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115CS 170 Efficient Algorithms and Intractable Problems Project 算法课业代做 Problem Statement Penguins are the last remaining species on Earth and they must figure out how to defend themselve...
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算法课业作业代写 Instructions The solutions should be typed, using proper mathematical notation. We cannot accept hand-written solutions. Here’s a short intro to
Consider the undirected, unweighted graph G = (V, E) with V = {s, u, v, w, x} and E = {su, sv, sw, uw, ux, vw}, and let T ⊆ E be T = {sv, sw, uw, ux}. This is pictured below with T represented by wide edges.
Carefully explain why T cannot be output by BFS with start vertex s for any choices of iteration order over neighborhoods in the algorithm.
Suppose we are given a finite, simple, connected, and weighted graph G(V, E, w), where the edge weights are non-negative. Suppose that the length of a path P is the product of the edge weights along P. Fix vertices s, t. Our goal remains to find a shortest path from s to t.
(a) Suppose we construct a new graph H(V, E, w’) that is identical to G, with the exception that w’ ({x, y}) = log(w({x, y})). That is, V (H) = V (G) and E(H) = E(G). So we have the same underlying graph, with the only difference being the edge weights. You may take as fact that P is a shortest s to t path in G if and only if P is a shortest s to t path in H.
Suppose now that we run Dijkstra’s algorithm on H, in order to fifind a shortest path from s to t in G. Is this approach valid? Justify your reasoning.
(b) Suppose now that the edge weights of G are all positive. That is, w({x, y}) > 0 for all edges {x, y} ∈ E(G). Let H(V, E, w’) be the graph corresponding to G, as defined in part (a). Is it now a valid approach to run Dijkstra’s algorithm on H, in order to find a shortest path from s to t in G? Justify your reasoning.
(c) Give conditions on the edge weights of G, so that it suffices to run Dijkstra’s algorithm on H, in order to find a shortest path from s to t in G. Clearly explain why your conditions are correct.
Suppose you want to drive from Town A to Town B along some fixed route of distance d, and you have a gas tank whose capacity will take you at most m miles along the route. Let 0 < d_{1} < d_{2} < … < d_{n}< d be the distances of each gas station along the route from Town A. (So the distance from A to gas station 2 is d_{2}; the distance between the first two gas stations is d_{2} − d_{1}.) Your goal is to get from A to B (equivalently, distance 0 to distance d) (a) without running out of gas, and (b) stopping as few times as possible.
The natural strategy most people use is a greedy one: go as far as you can, but refuel at gas station i if you don’t have enough to get you to gas station i+ 1. This strategy indeed minimizes the number of stops you need to make.
Consider a different greedy strategy, in which you stop at the nearest available gas station. Give an example (specify d, m, and the distances between the gas stations) showing that this strategy is not optimal. Show what this greedy algorithm does on your example, which subset of gas stations it outputs, and exhibit a strictly smaller set of gas stations that would still allow you to successfully complete the trip.
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