Suppose we have a list of positions, which is containing a list of coordinate points where some houses are located. If you want to make a service center at (xc, yc) such that the sum of Euclidean distance from any given point to (xc, yc) is minimum. So we have to find the sum of minimum distance.
So, if the input is like positions = [(10,11),(11,10),(11,12),(12,11)], then the output will be 4.0
To solve this, we will follow these steps −
numIter := 50
Define a function total() . This will take cx, cy, positions
total := 0.0
for each p in positions, do
x, y := p
total := total + Euclidean distance between (cx, cy) and (x, y)
return total
Define a function fy() . This will take x, positions
l := 0, r := 101
res := 0
for i in range 0 to numIter, do
y1 := l + (r - l) / 3
y2 := r - (r - l) / 3
t1 := total(x, y1, positions)
t2 := total(x, y2, positions)
res := minimum of t1 and t2
if t1 < t2, then
r := y2
otherwise,
l := y1
return res
Define a function fx() . This will take positions
l := 0, r := 101
res := 0
for i in range 0 to numIter, do
x1 := l + (r - l) / 3
x2 := r - (r - l) / 3
t1 := fy(x1, positions)
t2 := fy(x2, positions)
res := minimum of t1 and t2
if t1 < t2, then
r := x2
otherwise,
l := x1
return res
From the main method, return fx(positions)
Example
Let us see the following implementation to get better understanding
from math import sqrt
def solve(points):
numIter = 50
def total(cx, cy, positions):
total = 0.0
for p in positions:
x, y = p
total += sqrt((cx - x) * (cx - x) + (cy - y) * (cy - y))
return total
def fy(x, positions):
l, r = 0, 101
res = 0
for i in range(numIter):
y1 = l + (r - l) / 3
y2 = r - (r - l) / 3
t1 = total(x, y1, positions)
t2 = total(x, y2, positions)
res = min(t1, t2)
if t1 < t2:
r = y2
else:
l = y1
return res
def fx(positions):
l, r = 0, 101
res = 0
for i in range(numIter):
x1 = l + (r - l) / 3
x2 = r - (r - l) / 3
t1 = fy(x1, positions)
t2 = fy(x2, positions)
res = min(t1, t2)
if t1 < t2:
r = x2
else:
l = x1
return res
return fx(positions)
positions = [(10,11),(11,10),(11,12),(12,11)]
print(solve(positions))Input
[(10,11),(11,10),(11,12),(12,11)]
Output
4.0