graphin-c1.pnggraphin-c1.txt1 22 3 83 44 5.docx

1. Nov 2022
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graphin-c1.pnggraphin-c1.txt1 22 3 83 44 5.docx

• 1. graphin-c1.png graphin-c1.txt 1: 2 2: 3 8 3: 4 4: 5 5: 3 6: 7 7: 3 6 8 8: 1 9 9: 1 graphin-c2.jpg graphin-c2.txt 1: 2 9 2: 3 8 3: 4 4: 5 9 5: 3
• 2. 6: 7 7: 3 6 8 8: 1 9: graphin-DAG.png graphin-DAG.txt 1: 2 2: 3 8 3: 4 4: 5 5: 9 6: 4 7 7: 3 8 8: 9 9: CS 340 Programming Assignment III: Topological Sort
• 3. Description: You are to implement the Depth-First Search (DFS) based algorithm for (i) testing whether or not the input directed graph G is acyclic (a DAG), and (ii) if G is a DAG, topologically sorting the vertices of G and outputting the topologically sorted order. I/O Specifications: You will prompt the user from the console to select an input graph filename, including the sample file graphin.txt as an option. The graph input files must be of the following adjacency list representation where each xij is the j'th neighbor of vertex i (vertex labels are 1 through n): 1: x11 x12 x13 ... 2: x21 x22 x23 ... . . n: xn1 xn2 xn3 ... Your output will be to the console. You will first output whether or not the graph is acyclic. If the graph is NOT acyclic, then you will output the set of back edges you have detected during DFS. Otherwise, if the graph is acyclic, then you will output the vertices in topologically sorted order. Algorithmic specifications: Your algorithm must use DFS appropriately and run in O(E + V) time on any input graph. You will need to keep track of edge types and finish times so that you can use DFS for detecting cyclicity/acyclicity and topologically sorting if the graph is a
• 4. DAG. You may implement your graph class as you wish so long as your overall algorithm runs correctly and efficiently. What to Turn in: You must turn in a single zipped file containing your source code, a Makefile if your language must be compiled, appropriate input and output files, and a README file indicating how to execute your program (especially if not written in C++ or Java). Refer to proglag.pdf for further specifications. This assignment is due by MIDNIGHT of Monday, February 19. Late submissions carry a minus 40% per-day late penalty. Sheet1Name:Possible:Score:Comments:10Graph structure with adjacency list representationDFS16Correct and O(V+E) time10Detecting cycles, is graph DAG?Topological Sort16Correctness of Topo-Sort algorithm and output18No problems in compilation and execution? Non-compiling projects receive max total 10 points, and code that compiles but crashes during execution receives max total 18 points.700Total &"Helvetica,Regular"&12&K000000&P Sheet2 &"Helvetica,Regular"&12&K000000&P Sheet3 &"Helvetica,Regular"&12&K000000&P DFS and topological sort
• 5. CS340 Depth first search breadth depth Search "deeper" whenever possible *example shows discovery times Depth first search Input: G = (V,E), directed or undirected. No source vertex is given! Output: 2 timestamps on each vertex: v.d discovery time v.f finishing time These will be useful for other algorithms later on. Can also compute v.π 3 Depth first search Will methodically explore every edge. Start over from different vertices as necessary.
• 6. As soon as we discover a vertex, explore from it. Unlike BFS, which puts a vertex on a queue so that we explore from it later. DFS may repeat from multiple source nodes Unlike BFS, result is a forest of DFS trees Depth first search As DFS progresses, every vertex has a color: WHITE = undiscovered GRAY = discovered, but not finished (not done exploring from it) BLACK = finished (have found everything reachable from it) Discovery and finishing times: Unique integers from 1 to 2|V| For all v, v.d < v.f In other words, 1 ≤ v.d < v.f ≤ 2|V| DFS, recursive version DFS
• 7. Time complexity The procedure DFS-VISIT is called exactly once for each vertex v ∈ V , since the vertex u on which DFS-VISIT is invoked must be white and the first thing DFS-VISIT does is paint vertex u gray. During an execution of DFS-VISIT, the loop on lines 4–7 executes Adj[E] times = Θ(E). The running time of DFS is therefore Θ(V + E) Notice that BFS was O(V + E) because it was not certain that every vertex would be visited. Properties of DFS Discovery and finishing times have a parenthesis structure If we represent the discovery of vertex u with a left parenthesis “(u” and represent its finishing by a right parenthesis “u)”, then the history of discoveries and finishings makes a well-formed expression in the sense that the parentheses are properly nested. Parenthesis Structure For all u,v, exactly one of the following holds: 1. u.d < u.f < v.d < v.f or v.d < v.f < u.d < u.f (i.e., the intervals [u.d, u.f ] and [v.d, v.f] are disjoint) and neither of u and v is a descendant of the other. ( ) [ ] 2. u.d < v.d < v.f < u.f and v is a descendant of u. [ ( ) ] 3. v.d < u.d < u.f < v.f and u is a descendant of v. ( [ ] )
• 8. 10 Nesting of descendants Vertex v is a proper descendant of vertex u in the depth-first forest for a (directed or undirected) graph G if and only if u.d < v.d < v.f < u.f. White path theorem v is a descendant of u if and only if at time u.d, there is a path from u to v consisting of only white vertices. (Except for u, which was just colored gray.) Classification of edges Tree edge: in the depth-first forest. Found by exploring (u, v). Back edge: (u,v), where u is a descendant of v. Forward edge: (u,v), where v is a descendant of u, but not a tree edge. Cross edge: any other edge. Can go between vertices in same depth-first tree or in different depth-first trees. Classification of edges When we first explore an edge (u,v), the color of vertex v tells us something about the edge:
• 9. 1. WHITE indicates a tree edge, 2. GRAY indicates a back edge, and 3. BLACK indicates a forward or cross edge. Classification of edges Forward and cross edges never occur in a depth-first search of an undirected graph. In a depth-first search of an undirected graph G, every edge of G is either a tree edge or a back edge. Interview Questions Show how depth-first search works. Assume that the for loop of lines 5–7 of the DFS procedure considers the vertices in alphabetical order, and assume that each adjacency list is ordered alphabetically. Show the discovery and finishing times for each vertex, and show the classification of each edge. Solution
• 10. Interview Questions Show that using a single bit to store each vertex color suffices by arguing that the DFS procedure would produce the same result if line 3 of DFS-VISIT was removed. Rewrite the procedure DFS, using a stack to eliminate recursion. 18 Interview Questions Let G = (V,E) be a connected, undirected graph. Give an O(V+E)-time algorithm to compute a path in G that traverses each edge in E exactly once in each direction. Describe how you can find your way out of a maze if you are given a large supply of pennies.
• 11. 19 Interview Questions Depth first search will turn out to be an incredibly useful algorithm. Tell how DFS can help solve the following problems: Minimum spanning tree and all pair shortest path tree (when will DFS work?). Detecting a cycle in a graph Path Finding (find a path between two given vertices u and z) Topological Sorting Testing if a graph is bipartite Finding Strongly Connected Components Solving puzzles with only one solution, such as mazes. (DFS can be adapted to find all solutions to a maze by only including nodes on the current path in the visited set.) Topological Sort Directed acyclic graph (dag)
• 12. A directed graph with no cycles. A topological sort of a dag is a linear ordering of all its vertices such that if G contains an edge (u,v), then u appears before v in the ordering. An ordering of its vertices along a horizontal line so that all directed edges go from left to right. This is a different kind of sort than we have done in the past. Dag of Cardiovascular Disease A dag can be used to indicate causality. Useful in social sciences, epidemiology, etc. http://www.omicsonline.org/using-directed-acyclic-graphs-for- investigating-causal-paths-for-cardiovascular-disease-2155- 6180.1000182.php?aid=20947 Dag of course prerequisites Dags are used in scheduling, when one thing must happen
• 13. before another https://www.cs.northwestern.edu/academics/courses/311/html/gr aphs.html Dag of making pancakes Dags can model dependencies Topological Sort A directed edge (u,v) indicates that item u must be put on before item v
• 14. Topological Sort What is the time complexity? It is just a DFS with O(1) inserting to front of a linked list, so Θ(V+E) 27 Using DFS to detect a dag A directed graph G is acyclic if and only if a depth-first search of G yields no back edges
• 15. Interview Questions What is the ordering of vertices produced by a topological sort? Interview Questions Give an algorithm that determines whether or not a given undirected graph contains a cycle. Your algorithm should run in O(V) time, independent of |E|. Another way to perform topological sorting on a directed acyclic graph is to repeatedly find a vertex of in-degree 0, output it, and remove it and all of its outgoing edges from the graph. Explain how to implement this idea so that it runs in time O(V+E). What happens to this algorithm if G has cycles? 30
• 16. Strongly Connected Components A strongly connected component of a directed graph G = (V,E) is a maximal set of vertices C ⊆V such that for every pair of vertices u and v in C, we have both u↝v and v↝u; that is, vertices u and v are reachable from each other. Strongly Connected Components Component Graph Component Graph GSCC = (VSCC,ESCC). VSCC has one vertex for each SCC in G. ESCC has an edge if there’s an edge between the corresponding SCC’s in G. GSCC is a dag.
• 17. Transpose of a directed graph GT is the transpose of G. GT is G with all edges reversed. Can create GT in Θ(V+E) time if using adjacency lists. Do G and GT have the same strongly connected components? Compute Strongly Connected Components Why does it work? What does it mean that vertices are considered in order of decreasing finishing time?
• 18. decreasing finishing time = topological sort! 36 Interview Questions How can the number of strongly connected components of a graph change if a new edge is added? 37