Given an array, find all its elements that can become a leader

Question

I was successful in solving a challenge in codility, but my solution failed performance tests. How can I improve my solution?

Challenge:

Integers K, M and a non-empty array A consisting of N integers, not bigger than M, are given.

The leader of the array is a value that occurs in more than half of the elements of the array, and the segment of the array is a sequence of consecutive elements of the array.

You can modify A by choosing exactly one segment of length K and increasing by 1 every element within that segment.

The goal is to find all of the numbers that may become a leader after performing exactly one array modification as described above.

Write a function:

def solution(K, M, A)

that, given integers K and M and an array A consisting of N integers, returns an array of all numbers that can become a leader, after increasing by 1 every element of exactly one segment of A of length K. The returned array should be sorted in ascending order, and if there is no number that can become a leader, you should return an empty array. Moreover, if there are multiple ways of choosing a segment to turn some number into a leader, then this particular number should appear in an output array only once.

For example, given integers K = 3, M = 5 and the following array A:

A[0] = 2
A[1] = 1
A[2] = 3
A[3] = 1
A[4] = 2
A[5] = 2
A[6] = 3

the function should return [2, 3]. If we choose segment A[1], A[2], A[3] then we get the following array A:

A[0] = 2
A[1] = 2
A[2] = 4
A[3] = 2
A[4] = 2
A[5] = 2
A[6] = 3

and 2 is the leader of this array. If we choose A[3], A[4], A[5] then A will appear as follows:

A[0] = 2
A[1] = 1
A[2] = 3
A[3] = 2
A[4] = 3
A[5] = 3
A[6] = 3

and 3 will be the leader.

And, for example, given integers K = 4, M = 2 and the following array:

A[0] = 1
A[1] = 2
A[2] = 2
A[3] = 1
A[4] = 2

the function should return [2, 3], because choosing a segment A[0], A[1], A[2], A[3] and A[1], A[2], A[3], A[4] turns 2 and 3 into the leaders, respectively.

Write an efficient algorithm for the following assumptions:

• N and M are integers within the range [1..100,000];
• K is an integer within the range [1..N];
• Each element of array A is an integer within the range [1..M].

My Solution

def modify(segment):
   return [e+1 for e in segment]

def dominant(A):
   d = dict()
   lenOfHalfA = int(len(A)/2)
   domList = []

   for i in A:
       if not i in d:
           d[i] = 1
       else:
           d[i] = d[i]+1

   for key, value in d.items():
       if value > lenOfHalfA:
           domList.append(key)

   return domList


def solution(K, M, A):
   # write your code in Python 3.6
   dominantList = []

   x = 0
   while x <= len(A) - K:
       modifiedA = A[:]
       modifiedA[x:K+x] = modify(A[x:K+x])
       dominantList += dominant(modifiedA)
       x += 1

   return list(set(dominantList))

Answer 1

You can speed some things up by using more list and dict comprehensions, and by reducing your function calls. Some examples follow.

d = dict() vs d = {}: 131 ns vs 30 ns.

len_of_half_a = int(len(a)/2) vs len_of_half_a = len(a)//2: 201 ns vs 99 ns.

I used Python 3.8.1 for both tests.

Granted that this isn't much, but several of these tiny improvements could help you reach the target. You should see similar if not better performance increases by using list and dict comprehensions, e.g. for domList, d, and dominantList. And replacing your while x <= len(A) - K: with a range-based iterator should bump you up a little more.

And a final small note is that you should try to follow standards with your naming and ensure clear and obvious names. A is not a good name for a variable, nor is d, and python tends to use snake_case over camelCase.

Answer 2

Algorithm in O(N^2)

The current solution will be slow for large N, because it has a complexity of O(N^2) (It checks every element in the array for every possible position of the k adjusted elements => O(N * (N-K)) => O(N^2)).

There is an O(N) solution.

Consider that as the K-element segment "slides" along the array, there are only two elements whose value changes: the one entering and the one leaving the segment.

Something like:

def solution(K, M, A):
    threshold = len(A)//2
    counts = Counter(A)

    # increment count for each element in the initial window
    for head in A[:K]:
        counts[head] -= 1
        counts[head+1] += 1

    # initial leaders
    leaders = set(k for k,v in counts.items() if v > threshold)

    # slide window along the array adjusting the counts for 
    # elements that leave (tail) or enter (head) the window.
    # An element entering gets incremented, so 
    # count[element] -= 1 and count[element+1] += 1.
    # It is the reverse for element leaving.
    for tail, head in zip(A, A[K:]):
        counts[tail] += 1
        counts[tail + 1] -= 1

        counts[head] -= 1
        counts[head + 1] += 1

        if counts[tail] > threshold:
            leaders.add(tail)

        if counts[head + 1] > threshold:
            leaders.add(head + 1)

    return leaders

Answer 3

def modify(segment):
   return [e+1 for e in segment]

This function is used only in one place, and is only one line. That often means it's better to just inline it:

modifiedA[x:K+x] = modify(A[x:K+x])
# Becomes:
modifiedA[x:K+x] = [e+1 for e in A[x:K+x]]

Use meaningful variable names. No matter what your challenge says, K, M and A are not meaningful variable names. It also seems like you're not doing anything with that M, so why do we even pass it to the function?

In your dominant() function, you look like you want to use collections.Counter. For practice with a language it can be good to check how to set it up yourself, but sometimes we have a good solution ready-made and available, and much better tested than any individual developer could ever do.

With Counter, you can make it work like this:

from collections import Counter

def dominant(string):
    count = Counter(string)
    return [key for key, value in count.items() if value == count.most_common(1)[0][1]]

Yeah, it's as simple as that. You could split it out over multiple lines, for clarity:

from collections import Counter

def dominant(string):
    count = Counter(string)
    dom_list = []
    max_occurred = count.most_common(1)[0][1]
    for key, value in count.items():
        if value >= max_occurred:
            dom_list.append(key)
    return dom_list