There is one rather sneaky trick not mentioned here so far. We assume that you have no extra way to store data, but that is not strictly true. One way around your problem is to do the following horrible thing, which should not be attempted by anyone under any circumstances: Use the network traffic to … Read more
I think there are several questions buried in this topic: How do you implement buildHeap so it runs in O(n) time? How do you show that buildHeap runs in O(n) time when implemented correctly? Why doesn’t that same logic work to make heap sort run in O(n) time rather than O(n log n)? How do … Read more
Since I’m the current world record holder for the most digits of pi, I’ll add my two cents: Unless you’re actually setting a new world record, the common practice is just to verify the computed digits against the known values. So that’s simple enough. In fact, I have a webpage that lists snippets of digits … Read more
Use itertools.permutations from the standard library: import itertools list(itertools.permutations([1, 2, 3])) Adapted from here is a demonstration of how itertools.permutations might be implemented: def permutations(elements): if len(elements) <= 1: yield elements return for perm in permutations(elements[1:]): for i in range(len(elements)): # nb elements[0:1] works in both string and list contexts yield perm[:i] + elements[0:1] + … Read more
I’ll do my best to explain it here on simple terms, but be warned that this topic takes my students a couple of months to finally grasp. You can find more information on the Chapter 2 of the Data Structures and Algorithms in Java book. There is no mechanical procedure that can be used to … Read more
This is known as the ‘Hamming Weight’, ‘popcount’ or ‘sideways addition’. Some CPUs have a single built-in instruction to do it and others have parallel instructions which act on bit vectors. Instructions like x86’s popcnt (on CPUs where it’s supported) will almost certainly be fastest for a single integer. Some other architectures may have a … Read more
How to find time complexity of an algorithm You add up how many machine instructions it will execute as a function of the size of its input, and then simplify the expression to the largest (when N is very large) term and can include any simplifying constant factor. For example, lets see how we simplify … Read more
Tail-call optimization is where you are able to avoid allocating a new stack frame for a function because the calling function will simply return the value that it gets from the called function. The most common use is tail-recursion, where a recursive function written to take advantage of tail-call optimization can use constant stack space. … Read more
This link might be helpful to you, as it details the use of the Haversine formula to calculate the distance. Excerpt: This script [in Javascript] calculates great-circle distances between the two points – that is, the shortest distance over the earth’s surface – using the ‘Haversine’ formula. function getDistanceFromLatLonInKm(lat1,lon1,lat2,lon2) { var R = 6371; // … Read more
The following is an attempt to describe the Ukkonen algorithm by first showing what it does when the string is simple (i.e. does not contain any repeated characters), and then extending it to the full algorithm. First, a few preliminary statements. What we are building, is basically like a search trie. So there is a … Read more