The recursion unwinds and the best leaf value of the forgotten subtree (417) is backed up to Rimnicu Vilcea; then Fagaras is expanded, revealing a best leaf value of 450. Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu V. Sibiu Bucharest Craiova Pitesti Sibiu Bucharest Craiova Rimnicu V. 366 393 447 449 646 415 671 413 591 450 526 417 553 418 615 607 45
  • 46.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration (cont.) (c) The recursion unwinds and the best leaf value of the forgotten subtree (450) is backed up to Fagaras; then Rimnicu Vilcea is expanded. This time, because the best alternative path (through Timisoara) costs at least 447, the expansion continues to Bucharest. Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu V. Sibiu Bucharest Craiova Pitesti Sibiu Bucharest Craiova Rimnicu V. 366 393 447 449 646 415 671 413 591 450 526 417 553 418 615 607 46
  • 47.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Properties • Optimality • Like A*, optimal if h(n) is admissible • Time complexity • Difficult to characterize • Depends on accuracy of h(n) and how often best path changes • Can end up “switching” back and forth • Space complexity • Linear time: O(bd) • Other extreme to A* - uses too little memory even if more memory were available 47
  • 48.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Simplified Memory-bound A* – SMA* • Similar to A*, but delete the worst node (largest f -value) when memory is full • SMA* expands the (newest) best leaf and deletes the (oldest) worst leaf • SMA* also backs up the value of the forgotten node to its parent. • If there is a tie (equal f -values), delete the oldest nodes first • Simplified-MA* finds the optimal reachable solution given the memory constraint. • The depth of the shallowest goal node, is less than the memory size (expressed in nodes) • Time can still be exponential. 48
  • 49.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Algorithm function SMA*(problem) returns a solution, or failure frontier ← Make-Node(problem.Initial-State) loop do if frontier = ∅ then return failure node ← Pop(frontier) if problem.Goal-Test(node.State) then return Solution(node) s ← Next-Successor(node) if s is not a goal and is at maximum depth then s.f ← ∞ else s.f ← Max(s.g + s.h, node.f ) if all of node's successors have been generated then update node's f-cost and those of its ancestors if necessary if Successors(node) all in memory then remove node from frontier if memory is full then delete shallowest, highest f-cost node in frontier remove it from its parent's successor list insert its parent on frontier if necessary frontier ← Insert(s, frontier) 49
  • 50.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration • Find the lowest-cost goalnode with enough memory • Max Nodes = 3 • A: start node and D, F, I, J: goal nodes • Each edge/node is labelled with its step cost/heuristic value A B 10 G 8 C 10 D 10 H 8 I 16 E 10 F 10 J 8 K 8 12 5 5 5 2 5 0 0 5 0 0 50
  • 51.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Illustration (const.) • Consider only current f-cost A B G C D H I E F J K 12 15 13 25 18 35 30 24 29 20 24 51
  • 52.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Learning to search better • Could an agent learn how to search better? Yes • Metalevel state space: in which each state captures the internal (computational) state of a program that is searching in an object-level state space. • A metalevel learning algorithm learn from these experiences to avoid exploring unpromising subtrees • For example, the map of Romania problem • The internal state of the A* algorithm is the current search tree. • Each action in the metalevel state space is a computation step that alters the internal state; for example, each computation step in A* expands a leaf node and adds its successors to the tree 52
  • 53.
  • 54.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions The effect of heuristic accuracy on performance • Effective branching factor b∗ characterize the quality of a heuristic. • If the total number of nodes generated by A* for a particular problem is N and the solution depth is d, then b∗ is the branching factor that a uniform tree of depth d would have to have in order to contain N + 1 nodes N + 1 = 1 + (b∗ ) + (b∗ )2 + ... + (b∗ )d • b∗ varies across problem instances, but fairly constant for sufficiently hard problems • A well-designed heuristic would have a value of b∗ close to 1 ⇒ fairly large problems solved at reasonable cost 54
  • 55.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions The effect of heuristic accuracy on performance (cont.) Search Cost (nodes generated) Effective Branching Factor d IDS A*(h1) A*(h2) IDS A*(h1) A*(h2) 2 10 6 6 2.45 1.79 1.79 4 112 13 12 2.87 1.48 1.45 6 680 20 18 2.73 1.34 1.30 8 6384 39 25 2.80 1.33 1.24 10 47127 93 39 2.79 1.38 1.22 12 3644035 227 73 2.78 1.42 1.24 14 – 539 113 – 1.44 1.23 16 – 1301 211 – 1.45 1.25 18 – 3056 363 – 1.46 1.26 20 – 7276 676 – 1.47 1.27 22 – 18094 1219 – 1.48 1.28 24 – 39135 1641 – 1.48 1.26 Table 2: Comparison of the search costs and effective branching factors for the IDS and A* algorithms with h1, h2. Data are averaged over 100 instances of the 8-puzzle for each of various solution lengths d. 55
  • 56.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Dominance and Composite • If h2(n) ≥ h1(n) for all n (both admissible) then h2 dominates h1 and is better for search • A* using h2 will never expand more nodes than A* using h1 • Hence, it is generally better to use a heuristic function with higher values, provided it is consistent and that the computation time for the heuristic is not too long. • Given a collection of admissible heuristics {h1, ..., hm} is available for a problem and none of them dominates any of the others. • The composite heuristic function is defined as h(n) = max{h1(n), . . . , hm(n)} is consistent and dominates all component heuristics 56
  • 57.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Relaxed problems Concept 3 A problem with fewer restrictions on the actions is called a relaxed problem. • Prove that the angle in a semicircle is a right angle. A O B C 57
  • 58.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Generating admissible heuristics from relaxed problems • The state-space graph of the relaxed problem is a supergraph of the original state space because the removal of restrictions creates added edges in the graph. S a b c d G Figure 2: Original problem state-space S a b c d G Figure 3: Relaxed problem state-space 58
  • 59.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Generating admissible heuristics from relaxed problems (cont.) • Hence, the cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem. • Furthermore, because the derived heuristic is an exact cost for the relaxed problem, it must obey the triangle inequality and is therefore consistent. 59
  • 60.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Generating admissible heuristics from relaxed problems (cont.) Original problem of 8 puzzle • A tile can move from square A to square B if A is horizontally or vertically adjacent to B and B is blank We can generate three relaxed problems by removing one or both of the conditions Relaxed problems 1. A tile can move from square A to square B if A is adjacent to B → h2 2. A tile can move from square A to square B if B is blank → exercise 3. A tile can move from square A to square B → h1 60
  • 61.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Generating admissible heuristics from subproblems • Admissible heuristics can also be derived from the solution cost of a subproblem of a given problem • The cost of the optimal solution of this subproblem ≤ the cost of the original problem (lower bound). • More accurate than Manhattan distance in some cases Start State Goal State 1 2 3 4 6 8 5 2 1 3 6 7 8 5 4 Figure 4: A subproblem of the 8-puzzle instance. The task is to get tiles 1, 2, 3, and 4 into their correct positions, without worrying about what happens to the other tiles. 61
  • 62.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Goal fixed? • Assume that the goal state G is fixed • Using Uniform-Cost-Search to find the shortest path from G to the other states • The final g values can be considered as optimal h∗ values 62
  • 63.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Goal fixed? (cont.) 9 3 6 8 5 4 1 8 17 21 4 4 G b d h c e f 9 3 6 8 5 4 1 8 17 21 4 4 G g = 0 b g = 7 d g = 3 h g = 6 c g = 4 e g = 9 f g = 26 9 3 6 8 5 4 1 8 17 21 4 4 G h = 0 b h = 7 d h = 3 h h = 6 c h = 4 e h = 9 f h = 26 63
  • 64.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Pattern databases • Pattern databases (PDB) where store the exact solution costs for every possible subproblem instance • E.g., every possible configuration of the four tiles and the blank • The database itself is constructed by searching back from the goal and recording the cost of each new pattern encountered; the expense of this search is amortized over many subsequent problem instances • The complete heuristic is constructed using the patterns in the databases 64
  • 65.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Computing the Heuristic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 10 14 7 8 3 6 1 15 12 2 11 4 9 13 • 31 moves needed to solve red tiles • 22 moves need to solve blue tiles • Overall heuristic is maximum of 31 = max(31, 22) moves 65
  • 66.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Disjoint pattern databases • Limitation of traditional PDB: Take max → diminish returns on additional PDBs • Disjoint pattern databases: Count only moves of the pattern tiles, ignoring non-pattern moves. • If no tile belongs to more than one pattern, add their heuristic values. 66
  • 67.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Computing the Heuristic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 10 14 7 8 3 6 1 15 12 2 11 4 9 13 • 20 moves needed to solve red tiles • 25 moves needed to solve blue tiles • Overall heuristic is sum, or 45 = 20 + 25 moves 67
  • 68.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Performance • 15 Puzzle • 1000 × speedup vs. Manhattan distance • IDA* with the two DBs (the 7-tile database contains 58 million entries and the 8-tile database contains 519 million entries) solves 15-puzzles optimally in 30 milliseconds • 24 Puzzle • 12 million × speedup vs. Manhattan distance • IDA* can solve random instances in 2 days. • Requires 4 DBs (each DB has 128 million entries) • Without PDBs: 65,000 years 68
  • 69.
    Best-first search Greedy best-first search A* search Memory- bounded heuristicsearch Heuristic functions Learning heuristics from experience • Experience means solving a lot of instances of a problem. • E.g., solving lots of 8-puzzles • Each optimal solution to a problem instance provides examples from which h(n) can be learned • Learning algorithms • Neural nets • Decision trees • Inductive learning 69
  • 70.
    References Goodfellow, I., Bengio,Y., and Courville, A. (2016). Deep learning. MIT press. Lê, B. and Tô, V. (2014). Cở sở trí tuệ nhân tạo. Nhà xuất bản Khoa học và Kỹ thuật. Nguyen, T. (2018). Artificial intelligence slides. Technical report, HCMC University of Sciences. Russell, S. and Norvig, P. (2016). Artificial intelligence: a modern approach. Pearson Education Limited.