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Topic: DAG shortest paths

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	shortest path
		Definition: The problem of finding the shortest path in a graph from one vertex to another.
		For a map, it is to produce the (shortest) road distance from one city to another city, not which roads to take.
		Suppose predecessor[i][j] is k; then the shortest path ends with...
	www.nist.gov /dads/HTML/shortestpath.html (177 words)

	Shortest paths and topological ordering
		That there are two kinds of shortest paths: those formed by a single edge (s,t), and those in which the path from s to t goes through some other vertices; let's say x is the last vertex the path goes through before t.
		Then in the second case, the overall path must be formed by concatenating a path from s to x with edge (x,t).
		This defines a structure known as a "shortest path tree".
	www.ics.uci.edu /~eppstein/161/960208.html (1605 words)

	Shortest Paths Graph Algorithms > Underlying Principles (Site not responding. Last check: )
		Path relaxation is appropriate for all-pairs solutions where we maintain the lengths of the shortest paths that we have encountered between all pairs of vertices.
		If we define a path's weight to be the number of edges on that path, then Warshall's algorithm generalizes to Floyd's algorithm for finding all shortest paths in unweighted digraphs; it further generalizes to apply to networks, as we see in Section 21.3.
		Most shortest-paths algorithms also compute shortest paths from s to every vertex that is closer to s than to t (whether or not the vertex is on the path from s to t), although that is not a requirement (see Exercise 21.18).
	www.awprofessional.com /articles/article.asp?p=169575&seqNum=2 (2292 words)

	Dynamic programming - Wikipedia, the free encyclopedia
		Finding the shortest path in an acyclic graph using optimal substructure; a wavy line indicates a shortest path between the two vertices it connects
		For example, the shortest path to a goal from a vertex in an acyclic graph can be found by first computing the shortest path to the goal from all adjacent vertices, and then using this to pick the best overall path, as shown in Figure 1.
		That it is not a tree but a DAG indicates overlapping subproblems.
	en.wikipedia.org /wiki/Dynamic_programming (1506 words)

	Shortest Path
		The primary algorithm for finding shortest paths is Dijkstra's algorithm, which efficiently finds the shortest paths from a given vertex x to all n-1 other vertices.
		To find the shortest simple cycle, the easiest approach is to compute the lengths of the shortest paths from i to all other vertices, and then explicitly check whether there is an acceptable edge from each vertex back to i.
		The theoretically fastest algorithms known for single-source shortest path for positive edge weight graphs are variations of Dijkstra's algorithm with Fibonacci heaps [FT87].
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE162.HTM (1666 words)

	Algorithms, Yossi Azar
		Most of the course follows the above book.
		Graph Algorithms: Breadth first search, Depth first search, Topological sort, Strongly connected components, Biconnected components, Minimum spanning trees (Kruskal, Prim), Shortest path (Dijkstra, Bellman-Ford), All-pairs shortest path (Floyd-Warshall, Johnson), Flow algorithms (Ford-Fulkerson, Edmonds-Karp, Dinic).
		Here is a link to course notes written by the student Assaf Shtilman.
	www.cs.tau.ac.il /~azar/algs05-06.html (138 words)

	DAG HAMMARSKJÖLD: The UN years ...
		However, we cannot doubt that the main direction of the work of the United Nations, as determined by the purposes and principles of the Charter, indicates the path which the world must follow in order to preserve the achievements of the past and to lay a basis for a happier future.
		Dag Hammarskjöld has succeeded in finding solutions where none seemed to be in sight.
		At the end of the year, the Meditation Room, to which Dag Hammarskjöld gave a great deal of thought and personal attention was opened.
	www.un.org /depts/dhl/dag/time1957.htm (1797 words)

	Graphs concluded
		We can find shortest paths in a weighted graph (with non-negative weights) using the same algorithm, placing the paths in a priority queue that gives shorter paths higher priority.
		This algorithm can also be adapted to find the shortest paths from a given vertex to all other vertices; when we first visit a vertex, we record the path that got us there.
		There are two complications: first, vertices may be reached by more than one path, the list sorted (which is itself topologically sorted) acts as a visited list; second, the graph may have cycles, the list path records the path by which the algorithm arrived at the current node, and is used to check for cycles.
	homepages.inf.ed.ac.uk /mfourman/teaching/mlCourse/notes/graph-dag.html (730 words)

	biology - Glossary of graph theory
		Traditionally, a path is graph consisted of a sequence of successively incident edges and their endvertices, where the terminating vertices are distinct.
		The girth of a graph is the length of a shortest (simple) cycle in the graph; and the circumference, the length of a longest (simple) cycle.
		A directed path, or just a path when the context is clear, is an oriented simple path such that all arcs go the same direction, meaning all internal vertices have in- and out-degrees 1.
	www.biologydaily.com /biology/Glossary_of_graph_theory (5132 words)

	DAG Class Reference
		Returns the value of the shortest path from a node given as argument to the bottom the DAG (sinks).
		Despite the unicity of its value, the longest path in a DAG may not be unique.
		Before reading a DAG from a gl file, it is assumed that the ISA of the DAG defines all the opcodes ids of the nodes as present in the gl file.
	www.prism.uvsq.fr /users/touati/sw/DDG/classDDG_1_1DAG.html (2231 words)

	Single Source Shortest Path
		Shortest path problems are typically solved using a general method called a greedy algorithm.
		The current estimate is shortest path explored so far to the vertex.
		The longest path is the critical path, since we negated the edge weights, d should also be negated; the greatest weight (longest path) from source s is then to z.
	homepages.ius.edu /rwisman/C455/html/notes/Chapter24/SingleSource.htm (762 words)

	Shortest Paths Graph Algorithms > Shortest Paths in Acyclic Networks (Site not responding. Last check: )
		It is sometimes convenient to use the former to emphasize differences from unweighted DAGs that are implied by weights and the latter to emphasize differences from general networks that are implied by acyclicity.
		For shortest paths, the method is faster than Dijkstra's algorithm by a factor proportional to the cost of the priority-queue operations in Dijkstra's algorithm.
		The other algorithms for finding shortest paths in acyclic networks that are mentioned at the beginning of this section generalize the methods from Chapter 19 in a manner similar to the other algorithms that we have examined in this chapter.
	www.awprofessional.com /articles/article.asp?p=169575&seqNum=5 (2258 words)

	Multiple Shortest Paths in a Weighted DAG (Site not responding. Last check: )
		The problem is to find k directed paths starting at s, such that every node of G lies on at least one of those paths, and such that the sum of the weights of all the edges in the paths is minimized.
		This special case is too simple to be of much use (because there can be at most one path), but at least it gets you started.
		In this case, there is a solution whose time complexity is polynomial in n, the number of nodes in G. If you get this far and can go no farther, you have accomplished something.
	www.egr.unlv.edu /~larmore/Courses/CSC477/S99/Assignments/multpaths.html (155 words)

	Methods to Solve - Chapter 9. Graph (Site not responding. Last check: )
		It turns out that one can find the shortest paths from a given source to all points in a graph in the same time, hence this problem is called the Single-source shortest paths problem.
		The graph representing all the paths from one vertex to all the others must be a spanning tree - it must include all vertices.
		A more generalized single-source shortest paths algorithm which can find the shortest path in a graph with negative weighted edges.
	www.comp.nus.edu.sg /~stevenha/programming/prog_graph5.html (848 words)

	Directed Graphs
		It turns out that it as easy to find the shortest paths from a single source to all other vertices as it is to find the shortest path between any two vertices.
		It operates by enlarging the set of vertices `done' for which the shortest paths from the source are known.
		This is correct because any indirect path to this vertex would have to leave the source by an edge at least as long.
	www.csse.monash.edu.au /~lloyd/tildeAlgDS/Graph/Directed (1317 words)

	Intro to Algorithms: CHAPTER 25: SINGLE-SOURCE SHORTEST PATHS
		No path from s to a vertex on the cycle can be a shortest path--a lesser-weight path can always be found that follows the proposed "shortest" path and then traverses the negative-weight cycle.
		The representation we use for shortest paths is similar to the one we used for breadth-first trees in Section 23.2.
		Shortest paths are always well defined in a dag, since even if there are negative-weight edges, no negative-weight cycles can exist.
	www.personal.kent.edu /~mlu3/CSCourses/AdvAlgorithms/CLR-BOOK/books/book6/chap25.htm (9948 words)

	Lecture 18 - shortest path algorthms
		In a weighted graph, the weight of a path between two vertices is the sum of the weights of the edges on a path.
		Soon as we establish the shortest path from s to a new node x, we go through each of its incident edges to see if there is a better way from s to other nodes thru x.
		Notice that finding the shortest path between a pair of vertices (s,t) in worst case requires first finding the shortest path from s to all other vertices in the graph.
	www.cs.sunysb.edu /~algorith/lectures-good/node18.html (1276 words)

	Summary
		After the user issues a query, these shortest paths are dynamically combined to form a hierarchical outline of the context in which the search result occurs.
		For example, if we are looking for some computer science related web page, then the shortest path to this path computed from the root page of the local sub domain www.cs.iit.edu is much more meaningful than the path calculated from the root page of the main domain www.iit.edu.
		Once the documents have been found, the shortest paths are fetched are used to build a hierarchy starting from the root.
	ir.iit.edu /~dagr/IRCourse/Fall2000/Presentation/Group_4/Summary_04.html (2623 words)

	MAI: Lite Mat
		One of the most commonly used IGP's (Interior Gateway Protocols) in telecommunication networks is the OSPF (Open Shortest Path First) protocol with ECM (Equal Cost Multipath) distribution.
		By shortest path computations, router decide by themselves how to distribute the traffic.
		Necessary conditions, which flow patterns must fulfill for being identical to the shortest paths obtained from OSPF weights, are reported.
	math.liu.se /LiteMat/2004/v22-04 (1581 words)

	[No title] (Site not responding. Last check: )
		Print the sequence of all vertices along this shortest path from s to t.
		Print the length of the longest path from vertex s to vertex t.
		Print the length of the shortest path from vertex t to vertex s.
	cs.ua.edu /470/fall2001/progB.doc (228 words)

	Biopathways Graph Data Manager (BGDM)
		path existence queries Does there exist a (directed) path from node A to node B? The directed version of such queries are "subsumption queries" on taxonomies, i.e., does concept A subsume (encompass) concept B? These are very common queries, and are a component of subgraph isomorphism queries.
		Thus a complex query might ask to match the subgraphs SG1, and SG2, in a large graph G, then construct the shortest path between the matching subgraphs and finally compute the neighborhood of radius 2 (edges) around the shortest path.
		Algorithms include, shortest paths, minimum spanning tree, connected components, strongly connected components, etc. It is a generic library written in the style of the Standard Template Library for C++ (i.e., nodes can be arbitrary user defined data types).
	hpcrd.lbl.gov /staff/olken/graphdm/graphdm.htm (7829 words)

	COT 4400, Final Exam Examples (Site not responding. Last check: )
		Use the Bellman-Ford algorithm to find the shortest path between two nodes in a graph.
		Apply the DAG-Shortest-Paths algorithm to find the shortest path between two nodes in a DAG.
		Use Dijkstra's algorithm to find the shortest path between two nodes in a graph.
	www.cse.fau.edu /~roy/cot4400.02f/final-examples.html (207 words)

	Recursive descent parsing
		is a sum of weights of all edges on path p.
		d[v] for every vertex v in the graph that will be our upper bound estimate of the shortest path from s to v.
		Convergence property: If s to u followed by (u,v) edge is a shortest path from s to u and v, and if d[u] = b(s,u) at any time prior to relaxing edge (u,v), then d[v] = b(s,v) after relaxation.
	www.owlnet.rice.edu /~comp314/lec/week10/SSSP.htm (778 words)

	6. Experimental results
		All the considered algorithms explicitly store the shortest distances and the shortest paths of vertices from the source, and therefore distance and path queries can be always answered in optimal time.
		The correctness of the various implementations was verified by comparing the results of the algorithms subject to different inputs.
		In particular, we checked whether the shortest distances of vertices from the source, computed by the three algorithms on the same input graphs, were the same after each input update over arbitrary sequences of edge modifications.
	www.jea.acm.org /TURING/Vol3Nbr5/node6.html (2085 words)

	Feder, Tomas; Motwani, Rajeev; O'Callaghan, Liadan; Olston, Chris; Panigrahy, Rina: Computing Shortest Paths with ...
		We consider the problem of estimating the length of a shortest path in a DAG whose edge lengths are known only approximately but can be determined exactly at a cost.
		Initially, each edge e is known only to lie within an interval [l_e, h_e]; the estimation algorithm can pay c_e to find the exact length of e.
		In particular, we study the problem of finding the cheapest set of edges such that, if exactly these edges are queried, the length of the shortest path will be known, within an additive K > 0 that is given as an input parameter.
	dbpubs.stanford.edu:8090 /pub/2003-2 (304 words)

	WriteDesign On-Line - Quotes
		I speak as a citizen of the world, for the world as it stands aghast at the path we have taken.
		A leader is someone who has the courage to give power away and the ability to reinvent their role in pursuit of implementing a vision.
		Cultivate your sense of humor; laughter is the shortest distance between two people.
	www.writedesignonline.com /resources/quotes.html (17321 words)

	CS 332: Algorithms
		If were lucky and processes vertices on each shortest
		path from left to right, would be done in one pass
		Every path in a dag is subsequence of topologically
	www.cs.virginia.edu /~luebke/cs332.fall00/lecture20_files/slide0169.htm (59 words)

	Algorithms
		pseudocode for dijkstra's algorithm for generating the shortest path between vertices
		Give an algorithm for removing an arbitary element from a heap of size N. Describe your algorithm in Pseudo-code.
		Construct a DAG for evaluating a polynomial horner rule
	www.scribd.com /doc/5839 (1190 words)

	[No title]
		In the first stage, candidate start, donor, acceptor, and stop sites are scored using the SNoW learning architecture.
		These sites are then used to generate a directed acyclic graph in which each source-sink path represents a possible gene structure.
		Training sequences are used to optimize an edge weighting function so that the shortest source-sink path maximizes exon-level prediction accuracy.
	ismb01.cbs.dtu.dk /talks/session4.html (1601 words)

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