Given an array of integers, move all zeroes to the end of the array. Ideally, use no extra space, and keep the non-zero elements stable (meaning the order is preserved).
Attempt #1: Keep Moving Zeroes To The Right - Stable | O(N^2) Time | O(1) Space
A simple algorithm is to go through the array from left to right, and every time you encounter a zero, you search the rest of the array for a non-zero to swap with. However, this approach has a bad Time Complexity O(N^2) so it's not good.
Here's some pseudocode showing how you'd implement this:
Attempt #2: Place Elements Into New Array - Stable | O(N) Time | O(N) Space
An easy approach to this question is to create a new array, and place all non-zero elements into it. Then fill the remainder of the new array with zeroes. This approach takes O(N) extra space, so it's not ideal.
Here's some pseudocode showing how you'd implement this:
Attemp #3 (Best): Special Index Tracking - Stable | O(N) Time | O(1) Space
The ideal solution is a bit tricky. The idea is to go through the array and overwrite an old element if the current element is not zero. This is done by keeping track of an index for "where to write this current element too". This index is the "count" of the non-zero elements we saw so far. This is the ideal solution because the Time complexity is O(N) and the Space complexity is O(1).
Here's some C++ code for how you'd implement this:
The complete code for this project can be found on my algorithm questions and answers repo on github here.
To see the youtube video explaining this tutorial (which includes drawings that show how the recursion works), please click here.
The Problem - Repeatedly Finding The Minimum From A Range
Let's say that we have the following array.
What's the minimum value from index 0 to index 6? -> 1
What's the minimum value from index 0 to index 5? -> 2
What's the minimum value from index 1 to index 1? -> 8
What's the minimum value from index 4 to index 5? -> 4
It's easy to find the answer to these as a human looking at this small array, but how do we write a program to solve this?
Naive Algorithm - Use A For Loop
The simple thing to go through each number from index L to index R and check if the number at each index is smaller than the current minimum.
Here's some pseudocode that does that.
The running time of this function, used to query the array from range [L,R] is O(N), since we may be asked to query the entire array.
We don't have to do any extra work since we're given the array, so it costs O(1) extra space and O(1) time to build. And updating a value in the array costs O(1) time since we can access it's index directly.
Note: It's possible to get the query time to be smaller by memoizing results after we calculate them, but it then requires O(N^2) space/time to build the table to store the values.
These stats are pretty good, but it's possible to do even better using segment trees.
If you're asked to make multiple queries on subranges on an array, then segment trees are the ideal data structure to use.
The following sections will show how to build a segment tree structure, how to fill it with the appropriate values, how to query the segment tree for the value you need for a given subrange, and how to update the segment tree if a value of the original range changes.
Building An Empty Segment Tree
The first thing we need to do is make an empty tree. The segment tree is represented as an array, we just need to know what the size of the array is.
Here's the pseudocode to calculate the array size and create the array.
This array represents our segment tree. (Each index of the array is a node in our tree.)
Filling In The Segment Tree With Values
Now that we've built our segment tree, we need to fill it in with the values.
To fill in the segment tree, starting at the root, we'll say that the minimum of a node is the lesser value of its two children, and so we recursively search children for their minimums. The nodes that represent just one element (ex. [4,4]), however will say it's minimum is just the single value it represents from the original array.
Here's the pseudocode implementation for filling in the segment tree.
The running time of filling the values in the segment tree is O(N), since we visit each element of the original array once, and we touch each leaf of the segment tree once.
Querying The Segment Tree
After we've built a tree, we want to be able to search the tree for its values. (Which, in this case, is the minimum value between indices L and R.)
Similar to the technique we used before, we recursively search the tree, starting from the root, and if a node is within the range [L, R], then we check it's value as a valid minimum of the range.
The running time of this operation is O(log N). At worst, we'll have to go down the height of the tree twice, once for the left subtree, and one for the right subtree.
Updating A Value In The Segment Tree
If one of the elements of the original array needs to be updated with a new value, then we simply go through that element's direct path in the segment tree and update the minimum accordingly.
The running time is O(log N), since we only need to travel the height of the segment tree once. (The leaf node representing the updated index, up to the root of the tree.)
Adding/Deleting A Value In The Array
Basically, if you have to add a new value to the original array, then you have to construct a brand new segment tree. This ends up costing O(N) time to build the new tree. (Don't forget to delete the old tree.)
C++ Segment Tree Full Class Implementation
A full Segment Tree C++ implementation can be found on github here: https://github.com/srcmake/segment-tree
Note that the code was meant to be readable for this tutorial, and not necessarily "efficient". (Meaning some operations could be improved, or some code could be moved to functions. (Ex. Calculating the middle index.))
Here's a sample of the class definition (inside a file named segment-tree.hpp):
The code is similar to our pseudocode, with a few private variables included. (The definitions for each function can be found in the actual segment tree class file on github.
An example of using this class is...
Extending Segment Tree for Other Problems
This tutorial showed a Segment Tree designed for Range Minimum Queries (RMQ), specifically for storing integers, but of course segment trees can be extended for more than just finding the minimum.
We can also:
Segment trees are a data structure that comes up in algorithm competitions (and there's usually no time to code one from scratch), so it's better to have the code ready beforehand to copy and paste. So bookmark this blogpost, subscribe on youtube, and star the github repo.
Keep in mind that this tutorial was meant for demonstration purposes, and there are optimizations and edge cases we could've covered. Feel free to experiment with those on your own.
Like this content and want more? Feel free to look around and find another blog post that interests you. You can also contact me through one of the various social media channels.
To watch the youtube video on this topic (including a real demo of using cherry-picking) click here.
Introduction - When To Use Git Cherry Picking
When you work with multiple branches, sometimes there are commits in one branch that you want to copy to another branch, but you can't merge/rebase branches because you only want specific commits.
For example, let's say we had an application with certain foods in it, and let's say we had a branch named "all-foods" that had commits adding features for apples, spaghetti, bananas, cake, and cherries.
Now let's say we had a branch specifically for "fruits" that had apples, but we want to also add the work from the "all-foods" to add the "banana" and "cherry" features to the "fruits" branch. We can't merge the branches because then we'd have "spaghetti" and "cake" in our "fruits" branch. The way to solve this would be to use git cherry-picking.
1. First clone the project you're working on if you don't have it already.
2. Checkout the branch with the commit you want. (I'm not sure if this is totally required but it seems good.) You can also pull if your local branch is outdated.
3. Find the commit sha's you need by checking the commit history. (It's probably easier to just look at github/gitlab/bitbucket for this step, honestly.) Copy the commit sha's that you'll need.
4. Checkout the branch you want to add the commits too. You can also pull if your local branch is outdated.
5. Add the commits you want to add to this branch using the git cherry-pick command. If you have multiple commits, just separate the sha's with a space.
6. Optionally, create a new branch if you need too. (In case you can't directly use the branch you're currently on.)
7. Push your changes to the repository.
And that's it. Now your branch has commits in it from the other branch.
Git cherry-picking is useful for taking particular commits and copying them to another branch.
It's also a huge reason to want to keep commit histories clean by grouping changes of code logically into particular commits. (In our demo, if we added the features for bananas, spaghetti, cake, and apples all in the same commit, then cherry picking wouldn't have worked.)
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