Depth-first & breadth-first search

The textbook shows algorithms for Depth-First Search (DFS) and Breadth-First Search (BFS) applied to tree structures. Now we extend the algorithms to graph structures.

In our definition, BFS is a graph search algorithm that begins from a node (a "root" node) and explores all the neighboring nodes. Then for each of those nearest nodes, it explores their unexplored neighbor nodes, and so on, until all the nodes within the graph are explored.

DFS is a graph search algorithm that starts from a node (a "root" node) and explores as far as possible along each branch before backtracking (going back to a node that has been explored before). For an undirected and connected graph, such a process could produce a Depth-First Search Tree, in which the starting node serves as the root of the tree. Whenever a new unvisited node is explored for the first time during the DFS search process, it is attached to the DFS tree as a child to the node from which it is being reached.

Please note that our BFS and DFS algorithms are defined in a traditional way. The algorithms shown in the textbook try to find a goal node. In our algorithms, we are trying to explore all the nodes within the graph.

Please use the Figure 1 and answer the questions. The figure shows an undirected graph and all the edges have equal costs (e.g., the lengths of all edges are equal). For simplicity, when we have a number of alternate nodes to explore, we will select them in an alphabetical order. The problems can be easily solved by hand.

Assume node a is the starting point (the "root" node) from which the search starts.

1) Apply the BFS algorithm and show the output (list all the nodes by the order they are being explored)

2) Apply the DFS algorithm and show the output (list all the nodes by the order they are being explored)

3) Show the output from the previous question in the form of a DFS tree.

2: A* Search

The problem is described as follows. Consider an idealization of a problem where a robot has to navigate its way around obstacles. The goal is to find the shortest distance between two points on a plane that has convex polygonal obstacles. Figure 2 shows an example scene with seven polygonal obstacles where the robot has to move from the point start to the point end. Convince yourself that the shortest path from one polygon vertex to any other in the scene consists of straight-line segments joining some of the vertices of the polygon. (Note that the start and the end goal points may be considered polygons of size 0).

1) Suppose the state space (i.e., a set of possible locations for the robot) consists of all possible positions (x, y) in the plane. How many possible states are there? How many paths are there to the goal? Provide important assumptions for your answer.

2) Based on your solution for 1), can you find a better (and reasonable) state space for the problem? If the answer is yes, how large is the state space? If the answer is no, provide justifications why you think this is a good state space to implement your following searching algorithm.

3) Implement an algorithm to find the shortest path from the start node to the end node using an A* (A-star) heuristic search. Use the straight-line distance to the end node as a heuristic function. Show your pseudo code for this algorithm. Is this an admissible heuristic function? Why or why not?

4) Present the solutions for the following problem using the A* algorithm you implemented.

Polygon 1: ((220, 616), (220, 666), (251, 670), (272, 647))

Polygon 2: ((341, 655), (359, 667), (374, 651), (366, 577))

Polygon 3: ((311, 530), (311, 559), (339, 578), (361, 560), (361, 528), (336, 516))

Polygon 4: ((105, 628), (151, 670), (180, 629), (156, 577), (113, 587))

Polygon 5: ((118, 517), (245, 517), (245, 577), (118, 557))

Polygon 6: ((280, 583), (333, 583), (333, 665), (280, 665))

Polygon 7: ((252, 594), (290, 562), (264, 538))

Polygon 8: ((198, 635), (217, 574), (182, 574))

Start: (120, 650) End: (380, 560)

Note: This figure is for illustration purpose only. Positions of the polygons inside the figure may not reflect their actual coordinates.

5) Is it possible to solve the problem using a breadth-first or a depth-first search algorithm? If the answer is yes, briefly discuss your solutions. Otherwise please explain.