Solution of Quiz 2 - KFUPM Open Courseware - King Fahd ...

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King Fahd University of Petroleum and Minerals
College of Computer Science and Engineering
Information and Computer Science Department
ICS 381: Quiz #2-Section 02
Name:
ID Number:
Max. Grade (20 Points)
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Question 1: Let h be an admissible heuristic function. A best-first search algorithm
called Pathmax uses this function as follows. When it adds a new node N’ to the
search tree as a child of a node N, instead of computing f(N’) = g(N’) + h(N’),
Pathmax computes f(N’) = g(N’) + max{h(N’), h(N) - c(N,N’)}, where c(N,N’) is the
cost of the arc from N to N’.
‣ Explain what Pathmax is trying to do. (2 Points)
Answer:
1. Pathmax is trying to build a consistent heuristics out a non-consistent one. Clearly, if
h(N) - c(N, N’) > h(N’) at some node N’, then h(N) - c(N,N’) is a tighter (but still
optimistic) estimate of the cost to go from N’ to the goal than h(N’).
Question 2: The A * search algorithm has a major drawback in that it usually runs
out of memory before a solution is found. The algorithm called SMA * , or simplified
memory-bounded A * search, is an improvement on A * that tries to solve this. How?
Is it an optimal algorithm? (5 Points)
Idea
Expand the best leaf (just like A* ) until memory is full
When memory is full, drop the worst leaf node (the one with the highest f-value)
to accommodate new node.
• If all leaf nodes have the same f-value, SMA* deletes the oldest worst leaf and
expanding the newest best leaf.
Then, SMA* backs up the value of the forgotten node to its parent.
• In this way, the ancestor of a forgotten sub-tree knows the quality of the
best path in that sub-tree.
• With this info, SMA* regenerate the sub-tree only when all other paths
have been shown to look worst than the path it has forgotten.
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Dr. Tarek Helmy ICS 381: Quiz #2-Section 02

Question 3: In the “Four-Queens puzzle,” we try to place four queens on a 4x4 chess
board so that none can capture any other. (That is, only one queen can be on any
row, column, or diagonal of the array.) Suppose we try to solve this puzzle using the
following problem space: The start node is labeled by an empty 4x4 array; the
successor function creates new 4x4 arrays containing one additional legal placement
of a queen anywhere in the array; the goal predicate is satisfied if and only if there
are four queens in the array (legally positioned).
1. Give the 4- Queens Problem Formulation as an CSP (4 Points)
2. Solve the 4-Queens CSP by applying the Backtracking Search using:
Forward Checking and Constrains Propagation features (6 Points)
1
2 3 4
X1
{1,2,3,4}
X2
{1,2,3,4}
1
2
3
4
X3
{1,2,3,4}
X4
{1,2,3,4}
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Dr. Tarek Helmy ICS 381: Quiz #2-Section 02

Question 5: Game-Playing
If you are programming a game playing algorithm that uses the Minimax procedure and
alpha-beta pruning. So after your computer has done all this pruning, it is fair enough that
now it is your turn and you will do some of the alpha-beta pruning yourself. The tree
below indicates the complete Minimax tree for a particular problem (first move by MIN,
then MAX, and then MIN again). The number at each leaf p indicates the value of the
static evaluation function e(p) if it were computed at that leaf.
a) Now your job is to check the boxes under those leaves that do not need to be created
and evaluated thanks to the alpha-beta pruning.
b) Which next move (the left or right one) should MIN make, and why?
The left move, because it guarantees MIN a maximum score of -2 after 3 moves,
whereas the right move may lead to a worse result.
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Dr. Tarek Helmy ICS 381: Quiz #2-Section 02