Swamp Thing feels a disturbance in the Green. It seems that Anton Arcane has once again taken over the Rot and is spreading its evil all over the world. A lot of the world's capital cities are feeling the power of the Rot right now. It is Swamp Thing's duty to preserve a balance between the forces of the Green and the Rot. He must travel to each of the cities affected by the Rot as soon as possible. Luckily, he can use the Green's help to travel to cities through special bidirectional roads. However using the Green to travel from one city to another requires a certain amount of energy. Swamp Thing can only expend a certain amount of energy while travelling.Swamp Thing wants to travel to all of the world's cities in such an order that the sum of the distances he travels is minimized while the total energy he spends travelling through the Green remains under the maximum energy he can spend.If such a order of cities does not exist print "-1", else print the minimum distance required to be travelled by Swamp Thing.Note that Swamp Thing always starts from his swamp whose index is 1 in the input.Constraints:1 <= n <= 141 <= m <= n * (n - 1) / 21 <= E <= 1001 <= dist between cities <= 100000Example:INPUT FORMAT:Each input file with a line containing three integers, n - the number of cities, m - the number of roads between cities through the Green and E - the amount of energy Swamp Thing can spend.Then m lines follow, each describing a road. Each road is described in the following format:u v d e - this represents a bidirectional road from u to v having a distance of d and energy spent will be e.OUTPUT FORMAT:Print the minimum distance required to be traveled by Swamp Thing to visit all cities while spending energy <= E. If no such possible path exists, print -1.SAMPLE INPUT3 3 251 2 4 202 3 1 201 3 10 5SAMPLE OUTPUT11Explanation:Test #1: Since Swamp Thing has only 25 energy, his only choice is to go from 1 to 3 to 2, the sum of distances of which is 11Public Test Cases:# INPUT EXPECTED OUTPUT1 3 3 25 1 2 4 20 2 3 1 20 1 3 10 511
Question
Swamp Thing feels a disturbance in the Green. It seems that Anton Arcane has once again taken over the Rot and is spreading its evil all over the world. A lot of the world's capital cities are feeling the power of the Rot right now. It is Swamp Thing's duty to preserve a balance between the forces of the Green and the Rot. He must travel to each of the cities affected by the Rot as soon as possible. Luckily, he can use the Green's help to travel to cities through special bidirectional roads. However using the Green to travel from one city to another requires a certain amount of energy. Swamp Thing can only expend a certain amount of energy while travelling.Swamp Thing wants to travel to all of the world's cities in such an order that the sum of the distances he travels is minimized while the total energy he spends travelling through the Green remains under the maximum energy he can spend.If such a order of cities does not exist print "-1", else print the minimum distance required to be travelled by Swamp Thing.Note that Swamp Thing always starts from his swamp whose index is 1 in the input.Constraints:1 <= n <= 141 <= m <= n * (n - 1) / 21 <= E <= 1001 <= dist between cities <= 100000Example:INPUT FORMAT:Each input file with a line containing three integers, n - the number of cities, m - the number of roads between cities through the Green and E - the amount of energy Swamp Thing can spend.Then m lines follow, each describing a road. Each road is described in the following format:u v d e - this represents a bidirectional road from u to v having a distance of d and energy spent will be e.OUTPUT FORMAT:Print the minimum distance required to be traveled by Swamp Thing to visit all cities while spending energy <= E. If no such possible path exists, print -1.SAMPLE INPUT3 3 251 2 4 202 3 1 201 3 10 5SAMPLE OUTPUT11Explanation:Test #1: Since Swamp Thing has only 25 energy, his only choice is to go from 1 to 3 to 2, the sum of distances of which is 11Public Test Cases:# INPUT EXPECTED OUTPUT1 3 3 25 1 2 4 20 2 3 1 20 1 3 10 511
Solution
This problem is a variant of the Traveling Salesman Problem (TSP) with an additional constraint of energy. The TSP is a classic algorithmic problem in the field of computer science and operations research, focusing on optimization. In this problem, Swamp Thing wants to minimize the total distance traveled while also ensuring that the total energy spent does not exceed a certain limit.
Here are the steps to solve this problem:
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Create a graph where the cities are the nodes and the roads are the edges. The weight of each edge is the distance between the cities it connects, and each edge also has an associated energy cost.
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Use a modified version of the TSP algorithm to find the shortest path that visits all cities and does not exceed the maximum energy. This can be done using dynamic programming.
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The dynamic programming solution would involve creating a table where the rows represent the remaining energy and the columns represent the cities. The cell at the intersection of a row and a column would represent the minimum distance to reach that city with the remaining energy.
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Start filling the table from the bottom right corner, which represents the starting city with maximum energy. For each cell, calculate the minimum distance by considering all cities that can be reached from the current city with the remaining energy. Choose the city that results in the minimum distance.
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Continue this process until the table is completely filled. The cell at the top left corner of the table will represent the minimum distance to travel to all cities without exceeding the maximum energy.
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If the minimum distance is within the given constraints, print the minimum distance. If no such path exists that satisfies the constraints, print -1.
This solution assumes that the energy cost to travel between any two cities is the same in both directions, which may not be the case in a real-world scenario.
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