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	[No title] (Site not responding. Last check: 2007-10-29)
		Thus, an augmenting path is defined as one where you have a path from the source to the sink where every edge has a non-zero residual capacity.
		In the worst case, each augmenting path adds 1 to the flow of a network, and each search for an augmenting path takes O(E) time, where E is the number of edges in the graph.
		In this algorithm, the augmenting path suggested is the augmenting path with the minimal number of edges.
	longwood.cs.ucf.edu /courses/cop3530/spr2003/lectures/COP3530NetworkFlow01.doc (1407 words)

	Ford-Fulkerson algorithm - Wikipedia, the free encyclopedia
		By adding the flow augmenting path to the flow already established in the graph, the maximum flow will be reached when no more flow augmenting paths can be found in the graph.
		When the capacities are integers, the runtime of Ford-Fulkerson is bounded by O(E*f), where E is the number of edges in the graph and f is the maximum flow in the graph.
		This is because each augmenting path can be found in O(E) time and increases the flow by an integer amount which is at least 1.
	en.wikipedia.org /wiki/Ford-Fulkerson_algorithm (226 words)

	CMPSCI 311 Discussion Solutions #10, Fall 2003 (Site not responding. Last check: 2007-10-29)
		The first augmenting path goes from 1 to 3 to 2 to 4, and has size 1 because there is capacity 1 in the residual graph from 3 to 2.
		Our third augmenting path is exactly like the first, and the fourth is exactly like the second, leading to a flow of 2 over each of the four outer edges and a residual graph with 998 on each outer edge and still 1 from 2 to 3.
		We choose 998 more pairs of augmenting paths, with the first in each pair going from 1 to 3 to 2 to 4 and the second in each pair going from 1 to 2 to 3 to 4.
	www.cs.umass.edu /~barring/cs311/disc/10sol.html (965 words)

	Augmenting path (Site not responding. Last check: 2007-10-29)
		Burlington MA Bike Path Committee Goals, status, and activities of the Bike Path Committee, with map of planned bike path, members, and list of activities.
		The Thames Path Project The diary of a project to walk the length of the Thames Path National Trail.
		Critical Path Analysis Critical Path Analysis is an extremely effective method of analysing a complex project.
	www.serebella.com /encyclopedia/article-Augmenting_path.html (355 words)

	Maximum Flow1 (Site not responding. Last check: 2007-10-29)
		At each iteration, the flow value is increased by finding an "augmenting path" which is a path between the source and the sink along which more flow can be pushed and then augmenting the flow along this path.
		The bound on the Ford-Fulkerson method can be improved if the augmented path p is found by the breadth first search, i.e., the augmented path is a shortest path from s to t in the residual network where each edge has unit distance (weight).
		The shortest path from s to t using BFS is s, x, w, v, t and its capacity is 7.
	www1.math.luc.edu /~agouris/algos/MaxFlow.html (970 words)

	[No title]
		Note that the augmenting path allows edges of either direction to be used, as long as the edges are useful.
		If no augmenting path can be found, then it is not possible to increase the flow between s and t, which means that the flow must be maximum, and the algorithm stops.
		There may be dead-end paths in the layered network, which were there at the start, or have resulted after the removal of a vertex.
	www.cosc.canterbury.ac.nz /research/RG/alg/mf.txt (2887 words)

	[No title]
		It follows that the maximum flow of the original network is the sum of all the augmenting paths found by the iterations.
		The number of augmenting paths that must be found can be shown to be O(E V), where E is the number of edges and V is the number of vertices in the original flow network.
		It can be shown that the number of maximum flow augmenting paths that must be found is at most O(E log C) where E is the number of edges and C is the maximum capacity of any edge in the original flow network.
	people.deas.harvard.edu /~hc3/bospre/help/advanced/network_flows (469 words)

	Comments on Final Exam, UMBC CMSC 441, Fall 2001
		It might give you the longest shortest path from the root, but that is not the same as the longest path.
		The augmenting path with capacity 1 is straightforward.
		To get the right augmenting path with capacity r^2, you first go "forward" along the edge with capacity r, back along the edge with capacity 1 (which is now pointing backward) and then forward along the edge with capacity r^2.
	www.cs.umbc.edu /~chang/cs441.f01/finalcomments.shtml (791 words)

	3. Implementations
		For the bipartite matching problem, our augmenting path algorithms use a greedy algorithm to find an initial maximal matching (scanning vertices in their input order, we try to match each vertex in turn).
		This differs from the standard shortest augmenting path algorithm where the search is done from the source itself.
		Augmenting paths for the latter tend to be a little longer, but the regions of the graph examined during the search tend to be much smaller.
	www.jea.acm.org /ARTICLES/Vol3Nbr8/node3.html (1760 words)

	Lecture 23--shortest path algorithms; transitive closure. 11/19/97.
		Give a procedure for choosing augmenting paths for the network in Figure 332-9.1 for which the flow value increases by 1 unit with each augmenting path and on which the algorithm therefore must execute the loop M times.
		Find a shortest path from s to t based on these unit weights and use this path, with its original weights, as the augmenting path.
		Suppose there is a shortest path p from s to x and suppose x is the first vertex on the path for which the inequality holds.
	www.ececs.uc.edu /~cpurdy/lec332_9.html (1218 words)

	BackgroundMaterial (Site not responding. Last check: 2007-10-29)
		For the moment, we assume that there is at least one directed path in D from source s to sink t.
		We then augment the flow by increasing the flow on each forward edge by δ while decreasing the flow by δ on each backwards edge.
		Also, let m(x, y) denote the maximum number of pairwise disjoint paths in G from x to y (of course, the paths are allowed to have x and y in common - but no other points).
	www.math.gatech.edu /~trotter/Section2-LP.htm (2739 words)

	[No title]
		An augmenting path is a path in a residual network from source to sink which consists of edges that can admit more flow.
		If there is a path from s to t in the residual network, then it is an augmenting path and indicates where flow can increase.
		Thus, the augmenting path is shortest from s to t.
	ranger.uta.edu /~cook/aa/transcript/ln19f (653 words)

	Automatic Assignment of Bond Order (Site not responding. Last check: 2007-10-29)
		An augmenting path with respect to a matching M is a simple path abc...yz of nodes connected by edges in E such that nodes a and z are exposed, and edges bc, de,...
		Let P be the set of edges on an augmenting path p=[abc...yz] in a graph G with respect to the matching M.
		Let P be an augmenting path with endpoints two other exposed nodes v and w.
	www.chemcomp.com /Journal_of_CCG/Articles/bond.htm (945 words)

	augmenting path (Site not responding. Last check: 2007-10-29)
		Definition: A path with alternating free and matched edges that begins and ends with free vertices.
		Used to augment (improve or increase) a matching or flow.
		Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "augmenting path", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/augmentngpth.html (117 words)

	[No title]
		A path P is connected if its edges can be written in a sequence such that the signs of the edges alternate and each pair of consecutive edges shares at least one end- point.
		If M is a matching and P a path, then P is said to be an augmenting path of M if M + P exists and is also a matching.
		Such a path P is determined by a sequence of vertices x, x1, x2,..., xk, y where x1,..., xk are elements of domain(M1) and y is an element of R not in range(M1).
	people.deas.harvard.edu /~hc3/help/advanced/optimal_matchings (1043 words)

	Augmenting path -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-29)
		In (additional info and facts about graph theory) graph theory, the (A drawing illustrating the relations between certain quantities plotted with reference to a set of axes) graph
		from the source to the sink is called an augmented path of.
		This path can be used to increase the flow from to.
	www.absoluteastronomy.com /encyclopedia/a/au/augmenting_path.htm (75 words)

	No Title (Site not responding. Last check: 2007-10-29)
		Look for an augmenting path from u to v (not from s to t as we do at every iteration of the basic FordFulkerson method).
		If there is no such path, reduce the flow from s to u by augmenting the flow from u to s.
		That is, find an augmenting path from u to s and augment the flow along that path.
	ranger.uta.edu /~cook/aa/hw/qs10/qs10.html (423 words)

	Lecture 15 + 16 + 17
		Then either the path is traversing node i more than once, or it is finishing in a node i that was traversed before.
		A graph of degree is a composition of cycles and paths.
		The correct idea is to always look for paths in the associated network N(f) = (s, t, V, E(f), a), where for each edge (i, j) in E, E(f) has edges (i, j) and (j, i), with capacities b(i, j) - f(i, j) for the first and f(i, j) for the second.
	www.cs.umu.se /kurser/TDBC91/VT02/lec15.html (4514 words)

	Ford
		The first is a labeling process that searches for a flow augmenting path - i.e., a path from s to t for which f < c along all forward arcs and f > 0 along all backward arcs.
		Otherwise, no augmenting path exists, and optimality of the current flow is ensured by their theorem:
		After executing Routine A, we find there is no flow augmenting path, so the current flow has the maximum value of 3 units from s to t.
	www.geocities.com /nayan_vt/fordfulkerson_algo.htm (682 words)

	[No title]
		Exercise 27.2-2, you only have to show the series of augmenting paths, and the grading will be lenient.
		An augmenting path is a path that consists of edges
		Having found the path, I push the maximum flow I can, which is the minimum of all the residual flows I can find.
	www.cs.ucr.edu /~michalis/COURSES/99-141/maxflow.html (364 words)

	Advanced Algorithms Class Notes — October 18, 2000 - Girish Sheelvant (Site not responding. Last check: 2007-10-29)
		using paths that involve vertices from the set {1, 2, 3,......, i}.
		is shortest path from intermediate vertices {1, 2,.., k-1}.
		the paths then the worst case number of iterations will be 2000.
	www.msci.memphis.edu /~giri/7713/f00/Girish/Girish.html (961 words)

	The Blossom Algorithm for b -Matching
		The basic idea of the blossom algorithm is to grow an augmenting forest F until we determine that the forest is Hungarian or find an augmenting path.
		In the latter case we augment along this path, and continue afterwards with forest growing steps.
		By (B6), however, we can correct the matching within the blossom on an even length alternating path from the tip node of B to u, such that each node of B is perfectly matched.
	www.jea.acm.org /ARTICLES/Vol4Nbr7/node8.html (1816 words)

	Graph Theory Lecture Notes 11a
		A simple algorithm for finding a larger flow is to find a path from s to t such that all the arcs on the path have positive slack.
		A flow augmenting path in the (s,t)-network, is a path from s to t in the underlying graph, so that each forward arc has positive slack and each backward arc has a positive flow.
		Given a flow augmenting path, a larger flow is obtained by taking the smallest value of the positive slacks on forward arcs and the positive flows on backward arcs and adding this value to the flow on all forward arcs and subtracting it from the flow on all backward arcs of the path.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtaln11.html (1636 words)

	[No title]
		Let M be a matching in a graph G. An alternating path is a path in which the edges alternate between M and edges in G\M. Note that if we remove the edges of M from an alternating path, the remaining edges plus the other edges of M determine another matching.
		An alternating path is called an augmenting path if the resulting new matching is bigger than the original.
		Since M' has more edges than M one of the paths must have more edges from M' than from M. This path is an augmenting path, contradicting our assumption.
	faculty.washington.edu /~moishe/Notes/Graphs2.doc (844 words)

	Mathematical Programming Glossary - A (Site not responding. Last check: 2007-10-29)
		The Lagrangian augmented by a term that retains the stationary properties of a solution to the original mathematical program but alters the hessian in the subspace defined by the active set of constraints.
		The added term is sometimes called a penalty term, as it decreases the value of the augmented Lagrangian for x off the surface defined by the active constraints.
		In network flows, it is a path from a source (s) to sink (t) such that:
	carbon.cudenver.edu /~hgreenbe/glossary/A.html (2792 words)

	Intro to Algorithms: CHAPTER 27: MAXIMUM FLOW
		The shaded path in Figure 27.4(b) is an augmenting path.
		Before proving this, however, we obtain a simple bound for the case in which the augmenting path is chosen arbitrarily and all capacities are integers.
		It has no augmenting paths, and the flow â shown in (d) is therefore a maximum flow.
	www.personal.kent.edu /~mlu3/CSCourses/AdvAlgorithms/CLR-BOOK/books/book6/chap27.htm (13646 words)

	Augmenting path - Wikipedia, the free encyclopedia
		from the source s to the sink t is called an augmented path of f.
		This path can be used to increase the flow from s to t.
		This page was last modified 17:00, 18 September 2005.
	en.wikipedia.org /wiki/Augmenting_path (65 words)

	[No title]
		Property 12.7 If M and M* are two matchings their symmetric difference defines the subgraph G* (N, M (M*) with the property that every component is one of the six types shown in Figure 12.7.
		Augmenting path theorem Theorem 12.8 If a node p is unmatched in a matching M and this matching contains no augmenting path that starts at node p, then node p is unmatched in some maximum matching.
		Call new net Gc=(Nc,Ac) Contracted net Correctness 1.Show when there is an augmenting path in the contracted network we can find one in the original.
	www.pitt.edu /~banwork/3087/Ch12Notes01.doc (744 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		------------------- An augmenting path is an s->t path in the residual graph.
		If there is an augmenting path then the flow is not maximal, because you can push additional flow along the augmenting path.
		If the capacities are integers, then each augmentation increase the value of the flow by at least one, so the total time will be O(m f^*) Such a bad case really can happen: see the example on page 91.
	www.cs.ucdavis.edu /~rogaway/classes/731/00/lect27.txt (328 words)

	[No title]
		Clearly the flow from s to t can be augmented by the minimum capacity in the augmenting path.¡“ 2“ðH ð` ð0“”Þ½h¿ÿ ?ð ÿÿÿÌ33ÌÌÌÿ²²²î5ï åðÝÐð dðuð( ð ðdð¤¢ ðd ð0ðÉä¿¿ÀÿðPÀ ðD¨Initially, flow is 0¡ 2ð®¢ ðd ð0
		As we have proved earlier, the flow is the maximum flow.¡« 2«ð$¢ ð¨ ð0Ã¿¿ÀÿðÐpÖ ð,$ñD0 ð¨`Observe that an admissible path from the source to the destination is indeed the shortest path.
		But this change is restricted to the augmenting path only.
	www.seas.upenn.edu /~swati/lec7f01.ppt (648 words)

	Introduction to Algorithms \| Glossary
		By the definition of the residual network, each edge (u, v) on an augmenting path admits some additional positive flow from u to v without violating the capacity constraint on the edge.
		For the Hopcraft-Karp bipartite matching algorithm we say that a simple path P in G is an augmenting path with respect to M if it starts at an unmatched vertex in L, ends at an unmatched vertex in R, and its edges belong alternately to M and E - M.
		The bound on FORD-FULKERSON can be improved if we implement the computation of the augmenting path p with a breadth-first search, that is, if the augmenting path is a shortest path from s to t in the residual network, where each edge has unit distance (weight).
	highered.mcgraw-hill.com /sites/0070131511/student_view0/chapter26/glossary.html (1565 words)

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