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
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
Intro to Algorithms from MIT Press qualifies QuickSort as in-place – it sorts the elements within the array with at most a constant amount of them outside the array at any given time. At the end of the day, people will always have differing opinions (is Top-Down Memoization considered Dynamic Programming? Not to some “classical” folks), side with who you … Read more
(I’m assuming that this is just some kind of thought exercise, or even a trick homework/interview question, but I suppose I could imagine some bizarre scenario where you’re not allowed any heap space for some reason [some really bad custom memory manager? some bizarre runtime/OS issues?] while you still have access to the stack…) Breadth-first … 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 get the BigOh. As … Read more
You’ve got it, just keep going! And be careful with the indexes… To simplify a bit I’ll assume that N and M are > k, so the complexity here is O(log k), which is O(log N + log M). Pseudo-code: For the demonstration you can use the loop invariant i + j = k, but … Read more
I want to know what will be the best case for a bubble sort ? There may be a case wherein there may be no swapping for the say last 2 passes for example. I’m doing my program in C language. Suppose i have an array of 5 elements and i give the elements as … Read more