Map coloring. 4 colors. From coloring to CSP. Informed Search Methods. Constrained Satisfaction Search. Constrained Satisfaction Search.

Transcription

1 Constrained Satisfaction Search Informed Search Methods Example: 8-Queens Problem States: <V 1, V 2,, V 8 >, where V is the row occupied by the ith queen. Constraint satisfaction search Heuristics Best First Search Greedy Search A* Search Domains: {1, 2, 3, 4, 5, 6, 7, 8} Constraints: no-attack constraint { <1,3>, <1,4>, <1,5>, <2,4>. <2,5>, } where each element specifies a pair of allowable values for variables V i and V j. CIS 412, UMD, 2 Constrained Satisfaction Search Map coloring Types of Constraints Discrete (e.g. 8-queens) versus Continuous. Absolute constraints (violation rules out solution candidate) versus preferential constraints (e.g. programming). Other better search strategies Backtracking search, forward checking, arc consistency checking. constraint propagation. No color left CIS 412, UMD, 3 CIS 412, UMD, 4 4 colors Motivations Map coloring is typical CSP CSP problems Timetabling Scheduling New paradigm constraint programming From coloring to CSP CIS 412, UMD, 5 CIS 412, UMD, 6 1

2 Can we 2-color a triangle? Early failure CIS 412, UMD, 7 CIS 412, UMD, 8 Can we 2-color a triangle? Shortest Path Problems Common in real-life problems e.g. select routes for planes to minimise fuel use With planning and puzzles we saw that finding a plan or solving a puzzle becomes find a path But the relevant graphs are now implicit nodes and edges are only given by rules exponentially larger than occurs with explicit maps 15-puzzle has 16!/2 states (approx 10 13) CIS 412, UMD, 9 CIS 412, UMD, 10 Shortest Path Problems When the graph becomes too large blind search is no longer practical Problem is that blind search does not look in the right places To see this think of the search pattern of Uniform Cost Search On the next 2 slides we will see an example based on a map of Romania from AIMA These include an animation of how UCS performs the search, hence printed version will be a mess! CIS 412, UMD, 11 Study The optimal this map route of Romania. between the Note two the cities distances takes between us through cities and try and and. calculate the Press shortest space route to continue between with Sibiu the and slideshow.. Then press space to see the optimal route marked in red. 71 Oradea Lugoj 70 Mehadia Optimal route is (++101) = 278 miles Sibiu 99 Faragas Giurgui CIS 412, UMD, 12 2

3 ANIMATION OF UCS. Nodes Expanded 1.Sibiu 2. 3.Faragas Oradea GOAL!! 87 Fringe in RED with g Visited in BLUE 92 Optimal route is (++101) = 278 miles Sibiu 99 Faragas Lugoj Mehadia (F) 226 (R) (R,P) Giurgui [315 (R,P)] CIS 412, UMD, 13 Remarks We expanded and even though they are in totally the wrong direction! The search pattern was expanding concentric circles CIS 412, UMD, 14 Uniform Cost Search in Graphs Search Pattern: smaller cost nodes before larger Fringe in red Visited in blue Blind: still ignores s Heuristics in search What is heuristics? It uses heuristics, or rules-of-thumb to help decide which parts of a tree to examine. A heuristic is a rule or method that almost always improves the decision process. This region is basically wasted effort CIS 412, UMD, 15 CIS 412, UMD, 16 Heuristic Search Heuristic Search Desired Search Pattern: biased in direction of (s) Fringe in red Visited in blue When reaches will have had to explore less of the graph Reduces time and memory usage Want to achieve this pattern but stay complete optimal If bias the search too much then could miss s or miss shorter paths CIS 412, UMD, 17 CIS 412, UMD, 18 3

4 Heuristics for 8-tile puzzle Heuristics for 8-tile puzzle Two possible heuristic functions that never overestimate the number of steps to the are: 1. h 1 = the number of tiles that are in the wrong position. In figure 5.7, none of the 8 tiles is in the position, so that state would have h 1 = 8. h 1 is admissible heuristic, because it is clear that any tile that is out of the place must be moved at least once. 2. h 2 = the sum of distance of the tiles from their positions. Since no diagonal moves is allow, we use Manhattan distance or city block distance. h 2 is also admissible, because any move can only move 1 tile 1 step closer to the. The 8 tiles in the state give a Manhattan distance of h 2 = = 18 CIS 412, UMD, 19 CIS 412, UMD, 20 Heuristics for 8-tile puzzle More Realistic Picture In real problems the graphs are messy and the search pattern not clean What we think is correct direction might be ultimately wrong Standard solution to stay complete and optimal: A* search pronounced A star But first do a more general heuristic method: Best First Search CIS 412, UMD, 21 CIS 412, UMD, 22 Heuristic Search Best first search Suppose have some estimates, h, of the cost from a (fringe) node to the Want to favour small h Expand A first? g A A B g B h A h B CIS 412, UMD, 23 CIS 412, UMD, 24 4

5 CIS 412, UMD, 25 CIS 412, UMD, 26 Greedy search This is one of the simplest BestFS strategy. Heuristic function: h(n) - prediction of path cost lef to the. Greedy Search: To minimize the estimated cost to reach the. The node whose state is judged to be closest to the state is always expanded first. Two route-finding methods (1) Straight line distance; (2) minimum Manhattan Distance - movements constrained to horizontal and vertical directions. CIS 412, UMD, 27 CIS 412, UMD, 28 Greedy search Greedy search Noticed that the solution for A S F B is not optimum. It is 32 miles longer than the optimal path A S R P B. The strategy prefers to take the biggest bite possible out of the remaining cost to reach the, without worrying whether this is the best in the long run - hence the name greedy search. It seeks to minimise the estimated cost to the by expanding the node estimated to be closest to the state Greed is one of the 7 deadly sins, but it turns out that GS perform quite well though not always optimal. GS is susceptible to false. Consider the case of going from to Fagaras. will be consider before Vaului even though it is a dead end. CIS 412, UMD, 29 CIS 412, UMD, 30 5

6 Greedy Search Perfect Information Search Dives in what h says is the best direction Suppose that by magic (by an oracle in CS parlance) we knew the True minimum cost from any node N to the I.e. suppose we know h T (n) This direction is just an estimate and can be wrong In UCS we can expect to know g T (n) True min cost, f T (n) for going via n, f T (n) = g T (n) + h T (n) CIS 412, UMD, 31 CIS 412, UMD, 32 Comparisons 1. UCS: BestFS( g ) complete, optimal, high memory usage 2. Perfect-Info: BestFS( g T (n) + h T (n) ) complete, optimal, low memory usage 3. Greedy: BestFS( h ) incomplete, suboptimal, low memory usage A* Search Cannot implement Perfect Info! Try to get best combination use g(n) : best path cost so far use heuristic h(n), as do not have the true h T (n) A* Search is BestFS( g + h ) That is, define f(n) = g(n) + h(n) estimated minimum cost via n and use BestFS( f(n) ) CIS 412, UMD, 33 CIS 412, UMD, 34 Effects of h choices Two extreme cases 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. uses less memory but lose optimality. Usually when we say A* we exclude the last case, and imply that we have an: Admissible Heuristic: h(n) h T (n) we never over-estimate distance h(n) must provide a valid lower bound on cost to the the g term does not get swamped and misled, but the search pattern improves A* is complete and optimal, and memory usage can be much lower than UCS A* with Admissible Heuristic Complete and Optimal Search as for Uniform Cost Search Search Pattern seeks out s Reduced memory and time usage It is good on maps/graphs It is essential on planning and puzzles CIS 412, UMD, 35 CIS 412, UMD, 36 6

7 Admissible Heuristic Need an h Must underestimate actual distance via the roads to the Straight Line Distance (SLD) Roads cannot be shorter than this CIS 412, UMD, 37 Straight Line Distances to Town Fagaras Giurgiu Lugoj SLD Town Mehadai Oradea Sibiu SLD We can use straight line distances as an admissible heuristic as they will never overestimate the cost to the. This is because there is no shorter distance between two cities than the straight line distance. Press space to continue with the slideshow. CIS 412, UMD, 38 ANIMATION OF A*. Nodes Expanded 1.Sibiu Fagaras GOAL!! 71 Oradea 87 Annotations: Fringe in RED g+h=f Visited in BLUE +366= Optimal route is (++101) = 278 miles = =277 Sibiu 99 Fagaras could use? =273 Lugoj 177+=2 70 Mehadia =310 (F) =3(R) =278 (R,P) 120 Giurgui =4(R) CIS 412, UMD, 39 In terms of a search tree we could represent this as follows... Si =253 Maths: g + h = f 99 Press space to begin the search Ri 2 Pi =2 =273 Fa =277 Ar +366 =506 A* SEARCH TREE Cr =3 Bu = Bu =278 Cr =4 The state is achieved. In relation to path cost, A* has found the optimal route with 5 expansions. Press space to end. CIS 412, UMD, 40 Heuristic Searches - Example Problem Initial State Goal State Heuristic Searches - A* Algorithm Typical solution is about twenty steps Branching factor is approximately three. Therefore a complete search would need to search 3 20 states. But by keeping track of repeated states we would only need to search 9! (362,8) states But even this is a lot (imagine having all these in memory) CIS 412, UMD, 41 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 CIS 412, UMD, 42 7

8 Heuristic Searches - Possible Heuristics H 1 = the number of tiles that are in the wrong position (=7) H 2 = the sum of the distances of the tiles from their positions using the Manhattan Distance (=18) Both are admissible but which one is best? Test from 100 runs with varying solution depths Search Cost Depth IDS A*(h1) A*(h2) H 2 looks better as fewer nodes are expanded. But why? CIS 412, UMD, 43 CIS 412, UMD, 44 Effective Branching Factor Search EBF Cost Depth IDS A*(h 1) A*(h 2) IDS A*(h 1) A*(h 2) H 2 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 H 2 has a lower EBF and is therefore the better heuristic CIS 412, UMD, 45 Admissible Heuristics The restriction we mentioned on the previous slide for the h function is simply this: The h function must never overestimate the cost to reach the. Such an h is called an admissible heuristic. Another way of describing admissible functions is to say they are optimistic, as they always think the cost to the is less than it actually is. If we have an admissible h function, this naturally transfers to the function as f never overestimates the actual cost to the best solution through n. CIS 412, UMD, 46 Important Note Study The optimal this map route of Romania. between the Note two the cities distances takes us between through cities Sibiu, and try and calculate. the Press shortest space route to between continue with the and slideshow.. Then press space to see the optimal route marked in red. The following example demonstrating the A* search pattern is taken from the course textbook (Artificial Intelligence : A Modern Approach, Russell and Norvig, Prentice Hall, 1995, pages 95-99) If you are reading this book alongside your CIS 412 notes (and this is strongly recommended) you may notice some minor differences. For example, whereas in the book the authors expand back to cities they have just come from, we ignore this in order to make the animation a little clearer. Please note that this minor change does not affect the algorithm as travelling to a city and then travelling back to where you have just come from is not going to be included in the optimal solution. One other small point about the course textbook. Figure 4.4 on page has a minor error (at least, it certainly does in the 1995 edition). The expansion of Sibiu to Oradea reads + 3 = 526. It should read = 671. Fortunately, this error makes no difference to the way the algorithm operates. CIS 412, UMD, Oradea Optimal route is (+++101) = 418 miles 118 Sibiu 99 Faragas 111 Lugoj 70 Mehadia Giurgui CIS 412, UMD, 48 8

9 A* in action Straight Line Distances to Having established the optimal path between the two towns of and we can now test the efficiency of the A* search pattern in finding the same state. Remember that the A* search pattern is the union of two evaluation functions. In the following demonstration the A* search pattern evaluates the cost of the path so far (UCS), together with an admissible heuristic function based on the shortest line distance (SLD) between between the initial state and the location, such that hsld(n) = straight line distance between n and the location. Town Fagaras Giurgiu Lugoj SLD Town Mehadai Oradea Sibiu SLD The following is table of the straight line distances between some of the major Romanian cities and and the state,. CIS 412, UMD, 49 We can use straight line distances as an admissible heuristic as they will never overestimate the cost to the. This is because there is no shorter distance between two cities than the straight line distance. Press space to continue with the slideshow. CIS 412, UMD, 50 In Once We Press actual now have begin Fagaras space expand fact, just now with is to the is arrived is see the expanded Fagaras Sibiu algorithm initial at A* we. (that node search state we look will is, look is, () of is, for not of we. we for As the really expand the this Romanian that node The lowest recognise node revealed cost with the node with lowest of cost map the reaching, that with the lowest node. featured cost lowest the As the have node cost. lowest you in but cost. found Sibiu the AND from it can value As has previous see, has the. you a of cost of the we f of can f (or ). f ). slide. lowest ). f now of state ). gsee, Press 418. value) It have Note: just we value space If is there for has keeps two can space Throughout to 0 miles. continue f. the terminate is (The to expanding to lowest any continue The cost other straight value nodes. to animation search. lower the reach for lowest search. line One cost Sibiu f. If distance (The of all you cost these from cost look nodes (and from nodes to are back reach (based in labelled is this ( over case to on the miles, with f Sibiu ) there slides from until and f(n) is you it the one (or = finds g(n) will straight is hcheaper value) 317 a see + h(n). miles, that line node, is state 366 However,we the distance and AND Fagaras, solution miles. the it from straight has This with will ) the a has returned Sibiu line lowest be gives cost using distance to of f value the by 417) the total the of from abbreviations then of A* f. value state 418. So, we search we The is of need 253 ( must to pattern f other f, = to miles. g expand now + node h ( This hmove ) state 366 to ( it make in gives miles. is to Sibiu case Fagaras Sibiu the a miles. it total Press leads notation and of Fagaras This space to 393 expand a simpler gives better miles). to expand (2) a it. solution total Press the of space 415 initial to ) has miles). to state ), continue. an move is f of Press value fact than to this the space of optimal 418 node to We solution move and therefore solution. expand to we this have Press move node it. already space and to the expand to found. first continue it. Press with space node the to slideshow. and continue. expand it. Press space to continue F= F= 449 F= F= 366 Oradea Sibiu F= 417 F= F= 393 F= F= F= 413 F= 447 F= F= 526 F= F= 671 F= F= 415 F= Fagaras (2) F= F= 418 F= F= 450 Features of an A* Search 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 length of the search space for 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 search complexity is more serious. 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. A heuristic for which it holds is said to exhibit monotonicity. Because A* expands the leaf nodes of lowest f, we can see that an A* search fans out from the node, adding nodes in concentric bands of increasing f-cost. These bands can be modelled as contours in the state space. CIS 412, UMD, 51 CIS 412, UMD, 52 Map of Romania showing contours at f = 3, f = 400 and f = 420, with as the state. Note: Nodes inside a given contour have f-costs lower than the contour value. Press space to continue the slideshow. Oradea Sibiu Fagaras CIS 412, UMD, 53 9