We again consider directed graphs, also assuming that edges carry a positive “le
ID: 3674171 • Letter: W
Question
We again consider directed graphs, also assuming that edges carry a positive “length”; this time our goal is to reverse the edges. That is, given a graph (V, E) where the nodes in V are numbered 1..n, we want to construct a graph (V, E ) such that for all i and k in V, E contains an edge with length d from i to k i E contains an edge with length d from k to i.
Your task is to write pseudo-code to implement this specication, and analyze the algorithm’s running time, as a function of n and of a (the number of edges in E). You should do that
1. When the graph is represented as an adjacency matrix A (that is, A(i, k) = d if the graph has an edge from i to k with length d; if there is no edge from i to k, A(i, k) = 0)
2. When the graph is represented as adjacency lists L (that is, for each i 1..n, L[i] is a list that contains the edges from i, where an edge is a pair (k, d) with k the target and d the length of the edge;you may assume a function Cons that adds an edge to a list of edges.)
Explanation / Answer
This non-recursive implementation is similar to the non-recursive implementation of depth-first search, but differs from it in two ways:
1. it uses a queue instead of a stack and
2. it checks whether a vertex has been discovered before enqueueing the vertex rather than delaying this check until the vertex is dequeued from the queue.
The distance attribute of each vertex (or node) is needed for example when searching for the shortest path between nodes in a graph. At the beginning of the algorithm, the distance of each vertex is set to INFINITY, which is just a word that represents the fact that a node has not been reached yet, and therefore it has no distance from the starting vertex. We could have used other symbols, such as -1, to represent this concept.
The parent attribute of each vertex can also be useful to access the nodes in a shortest path, for example by backtracking from the destination node up to the starting node, once the BFS has been run, and the predecessors nodes have been set.
The NIL is just a symbol that represents the absence of something, in this case it represents the absence of a parent (or predecessor) node; sometimes instead of the word NIL, words such as null, none or nothing can also be used.
Note that the word node is usually interchangeable with the word vertex.
Breadth-first search produces a so-called breadth first tree. You can see how a breadth first tree looks in the following example.
Example[edit]
The following is an example of the breadth-first tree obtained by running a BFS starting from Frankfurt:
An example map of Germany with some connections between cities
The breadth-first tree obtained when running BFS on the given map and starting inFrankfurt
Analysis[edit]
Time and space complexity[edit]
The time complexity can be expressed as ,[5] since every vertex and every edge will be explored in the worst case. is the number of vertices and is the number of edges in the graph. Note that may vary between and , depending on how sparse the input graph is.
When the number of vertices in the graph is known ahead of time, and additional data structures are used to determine which vertices have already been added to the queue, the space complexity can be expressed as , where is the cardinality of the set of vertices (as said before). If the graph is represented by an adjacency list it occupies [6] space in memory, while an adjacency matrix representation occupies .[7]
When working with graphs that are too large to store explicitly (or infinite), it is more practical to describe the complexity of breadth-first search in different terms: to find the nodes that are at distance d from the start node (measured in number of edge traversals), BFS takes O(bd + 1) time and memory, where b is the "branching factor" of the graph (the average out-degree).[8]:81
Completeness and optimality[edit]
In the analysis of algorithms, the input to breadth-first search is assumed to be a finite graph, represented explicitly as an adjacency list or similar representation. However, in the application of graph traversal methods in artificial intelligence the input may be an implicit representation of an infinite graph. In this context, a search method is described as being complete if it is guaranteed to find a goal state if one exists. Breadth-first search is complete, but depth-first search is not: when applied to infinite graphs represented implicitly, it may get lost in parts of the graph that have no goal state and never return
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