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#5 Homework UCSD Winter 2002, CSE 101, 3/9/2002 Question 1. Best-first search is a general search algorithm that always considers (expands) the estimated best partial solution. This is typically implemented with a priority queue. One well-known best-first search algorithm estimates the cost of a partial solution as the cost of getting to the node that represents that partial solution, plus an estimate of the cost of getting from that partial solution to a complete solution ("the goal"). More formally: g(n) = the cost of path from the initial state to state() h(n) = estimated cost of the cheapest path from the state represented by the node n to a goal state f(n) = g(n) + h(n) = estimated cost of the cheapest solution through n Given a grid graph G representing the 2-dimensional Manhattan grid (i.e., integer lattice) with all edges having unit cost, we wish to find the shortest path from node s to node t. Let g(n) be the path length from node s to node n, and let h(n) be the Manhattan distance between node n and t, where the Manhattan distance between nodes a and b is defined as d(a,b) = |ax - bx| + |ay - by|. (a) Show with contour lines the order in which nodes in G are expanded by the algorithm. Justify your drawing with a few sentences. Solution: The algorithm described above for historical reasons is known in the literature as the A* algorithm. To see the behavior of this algorithm we have simulated its search pattern on a grid graph. The task of the algorithm was to find a shortest path from the start node s the goal node t. In the pictures below s is represented as the lighter dot at the center and t is represented as the darker dot in the top right corner. From left to right the grayscale images represent g, h, and f values respectively. The values, according to the problem specifications were calculated in Manhattan distances, and darker shades represent lower values. Looking at the left most picture we can see that in the upper right quadrant all the values are the same and are lowest overall. Since goal node t is present in this region (hence it will be found), and since in the other regions all the points have higher f values, the only nodes that will be expanded must be in this upper right quadrant. Because of the grid structure of G, and because A* can move between nodes that are adjacent to each other, the contour lines will all be concentric around s and will gradually reach out towards t. These contour lines are shown in the left most picture. (b) Give an estimate of how many nodes are expanded by the algorithm as a function of whatever parameters you feel appropriate (e.g., d(a,b), |ax - bx|, |ay - by|, etc.). Solution: From the rightmost image above one can see that the f values in the upper right quadrant are the lowest and are all equal to each other. Since G, on which the algorithm operates, is a grid graph with a simple FIFO order implementation of the priority queue nodes closest to s are expanded first. Hence, by the time the algorithm reaches the goal state t, most of the nodes in the upper right quadrant will be expanded. Thus the number of nodes expanded is O( |sx tx| * |sy ty| ). (c) Does your estimate in (b) depend on how the priority queue is implemented? Describe an implementation of the priority queue that allows the search algorithm to reach t while expanding the minimum possible number of nodes. Solution: The estimate for the number of nodes expanded given in part b does depend on the implementation of the priority queue. Note that the heuristic value h is a perfect estimate of the cost to the goal node. Hence if all the f values would be all distinct (which is clearly not the case in the upper left quadrant) then A* would straight march to the goal node t and will expand an optimal number of nodes. If we implemented the priority queue in the best first search such that equal values are processed in a LIFO order, the two alternative routes for A* would be a straight marsh to the goal and are shown in the picture below. The actual route taken depends on in what order the children of an expanding node are processed. Question 2. Prove the optimality of the above algorithm, given the assumptions that (1) the h function is admissible, and that (2) the f = g + h function is monotonic. Intuitively, admissible means acceptable or reasonable. Formally, h(n) is admissible if it is always true that: h(n) = estimated cost of cheapest path from state(n) to goal <= actual cost of cheapest path from state(n) to goal Solution: Adopted from Professor Charles Elkan at UCSD Proof of optimality: By definition, the next node expanded by A* is always one of the remaining unexpanded nodes that has the lowest f = g + h value (maybe the equal-lowest f value). When A* expands a goal node, it first checks and discovers that it is indeed a goal node. Now the h value of a goal node is zero, so its f value is its g value, which is exactly the cost of the path to this node. Call this cost c*. So at the instant that A* finds a goal node, every other node in the priority queue, say node n, must have f(n) >= c*. For each such node n, f(n) is an under-estimate of the true cost c' of reaching the cheapest goal node reachable through n. Therefore every other goal node has cost c' >= f(n) >= c*. This proves that A* is optimal. Question 3. Imagine that there are obstacles in the search space, i.e., vertices in G that cannot be visited. (Such obstacles might represent real obstacles, e.g., in a VLSI chip routing problem.) (a) Draw contour lines as in Question 1 to represent the order of nodes visited or expanded by the algorithm in Question 1, when obstacle an is placed in the close vicinity of s. To be specific, suppose that s is at (0,0); and that the obstacle consists of all vertices on the line segments (-3,5) to (5,5), and (5,-3) to (5,5). Solution: In the left half of Figure 1, we show contour lines that represent the order in which nodes are expanded by the algorithm. Again, the particular order of expansions depends on the adjacency characteristics in the grid graph G. It is important to note that none of the nodes outside of the big rectangular area are expanded because they have higher f values than any one of the nodes inside the rectangle. Nodes adjacent to the outside of the big rectangular area are visited but not expanded. (b) Draw contour lines as in Question 1 to represent the order of nodes visited or expanded by the algorithm in Question 1 when an obstacle is placed in the close vicinity of t. Specifically, assume a symmetric situation to that given in part (a). Solution: Reasoning is identical to the reasoning presented in part a. Contour lines for this situation are shown in the right half of Figure 1. An important thing to note is that the two regions around t (upper right and lower left have higher f values than the other surrounding values. This is true since for all points in this region it is true that they cannot by on an optimal path. For this reason they will not be expanded on by A*. t k k-1 13 10 9 8 7 11 12 15 14 8 9 11 12 10 13 14 t 14 13 12 11 7 1 5 4s 6 7 11 10 89 2 3 13 2 1 4 3 8 6 5 12 10 9 6 s Figure 1: Contour lines for finding the shortest path from s to t with an object around s (left), and with an object around t (right). Order of expansions is indicated by the numbers next to the contour lines. Question 4. Suppose we simultaneously run the above algorithm starting from s (to find the shortest path to t), and starting from t (to find the shortest path to s). (a) What do the contour lines look like? Do you think that, in general, this "bidirectional" approach is more efficient than the original "uni-directional" approach? Explain your thinking. Solution: Contour lines from s to t are identical to that presented in the right most picture in question 1. Contour lines from t to s are symmetrical and are approaching s. They are shown on the left half of figure 2. t 2 3 4 5 6 5 4 1 2 3 1 t 6 3 2 1 4 5 4 3 2 1 5 s s Figure 2: Contour lines for the bi-directional approach. Contour lines on the left are for a perfect heuristic, while on the right for a more general case (i.e.: a less perfect heuristic). As one can see from the above figure, the number of nodes expanded by the bidirectional approach in our case (left) is exactly equal to the nodes expanded by the uni-directional approach. In a more general case, when the heuristic function h used by the algorithm is not perfect the number of nodes expanded by the bi-directional approach is less, however, as we will see in the part (b), the implementation overhead to find a solution to the original problem outweigh the benefits of this approach. Contour lines for this more general case are shown on the right half of figure 2. (b) Is it true that when the two expanding wave fronts first meet, you have found a shortest path from s to t? If yes, justify your answer. If no, then under what condition have we actually found a shortest s-t path? Solution: As we have mentioned earlier the heuristic function h used by the algorithm is a perfect heuristic or estimator. In this special case it is true that when the two expanding wave fronts meet in the bi-directional approach the algorithm found an optimal solution to the original problem. Assuming the two expanding wave fronts meet at some node n, an optimal solution can be constructed as the path s to n and the path from t to n (in reverse direction). In the more general case, when we do not have a perfect heuristic, that is all we know is that h is an under-estimator, the condition for finding an optimal solution is that for a given node n either gs(n) = ht(n), or gt(n) = hs(n), where the subscripts denote the direction of the search. In both of these cases we have found an optimal solution, in other words an optimal path to the goal node, since both ht and hs are under-estimators and gt and gs are actual partial path costs. As we can see, in order to take advantage of the reduction of expanded nodes by this bi-directional approach, one has to constantly check for the conditions above. This inevitable induces an implementation cost that may overweigh the benefits of this approach. Question 5. Given the following branch-and-bound tree, where the numbers at the nodes represent the incremental costs of visiting that node (i.e., growing the partial solution vector), and the leaves represent complete solutions. Trace the execution sequence of depth-first branch-and-bound (DFBB) by showing the sequence of visited nodes. Solution: The nodes that are visited by DFBB are gray while the unexpanded nodes are white. The diagonal lines show where the search tree was pruned by DFBB. Leaves in the search tree represent goal nodes or full solutions, while internal nodes represent partial solutions. In this context visited means that the cost of the partial or full solution was evaluated. Pruning of the search tree happens when the cost of a partial solution becomes more than the cost of the so far best full solution. The boldly outlined nodes represent partial solutions that were fully expanded (i.e.: all the operators of the problem were applied to a node and all its successors were returned). 0 2 12 3 4 7 1 3 1 6 1 1 1
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Path: UCSD >> CSE >> 101 Winter, 1998
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Path: UCSD >> CSE >> 5 Fall, 2008
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Path: UCSD >> CS >> 008 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> CSE >> 237 Fall, 2008
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Path: UCSD >> COGS >> 260 Winter, 2007
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Path: UCSD >> GS >> 09 Fall, 2008
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GS09GermanyHIEU152.pdf
Path: UCSD >> GS >> 09 Fall, 2008
Description: Course Overview and Outline HIEU 152 The Worst of Times: Everyday Life in Authoritarian and Dictatorial Societies Prof. Patrick H. Patterson, Dept. of History Aims and Scope of the Class: This course is an outgrowth of one of my primary areas of rese...
GS09GermanyMMW6.pdf
Path: UCSD >> GS >> 09 Fall, 2008
Description: Course Overview and Outline Making of the Modern World 6: The Twentieth Century and Beyond Summer 2009 - Global Seminar in Berlin Prof. Patrick H. Patterson, Dept. of History Aims and Scope of the Class: This course will cover a number of the most i...
OL2000.pdf
Path: UCSD >> DSS2 >> 2 Fall, 2008
Description: Original Source: http:/www.officetutorials.com/ Introduction to Microsoft Outlook 2000 Mail Created: 10 September 2001 Starting Outlook 2000 In this Microsoft Outlook 2000 tutorial, well discuss a number of the basic procedures used in creating, ed...
