## Thursday, November 3, 2016

### Optimum location of point to minimize total distance - GeeksforGeeks

Optimum location of point to minimize total distance - GeeksforGeeks
Given a set of points as and a line as ax+by+c = 0. We need to find a point on given line for which sum of distances from given set of points is minimum.

If we take one point on given line at infinite distance then total distance cost will be infinite, now when we move this point on line towards given points the total distance cost starts decreasing and after some time, it again starts increasing which reached to infinite on the other infinite end of line so distance cost curve looks like a U-curve and we have to find the bottom value of this U-curve.
As U-curve is not monotonically increasing or decreasing we can’t use binary search for finding bottom most point, here we will use ternary search for finding bottom most point, ternary search skips one third of search space at each iteration, you can read more about ternary search here.
So solution proceeds as follows, we start with low and high initialized as some smallest and largest values respectively, then we start iteration, in each iteration we calculate two mids, mid1 and mid2, which represent 1/3rd and 2/3rd position in search space, we calculate total distance of all points with mid1 and mid2 and update low or high by comparing these distance cost, this iteration continues untill low and high become approximately equal.
`//  structure defining a point`
`struct` `point`
`{`
`    ``int` `x, y;`
`    ``point()  {}`
`    ``point(``int` `x, ``int` `y) : x(x), y(y)   {}`
`};`

`//  structure defining a line of ax + by + c = 0 form`
`struct` `line`
`{`
`    ``int` `a, b, c;`
`    ``line(``int` `a, ``int` `b, ``int` `c) : a(a), b(b), c(c) {}`
`};`

`//  method to get distance of point (x, y) from point p`
`double` `dist(``double` `x, ``double` `y, point p)`
`{`
`    ``return` `sqrt``(sq(x - p.x) + sq(y - p.y));`
`}`

`/*  Utility method to compute total distance all points`
`    ``when choose point on given line has x-cordinate`
`    ``value as X   */`
`double` `compute(point p[], ``int` `n, line l, ``double` `X)`
`{`
`    ``double` `res = 0;`

`    ``//  calculating Y of choosen point by line equation`
`    ``double` `Y = -1 * (l.c + l.a*X) / l.b;`
`    ``for` `(``int` `i = 0; i < n; i++)`
`        ``res += dist(X, Y, p[i]);`

`    ``return` `res;`
`}`

`//  Utility method to find minimum total distance`
`double` `findOptimumCostUtil(point p[], ``int` `n, line l)`
`{`
`    ``double` `low = -1e6;`
`    ``double` `high = 1e6;`

`    ``// loop untill difference between low and high`
`    ``// become less than EPS`
`    ``while` `((high - low) > EPS)`
`    ``{`
`        ``// mid1 and mid2 are representative x co-ordiantes`
`        ``// of search space`
`        ``double` `mid1 = low + (high - low) / 3;`
`        ``double` `mid2 = high - (high - low) / 3;`

`        ``//`
`        ``double` `dist1 = compute(p, n, l, mid1);`
`        ``double` `dist2 = compute(p, n, l, mid2);`

`        ``// if mid2 point gives more total distance,`
`        ``// skip third part`
`        ``if` `(dist1 < dist2)`
`            ``high = mid2;`

`        ``// if mid1 point gives more total distance,`
`        ``// skip first part`
`        ``else`
`            ``low = mid1;`
`    ``}`

`    ``// compute optimum distance cost by sending average`
`    ``// of low and high as X`
`    ``return` `compute(p, n, l, (low + high) / 2);`
`}`

`//  method to find optimum cost`
`double` `findOptimumCost(``int` `points[N][2], line l)`
`{`
`    ``point p[N];`

`    ``//  converting 2D array input to point array`
`    ``for` `(``int` `i = 0; i < N; i++)`
`        ``p[i] = point(points[i][0], points[i][1]);`

`    ``return` `findOptimumCostUtil(p, N, l);`
`}`

https://en.wikipedia.org/wiki/Ternary_search

```def ternarySearch(f, left, right, absolutePrecision):
'''
left and right are the current bounds;
the maximum is between them
'''
if abs(right - left) < absolutePrecision:
return (left + right)/2

leftThird = (2*left + right)/3
rightThird = (left + 2*right)/3

if f(leftThird) < f(rightThird):
return ternarySearch(f, leftThird, right, absolutePrecision)
else:
return ternarySearch(f, left, rightThird, absolutePrecision)```

```def ternarySearch(f, left, right, absolutePrecision):
"""
Find maximum of unimodal function f() within [left, right]
To find the minimum, revert the if/else statement or revert the comparison.
"""
while True:
#left and right are the current bounds; the maximum is between them
if abs(right - left) < absolutePrecision:
return (left + right)/2

leftThird = left + (right - left)/3
rightThird = right - (right - left)/3

if f(leftThird) < f(rightThird):
left = leftThird
else:
right = rightThird
```

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