lec12 - Lecture 12 Algorithms on graphs Breadth first depth first searches Shortest path in unweighted graphs Greedy algorithms Djikstras algorithm for

lec12 - Lecture 12 Algorithms on graphs Breadth first depth...

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Page 1 of 35 CSE 100, UCSD: LEC 12 Lecture 12 Algorithms on graphs Breadth first, depth first searches Shortest path in unweighted graphs Greedy algorithms Djikstra’s algorithm for shortest path in weighted graphs Reading: Weiss, Chapter 9, 10
Page 2 of 35 CSE 100, UCSD: LEC 12 Shortest path problems Suppose graph vertices represent computers, and graph edges represent network links between computers, and edge weights represent communications times... ... then a shortest-path algorithm can find the fastest route to send email between one computer and another Suppose graph vertices represent cities, and graph edges represent airline routes between cities, and edge weights represent travel costs ... ... then a shortest-path algorithm can find the cheapest route to travel by air between one city and another Many, many other examples... We will look at shortest-path algorithms in unweighted and weighted graphs These algorithms will find the shortest path from a “source” or “start” vertex to every other vertex in the graph (Often you may want only the shortest path from a source vertex to one particular destination vertex... but there is no known algorithm to do that with better worst-case time cost than a good algorithm to find shortest path to every other vertex!)
Page 3 of 35 CSE 100, UCSD: LEC 12 The unweighted shortest path problem Input: an unweighted directed graph G = (V,E) and a “source vertex” s in V Output: for each vertex v in V, a representation of the shortest path in G that starts at s and ends at v What is the best approach to this problem?
Page 4 of 35 CSE 100, UCSD: LEC 12 Search in graphs The shortest-path problem is a search problem in a graph: starting at a vertex s , you are searching for the shortest path to another vertex v When searching in a graph, three important approaches are: depth-first search breadth-first search best-first search These approaches can be applied to many different search problems We’ll consider applying depth-first and breadth-first to the unweighted shortest-path problem. (Best-first will arise in the weighted shortest-path case)
Page 5 of 35 CSE 100, UCSD: LEC 12 Breadth-first search Breadth-first search in a graph visits a node; and then all the nodes adjacent to that node; then all the nodes adjacent to those nodes; etc. A level-order traversal of a tree is a breadth-first search: 1 2 3 4 5 6 7 8 10 9
Page 6 of 35 CSE 100, UCSD: LEC 12 Depth first search Depth first search in a graph visits a node; and then recursively does depth-first search from each of the nodes adjacent to that node A pre-order traversal of a tree is a depth-first search: 1 2 7 3 8 9 4 6 5 10
Page 7 of 35 CSE 100, UCSD: LEC 12 Depth-first search for shortest paths Consider searching for shortest paths with start vertex V0 in this unweighted graph, using depth-first search: V0 V1 V4 V2 V3
Page 8 of 35 CSE 100, UCSD: LEC 12 Depth-first search for shortest paths: frame 1 Immediately we have the shortest path from V0 to V0, following 0 edges.

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