Computer Engineering(CE)

Question 1: Algorithms

August 2013

Problem 2.

Suppose your company develops and manages construction of boat launching docks along a downstream stretch of the Wabash river. This stretch runs north-south for $L$ miles within the State of Indiana. The possible sites for docks are given by numbers $x_1 < x_2 < x_3 < ... < x_n$, each in the interval $[0,L]$, specifying their positions in miles measured from the northern end of this stretch of the Wabash river. If your company constructs a dock at position $x_i$, it receives a revenue of $r_i >0$. Regulations imposed by the Indiana Department of Water Resource Management require that no two docks should be built within a distance of less than 5 miles from each other. Your company plans to construct docks at a subset of the potential sites so as to maximize the total revenue., subject to this distance restriction. For example, suppose $L=20$ and $n=5$ with potential sites given by $\lbrace x_1, x_2, x_3, x_4, x_5 \rbrace = \lbrace 6,7,12,13,14\rbrace$ and $\lbrace r_1, r_2, r_3, r_4, r_5 \rbrace = \lbrace 5,6,5,3,1\rbrace$. Then the best solution is to construct docks at locations $x_1$ and $x_3$ to achieve a revenue of 10. Describe a dynamic programming formulation to find a solution for this optimization problem. Compute the complexity of solving your dynamic programming formulation of this problem.

Share and discuss your solution below.

### Solution 1

This problem can be solved using dynamic programming. For each dock $x_i$, compute the revenue from $x_1$ to $x_i$, if $x_i$ is selected, and the revenue for remaining docks $x_{i+1}$ to $x_n$, if $x_i$ is not selected.

$R[i]$: denote the total revenue using only sites $x_1, \dots , x_i$.

$L[i]$: denote $i$ with the greatest value such that $x_i$ is used for the solution in $R[i]$.

Initially, $L[<=0]=0$, $R[<=0]=0$, $x_0 = -\infty$, $r_0=0$. The pseudo code for dynamic programming is showing below. Note we use bottom up to fill up $R$ and $L$.

In the end of the program, $R[n]$ will be the maximum revenue, and $L[n]$, $L[L[n]]$, ... will be the indices of locations to choose.

### Solution 2

Sub-problem: after picking one location $q$, need to find the max revenue of the remaining docks whose location is greater than 5 miles away from $q$. Bottom-up-dock-revenue.

Let $r[0...n]$ be a new array, $r=0$. The pseudo code is show below: Using dynamic program and bottom up to fill in the solution is a good approach. For each dock, we need to find the greater of either selecting this dock and have the revenue for this dock, while any other docks within 5 miles will not be selected; or not selecting the dock and having the remaining docks that includes the docks within 5 miles away to achieve their best revenue. In this solution, it is not clear what $p[i]$ is represent for. In addition, the docks what within 5 miles away could be on both sides. This solution only consider one side. 