Given the following minimum cost flow network find the optimal cost flow. (Here red, blue and cyan color represent supply, demand and junction nodes, respectively. Numbers associated with these nodes show supply and demand amount. Numbers associated with edges represent per unit cost if that edge is chosen for transport. 3 -1 3 -4 2 1 3 -5 Figure 3: Minimum cost flow network
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- Dijkstra’s algorithm computes the least-cost path from one node (the source, which we will refer to as ?u) to all other nodes in the network. Dijkstra’s algorithm is iterative and has the property that after the k-th iteration of the algorithm, the least-cost paths are known to k destination nodes, and among the least-cost paths to all destination nodes, these k paths will have the k smallest costs. Let - D(v): cost of the least-cost path from the source node to destination ?v as of this iteration of the algorithm. - p(v): previous node (neighbor of v) along the current least-cost path from the source to v. - N′: a subset of nodes; v is in N′ if the least-cost path from the source to ?v is definitely known. The centralized routing algorithm consists of an initialization step followed by a loop. The number of times the loop is executed is equal to the number of nodes in the network. Upon termination, the algorithm will have calculated the shortest paths from the source…Consider the following network: (a) Which of the following most accurately describes the connectedness of this network? Strongly Weakly Disconnected None of the above (b) When discussing path lengths on a weighted graph, one must first define how the weights are related to the length of a path between two nodes is then the sum of the distances of the links in that path. Consider the previous network and assume that the link weights represent distances. Using this distance metric, what is the shortest path between nodes 1 and 6? (c) A common way to define the distance between two nodes is the inverse (or reciprocal) of the link weight. Consider the previous network and assume that the distance between two adjacent nodes is defined as the reciprocal of the link weight. Using this distance metric, what is the shortest path between node 1 and 6?Dynamic ProgrammingConsider the following network below: (a) Label all of the nodes such that for every arc, (i,j), where i is the start and j the end node for the arc, the labels satisfy i < j.(b) Why can we use the node labeling algorithm to label the nodes in this network? (c) Use forward DP to find the shortest path from your first labeled node to every node in the network.(d) What fundamental characteristic of dynamic programs allow us to breakdown large-sized problems into a series of smaller-sized problems? What does this characteristic establishes?
- Algorithm : (1. Single-destination Shortest Path, 2. Bellman-Ford, 3. Negative-Weight Cycles), Dynamic Programming, RNA Secondary Structure Consider n airports, labelled 1 through n. Each airport maintains a number of direct flights to other airports, represented by an edge with an associated cost. Since travel is currently restricted, one may only book direct flights—no airports will perform any transfers. However, things are beginning to open up again and airports are being slowly cleared to process transfers. These are done in order of label, so the first airport to allow transfers is 1, followed by 2, and so on. Once an airport is allowed to take transfers, flights through that airport may be booked. For example, consider the scenario depicted in Figure 1. When all airports are closed to transfers, flights may only be booked between airports connected by an edge. Thus, a flight from airport 5 to airport 2 may not be booked. However, once airport 1 is allowed to perform…Please answer the following question in detail and explain all the proofs and assumptions for all parts. The question has three parts, (a), (b) and (c). Iterative lengthening search is an iterative analogue of uniform-cost search. The basic idea is to use increasing limits on path cost. If a node is generated whose path cost exceeds the current limit, it is immediately discarded. For each new iteration, the limit is set to the lowest path cost of any node discarded in the previous iteration. (a) Show that this algorithm is optimal for general path costs. You may assume that all costs are integers (this is not a loss of generality if the search space is finite). You may wish to consider the minimal path cost C; what happens when we set the path cost to be some limit l < C? (b) Consider a uniform tree with branching factor b, solution depth d, and unit step costs (each action costs one unit). How many iterations will iterative lengthening require? (c) (7 points) Now consider the…16. A 1 Ꭰ Step 0 1 2 3 4 5 1 N' 1 Consider the following network. 7 5 --F 1) With the indicated link costs, use Dijkstra's shortest-path algorithm to compute the shortest path from D to all network nodes. Show how the algorithm works by filling in the following table, where d(x) is the cost of the least-cost path from the source node D destination x as of the current iteration of the algorithm, p(x) is the previous node (neighbor of x) along the current least-cost path from the source D to x, N' is the subset of nodes whose shortest path from the source D has definitely been known. d(A),p(A) d(B),p(B) d(C),p(C) d(E),p(E) d(F),p(F) -B 2) According to the table computed in 1), show the shortest path tree rooted at D. 3) According to the results from 1) and 2), create the forwarding table, in which each row consists of two fields, destination and next hop.
- How exactly would I be able to paraphrase this pseudo-code so that it can make a bit more sense? 1.Initialtzation: generate a random solution X ← RandomSolution() 2. Solve the traffic assignment problem at equilibrium for each link: (flow, cost) ← NonLinearComplementary(X) 3. Evaluate the objective value for the current solution: f (X) ← EvaluateObj Val (f low, cost) 4. T ← Tmax 5 while (T > Tmin) do 6 I ← 1 7 while (t < ltermax)do 8 X' ← FindNeighborSolution(X) 9 (flow,cost) ← NonLinearComplementary(X') 10 f(X') ← EvaluateObf Val (f low, Cost) 11 Δ ←(f(x) - f(x')) 12 if(Δ<0)then 13 BoltzmannValue ← exp(-Δ/T) 14 if (BoltzmannValue < Random (0,1)) then 15 Accept the new solution: X ← X' 16 end If 17 Else I8 Accept the new solution: X ← X' 19 end if 20 i ← i+1 21 end while 22 T ←T ∞ 23 end while ________You are currently building an set of office network infrastructure. When connecting two computers, assume that the chance of a network outage drops by a factor of 10. That is, two computers connected to each other have a 10% chance of network failure, while three computers have a 1% chance. How would you determine how resilient your network is given a graph of all computers and their connections to one another? O Determine the k-regularity of the graph, use this value of k to calculate network resiliency. O Determine if there is a Hamiltonian path within your network, use the length of this path to calculate network resiliency. 4 Invert the graph, find the vertex with the smallest degree, and use this degree to calculate network resiliency. O Find a k-coloring such that k is minimized, us this k to calculate network resiliency.C. A 2 B 5 с 10 3 D 4 E 11 F i. What is the shortest cost or distance to travel from node A to node F? ii. Give the weight of each edges that is traversed to get from A to F. [ iii. What would be the longest path from node A to node B?
- where a is the start node and e is the goal node. The pair [f, h] at each node indicates the value of the f and h functions for the path ending at that node. Given this information, what is the cost of each path? The cost < a, c >= 2 is given as a hint. Is the heuristic function h admissible? ExplainLet a network be given by nodes V = {S, A, B,C, D,T} and arcs, capacities e and a flow f according to the table: SA SC AB Bт АC DC BD DT AT CT 3 ? 4 1 1 ? 3 4 3 5 ? 3 9. 3 5 1 1 (a) Complete f and e such that f is an admissible flow from S to T. Justify your answer. What is the value of the flow?2. Determine the sets N and A for each network 3. Draw the network defined by N = {1, 2, 3, 4, 5, 6} A = {(1,2), (1,5), (2,3), (2,4), (3,4), (3,5), (4,3), (4,6), %3D