slalom_skiing.py

# -*- coding: utf-8 -*-
"""
Author: Niall O'Connor

# https://app.codility.com/programmers/lessons/90-tasks_from_indeed_prime_2015_challenge/slalom_skiing/



You are a skier participating in a giant slalom. The slalom track is located on 
a ski slope, goes downhill and is fenced by barriers on both sides. The barriers 
are perpendicular to the starting line located at the top of the slope. There 
are N slalom gates on the track. Each gate is placed at a distinct distance from 
the starting line and from the barrier on the right-hand side (looking downhill).

You start from any place on the starting line, ski down the track passing as 
many gates as possible, and finish the slalom at the bottom of the slope. 
Passing a gate means skiing through the position of the gate.

You can ski downhill in either of two directions: to the left or to the right. 
When you ski to the left, you pass gates of increasing distances from the right 
barrier, and when you ski to the right, you pass gates of decreasing distances 
from the right barrier. You want to ski to the left at the beginning.

Unfortunately, changing direction (left to right or vice versa) is exhausting, 
so you have decided to change direction at most two times during your ride. 
Because of this, you have allowed yourself to miss some of the gates on the way 
down the slope. You would like to know the maximum number of gates that you can 
pass with at most two changes of direction.

The arrangement of the gates is given as an array A consisting of N integers, 
whose elements specify the positions of the gates: gate K (for 0 ≤ K < N) is at 
a distance of K+1 from the starting line, and at a distance of A[K] from the 
right barrier.

For example, consider array A such that:
  A[0] = 15
  A[1] = 13
  A[2] = 5
  A[3] = 7
  A[4] = 4
  A[5] = 10
  A[6] = 12
  A[7] = 8
  A[8] = 2
  A[9] = 11
  A[10] = 6
  A[11] = 9
  A[12] = 3

The picture above illustrates the example track with N = 13 gates and a course 
that passes eight gates. After starting, you ski to the left (from your own 
perspective). You pass gates 2, 3, 5, 6 and then change direction to the right. 
After that you pass gates 7, 8 and then change direction to the left. Finally, 
you pass gates 10, 11 and finish the slalom. There is no possible way of passing 
more gates using at most two changes of direction.

Write a function:

    def solution(A)

that, given an array A consisting of N integers, describing the positions of the 
gates on the track, returns the maximum number of gates that you can pass during 
one ski run.

For example, given the above data, the function should return 8, as explained 
above.

For the following array A consisting of N = 2 elements:
  A[0] = 1
  A[1] = 5

the function should return 2.

Write an efficient algorithm for the following assumptions:

        N is an integer within the range [1..100,000];
        each element of array A is an integer within the range [1..1,000,000,000];
        the elements of A are all distinct.

69%  solution https://app.codility.com/demo/results/trainingXCYKCB-J67/
100% solution https://app.codility.com/demo/results/trainingT58YJB-DWW/
"""

import time

def binary_search(L, target):
    """
    The point at which the target should be inserted is returned.
    """
    lower = 0
    upper = len(L)-1
    while lower <= upper:
        mid = (lower + upper) // 2
        if target > L[mid]:
            lower = mid + 1
        else: # target < L[mid]
            upper = mid - 1
    return lower 
            


def LIS(A):
    lis = []
    for num in A:
        # determine where the number should be inserted in the lis.
        insert_idx = binary_search(lis, num)
        if insert_idx < len(lis):
            # If the number less than any encountered number then it:
            # 1. Replaces a previous part of the LIS
            # 2. Is a new starting point for the Array.
            # Because we are not tasked with only finding the LEN of the LIS,
            # it is safe to over write a previous value as long as it doesn't
            # alter the list length incorrectly.
            lis[insert_idx] = num
        else:
            # In the case of a new current max value, append it and increase
            # the length of the lis.
            lis.append(num)
    return len(lis)

def solution(A):
    """
    A mirror represents a turn on the slope.
    We are permitted a max of two turns on this problem.

    VITAL!! You must start in the left direction.
    """
    mirror_limit = max(A)
    total_mirrors = []
    # Because we are travelling left it is essential that the original mirror is last.
    for i in A:
        total_mirrors += [
            (mirror_limit * 2) + i, # second mirror
            (mirror_limit * 2) - i, # first mirror
            i,
        ]
    return LIS(total_mirrors)

if __name__ == '__main__':
    tests = (
        #( expected, args )
        (7, ([9, 5, 6, 3, 4, 10, 4, 7, 8, 9],)),
        (8, ( [15, 13, 5, 7, 4, 10, 12, 8, 2, 11, 6, 9, 3],)),
    )
    for expected, args in tests:
        tic = time.perf_counter()
        res = solution(*args)
        toc = time.perf_counter()
        
        if expected is None:
            print(f'SPEED-TEST {len(args[0])} args finished in {toc - tic:0.8f} seconds')
            continue # This is just a speed test
        print(f'ARGS produced {res} in {toc - tic:0.8f} seconds')
        try:
            assert(expected == res)
        except AssertionError as e:
            print(f'ERROR {args} produced {res} when {expected} was expected!\n')