The link is here. A: 10 B: 15 C: 20 D: 25 Relationship between ßows and cuts Flow value lemma. The function maxflow will return the value of the maximal flow. In the original flow network we are not allowed to send any flow from $A$ to $D$. The max-flow min-cut theorem goes even further. Complexity theory, randomized algorithms, graphs, and more. The pseudo-code for this method is quite short; however, there are some functions that bear further discussion. As noted in the pseudo-code, they are calculated at every step so that augmenting paths can be found from the source to the sink. Residual graphs show the differences between various paths' capacity and their current flow and allow future augmenting paths to be calculated accurately. Level Graph 373F20 - … Ford{Fulkerson algorithm Ford{Fulkerson augmenting path algorithm. Material flowing through a transportation network; ! The total number of cars sent is now 7, and it is the maximum flow. While there is a path, an augmenting capacity is in each edge of the path, so a flow can be calculated on line 98. Lowest Common Ancestor - Farach-Colton and Bender algorithm; Solve RMQ by finding LCA; Lowest Common Ancestor - Tarjan's off-line algorithm; Flows and related problems. This concept is used in Ford–Fulkerson algorithm which computes the maximum flow in a flow network. ・How to compute a mincut? Ford-Fulkerson algorithm, but you are not required to read the proofs. In this post, Goldberg’s “generic” maximum-flow algorithm is discussed that runs in O(V 2 E) time. Residual capacity is defined as the new capacity after a given flow has been taken away. 1. v should not be used. Is such a path is found, then we can add increase the flow along these edges. A good analogy for a flow network is the following visualization: The flow, 7, is a maximum flow. Ford–Fulkerson algorithm Ford–Fulkerson augmenting path algorithm. Log in. Add this path-flow to flow. All of the above suggests the famous Ford-Fulkerson algorithm for network ow. ... Ford-Fulkerson Algorithm (1956) This algorithm is sometimes referred to as a method because parts of its protocol are not fully specified and can vary from implementation to implementation. Let $C$ be the smallest residual capacity of the edges in the path. The only ambiguous point is the 'foward edge' terminology on line 8. Then, it creates the new edge and a corresponding returning edge with capacity 0. The Ford-Fulkerson algorithm (see, e.g., Section 8.2 in Jeremy L. Martin's Lecture Notes on Algebraic Combinatorics, or Section 4.3 in Lex Schrijver's A Course in Combinatorial Optimization, ... but I don't know how to make this intuition precise. It is a recursive function that walks through the flow network starting at a given vertex and calculates residual capacity for each edge. On line 7, the Edge class is defined. Proof First, there are some important initial logical steps to proving that the maximum flow of any network is equal to the minimum cut of the network. 23 Start with 0 flow. Ford-Fulkerson Algorithm 11 Define an augmenting path to be a path from → in the residual graph (using edges of non-zero weight) Overview: Repeatedly add the flow of any augmenting path Ford-Fulkerson max-flow algorithm: • Initialize =0for all ∈ • Construct the residual network Finding this maximal flow of a flow network is the problem that we want to solve. The given graph is directed and there is an edge from A to B with capacity y. Ford Fulkerson Algorithm . E. rì Find an s!t path P where each edge has f(e) < c(e). Time Complexity: Time complexity of the above algorithm is O(max_flow * E). Approach 1: (not the Ford-Fulkerson but equivalent) If C = 1, Ford-Fulkerson runs in O(mn) time. Now, this class can calculate its own maximum flow. rì Start with f(e) = 0 for each edge e ! We can find the path $s - A - B - t$ with the residual capacities 7, 5 and 8. A residual capacity of an directed edge is the capacity minus the flow. The network dictionary is a data structure that maps every vertex's name to all of the edges coming out of the corresponding vertex. The Ford-Fulkerson algorithm is a classic algorithm that computes the maximum flow in a network. Possible approaches are using DFS or BFS which both work in $O(E)$. Let's define one more thing. This time complexity is better than O(E 2 V) which is time complexity of Edmond-Karp algorithm (a BFS based implementation of Ford-Fulkerson). The analysis of Ford-Fulkerson depends heavily on how the augmenting paths are found. We run a loop while there is an augmenting path In this episode an intuitive maximum flow explanation is presented that avoids the typical formalization of maximum flow. One possible minimum cut is the following: While there is an augmenting path 3. do augment flow as much as legally possible More detailed description on selecting paths. Perhaps the most well-known algorithm which uses augmenting paths to find a maximum flow is the Ford-Fulkerson algorithm. Now, a new augmenting path must be found (((the top-most path can never be used again because the edge (B, T) was erased).).). Augment ow as long as it is possible while there exists an s-t path P in G f do f = Augment( f, P ) update G f end while return f Relatemaximum owtominimum cut Clicker Question 1 What is the value of the ow across the black-white cut? 3) Return flow. In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford-Fulkerson algorithm. We keep on searching for augmenting paths and increasing the flow. It is easy to see that the following equation holds: Algorithm: 1.Find a path from s to t and route max ow through this path 2.Update capacities by updating the residual graph 3.Repeat 1 and 2 until algorithm halts (no path from s to t is found) Explanation: - Intuition: The back edges in residual graph can undo suboptimal ows in some edges for future iterations. The FlowNetwork class defined on line 15 has two attributes. CS 31: Algorithms (Spring 2019): Lecture 16 Date: 16th May, 2019 Topic: Graph Algorithms 6: The Ford-Fulkerson Algorithm Disclaimer: These notes have not gone through scrutiny and in all probability contain errors. So we need to clean it up a little. Ford-Fulkerson Algorithm 6 Define an augmenting path to be a path from → in the residual graph (using edges of non-zero weight) Overview: Repeatedly add the flow of any augmenting path Ford-Fulkerson max-flow algorithm: • Initialize =0for all ∈ • Construct the residual network 13 0 / 2 0 / 10 0 / 6 0 / 10 0 / 4 0 / 8 0 / 10 0 / 9 0 0 / 10 ßow network G and ßow f s t Sign up to read all wikis and quizzes in math, science, and engineering topics. If it is a reversal of an original edge, it is called a 'backwards edge.'. In each junction all the incoming water has to be distributed to the other pipes. the edges should not be used to transmit commodity. This time we find the path $s - D - C - B - t$ with the residual capacities 1, 2, 3 and 3, and we increase by 1. There cannot flow more water through a pipe than its capacity. 3) Return flow. Each augmenting step causes at least one edge to be deleted from the residual graph. 2) While there is a augmenting path from source to sink. I chose the olive-colored path from a -> z to begin the algorithm. This pseudo-code is not written in any specific computer language. It was discovered in 1956 by Ford and Fulkerson. Line 8: reducing flow.) In other words, for a given edge (u,v)(u, v)(u,v), the residual capacity, cfc_fcf is defined as. Then we might send 3 cars along the bottom path for a total of 5 cars. Intuition: in a max flow, the min cut better fill up, and this is the bottleneck. Residual graphs are an important middle step in calculating the maximum flow. • Start with f (e) = 0for each edge e 2E. The Ford-Fulkerson algorithm repeatedly nds an s-tpath P in the current residual graph G f, and augments along pas much as possible subject to the capacity constraints of c2016, Tim Roughgarden. Sign up, Existing user? Finding the augmenting path inside the while loop takes O(V+E′),O\big(V + E^{'}\big),O(V+E′), where E′E^{'}E′ is the set of edges in the residual graph. ・After two more, some residual capacities are r3 and r4. Vertices, which are pipe junctions, have two interesting properties. Ford-Fulkerson algorithm: “undo” functionality Key observation: When constructing a flowf, one might commit errors on some edges, i.e. 00:00 - Welcome 01:05 - Why Intuition? Then, those residual capacities are used to make a residual network, GfG_fGf. calculateMaxFlow() on line 91 is what calls getPath with the correct parameters and adds up the max flow. It is well-known that if irrational arc capacities are allowed, the algorithm does not necessarily terminate. We use the same flow network as above. Taking the original intuition network and adding labels to the vertices, we now have this: It was shown that 2 units of flow can be pushed along the top-most path initially. However, the algorithm does satisfy the following properties. The basic intuition behind this fact is as follows. Intuition: keep augmenting flow along an augmenting path until there is no augmenting path. The resulting graph is as follows: Finally, a flow of 2 can be sent along the path [(S, A), (A, B), (B, C), (C, D), (D, T)] because the minimum residual capacity along that path is 2. the edges should not be used to transmit commodity. There must also be a source vertex and sink vertex to understand the beginning and end of the flow network. Ford-Fulkerson 方法——最大流问题 最大流&&最小费用最大流&&最大二分匹配. Finding the max flow of an undirected graph with Ford-Fulkerson. History. Practice math and science questions on the Brilliant iOS app. Once there doesn't exists an augmenting path any more, the flow is maximal. The vertices act as junctions, where water comes out of some pipes, and distributes it in some way to other pipes. Obviously we cannot send more flow from $s$ to $t$ than the capacity of any $s$-$t$-cut. If this method is used, Ford-Fulkerson runs in polynomial time. 2. If all flows are integers, then the while loop of Ford-Fulkerson is run at most ∣f∗∣|f^{*}|∣f∗∣ times, where f∗f^{*}f∗ is the maximum flow. Intialize flows to zero. material originates at source and is sent to sink. The intuition goes like this: as long as there is a path from the source to the sink that can take some flow the entire way, we send it. The basic intuition behind this fact is as follows. •Today we’re going to just start with the proof, and this will inspire the algorithm. No. The flow network is created, so its vertices and network can be inspected. And the length of a simple paths is bounded by $V$. The abstract description (F-F algorithm) : 1. • Find an s t path P in the residual network G f. • Augment ow along path P. • Repeat until you get stuck. rì Augment flow along path P. rì Repeat until you get stuck. It includes the reversed edge $(A, D)$. Conservation of flow. Ford-Fulkerson algorithm Questions. The algorithm was first published by Yefim Dinitz in 1970, and later independently published by Jack Edmonds and Richard Karp in 1972. I do hope that some of you will nd them interesting. Two things happen: When a new residual graph is constructed with these new edges, any edges with a residual capacity of 0—like (B, T)—are not included. This residual capacity is used to decide how much flow to send along a given path in the next function. The bottom path can be chosen and flow of 3 can be sent along it. This can be illustrated by the following graphic. Ford-Fulkerson Algorithm for Max Flow Problem version 1.0.0.0 (2.54 KB) by Karl Ezra Pilario An Edmonds-Karp implementation to solve the Max-flow Min-cut Proble ; Dinic's algorithm for Maximum Flow - GeeksforGeeks . Ford-Fulkerson algorithm ... Ford-Fulkerson Recap 373F20 - Nisarg Shah 4 •Define the residual graph ... o Intuition: Keep only the edges useful for shortest paths. cf(u,v)=c(u,v)−f(u,v).c_f(u, v) = c(u, v) - f(u, v).cf(u,v)=c(u,v)−f(u,v). Please notify errors on Piazza/by email to deeparnab@dartmouth.edu. For example, the function on line 32, getVertex() takes in a vertex name and goes to find the entire Vertex object in FlowNetwork's vertices attribute. This time we find the augmenting path $s - A - D - C - t$ with the residual capacities 2, 3, 1 and 2. To illustrate the above python implementation, we can construct the exact same graph from the Intuition section. The Ford-Fulkerson algorithm assumes that the input will be a graph, GGG, along with a source vertex, sss, and a sink vertex, ttt. The max-flow min-cut theorem states that flow must be preserved in a network. The Edmonds-Karp algorithm is an implementation of the Ford-Fulkerson method for computing a maximal flow in a flow network. 2. While there is an augmenting path 3. do augment flow as much as legally possible More detailed description on selecting paths. >Described the Ford-Fulkerson algorithm – starts with a feasible flow (all zeros) and improves it (by augmentations) – essentially optimal if max capacity into t is O(1) ... >Intuition: flow from source has to cross the cutto get to the sink addVertex() on line 50 adds a vertex to the graph after it checks various error cases to make sure the vertex can be added. Push-Relabel approach is the more efficient than Ford-Fulkerson algorithm. the set of all vertices that can be reached from $s$ in the residual graph (using edges with positive residual capacity), and the set of all the other vertices. Email: tim@cs.stanford.edu. Review: Ford-Fulkerson algorithm. A. Therefore the maximum flow is bounded by the minimum cut capacity. Review: Ford-Fulkerson Algorithm. Discussion of efficiency of Ford-Fulkerson implementation, and of the Edmonds-Karp algorithm. The Ford-Fulkerson algorithm is an algorithm that tackles the max-flow min-cut problem. Pf. This class has a starting vertex (just the name of the vertex, not the class instance), an ending vertex, a capacity, a flow, and a pointer to the return edge which is used in the residual graph. s u v t 1 = 1 0= 1 1 = 1 1 = 1 0= 1 jfj = 1 To improve the current flowf, we should work out ways to or in more detail (Line 7: adding flow. 11/10/2017 Ford-Fulkerson Algorithm | Brilliant Math & Science routing algorithms for the next generation wireless ... Dijkstra and Ford-Fulkerson algorithms and can be implemented in the cases of unicast and multicast transmission respectively. Intuition: A random walk of nding augmenting paths in Ford Fulkerson algorithm will be very fast if there are lots of short paths. Ford-Fulkerson Algorithm The algorithm calculates the max-flow of a directed graph G(V,E). Intialize flows to zero. Intuition: We’re choosing the wrong paths! $$\sum_{(v, u) \in E} f((v, u)) = \sum_{(u, v) \in E} f((u, v))$$ Here is the difference before and after calling fn.calculateMaxFlow(): If you draw out all of the edges, you will see that the graph you've created is the same that was made in the residual graphs section. Then we increase the flow in the following way: We keep doing this until there are no more augmenting paths. Let us specify in more detail, what increasing the flow along an augmenting path means. Runtime depends on what costs can be, and method used to find paths. how much water can we push through the pipes from the source to the sink. The Ford-Fulkerson algorithm is an algorithm that tackles the max-flow min-cut problem. v should not be used. A minimum cut can be found after performing a maximum flow computation using the Ford-Fulkerson method. Step by step instructions showing how to run Bellman-Ford on a graph.The theory behind Bellman-Ford: https://www.youtube.com/watch?v=9PHkk0UavIM.Sources: 1. When this happens, only three edges are affected: (S, A), (A, B), and (B, T). Then, we start the important Ford-Fulkerson method and continue to calculate flow while there still is a path. The subsequent push-relabel algorithms extend this intuition by not recomputing shortest path distances from scratch every time. s 1 2 t C C 0 0 0 0 0 C C 1s 1 2 C C 0 0 0 0 X0 1 C C X X X 1 1 1 X X 1 X1 X 0 1 m, n, and log C The value of a flow of a network is the sum of all flows that gets produced in source $s$, or equivalently of the flows that are consumed in the sink $t$. Now sending a flow of x units (x < y) from A to B is equivalent to sending -x units from B to A, so we add a back edge from B to A of capacity x units. Residual Graph (Max - Flow) - Intuition and correctness. The intuition behind the algorithm is quite simple (even though the implementation details can obscure this). This is because the flow is increased, at worst, by 1 in each iteration. ... Intuition. The Ford–Fulkerson method proceeds in exactly the same way as the greedy approach described above, but it only stops when there are no more augmenting paths in the residual graph (not in the original network). along the edges whose residual capacity is positive. For example, the edge u! 1. Therefore we can increase the flow by 3 and we get a flow of 8 for the network. Then we look for an augmenting path from $s$ to $t$. yDepartment of Computer Science, Stanford University, 474 Gates Building, 353 Serra Mall, Stanford, CA 94305. As noted in the introduction to this wiki, this algorithm is often called a method because the method of finding augmenting paths in a residual graph is not fully specified. rì Start with f(e) = 0 for each edge e ! It is well-known that if irrational arc capacities are allowed, the algorithm does not necessarily terminate. This concept is used in Ford–Fulkerson algorithm which computes the maximum flow in a flow network. The capacity of an edge is the total weight it can carry whereas an edge's flow is the weight it is currently carrying. ・After two more augmenting paths, some residual capacities are r2 and r3. This also defines the residual capacity for all reversed edges. An augmenting path is simple path in the residual graph, i.e. Max flow in simple weighted graph with no specified source or sink. s u v t 1 = 1 0= 1 1 = 1 1 = 1 0= 1 jfj = 1 To improve the current flowf, we should work out ways to Add this path-flow to flow. >Ford-Fulkerson >Other Algorithms Outline for Today ... – intuition: -excess flow leaving s cannot pool at any other node, so it ends up at t – proof >sum of excess flows of all nodes is 0 – every edge flow appears twice —once incoming (+) and once outgoing (-) —so sum is 0 Toward a max-ßow algorithm Greedy algorithm. During the algorithm the matrix capacity will actually store the residual capacity of the network. Ford{Fulkerson algorithm Ford{Fulkerson augmenting path algorithm. Edmonds-Karp algorithm is just an implementation of the Ford-Fulkerson method that uses BFS for finding augmenting paths. ・Does FF always terminate? This is the main idea behind the Ford–Fulkerson method. Ford-Fulkerson: Exponential Number of Augmentations Q. Each road can hold a certain number of cars. Email: tim@cs.stanford.edu. The Ford-Fulkerson method works as follows. Ford–Fulkerson algorithm: pathological example Intuition. Imagine a flow network that is just a traffic network of cars. The most famous algorithm is the Ford-Fulkerson algorithm, named after the two scientists that discovered the max-flow min-cut theorem in 1956. A priori, there is no reason for such a map even to be continuous (e.g., if it chooses between two parallel edges based on which one has bigger capacity, then it won't be continuous). The Ford-Fulkerson algorithm is used to detect maximum flow from start vertex to sink vertex in a given graph. $\endgroup$ – Evan Jenkins Apr 8 '18 at 16:52 3 s 5 t 15 10 15 16 9 15 6 8 10 4 15 4 10 10 capacity assume all nodes are reachable from s A.Artale AlgorithmsforDataProcessing. However, after this, I'm stuck. In the context of flow analysis, there is only an interest in considering how units are transferred between nodes in a holistic sense. Ford fulkerson explained Ford-Fulkerson Algorithm for Maximum Flow Problem . 2) While there is a augmenting path from source to sink. They are explained below. Given a flow f in a graph G, define the residual graph G f on the same verticeswith (residual) capacities defined as follows: [“Forward edges”] for each edge e in G with f(e) < c. e, include e in G f with capacity c e. – f(e) [“Backward edges”] for each e in G with f(e) > 0, include reverse edge e’ in G. f. with capacity f(e) The generic push-relabel algorithm has a rather different intuition. If all capacities are integers, then there exists a max flow f for which every flow value f(e) is an integer. 07maxflow-Part2.pdf - Ch7 Recap Summary \u2751 Definition of Network Flow \u2751 Definition of Cut and its capacity \u2751 Weak Duality v(f \u2264 cap(A B for any It cannot magically disappear or appear. It should be noted, that the Ford-Fulkerson method doesn't specify a method of finding the augmenting path. E. rì Find an s!t path P where each edge has f(e) < c(e). That is, given a network with vertices and edges between those vertices that have certain weights, how much "flow" can the network process at a time? Ford-Fulkerson Algorithm (((Graph GGG, source sss, sink t):t):t): Basically, what this simplified version says is that as long as there is a path from the source to the sink that can handle more flow, send that flow. But because we already have a flow of 3 from $D$ to $A$ this is possible. In the context of flow analysis, there is only an interest in considering how units are transferred between nodes in a holistic sense. We represent edges as water pipes, the capacity of an edge is the maximal amount of water that can flow through the pipe per second, and the flow of an edge is the amount of water that currently flows though the pipe per second. The path is grown until it reaches the end of the flow network; at that point the base case for the recursion is called on line 82 and the function ends. However, these attributes contain full objects—the vertices attribute contains Vertex objects and the network attribute contains Edge objects. A maximal flow is a flow with the maximal possible value. This motivates the first flow condition. The vertices attribute is a list of all of the instances of vertices in the graph. This gives a flow of 5 for the network. When a residual graph, GfG_fGf, is created, edges can be created that go in the opposite direction when compared to the original graph. (I'm told that CLRS has a formal proof.) Level Graph 373F20 - … Finally, we can send 2 more cars along the top path for two edges, send them down to bottom path and through to the sink. 1 The Ford Fulkerson Algorithm we update $f((u, v)) ~\text{+=}~ C$ and $f((v, u)) ~\text{-=}~ C$ for every edge $(u, v)$ in the path. Finally, we only need to calculate the total flow coming out of the source because this is the exact amount of flow through the entire network (since we only have one source). Then, another path is calculated and the process resumes. $$\sum_{(s, u) \in E} f((s, u)) = \sum_{(u, t) \in E} f((u, t))$$. Push-Relabel approach is the more efficient than Ford-Fulkerson algorithm. Show that the depth of a BFS tree can't be larger than the depth of a DFS tree while they're operate on the same vertex. By induction on the size of B. Is generic Ford-Fulkerson algorithm polynomial in input size? However, ... Intuition and correctness. A network is a directed graph $G$ with vertices $V$ and edges $E$ combined with a function $c$, which assigns each edge $e \in E$ a non-negative integer value, the capacity of $e$. It then adds the new edge to the network map on line 77 and adds the return edge to the same map on line 79. Pf. 23 FORD–FULKERSON(G) While there exists an augmenting path: - find an augmenting path - compute bottleneck capacity $$f(e) \le c(e)$$. Multiple algorithms exist in solving the maximum flow problem. So, the runtime of Ford-Fulkerson is O(E⋅∣f∗∣).O\big(E \cdot |f^{*}|\big).O(E⋅∣f∗∣). The source $s$ is origin of all the water, and the water can only drain in the sink $t$. The following image show a flow network. As before, directed edges correspond to pipes. Algorithm: 1.Find a path from s to t and route max ow through this path 2.Update capacities by updating the residual graph 3.Repeat 1 and 2 until algorithm halts (no path from s to t is found) Explanation: - Intuition: The back edges in residual graph can undo suboptimal ows in some edges for future iterations. The “Follow the Regularized Leader” algorithm stems from the online learning setting, where the learning process is sequential. That flow is added to the forward edges and subtracted from the reverse edges. The algorithm then continually improves f by searching for an augmenting path ˇ, and using this path to improve f, … We have found the maximal flow. It should be noted that if there is a flow along some directed edge $(u, v)$, than the reversed edge has capacity 0 and we can define the flow of it as $f((v, u)) = -f((u, v))$. adj is the adjacency list of the undirected graph, since we have also to use the reversed of directed edges when we are looking for augmenting paths. The algo-rithm begins by initializing the ow fto the all-0 ow, that is, the ow that is 0 along every edge. After each iteration we have a valid flow, and the flow on each edge is an integer. rì Augment flow along path P. rì Repeat until you get stuck. Ford-Fulkerson Algorithm The algorithm calculates the max-flow of a directed graph G(V,E). (S, A) and (A, B) are affected in the same way because they have the same capacity. This also motivates the second flow condition. Here is a version of the pseudo-code that explains the flow augmentation more in depth: This algorithm is more well-defined. In the visualization with water pipes, the problem can be formulated in the following way: Ford-Fulkerson algorithm ... Ford-Fulkerson Recap 373F20 - Nisarg Shah 4 •Define the residual graph ... o Intuition: Keep only the edges useful for shortest paths. Ask Question Asked 6 years, 9 ... it is directed as shown below. In the following image you can see the minimum cut of the flow network we used earlier. In the image above, we could start by sending 2 cars along the topmost path (because only 2 cars can get through the last portion). And the sum of the incoming flow of a vertex $u$ has to be equal to the sum of the outgoing flow of $u$ except in the source and sink vertices. The Ford-Fulkerson algorithm was eventually improved upon by the Edmonds-Karp algorithm, which does the same thing in O(V2⋅E)O\big(V^2 \cdot E\big)O(V2⋅E) time, independent of the maximum flow value. The algorithm terminates in at most v(f*) nC iterations. In this setting, an online player makes a decision in every round and suffers a loss. yDepartment of Computer Science, Stanford University, 474 Gates Building, 353 Serra Mall, Stanford, CA 94305. 13 0 / 2 0 / 10 0 / 6 0 / 10 0 / 4 0 / 8 0 / 10 0 / 9 0 0 / 10 ßow network G and ßow f s t The following image show the maximal flow in the flow network. Therefore, the loop is complete. The intuition is, that every time we find an augmenting path one of the edges becomes saturated, and the distance from the edge to $s$ will be longer, if it appears later again in an augmenting path. • Find an s t path P in the residual network G f. • Augment ow along path P. • Repeat until you get stuck. What happens to the 'capacity' and 'flow' attributes of the edges in 'getEdges()' when 'fn.calculateMaxFlow()' is called? It only has a name, which we use for edge creation (and debugging), whether or not the vertex is a source or a sink.
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