The following function prints the powers of 2 from 1 through n (inclusive). This means that the function will compute 2^i where i = 1, 2, …, n, in other words, i can have values from 1 up to and including the value n. i.e n is Included in Inclusive If, on the other hand, your book had said: The following function prints the powers of … Read more
Your algorithm is correct, how about something more elegant:
The absolute best case happens when we just check every element and see that the data’s already sorted. This will result in n-1 comparisons and thus the leaf will have a depth of n-1. Practically, this happens for insertion sort (which isn’t all that good otherwise though). Does it change depending on the algorithm? Absolutely. The best case of an … Read more
Here’s what MAX-HEAPIFY does: Given a node at index i whose left and right subtrees are max-heaps, MAX-HEAPIFY moves the node at i down the max-heap until it no longer violates the max-heap property (that is, the node is not smaller than its children). The longest path that a node can take before it is in the proper position … Read more
Use Prim’s algorithm when you have a graph with lots of edges. For a graph with V vertices E edges, Kruskal’s algorithm runs in O(E log V) time and Prim’s algorithm can run in O(E + V log V) amortized time, if you use a Fibonacci Heap. Prim’s algorithm is significantly faster in the limit when you’ve got a really dense graph with many … Read more
The main purpose of the Big-O notation is to let you do the estimates like the ones you did in your post, and decide for yourself if spending your effort coding a typically more advanced algorithm is worth the additional CPU cycles that you are going to buy with that code improvement. Depending on the … Read more
Recall that in Dijkstra’s algorithm, once a vertex is marked as “closed” (and out of the open set) – the algorithm found the shortest path to it, and will never have to develop this node again – it assumes the path developed to this path is the shortest. But with negative weights – it might not … Read more
Let’s first look at the n^2 algorithm: Now the improvement happens at the second loop, basically, you can improve the speed by using binary search. Besides the array dp[], let’s have another array c[], c is pretty special, c[i] means: the minimum value of the last element of the longest increasing sequence whose length is … Read more
this turns the recursion in to a loop:
If the input matrix is guaranteed to describe transitive connectivity, it has a peculiar form that allows for an algorithm probing only a subset of the matrix elements. Here is an example implementation in Python: An optimized version of the same algorithm (less readable, but faster and more easily translatable into other languages) is the following: