# 算法代写｜COMP3027 & COMP3927 Algorithm Design Assignment 3

Problem 1. (20 points)

To begin, let us consider the Ford-Fulkerson algorithm. Consider below the flow

network G,c in Figure 1 and the current s−t flow f in Figure 2. Your task is to run one iteration of the Ford-Fulkerson algorithm on the flow network.

1. a)  Draw the residual graph associated with the flow f. Specify the residual capacity of every edge. You are allowed to draw this by hand and insert a photo (Alternatively, there is a LaTeX template on the last two pages of this assignment that you can use).
2. b)  Specify an s − t path used to update the flow.
3. c)  After the update, is the new flow through the network a max flow? Explain

why.

4. d)  Specify a Min-Cut associated with graph G.

Problem 2. (80 points) (COMP3027 only)

You and your friends have decided to go on a great go-karting adventure. You’ve

heard that there is to be a symposium for algorithms and computing theory (SACT for short) in the bush (with music!), and since your good friend André owns k go-karts, you decide that the most fun way to get there is via an epic four wheel journey. There is only one issue – the go-karts can only go so far before they need to be refueled, meaning you need to design a trip that allows for picking up fuel every 10km. Even worse, you have realised that all these stops only have enough fuel for one go-kart, meaning that each individual go-kart will need to refuel at different stops. Formally, the SACT Go-Kart Problem is the following; given a positive integer k, n (2D) Cartesian points describing fuel stations and both a starting point S and end point T in the Cartesian plane, determine if k friends have valid paths from S to T. Note that a valid path involves stopping at a fuel station every 10km or less.

Figure 3: Map including go-karts start location, fuel stations, and the bush sympo- sium. A go-kart cannot travel 10km without visiting a fuel station.

As a keen algorithm’s student (after all, you’re trying to get to an algorithms conference!), you realise that the SACT Go-Kart Problem can potentially be solved through the use of flow network algorithms!

Figure 4: Map showing paths the go-karts could take to get to the symposium. Note that a fuel station can only be visited by one car, making three the highest number of paths from the starting location to the bush symposium.

Your task is to design a reduction from the SACT Go-Kart Problem above to an instance of Max Flow. For full marks, your reduction should be as fast as possible.

1. a)  Describe how you translate an instance of the SACT Go-Kart Problem into an instance of Max Flow. In particular, you need to describe how you construct the flow network H based on the input of fuel stations, start and end points.
2. b)  State which algorithm you use to compute the max flow in H.
3. c)  Describe how you translate the output of the algorithm in (b) into a solution for the SACT Go-Kart Problem. In particular, describe how you determine if all k people are able to make the bush conference, and what their paths would be.
4. d)  Prove the correctness of your reduction. In particular, you need to prove that there exists an integer F for which the following two statements hold:

• If the max flow is at least F, then your translation procedure in (c) pro- duces feasible paths for all k people.

• If there are feasible paths for all k people, then the max flow is at least F.

5. e)  Prove the time complexity of your entire reduction, taking into account the steps in parts (a), (b) and (c). Make sure that your time complexity is stated in terms of n and k.