Dynamic Programming - Highway Billboard Problem | Algorithms
Objective: Suppose you're managing construction of billboards on the Rocky & Bullwinkle Memorial Highway, a heavily traveled stretch of road that runs west-east for M miles. The possible sites for billboards are given by numbers x1 < x2 < · · · < xn, each in the interval [0, M], specifying their position in miles measured from the western end of the road. If you place a billboard at position xi , you receive a revenue of ri > 0.
Regulations imposed by the Highway Department require that no two billboards be within five miles or less of each other. You'd like to place billboards at a subset of the sites so as to maximize your total revenue, subject to this restriction.
Problem Source: https://cgi.csc.liv.ac.uk/~martin/teaching/comp202/Exercises/Dynamic-programming-exercises-solution.pdf
Example:
int[] MR = new int[distance + 1];
//Next billboard which can be used will start from index 0 in billboard[]
int nextBillBoard = 0;
//example if milesRes = 5 miles then any 2 bill boards has to be more than
//5 miles away so actually we can put at 6th mile so we can add one mile milesRes
milesRes = milesRes + 1; // actual minimum distance can be between 2 billboards
MR[0] = 0;
for (int i = 1; i <= distance; i++) {
//check if all the billboards are not already placed
if(nextBillBoard < billboard.length){
//check if we have billboard for that particular mile
//if not then copy the optimal solution from i-1th mile
if (billboard[nextBillBoard] != i) {
//we do not have billboard for this particular mile
MR[i] = MR[i - 1];
} else {
//we do have billboard for this particular mile
//now we have 2 options, either place the billboard or ignore it
//we will choose the optimal solution
if(i>=milesRes){
MR[i] = Math.max(MR[i - milesRes] + revenue[nextBillBoard], MR[i - 1]);
}else{
//there are no billboard placed prior to ith mile
//we will just place the billboard
MR[i] = revenue[nextBillBoard];
}
nextBillBoard++;
}
}else{
//All the billboards are already placed
//for rest of the distance copy the previous optimal solution
MR[i] = MR[i - 1];
}
}
//System.out.println(Arrays.toString(MR));
return MR[distance];
}
Read full article from Dynamic Programming - Highway Billboard Problem | Algorithms
Objective: Suppose you're managing construction of billboards on the Rocky & Bullwinkle Memorial Highway, a heavily traveled stretch of road that runs west-east for M miles. The possible sites for billboards are given by numbers x1 < x2 < · · · < xn, each in the interval [0, M], specifying their position in miles measured from the western end of the road. If you place a billboard at position xi , you receive a revenue of ri > 0.
Regulations imposed by the Highway Department require that no two billboards be within five miles or less of each other. You'd like to place billboards at a subset of the sites so as to maximize your total revenue, subject to this restriction.
Problem Source: https://cgi.csc.liv.ac.uk/~martin/teaching/comp202/Exercises/Dynamic-programming-exercises-solution.pdf
Example:
int[] x = {6, 7, 12, 13, 14}; int[] revenue = {5, 6, 5, 3, 1}; int distance = 20;
int milesRestriction = 5;
Output: Maximum revenue can be generated :10 ( x1 and x3 billboard will be placed)public int maxRevenue(int[] billboard, int[] revenue, int distance, int milesRes) {
int[] MR = new int[distance + 1];
//Next billboard which can be used will start from index 0 in billboard[]
int nextBillBoard = 0;
//example if milesRes = 5 miles then any 2 bill boards has to be more than
//5 miles away so actually we can put at 6th mile so we can add one mile milesRes
milesRes = milesRes + 1; // actual minimum distance can be between 2 billboards
MR[0] = 0;
for (int i = 1; i <= distance; i++) {
//check if all the billboards are not already placed
if(nextBillBoard < billboard.length){
//check if we have billboard for that particular mile
//if not then copy the optimal solution from i-1th mile
if (billboard[nextBillBoard] != i) {
//we do not have billboard for this particular mile
MR[i] = MR[i - 1];
} else {
//we do have billboard for this particular mile
//now we have 2 options, either place the billboard or ignore it
//we will choose the optimal solution
if(i>=milesRes){
MR[i] = Math.max(MR[i - milesRes] + revenue[nextBillBoard], MR[i - 1]);
}else{
//there are no billboard placed prior to ith mile
//we will just place the billboard
MR[i] = revenue[nextBillBoard];
}
nextBillBoard++;
}
}else{
//All the billboards are already placed
//for rest of the distance copy the previous optimal solution
MR[i] = MR[i - 1];
}
}
//System.out.println(Arrays.toString(MR));
return MR[distance];
}
Track the actual solution: To track the actual solution use MR[]. Start traversing from the end towards start. Whenever value changes means billboard has been place at the location. Note the billboard and jump 5 miles back (no two billboards be within five miles or less of each other) and again traverse backwards and so on.
http://www.geeksforgeeks.org/highway-billboard-problem/
Let maxRev[i], 1 <= i <= M, be the maximum revenue generated from beginning to i miles on the highway. Now for each mile on the highway, we need to check whether this mile has the option for any billboard, if not then the maximum revenue generated till that mile would be same as maximum revenue generated till one mile before. But if that mile has the option for billboard then we have 2 options:
1. Either we will place the billboard, ignore the billboard in previous t miles, and add the revenue of the billboard placed.
2. Ignore this billboard. So maxRev[i] = max(maxRev[i-t-1] + revenue[i], maxRev[i-1])
1. Either we will place the billboard, ignore the billboard in previous t miles, and add the revenue of the billboard placed.
2. Ignore this billboard. So maxRev[i] = max(maxRev[i-t-1] + revenue[i], maxRev[i-1])
Time Complexity: O(M), where M is distance of total Highway.
Auxiliary Space: O(M).
Auxiliary Space: O(M).
int
maxRevenue(
int
m,
int
x[],
int
revenue[],
int
n,
int
t)
{
// Array to store maximum revenue at each miles.
int
maxRev[m+1];
memset
(maxRev, 0,
sizeof
(maxRev));
// actual minimum distance between 2 billboards.
int
nxtbb = 0;
for
(
int
i = 1; i <= m; i++)
{
// check if all billboards are already placed.
if
(nxtbb < n)
{
// check if we have billboard for that particular
// mile. If not, copy the previous maximum revenue.
if
(x[nxtbb] != i)
maxRev[i] = maxRev[i-1];
// we do have billboard for this mile.
else
{
// We have 2 options, we either take current
// or we ignore current billboard.
// If current position is less than or equal to
// t, then we can have only one billboard.
if
(i <= t)
maxRev[i] = max(maxRev[i-1], revenue[nxtbb]);
// Else we may have to remove previously placed
// billboard
else
maxRev[i] = max(maxRev[i-t-1]+revenue[nxtbb],
maxRev[i-1]);
nxtbb++;
}
}
else
maxRev[i] = maxRev[i - 1];
}
return
maxRev[m];
}