Question

Give a dynamic programming algorithm that runs within the time complexity. Also give the space complexity of the algorithm. Please

Given a directed graph with non-negative integer edge weights, a pair of vertices s and t, and integers K and W, describe a dynamic-programming algorithm for deciding whether there exists a path from s to t that has total weight W and uses exactly K edges. Your algorithm should run in time O(nm)WK). Analyze the time- and space-complexity of your solution. Hint: You will have to define a three-dimensional table for the recurrence relation.

Answer 1

Answer:

et total number of vertices be n and total number of edges be m.

Let dp[n][n][K] be the dynamic programming array where dp[i][j][p] stores the weight of the path between vertex i and vertex j using exactly p edges.

Initially, dp[i][j][p] = infinity for all i, j, p

for i in 1 to n:

for j in 1 to n:

for p in 1 to K:

if (exists path between i and j using p vertices):

dp[i][j][p] = path_weight(i, j)

if(i == s and j == t and dp[i][j][p] == W) return true;

return false

Time complexity == O((n+m)WK)

Space complexity = O(n^2. K)

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