From the paper that OP links, the following string:

Insurgents killed in ongoing fighting

Yields:

2-skip-bi-grams = {insurgents killed, insurgents in, insurgents ongoing, killed in, killed ongoing, killed fighting, in ongoing, in fighting, ongoing fighting}

2-skip-tri-grams = {insurgents killed in, insurgents killed ongoing, insurgents killed fighting, insurgents in ongoing, insurgents in fighting, insurgents ongoing fighting, killed in ongoing, killed in fighting, killed ongoing fighting, in ongoing fighting}.

With slight modification to NLTK's ngrams code (https://github.com/nltk/nltk/blob/develop/nltk/util.py#L383):

from itertools import chain, combinationsimport copyfrom nltk.util import ngramsdef pad_sequence(sequence, n, pad_left=False, pad_right=False, pad_symbol=None):    if pad_left:        sequence = chain((pad_symbol,) * (n-1), sequence)    if pad_right:        sequence = chain(sequence, (pad_symbol,) * (n-1))    return sequencedef skipgrams(sequence, n, k, pad_left=False, pad_right=False, pad_symbol=None):    sequence_length = len(sequence)    sequence = iter(sequence)    sequence = pad_sequence(sequence, n, pad_left, pad_right, pad_symbol)    if sequence_length + pad_left + pad_right < k:        raise Exception("The length of sentence + padding(s) < skip")    if n < k:        raise Exception("Degree of Ngrams (n) needs to be bigger than skip (k)")        history = []    nk = n+k    # Return point for recursion.    if nk < 1:         return    # If n+k longer than sequence, reduce k by 1 and recur    elif nk > sequence_length:         for ng in skipgrams(list(sequence), n, k-1):            yield ng    while nk > 1: # Collects the first instance of n+k length history        history.append(next(sequence))        nk -= 1    # Iterative drop first item in history and picks up the next    # while yielding skipgrams for each iteration.    for item in sequence:        history.append(item)        current_token = history.pop(0)              # Iterates through the rest of the history and         # pick out all combinations the n-1grams        for idx in list(combinations(range(len(history)), n-1)):            ng = [current_token]            for _id in idx:                ng.append(history[_id])            yield tuple(ng)    # Recursively yield the skigrams for the rest of seqeunce where    # len(sequence) < n+k    for ng in list(skipgrams(history, n, k-1)):        yield ng

Let's do some doctest to match the example in the paper:

>>> two_skip_bigrams = list(skipgrams(text, n=2, k=2))[('Insurgents', 'killed'), ('Insurgents', 'in'), ('Insurgents', 'ongoing'), ('killed', 'in'), ('killed', 'ongoing'), ('killed', 'fighting'), ('in', 'ongoing'), ('in', 'fighting'), ('ongoing', 'fighting')]>>> two_skip_trigrams = list(skipgrams(text, n=3, k=2))[('Insurgents', 'killed', 'in'), ('Insurgents', 'killed', 'ongoing'), ('Insurgents', 'killed', 'fighting'), ('Insurgents', 'in', 'ongoing'), ('Insurgents', 'in', 'fighting'), ('Insurgents', 'ongoing', 'fighting'), ('killed', 'in', 'ongoing'), ('killed', 'in', 'fighting'), ('killed', 'ongoing', 'fighting'), ('in', 'ongoing', 'fighting')]

But do note that if n+k > len(sequence), it will yield the same effects as skipgrams(sequence, n, k-1) (this is not a bug, it's a fail safe feature), e.g.

>>> three_skip_trigrams = list(skipgrams(text, n=3, k=3))>>> three_skip_fourgrams = list(skipgrams(text, n=4, k=3))>>> four_skip_fourgrams  = list(skipgrams(text, n=4, k=4))>>> four_skip_fivegrams  = list(skipgrams(text, n=5, k=4))>>>>>> print len(three_skip_trigrams), three_skip_trigrams10 [('Insurgents', 'killed', 'in'), ('Insurgents', 'killed', 'ongoing'), ('Insurgents', 'killed', 'fighting'), ('Insurgents', 'in', 'ongoing'), ('Insurgents', 'in', 'fighting'), ('Insurgents', 'ongoing', 'fighting'), ('killed', 'in', 'ongoing'), ('killed', 'in', 'fighting'), ('killed', 'ongoing', 'fighting'), ('in', 'ongoing', 'fighting')]>>> print len(three_skip_fourgrams), three_skip_fourgrams 5 [('Insurgents', 'killed', 'in', 'ongoing'), ('Insurgents', 'killed', 'in', 'fighting'), ('Insurgents', 'killed', 'ongoing', 'fighting'), ('Insurgents', 'in', 'ongoing', 'fighting'), ('killed', 'in', 'ongoing', 'fighting')]>>> print len(four_skip_fourgrams), four_skip_fourgrams 5 [('Insurgents', 'killed', 'in', 'ongoing'), ('Insurgents', 'killed', 'in', 'fighting'), ('Insurgents', 'killed', 'ongoing', 'fighting'), ('Insurgents', 'in', 'ongoing', 'fighting'), ('killed', 'in', 'ongoing', 'fighting')]>>> print len(four_skip_fivegrams), four_skip_fivegrams 1 [('Insurgents', 'killed', 'in', 'ongoing', 'fighting')]

This allows n == k but it disallow n > k as shown in the lines :

if n < k:        raise Exception("Degree of Ngrams (n) needs to be bigger than skip (k)")    

For understanding sake, let's try to understand the "mystical" line:

for idx in list(combinations(range(len(history)), n-1)):    pass # Do something

Given a list of unique items, combinations produce this:

>>> from itertools import combinations>>> x = [0,1,2,3,4,5]>>> list(combinations(x,2))[(0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)]

And since the indices of a list of tokens is always unique, e.g.

>>> sent = ['this', 'is', 'a', 'foo', 'bar']>>> current_token = sent.pop(0) # i.e. 'this'>>> range(len(sent))[0,1,2,3]

It's possible to compute the possible combinations (without replacement) of the range:

>>> n = 3>>> list(combinations(range(len(sent)), n-1))[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]

If we map the indices back to the list of tokens:

>>> [tuple(sent[id] for id in idx) for idx in combinations(range(len(sent)), 2)[('is', 'a'), ('is', 'foo'), ('is', 'bar'), ('a', 'foo'), ('a', 'bar'), ('foo', 'bar')]

Then we concatenate with the current_token, we get the skipgrams for the current token and context+skip window:

>>> [tuple([current_token]) + tuple(sent[id] for id in idx) for idx in combinations(range(len(sent)), 2)][('this', 'is', 'a'), ('this', 'is', 'foo'), ('this', 'is', 'bar'), ('this', 'a', 'foo'), ('this', 'a', 'bar'), ('this', 'foo', 'bar')]

So after that we move on to the next word.

EDITED

The latest NLTK version 3.2.5 has the skipgrams implemented.

Here's a cleaner implementation from @jnothman from the NLTK repo: https://github.com/nltk/nltk/blob/develop/nltk/util.py#L538

def skipgrams(sequence, n, k, **kwargs):    """    Returns all possible skipgrams generated from a sequence of items, as an iterator.    Skipgrams are ngrams that allows tokens to be skipped.    Refer to http://homepages.inf.ed.ac.uk/ballison/pdf/lrec_skipgrams.pdf    :param sequence: the source data to be converted into trigrams    :type sequence: sequence or iter    :param n: the degree of the ngrams    :type n: int    :param k: the skip distance    :type  k: int    :rtype: iter(tuple)    """    # Pads the sequence as desired by **kwargs.    if 'pad_left' in kwargs or 'pad_right' in kwargs:    sequence = pad_sequence(sequence, n, **kwargs)    # Note when iterating through the ngrams, the pad_right here is not    # the **kwargs padding, it's for the algorithm to detect the SENTINEL    # object on the right pad to stop inner loop.    SENTINEL = object()    for ngram in ngrams(sequence, n + k, pad_right=True, right_pad_symbol=SENTINEL):    head = ngram[:1]    tail = ngram[1:]    for skip_tail in combinations(tail, n - 1):        if skip_tail[-1] is SENTINEL:            continue        yield head + skip_tail

[out]:

>>> from nltk.util import skipgrams>>> sent = "Insurgents killed in ongoing fighting".split()>>> list(skipgrams(sent, 2, 2))[('Insurgents', 'killed'), ('Insurgents', 'in'), ('Insurgents', 'ongoing'), ('killed', 'in'), ('killed', 'ongoing'), ('killed', 'fighting'), ('in', 'ongoing'), ('in', 'fighting'), ('ongoing', 'fighting')]>>> list(skipgrams(sent, 3, 2))[('Insurgents', 'killed', 'in'), ('Insurgents', 'killed', 'ongoing'), ('Insurgents', 'killed', 'fighting'), ('Insurgents', 'in', 'ongoing'), ('Insurgents', 'in', 'fighting'), ('Insurgents', 'ongoing', 'fighting'), ('killed', 'in', 'ongoing'), ('killed', 'in', 'fighting'), ('killed', 'ongoing', 'fighting'), ('in', 'ongoing', 'fighting')]

Although this would part entirely from your code and defer it to an external library; you can use Colibri Core (https://proycon.github.io/colibri-core) for skipgram extraction. It's a library written specifically for efficient n-gram and skipgram extraction from big text corpora. The code base is in C++ (for speed/efficiency), but a Python binding is available.

You rightfully mentioned efficiency, as skipgram extraction quickly shows exponential complexity, which may not be a big issue if you only pass a sentence as you did in your input_list, but becomes problematic if you release it on large corpus data. To mitigate this you can set parameters such as an occurrence threshold, or demand each skip of a skipgram to be fillable by at least x distinct n-grams.

import colibricore#Prepare corpus data (will be encoded for efficiency)corpusfile_plaintext = "somecorpus.txt" #input, one sentence per lineencoder = colibricore.ClassEncoder()encoder.build(corpusfile_plaintext)corpusfile = "somecorpus.colibri.dat" #corpus outputclassfile = "somecorpus.colibri.cls" #class encoding outputencoder.encodefile(corpusfile_plaintext,corpusfile)encoder.save(classfile)#Set options for skipgram extraction (mintokens is the occurrence threshold, maxlength maximum ngram/skipgram length)colibricore.PatternModelOptions(mintokens=2,maxlength=8,doskipgrams=True)#Instantiate an empty pattern model model = colibricore.UnindexedPatternModel()#Train the model on the encoded corpus file (this does the skipgram extraction)model.train(corpusfile, options)#Load a decoder so we can view the outputdecoder = colibricore.ClassDecoder(classfile)#Output all skipgramsfor pattern in model:     if pattern.category() == colibricore.Category.SKIPGRAM:         print(pattern.tostring(decoder))

There's a more extensive Python tutorial about all this on the website.

Disclaimer: I'm the author of Colibri Core