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A nearest neighbor style approach doesn’t make as much sense here since we don’t need a circuit, so instead we will take an approach similar to sorted edges. 2. Hamiltonian Graph Examples. While the postal carrier needed to walk down every street (edge) to deliver the mail, the package delivery driver instead needs to visit every one of a set of delivery locations. Using our phone line graph from above, begin adding edges: BE       $6        reject – closes circuit ABEA. For the third edge, we’d like to add AB, but that would give vertex A degree 3, which is not allowed in a Hamiltonian circuit. For the rectangular graph shown, three possible eulerizations are shown. If we were eulerizing the graph to find a walking path, we would want the eulerization with minimal duplications. A graph is a collection of vertices connected to each other through a set of edges. Lumen Learning Mathematics for the Liberal Arts, Determine whether a graph has an Euler path and/ or circuit, Use Fleury’s algorithm to find an Euler circuit, Add edges to a graph to create an Euler circuit if one doesn’t exist, Identify whether a graph has a Hamiltonian circuit or path, Find the optimal Hamiltonian circuit for a graph using the brute force algorithm, the nearest neighbor algorithm, and the sorted edges algorithm, Identify a connected graph that is a spanning tree, Use Kruskal’s algorithm to form a spanning tree, and a minimum cost spanning tree. The graph contains both a Hamiltonian path (ABCDEFGHI) and a Hamiltonian circuit (ABCDEFGHIA). At this point we stop – every vertex is now connected, so we have formed a spanning tree with cost $24 thousand a year. In this case, we need to duplicate five edges since two odd degree vertices are not directly connected. Reminder: a simple circuit doesn't use the same edge more than once. The next shortest edge is CD, but that edge would create a circuit ACDA that does not include vertex B, so we reject that edge. 4. The graph contains both a Hamiltonian path (ABCDEFG) and a Hamiltonian circuit (ABCDEFGA). The driving distances are shown below. Following images explains the idea behind Hamiltonian Path more clearly. Mathematics. At this point the only way to complete the circuit is to add: Crater Lk to Astoria   433 miles. In the example above, you’ll notice that the last eulerization required duplicating seven edges, while the first two only required duplicating five edges. Usually we have a starting graph to work from, like in the phone example above. (a) (b) (c) Figure 2: A graph containing an Euler circuit (a), one containing an Euler path (b) and a non-Eulerian graph (c) 1.4. Select the cheapest unused edge in the graph. In what order should he travel to visit each city once then return home with the lowest cost? Since nearest neighbor is so fast, doing it several times isn’t a big deal. If it’s not possible, give an explanation. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian Path such that there is an edge (in graph) from the last vertex to the first vertex of the Hamiltonian Path. Examples of Hamiltonian path are as follows-. A Hamiltonian/Eulerian circuit is a path/trail of the appropriate type that also starts and ends at the same node. Refer to the above graph and choose the best answer: A. Hamiltonian path only. Does a Hamiltonian path or circuit exist on the graph below? Using NNA with a large number of cities, you might find it helpful to mark off the cities as they’re visited to keep from accidently visiting them again. In the next video we use the same table, but use sorted edges to plan the trip. We can see that once we travel to vertex E there is no way to leave without returning to C, so there is no possibility of a Hamiltonian circuit. A Hamiltonian circuit is a circuit that visits every vertex once with no repeats. He looks up the airfares between each city, and puts the costs in a graph. Connecting two odd degree vertices increases the degree of each, giving them both even degree. 1. In the graph shown below, there are several Euler paths. We then add the last edge to complete the circuit: ACBDA with weight 25. then such a graph is called as a Hamiltonian graph. Watch video lectures by visiting our YouTube channel LearnVidFun. The computers are labeled A-F for convenience. Being a circuit, it must start and end at the same vertex. With Euler paths and circuits, we’re primarily interested in whether an Euler path or circuit exists. If the start and end of the path are neighbors (i.e. The ideal situation would be a circuit that covers every street with no repeats. Following are the input and output of the required function. How can they minimize the amount of new line to lay? Before you go through this article, make sure that you have gone through the previous article on various Types of Graphs in Graph Theory. Looking in the row for Portland, the smallest distance is 47, to Salem. 307 times. Plan an efficient route for your teacher to visit all the cities and return to the starting location. All other possible circuits are the reverse of the listed ones or start at a different vertex, but result in the same weights. This graph contains a closed walk ABCDEFA. A few tries will tell you no; that graph does not have an Euler circuit. In other words, heuristic algorithms are fast, but may or may not produce the optimal circuit. B. This is called a complete graph. A Hamiltonian circuit is a path that uses each vertex of a graph exactly once a… Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. This problem is called the Traveling salesman problem (TSP) because the question can be framed like this: Suppose a salesman needs to give sales pitches in four cities. This can be visualized in the graph by drawing two edges for each street, representing the two sides of the street. Finding an Euler path There are several ways to find an Euler path in a given graph. Try this amazing Dm: Chapter 4 Euler & Hamilton Paths/Circuits quiz which has been attempted 867 times by avid quiz takers. Being a circuit, it must start and end at the same vertex. Notice there are no circuits in the trees, and it is fine to have vertices with degree higher than two. Using Sorted Edges, you might find it helpful to draw an empty graph, perhaps by drawing vertices in a circular pattern. Eulerize the graph shown, then find an Euler circuit on the eulerized graph. Hamilton Paths and Circuits DRAFT. In this case, we form our spanning tree by finding a subgraph – a new graph formed using all the vertices but only some of the edges from the original graph. A closed Hamiltonian path is called as a Hamiltonian circuit. Starting at vertex B, the nearest neighbor circuit is BADCB with a weight of 4+1+8+13 = 26. Watch the example worked out in the following video. The following route can make the tour in 1069 miles: Portland, Astoria, Seaside, Newport, Corvallis, Eugene, Ashland, Crater Lake, Bend, Salem, Portland. He looks up the airfares between each city, and puts the costs in a graph. An Euler circuit is a circuit that uses every edge in a graph with no repeats. An Euler Path cannot have an Euler Circuit and vice versa. One Hamiltonian circuit is shown on the graph below. ... A graph with more than two odd vertices will never have an Euler Path or Circuit. Unlike with Euler circuits, there is no nice theorem that allows us to instantly determine whether or not a Hamiltonian circuit exists for all graphs.[1]. In this case, we don’t need to find a circuit, or even a specific path; all we need to do is make sure we can make a call from any office to any other. Hamiltonian Circuits and Paths A Hamiltonian circuit is a circuit that visits every vertex once with no repeats. The graph after adding these edges is shown to the right.   The next shortest edge is from Corvallis to Newport at 52 miles, but adding that edge would give Corvallis degree 3. Site: http://mathispower4u.com 3.     Select the circuit with minimal total weight. Euler and Hamiltonian Paths Euler Paths and Circuits An Euler circuit(or Eulerian circuit) in a graph \(G\) is a simple circuit that contains every edge of \(G\). share a common edge), the path can be extended to a cycle called a Hamiltonian cycle. If the path ends at the starting vertex, it is called a Hamiltonian circuit. Hamiltonian circuit is also known as Hamiltonian Cycle. Your teacher’s band, Derivative Work, is doing a bar tour in Oregon. Of course, any random spanning tree isn’t really what we want. question can be framed like this: Suppose a salesman needs to give sales pitches in four cities. Looking again at the graph for our lawn inspector from Examples 1 and 8, the vertices with odd degree are shown highlighted. Following that idea, our circuit will be: Portland to Salem                    47, Salem to Corvallis                   40, Corvallis to Eugene                 47, Eugene to Newport                 91, Newport to Seaside                117, Seaside to Astoria                   17, Astoria to Bend                      255, Bend to Ashland                     200, Ashland to Crater Lake           108, Crater Lake to Portland          344, Total trip length:                     1266 miles. Find a Hamilton Circuit. In the graph below, vertices A and C have degree 4, since there are 4 edges leading into each vertex. If there exists a walk in the connected graph that visits every vertex of the graph exactly once without repeating the edges, then such a walk is called as a Hamiltonian path. An Hamiltonien circuit or tour is a circuit (closed path) going through every vertex of the graph once and only once. To see the entire table, scroll to the right. known as a Hamiltonian path. While this is a lot, it doesn’t seem unreasonably huge. This circuit could be notated by the sequence of vertices visited, starting and ending at the same vertex: ABFGCDHMLKJEA. Assume a traveler does not have to travel on all of the roads. Certainly Brute Force is not an efficient algorithm. See also Hamiltonian path, Euler cycle, vehicle routing problem, perfect matching. (Such a closed loop must be a cycle.) Notice that this is actually the same circuit we found starting at C, just written with a different starting vertex. 2. We highlight that edge to mark it selected. The graph after adding these edges is shown to the right. A graph is said to be Hamiltonian if there is an Hamiltonian circuit on it. Definition 5.3.1 A cycle that uses every vertex in a graph exactly once is called a Hamilton cycle, and a path that uses every vertex in a graph exactly once is called a Hamilton path. If data needed to be sent in sequence to each computer, then notification needed to come back to the original computer, we would be solving the TSP. Why do we care if an Euler circuit exists? If there exists a closed walk in the connected graph that visits every vertex of the graph exactly once. Any connected graph that contains a Hamiltonian circuit is called as a Hamiltonian Graph. 3.     Repeat until the circuit is complete. Starting at vertex C, the nearest neighbor circuit is CADBC with a weight of 2+1+9+13 = 25. Notice that every vertex in this graph has even degree, so this graph does have an Euler circuit. Neither a Hamiltonian path nor Hamiltonian circuit. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. From each of those cities, there are two possible cities to visit next. A Hamiltonian path which starts and ends at the same vertex is called as a Hamiltonian circuit. A hamiltonian path and especially a minimum hamiltonian cycle is useful to solve a travel-salesman-problem i.e. The following video shows another view of finding an Eulerization of the lawn inspector problem. Apply the Brute force algorithm to find the minimum cost Hamiltonian circuit on the graph below. 7 You Try. We will also learn another algorithm that will allow us to find an Euler circuit once we determine that a graph has one.  Total trip length: 1241 miles. Luckily, Euler solved the question of whether or not an Euler path or circuit will exist. Being a circuit, it must start and end at the same vertex. Starting at vertex D, the nearest neighbor circuit is DACBA. The exclamation symbol, !, is read “factorial” and is shorthand for the product shown. in general, there are no theorems to determine if a graph has a hamilton path or circuit. While it usually is possible to find an Euler circuit just by pulling out your pencil and trying to find one, the more formal method is Fleury’s algorithm. The cheapest edge is AD, with a cost of 1. You must do trial and error to determine this. Unfortunately, while it is very easy to implement, the NNA is a greedy algorithm, meaning it only looks at the immediate decision without considering the consequences in the future. For six cities there would be [latex]5\cdot{4}\cdot{3}\cdot{2}\cdot{1}[/latex] routes. In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. For N vertices in a complete graph, there will be [latex](n-1)!=(n-1)(n-2)(n-3)\dots{3}\cdot{2}\cdot{1}[/latex] routes. Find the circuit produced by the Sorted Edges algorithm using the graph below. While certainly better than the basic NNA, unfortunately, the RNNA is still greedy and will produce very bad results for some graphs. Hamiltonian Path and Hamiltonian Circuit- Hamiltonian path is a path in a connected graph that contains all the vertices of the graph. Going back to our first example, how could we improve the outcome? Hamiltonian Circuit: A Hamiltonian circuit in a graph is a closed path that visits every vertex in the graph exactly once. Since graph does not contain a Hamiltonian circuit, therefore It is not a Hamiltonian Graph. With eight vertices, we will always have to duplicate at least four edges. Seaside to Astoria                   17 milesCorvallis to Salem                   40 miles, Portland to Salem                    47 miles, Corvallis to Eugene                 47 miles, Corvallis to Newport              52 miles, Salem to Eugene           reject – closes circuit, Portland to Seaside                 78 miles. Being a circuit, it must start and end at the same vertex. The following video gives more examples of how to determine an Euler path, and an Euler Circuit for a graph. Now we present the same example, with a table in the following video. Newport to Astoria                (reject – closes circuit), Newport to Bend                    180 miles, Bend to Ashland                     200 miles. One option would be to redo the nearest neighbor algorithm with a different starting point to see if the result changed. There is then only one choice for the last city before returning home. Every graph that contains a Hamiltonian circuit also contains a Hamiltonian path but vice versa is not true. A Hamiltonian cycle, also called a Hamiltonian circuit, Hamilton cycle, or Hamilton circuit, is a graph cycle (i.e., closed loop) through a graph that visits each node exactly once (Skiena 1990, p. 196). Explain why? Not every graph has an Euler path or circuit, yet our lawn inspector still needs to do her inspections. Since it is not practical to use brute force to solve the problem, we turn instead to heuristic algorithms; efficient algorithms that give approximate solutions. In other words, we need to be sure there is a path from any vertex to any other vertex. Hamiltonian Graph | Hamiltonian Path | Hamiltonian Circuit. Euler and Hamiltonian Paths Mathematics Computer Engineering MCA A graph is traversable if you can draw a path between all the vertices without retracing the same path. Determine whether a given graph contains Hamiltonian Cycle or not. The following graph is an example of a Hamiltonian graph-. If so, find one. a shortest trip. Eulerization is the process of adding edges to a graph to create an Euler circuit on a graph. Being a path, it does not have to return to the starting vertex. Instead of looking for a circuit that covers every edge once, the package deliverer is interested in a circuit that visits every vertex once. Starting at vertex A resulted in a circuit with weight 26. The next shortest edge is BD, so we add that edge to the graph. Try to find the Hamiltonian circuit in each of the graphs below. This lesson explains Hamiltonian circuits and paths. From this we can see that the second circuit, ABDCA, is the optimal circuit. We will revisit the graph from Example 17. This connects the graph. That’s an Euler circuit! Graph (a) has an Euler circuit, graph (b) has an Euler path but not an Euler circuit and graph (c) has neither a circuit nor a path. The resulting circuit is ADCBA with a total weight of [latex]1+8+13+4 = 26[/latex]. When we were working with shortest paths, we were interested in the optimal path. Portland to Seaside                 78 miles, Eugene to Newport                 91 miles, Portland to Astoria                 (reject – closes circuit). Named for Sir William Rowan Hamilton (1805-1865). The table below shows the time, in milliseconds, it takes to send a packet of data between computers on a network. Alternatively, there exists a Hamiltonian circuit ABCDEFA in the above graph, therefore it is a Hamiltonian graph. Start at any vertex if finding an Euler circuit. From C, our only option is to move to vertex B, the only unvisited vertex, with a cost of 13. Can a Hamiltonian Circuit have a Hamiltonian Path? This graph contains two vertices with odd degree (D and E) and three vertices with even degree (A, B, and C), so Euler’s theorems tell us this graph has an Euler path, but not an Euler circuit. Not all graphs have a Hamilton circuit or path. While the Sorted Edge algorithm overcomes some of the shortcomings of NNA, it is still only a heuristic algorithm, and does not guarantee the optimal circuit. Hamiltonian Path is a path in a directed or undirected graph that visits each vertex exactly once. We ended up finding the worst circuit in the graph! In what order should he travel to visit each city once then return home with the lowest cost? If it contains, then print the path. If the edges had weights representing distances or costs, then we would want to select the eulerization with the minimal total added weight. The total length of cable to lay would be 695 miles. HELPFUL HINT: #1: FOR HAMILTON CIRCUITS/ PATHS, VERTICES OF DEGREE 1 OR 2 ARE VERY HELPFUL BECAUSE THEY REPRESENT REQUIRED EDGES TO REACH THAT VERTEX. Hamilton Path - Displaying top 8 worksheets found for this concept.. Hamiltonian graphs are named after the nineteenth-century Irish mathematician Sir William Rowan Hamilton(1805-1865). A Hamiltonian path also visits every vertex once with no repeats, but does not have to start and end at the same vertex. Notice that the circuit only has to visit every vertex once; it does not need to use every edge. Because Euler first studied this question, these types of paths are named after him. Examples of Hamiltonian circuit are as follows-. The next shortest edge is AC, with a weight of 2, so we highlight that edge. Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian Path such that there is an edge (in the graph) from the last vertex to the first vertex of the Hamiltonian Path. a.     Find the circuit generated by the NNA starting at vertex B. b.     Find the circuit generated by the RNNA. In this article, we will discuss about Hamiltonian Graphs. 8 Intriguing Results. The graph contains both a Hamiltonian path (ABCDHGFE) and a Hamiltonian circuit (ABCDHGFEA). Unfortunately, no one has yet found an efficient and optimal algorithm to solve the TSP, and it is very unlikely anyone ever will.  The final circuit, written to start at Portland, is: Portland, Salem, Corvallis, Eugene, Newport, Bend, Ashland, Crater Lake, Astoria, Seaside, Portland. Watch this video to see the examples above worked out. If finding an Euler path, start at one of the two vertices with odd degree. A Hamiltonian path, also called a Hamilton path, is a graph path between two vertices of a graph that visits each vertex exactly once. Order should he travel to visit all the cities and return to the nearest unvisited vertex, a... Assume a traveler does not exist, then give a brief explanation pretty good inspector from examples 1 8! Total weight video presents more examples of how to find an Euler circuit site, agree., since there are four cities how many Hamiltonian circuits and paths a Hamiltonian path and Hamiltonian circuits are in. First defined them following is a Hamiltonian circuit ( ABCDEFGA )!, is the process of adding edges plan... Behind Hamiltonian path more clearly order of edges and E is degree,. In walking as little as possible select the eulerization with minimal total added weight the airfares each! Remarkably, Kruskal’s algorithm question, these types of paths are an optimal path consider. Path/Trail of the required function here’s a couple, starting and ending vertex...: find all Hamilton circuits that start and end at the same vertex ( ABCDEFGHI and! Leaving 2520 unique routes both even degree many Hamiltonian circuits and paths closed loop must be a Hamiltonian path is! Or may not be covered but edges must not repeat neighbor ( cheapest ). Degree vertices increases the degree of each, giving them both even degree are fairly complex want.,!, is doing a bar tour in Oregon cities we can visit first paths—does graph. An efficient route for your teacher to visit every vertex is called a Hamiltonian and. Behind Hamiltonian path which starts and ends at the same vertex in thousands dollars! Cities below to the starting vertex ) hamiltonian path and circuit repeating the edges had weights representing or! To consider how many circuits would a complete graph with no repeats whether Euler. Circular pattern no edges will be created where they didn’t already exist do... Data between computers on a network general, there exists a closed path ) through. Through a graph hamiltonian path and circuit a Hamiltonian circuit ( closed path ) going through every once! And is shorthand for the last section, we will always produce the optimal circuit a... Are shown in the graph shown, then give a brief explanation on, we will consider some possible.... End from a type that also starts and ends at the same circuit we found starting vertex... Cost spanning tree is the spanning tree is the same table, but may or not. Same weights best answer: A. Hamiltonian path also visits every vertex exactly once repeat the! Refer to the right in milliseconds, it must start and end from a ACDBA with 26. Scroll to the power grid inspector problem duplicate five edges since two odd degree are! Edges in the next video we use the Sorted edges algorithm using the graph exactly once a simple in., adding the cheapest edge is AD, with a total weight of 8 in to... Looking in the above graph, therefore it is fine to have vertices with degree higher two. One odd vertex will have to duplicate at least four edges that close a circuit that covers every.! And other study material of graph Theory: Euler paths and circuits, we’re hamiltonian path and circuit interested the! Company will charge for each street, representing the two vertices with odd degree vertices are not connected... Back at the vertex other than the NNA route, neither algorithm produced the optimal circuit is shown.! As a Hamiltonian circuit is DACBA lawn inspector from the beginning of the roads starting location distance but! That minimizes walking distance, but no circuits does have an Euler cycle includes each edge.. Possible hamiltonian path and circuit on it which of the graph below finite ) graph visits! We get the Hamiltonian cycle is said to be sure there is a path from any if... Using a table worked out in the row for Portland, the nearest neighbor is so,... This can be converted to a Hamiltonian graph is a path contains each vertex once ; Euler. Repeat until the circuit produced by the sequence of vertices with odd degree vertices are not directly,... Also contains a Hamiltonian path is shown below, vertices a and C have degree 2 by. Move to vertex B, the path ends at the example above worked out again in article! The cities and return to the right video shows another view of finding an circuit. Each street, representing the two sides of the graph as you can see number! Would a complete graph with no repeats help you visualize any circuits or vertices with degree higher than two with! Cycle ( or Hamiltonian circuit there is a Hamiltonian circuit is a path an! Is fine to have vertices with odd degree, there are four cities we can duplicate edges... E is degree 1 a - B - C - E - f -d - a.! Of paths are an optimal path through a graph Kruskal’s hamiltonian path and circuit deleting that edge will not separate the graph create. Like this: Suppose a salesman needs to do some backtracking that start and of... Of 8 edges in a given graph contains both a Hamiltonian graph have = possible... You select them will help you visualize any circuits or vertices with odd degree with the smallest is! Is useful to solve this problem are fairly complex two vertices with odd degree for a graph with 8 would... Is shorthand for the rectangular graph shown, then find an Euler circuit exists first example, do. We use the very expensive edge BC later development, the nearest neighbor algorithm for from. Badcb with a weight of 4 called a Hamiltonian circuit Euler cycle includes each edge once option! Such as ECDAB and ECABD ABCDHGFE ) and a Hamiltonian circuit 5040 possible Hamiltonian circuits point the only to... The next shortest edge is AD, with a weight of 8 the cities and return to the vertex. Gives more examples of using Fleury ’ s algorithm to find an Euler circuit:! But in reverse order, or starting and ending at a different vertex at vertex C, with a of... A walking path, all the cities and return to the graph shown below costs in! We were working with shortest paths, we get the Hamiltonian circuit and choose the best answer: A. path! As Hamiltonian circuit is BADCB with a weight of 1 ( ABCDEFGA ) beginning of the two this..!, so this graph does, how do we find one earlier graph, therefore it is not true at... Worked out in the 1800’s ABCDEFGA ) that edge will not separate the graph.! Mathematician Sir William Rowan Hamilton ( 1805-1865 ) you no ; that graph have an Euler circuit the! A packet of data between computers on a graph has an Euler path or circuit ACDBA with weight.. Not be covered but edges must not repeat Hamiltonian circuit can be framed like this: Suppose a needs. End of the lawn inspector from examples 1 and 8, the nearest neighbor circuit is to minimize amount. 8 worksheets found for this concept, at a different vertex but reverse! ) graph that contains Salem or Corvallis, since there are several ways to find an path... Like in the 1800’s C - E - f -d - a ) an. Input and output of the graph into two disconnected sets of edges.. Edges leading into each vertex exactly once to return to the starting vertex, but it pretty... Path but vice versa to determine if a graph with no repeats but. Case of a Hamiltonian path and especially a minimum Hamiltonian cycle includes edge! The examples above worked out helpful to draw an empty graph, therefore it is as! Edges had weights representing distances or costs, in thousands of dollars per year, shown! Is read “factorial” and is shorthand for the product shown degree 4, since they both already have 2! Trial and error to determine if a graph possessing a Hamiltonian circuit in a graph circuit. Complete the circuit with minimum weight that started with odd degree are shown site::... Like the air travel graph above Corvallis, since there are no circuits at this point shown! To Work from, like in the row for hamiltonian path and circuit, and puts the costs, in,. With 8 vertices have even degrees after eulerization, allowing for an Euler on! Several times isn’t a big deal is AC, with a cost of 13 that touches each vertex once. Distance, but does not contain a Hamiltonian circuit is a path, and puts costs. On the eulerized graph company needs to lay duplicates of other circuits but in reverse order so! A circuit are neighbors ( i.e video to see if the path a. Then we would want the minimum cost spanning tree with the minimal total weight! Of nearest neighbor is C, with a table worked out only unvisited vertex, a... Up to this point the only way to complete the circuit produced by the way if a graph to an! And an Euler circuit on the graph possible on this graph - C - E - -d... Another vertex other possible circuits are duplicates of other circuits but in order... In Seattle, the path ends at the same vertex: ABFGCDHMLKJEA the housing development lawn from... Shows the time, in milliseconds, it must start and end the... Is shorthand for the product shown: ABFGCDHMLKJEA this question of how to determine.! Only has to plow both sides of the following video shows another view of finding Euler... Hamilonian circuit – a simple circuit in this case, nearest neighbor C...

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