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**Semir Elezovikj Mo'taz Abdul Aziz Ali Al-Hami Howard Liu**

Group 5 Semir Elezovikj Mo'taz Abdul Aziz Ali Al-Hami Howard Liu

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**Heuristic Function H(n)**

F(n) = G(n) + H(n) How to decide the heuristic function H(n) in this project? Our choice for this project: for two points (x1, y1) and (x2, y2): H(n) = Diagonal Distance = max(abs(x1-x2),abs(y1-y2))

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**Consider a 3x6 map (in terms of edge length):**

1. Costs are assigned to nodes but not edges. 2. Paths between two nodes can be vertical, horizontal, or diagonal. 3. costs are random numbers between 1 and 11.

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**To find the minimum cost for a 3x6 map:**

1. Assume all nodes have the lowest cost: 1. 2. Find the shortest path in this map. 3. The cost of this shortest path is the lowest cost of all possible “random number costs” 3x6 maps.

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**1. By brute force, we can find many shortest paths**

1. By brute force, we can find many shortest paths. Here are four of them. 2. They all consist of the start node, 3 diagonal moves, and 3 vertical moves. 3. They all have the lowest cost: = 7. 4. The cost 7 should be the minimum cost of all possible “random number costs” 3x6 maps.

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**1. In a 3x3 square, the number of diagonal moves is equal to its side length 3.**

2. In fact, 7 = the longer length of the rectangle + start node = 3. Therefore, we can conclude the minimum cost of a map is its longer side length + 1. This is an admissible heuristic function. H(n) <= H*(n). 5. We can ignore the 1 cost of start node, since it is already counted in G(n). 6. Formula: for two points (x1, y1) and (x2, y2), H(n) = max(abs(x1-x2), abs(y1-y2)) 7. This is called “Diagonal Distance”, “Chessboard Distance”, or “Chebyshev Distance”.

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**Speed Up the Program Our strategies:**

Avoid repeatedly changing matrix size in a big loop. Avoid repeatedly calling sub-functions in a big loop. Since the map size is not very big, a simple algorithm is faster than a fancy algorithm. In our case, the simple linear search for minimum is faster than a complicated implementation of priority queue (min-heap) because the linear search has less data movements.

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**Pseudo code (found on Wikipedia)**

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