Codility ‘NumberOfDiscIntersections’ 2010 Beta Solution

Short Problem Definition:

Compute intersections between sequence of discs.

Given an array A of N integers, we draw N discs in a 2D plane such that the I-th disc is centered on (0,I) and has a radius of A[I]. We say that the J-th disc and K-th disc intersect if J ≠ K and J-th and K-th discs have at least one common point.

Link

Number of Disc Intersections

Complexity:

expected worst-case time complexity is O(N*log(N))

expected worst-case space complexity is O(N)

Execution:

In my mind this problem is very similar to the Codility Fish. You keep count of how many not-yet-ended disks there are. Every beginning has to cross all not-yet-ended disks. The best explanation on the web was written by Luca. His solution is much better than my C-style double-checked while loop. I therefore created a mixture of both 🙂

  • Keep in mind that any pair of circles can intersect only once.
  • If 1+ beginnings have the same point as 1+ endings. You first have to process the beginnings. The old and new ones have per definition a meeting point.
Solution:
def solution(A):
    circle_endpoints = []
    for i, a in enumerate(A):
        circle_endpoints += [(i-a, True), (i+a, False)]

    circle_endpoints.sort(key=lambda x: (x[0], not x[1]))

    intersections, active_circles = 0, 0

    for _, is_beginning in circle_endpoints:
        if is_beginning:
            intersections += active_circles
            active_circles += 1
        else:
            active_circles -= 1
        if intersections > 10E6:
            return -1

    return intersections

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  • João Prudêncio

    Isn’t the time complexity here nlogn + n ?
    Since nlogn is for sorting an you also iterate through all the array.

  • BruceFromSeattle

    I submit 2 solutions (in C#, sorry) which are 100% in agreement and 100% Codility correct; 2nd one is 100% Codility performant.

    static int solution(int[] A) // time O(N^2), space O(1)
    { // simple-minded way: check if each (unordered) pair is intersecting
    int n = A.Length;
    int numIntersecs = 0;
    for (long i = 0; i < n – 1; i++)
    for (long j = i + 1; j < n; j++)
    if (i – A[i] <= j + A[j] && j – A[j] <= i + A[i]) //just touching is intersection
    if (numIntersecs == 10E6)
    return -1;
    else
    numIntersecs++;
    return numIntersecs;
    }
    static int solution(int[] A) // time O(N*Log(N)), space O(N)
    { // decrement max number of pairs by number of non-intersecting pairs
    long n = A.Length;
    if (n < 2)
    return 0;
    long numIntersecs = n * (n – 1) / 2; // unordered pairs, initialized to max possible
    long[] hiVals = new long[n];
    long[] loVals = new long[n];
    for (long i = 0; i < n; i++)
    {
    hiVals[i] = i + A[i]; // high value of disk edge (along x-axis)
    loVals[i] = i – A[i]; // low value of disk edge (along x-axis)
    }
    Array.Sort(hiVals);
    Array.Sort(loVals);
    int jLo = 0; // initialize inner iterator only once
    for (int iHi = 0; iHi < n; iHi++)
    {
    for ( ; jLo hiVals[iHi]) // disks don’t intersect if one’s lo > other’s hi
    {
    numIntersecs -= n – jLo; // decrement by the num of lo values > this hi value
    break; // don’t increment iterator, check this low value again next time
    }
    }
    if (jLo == n)
    break; // no more low values > high values
    }
    if (numIntersecs > 10E6)
    return -1;
    else
    return (int) numIntersecs;
    }