Rolling median solution

Question

I was implementing a rolling median solution and was not sure why my Python implementation was around 40 times slower than my C++ implementation.

Here are the complete implementations:

C++

#include <iostream>
#include <vector>
#include <string.h>
using namespace std;

int tree[17][65536];

void insert(int x) { for (int i=0; i<17; i++) { tree[i][x]++; x/=2; } }
void erase(int x) { for (int i=0; i<17; i++) { tree[i][x]--; x/=2; } }
int kThElement(int k) {
    int a=0, b=16;
    while (b--) { a*=2; if (tree[b][a]<k) k-=tree[b][a++]; }
    return a;
}

long long sumOfMedians(int seed, int mul, int add, int N, int K) {
    long long result = 0;
    memset(tree, 0, sizeof(tree));
    vector<long long> temperatures;
    temperatures.push_back( seed );
    for (int i=1; i<N; i++)
      temperatures.push_back( ( temperatures.back()*mul+add ) % 65536 );
    for (int i=0; i<N; i++) {
      insert(temperatures[i]);
      if (i>=K) erase(temperatures[i-K]);
      if (i>=K-1) result += kThElement( (K+1)/2 );
    }
    return result;
}

// default input
// 47 5621 1 125000 1700
// output
// 4040137193

int main()
{   
  int seed,mul,add,N,K;
  cin >> seed >> mul >> add >> N >> K;
  cout << sumOfMedians(seed,mul,add,N,K)  << endl;
  return 0;
}

Python

    def insert(tree,levels,n):
            for i in xrange(levels):
                    tree[i][n] += 1
                    n /= 2
    def delete(tree,levels,n):
            for i in xrange(levels):
                    tree[i][n] -= 1
                    n /= 2
    
    def kthElem(tree,levels,k):
            a = 0
            for b in reversed(xrange(levels)):
                    a *= 2
                    if tree[b][a] < k:
                            k -= tree[b][a]
                            a += 1
            return a
    
    def main():
            seed,mul,add,N,K = map(int,raw_input().split())
            levels = 17
            tree = [[0] * 65536 for _ in xrange(levels)]
            temps = [0] * N
            temps[0] = seed
            for i in xrange(1,N):
                    temps[i] = (temps[i-1]*mul + add) % 65536
            result = 0
            for i in xrange(N):
                    insert(tree,levels,temps[i])
                    if (i >= K):
                            delete(tree,levels,temps[i-K])              
                    if (i >= K-1):
                            result += kthElem(tree,levels,((K+1)/2))
    
            print result
    
    # default input
    # 47 5621 1 125000 1700
    # output
    # 4040137193
    main()

On the above mentioned input (in the comments of the code), the C++ code took around 0.06 seconds while Python took around 2.3 seconds.

Can some one suggest the possible problems with my Python code and how to improve to less than 10x performance hit?

I don't expect it to be anywhere near C++ implementation but to the order of 5-10x. I know I can optimize this by using libraries like NumPy (and/or SciPy). I am asking this question from the point of view of using Python for solving programming challenges. These libraries are usually not allowed in these challenges. I am just asking if it is even possible to beat the time limit for this algorithm in Python.

In case somebody is interested, the C++ code is borrowed from floating median problem here.

Answer 1

Integer math in tight loops is an area where C++ will definitely run circles around Python. I think your expectations are too high. However, I'd like to point out a couple of things for you, even if they are not enough to reach your goal.

Avoid indexing when you can use direct iteration instead. For example, insert can be written like this and Python won't have to deal with i:

def insert(tree,n):
     for level in tree:
         level[n] += 1
         n /= 2

Python does not optimize constant subexpressions out of loops, so you'll want to compute (K+1)/2 before the main loop and store it in a variable.

Answer 2

A 40x speed difference between C++ and Python for this sort of work is not uncommon. Implementing the optimizations by @JanneKarilla helps. Additionally you can ommit both if statements by looping in two segments, first from 0 to K, then from K to N, which shaves off a fair amount of time.

The fastest that I can come up with uses the "maintain a sorted array" approach and is about 4.5 x faster than your original code for the input given:

from collections import deque
from bisect import bisect_left, insort

def rolling_median(iterable, K):
    # taking iterator so it works on sequences and generators
    it = iter(iterable)

    # a deque has optimized access from its endpoints
    # we consume and store the first K values
    deq = deque(next(it) for _ in xrange(K))

    # initialize the sorted array
    sor = sorted(deq)

    # index of median in sorted list
    i = (K+1)//2 -1

    yield sor[i]
    for r in it:
        # `deq` keeps track of chrological order
        out = deq.popleft()
        deq.append(r)
        # `sor` keeps track of sortedness
        sor.pop(bisect_left(sor, out))
        insort(sor, r)
        yield sor[i]

def RNG(seed, mul, add, N):
    tk = seed
    for _ in xrange(N):
        yield tk
        tk = (tk * mul + add) % 65536

def main(seed=47, mul=5621, add=1, N=125000, K=1700):
    rand = RNG(seed, mul, add, N)
    return sum(rolling_median(rand, K))

It's possible to shave off some more milisec I'm sure, but I wanted to write a nice and general rolling median function :)