Often the most crucial part of a system is the database. Databases manage and handle the storage of our business data, an inefficient database can impact the overall performance of our system. There are, however, a few ways we can improve database performance, one such way is defining appropriate indexes.

This article describes indexes in Postgres and some of the lesser known indexes it uses, such as GIN and BRIN indexes.

Why do we use indexes.

We use indexes in postgres to optimise how we scan the data we have stored in the database. Similar to how binary search can optimise how quickly we find things in a dictionary (see my post on binary search).

We can use an index to inform the database on how to structure data in a way that allows us to find what we are looking for effectively i.e. fewer operations, fewer i/o and fewer comparisons leading to lower CPU usage and better performance.

What is an index.

An index is a data structure which allows us to minimise the number of instructions performed by the database in order to lookup table rows or improve the speed at which a result is sorted.

In the case of sorting if there's an index on a column that you require results to be ordered by, the query can scan the index to return the results in sorted order, if there is no index on the column the data must first be fetched and then sorted in a separate step.

Indexes aren't always good

Over doing the number of indexes on a table can lead to lower write performance as every update or insert into the table requires updating the index. Indexes also require disk space, so having unnecessary indexes will contribute to the rate at which your disk usage grows.

B-Tree

The default index type Postgres uses is a B-Tree. This data structure is similar to a binary tree except instead of having a node with only two pointers a node can contain many pointer. The number of pointers is determined by Postgres's default block size, which is 8kb. A single node will store as many sorted keys as it can until it reaches this size limit. Postgres refers to these nodes as pages.

Having a smaller index keys can improve the performance of your index as you'll be able to store more keys within the page limit. Resulting in fewer i/o instructions as fewer pages are required to be read from disk.

In postgres leaf nodes will point to the physical location of the row on disk and the intermediary nodes (or internal pages) will point to the nodes on the next level down the tree.

There's a neat animation that can help visualise b-trees: https://btree.app/

Partition Indexes

We can optimise the indexes we create by understanding usage patterns and having some intuition about the system and the business. One such optimisation is the use of partition indexes, these are handy when you have a large table but only a subset of the data is used most frequently.

As an example, we can imagine an online order system, it's quite likely that once an order is fulfilled the status transitions to "complete" and this order isn't likely to be accessed as frequently as the orders that are still in progress. We can partition our index so that it contains only unfulfilled orders which will be a significantly smaller portion of our overall orders.

CREATE INDEX idx_uncomplete_orders_customer ON
orders(customer_id) WHERE status != 'complete';

We also have to include this WHERE filter in our queries if we wish to make use of this index.

Covering Index

A covering index allows us to add columns to the index's leaf nodes and thus avoids the lookup and reading of the entire the table row from disk. Essentially everything that the query needs is available on the index, reducing the number of operations required to complete the query.

As an example if we typically request a user's first name and last name with an email we can write an index on the email that includes the first and last name.

CREATE INDEX idx_user_email_include
ON users (email) INCLUDE (firstname, lastname);

We have to bare in mind that we should only include columns which change as frequently or less frequently than the key we are indexes otherwise we are duplicating data and increasing the write overhead. This isn't an issue for columns that rarely change.

More on covering indexes see "INCLUDE".

Gin Index

GIN stands for Generalised Inverted Index, similar to the inverted index that Lucene is built on, the same Lucene that Elasticsearch is built on. The slight difference is that this inverted index is generalised so it expects items that are indexed to be composed of multiple values the same way a sentence is composed of multiple words, with the goal to support full-text search.

An index entry is created for each element in the composite column and the entry points to a list of matching locations. So as an example a row that has a column with the value "Here's is a blog post" will create an index entry for each word in the value, (e.g. "blog") and then each entry will point to the rows that contain "blog" inside the composite column.¹

ALTER TABLE blogposts
ADD COLUMN search_vector tsvector;

CREATE INDEX idx_blogposts_search ON blogposts
USING GIN (search_vector);

GIN indexes aren't limited to just text, they are generalised so they can be used with any composite type, JSONB for example.

BRIN Index

BRIN stands for Block Range Index. These are indexes that specialise in handling very large tables in which certain columns have some natural correlation to their physical storage location. Essentially if each row in the table can be grouped in some manner to the blocks on disk then we have a potential use-case for the BRIN Index.

GPS tracking points and log tables are good examples of data with this natural correlation, if you have a table that is small or data is updated or inserted out of chronological order then they won't form a good fit for BRIN indexes.

Instead of storing pointers to entry locations in the B-Tree leaf nodes, BRIN indexes store summary information for the ranges that the blocks on disk are ordered. This allows for a smaller overhead compared to the default B-Tree which stores all entries and their corresponding locations.

CREATE INDEX idx_transactions_created_at_brin
ON transactions using BRIN (created_at);

This can be used to optimise queries that rely on fetching rows between certain ranges.

SELECT * FROM orders WHERE
created_at BETWEEN '2024-01-01' AND '2024-12-31';

Anything that loads faster than 200ms is instantaneous for humans, anything slower and we would perceive the delay. Some systems are built to respond quicker than this, business cost can be sensitive to latency and every ms can make a difference.

How do we develop highly performant systems? I happened to share a train ride with Mark Shannon, who at the time was leading a team at Microsoft that had one goal: Make Python Fast.

I asked him, how does one make code more performant? To which he responded.

Make it do less.

Here's what doing less looks like:

Loop Invariant

A simple example of doing less in order to achieve better performance is with loop invariant conditions.

We loop over arrays while we code and make calls to other functions while looping. During a refactor or while we move code around we might not realise that some of the variables we are computing are constant.

def expensive(trades: list[Trade], n: int):
    rolling_sum = 0
    for trade in trades:
        beta = compute_beta(n)
        rolling_sum += beta * trade.risk

    return rolling_sum / len(trades)

You'll notice in the example that I am looping over the list trades and computing a beta value, but this beta value is computed with the integer n which is consistent within the scope of this function. If we move the computation of beta to a point before we've entered loop we avoid having to compute beta for each trade in trades.

def expensive(trades: list[Trade], n: int):
    rolling_sum = 0
    beta = compute_beta(n)
    for trade in trades:
        rolling_sum += beta * trade.risk

    return rolling_sum / len(trades)

Now beta is computed at most once within this method scope, we have essentially achieved doing less. If the compute_beta method incurs an expensive overhead and we are looking at a lengthy trades list this subtle oversight would have a significant impact on our latency.

Python

There are several layers of abstraction in Python and there are ways we can code that allow us to utilise computation in C more effectively. One such example of this: using list comprehension over a vanilla for-loop.

If we wish to understand the number of steps the machine is doing for a method, Python allows us to disassemble bytecode using import dis. You might notice that there are additional LOAD_ATTR instructions in a vanilla for-loop so defaulting to a list comprehension avoids this overhead and allows us to do less.

Databases

Relying on external systems can be expensive, in some cases we might query the same system several times in order to complete an operation. If we have a distributed system or intend to scale our server horizontally we are increasing the number of things that open and close connections to the database. Each time we open/close a connection the CPU has to process these instructions, this overhead can be significant when we are working in low latency environments. To get around this we introduce a connection pool between the database and our server, the connection pool manages long lived connections on the database and frees up the database to focus on database things.

A common anti-pattern involving a database is having your ORM make N+1 queries. For this to occur we have a 1-many entity relationship. If we ask the database to return a group of n countries and then for each country we ask for all cities in that country we are making N (number of countries in our initial query) + 1 (the initial query) total requests to the database. These can be tricky to spot since the individual query could be performant, but the accumulative overhead and latency of all these queries can cause problems.

We get around this by either asking for all countries and then ask for all cities given a set of country ids and perform the mapping in memory on the server, or we can increase the complexity of the query using a JOIN and allow the database to make the mapping. Either way we avoid the back and forth overhead of making N+1 queries, effectively doing less.

When we can't do less

Sometimes we can't do less, but we can decide when work is done. There are situations that allow us to pre compute results and provide performant computation at the cost of a slower start time.

Chess programming is an area which relies on high performance, the quick our program the further into the future we can look on each move during the game. One way we speed up a chess engine is by precomputing the movesets for every piece on the board for every possible position they might be in before the game starts. This allows us to look up all possible board positions for a piece during the game instead of computing only when we need it.¹

In 3b1b's wordle solver he actually computes a pattern matrix for each word and saves it to a file in order to avoid having to incur the overhead on each execution of his wordle solver.

Constant Folding

Lastly there's an interesting optimisation that compilers make which is also utilised in python called Constant Folding.

When python is interpreted it parses the AST before executing. During this AST construction it finds expressions who's outcome is constant and computes the constant at this point instead of computing the expression every time it is referenced. As an example the variable x = 24 * 60 will be converted to 1440 before compiling the bytecode instructs for the machine.

So in all cases, if we want a computer to be highly performant we just have to make it do less.

Further Reading

• perflint try running this on your python codebase to check for performance optimisations.

Low performing teams have a lot in common. These teams often lack some of the basics. Low context, no direction and no understanding of purpose. We can summarise these basics into one thing: a lack of leadership.

Reflecting on failing teams, I've identified how I would nudge/how I've nudged a group of under performers into a team that delivers impact. Often there's a single missing ingredient and the team needs a catalyst for change. This can come from within the team and doesn't have to come from a leader.

Most failure patterns occur in companies that rely on organic organisation, as they've determined that autonomy is at odds with any intentional organisation design.

Things that are done with an intent in mind will always lead to a better outcome, otherwise you're just leaving it up to chance. We can have both, autonomy and organisation.

Failure case 1.

Everyone's a leader.

Communication is good, avoiding a decision is bad. In teams where everyone thinks/tries to be a leader the team behaves as if it were leaderless.

Individual contributors will state what they believe ought to be the best direction but it ends without consensus. These meetings are pointless as members stick to the idea that they've presented, and a team with 5 members will have 5 directions.¹

This pattern is hard to notice since they usually come to light when you're in the middle of a stressful situation and everyone is sharing an opinion as a means to turn the ship around. It's also easy to be absorbed into giving your own bit of advice in order to feign contribution.

The members present a large amount of information and set no goal, or goals are highly individualistic. There's no point in calling yourself a team if there's no underlying vision/goal to achieve. You could now say this is hardly even a team and you would have saved time by not having a meeting at all.

The best way to address this situation is to take a step back and be honest. "I'm not going to remember half of this meeting." Then direct the attention to someone that is meant to be taking a leadership role and say: "Maybe x can summarize these thoughts into one or two points and we can focus on this, so we leave this meeting with the same aim". What ever is said next will certainly be remembered.

If you are the leader, then don't let your team leave more uncertain about how they should address the situation than when they arrived. A mature leader might even recognise that there's a team mate that's probably the most knowledgable and should use the opportunity to direct them to state the focus.

It's similar to a half time chat, you can't have 11 people in the team telling you what you should do better in the next half. That's not how a rally works.

Agreeing to commit to a direction and a goal is all part of being a team. Someone needs to stand up in moments when there's dysfunction.

Failure Case 2

No one is a leader.

Communication is good. Some teams are just sitting around waiting to be told what to do, if there's a sit around culture then nothing impactful gets done.

It's quite easy to tell if you're in a team that lacks leadership since no one will know what work should be done or how their own work fits into the bigger picture of the organisation.

Teams without a leader will lack context and purpose which are all traits that lead to lower motivation. Assuming that people will just know what to do, without forming some consensus is naive and might lead the team to doing pointless work.

Turning these kind of teams around requires a little more diagnosis of the no-leader symptom.

1. Why is no one grabbing initiative?
2. What are the risks involved in grabbing initiative?
3. Is fear involved?
4. Do we provide positive examples that initiative is rewarded?

You might come to realise that no one want's to take on the risk that comes from taking initiative, especially if there's only downside and no upside. Initiative should always be rewarded in organisations even if the wrong initiative was taken. Learning from mistakes is always better than sitting and doing nothing.²

Diagnosing why the team is behaving this way can lead you to notice that your org is not incentivising the correct behaviours.

There can be other reasons, like a lack of experience in taking ownership. There could also be a lack of hiring with intent leading to a skill misalignment. What ever it is, if there's a team that appears leaderless then there's a problem that needs fixing.

Leaders

In truth leaders are in a team not to be dictators, but to hear the concerns of everyone and distill it down to a direction and focus. Getting the whole team aiming at the same target can shift a low performing team to a high performing team over night.

Having a team aim all over the place will result in weak performance.

Motivating teams, is about context and purpose - you can leave them to do what they want as long as they have context and purpose. Without this they're directionless and will not meet expectations.

Some situations are a prime ground for someone to take ownership, just make sure you're awarding this behaviour.

Binary search is quite a simple algorithm to explain and thus it is often one of the first algorithms computer scientists learn. It's also occasionally brought up in software engineering interviews as it can be applied when solving optimisation problems.

The traditional explanation of binary search is flawed and acts as a barrier to its application. Binary search is not an algorithm that believes a set can be divided into three groups, e.g. these numbers are too cold, these number are too hot and this number is just right.

This goldilocks framing of binary search forces us to look for situations where the problem contains an array of porridge ordered by temperature and we are in search of the one falls into our preferred consumption temperature.

Binary search is an algorithm that is optimal at finding state transitions and framing it as such opens up the application of binary search as we no longer approach problems like goldilocks and instead; finding the best porridge becomes finding the point at which the ordered bowls of porridge transition from being tepid to scolding.¹

The Basics

Commonly, binary search is introduced with the example of finding a word in a dictionary. As an example say we are tasked to find the word "Hippo":

The brute force approach is to start at A, and flick through the pages until you get to the page that contains "Hippo". This approach works, however the time complexity of this algorithm is O(n); where n is the number of pages in the dictionary. So in the worst case, "Zebra" or something, You're looking at all n pages.

Pretend H is Hippo:

n operations is not inherently bad. If you have a list of 10 words. n operations isn't going to harm you. It starts to get a bit much when you have a billion words and it's not practical when you have n words to find among a billion words.

Here's where we can apply binary search. If we want to find the word "Hippo". We start by jumping to a random page, or we open the book in the middle.² If the dictionary is 260 pages the middle will land us on 130 which will be M. We know that H is before M so we can throw away half the dictionary. M->Z, so from a single operation we've already chopped the worst case scenario to n/2. We don't stop here, now we go to page 65 (half of 130) which lands us somewhere between F and G we know H is after G so now we can pick something in between G -> M = J. We continue doing this, narrowing down the pages, until we land on the page that contains Hippo.

In the worst-case scenario, when we are looking in a dictionary of 260 pages, we do 9 operations. If we relied on the previous algorithm we would do 260 operations in the worst-case.

The Weaknesses

This explanation does a good job to help us understand the algorithm but it overcomplicates binary search as it breaks down the operations that binary search is doing into three parts.

1. Is the current word the word we are looking for?
2. Is the word we are looking for ahead of the current word?
3. Is the word we are looking for behind the current word?

These questions may help inform us as to how the algorithm behaves, but we can simplify it to the single binary question.

When does the word we are looking for no longer appear after our current word?

Instead of imaging the sequence as the words that might appear on each page, we can picture the sequence as the outcome of the statement: The word I'm looking for is after the current word.

As a result this frames binary search as a solution to finding a state transition. It is at the point when our word not longer appears after the current word that we find the word we are looking for.

Git Bisect

Binary search is used to find bugs. If you're familiar with git then you might have heard of git bisect. For those unfamiliar with git imagine a timeline with each point in that timeline being a change to a file. So every time the file is saved it adds a new node to the timeline.

Now instead of a single file, git works across a whole software project and each save is being noted down on the timeline. A change is what is being added and what is being removed, so the accumulated sum of all these changes is the current state of the project.

On large teams and complex projects bugs can occur after several changes have been made to the code. If we know that the code is broken in its current state but aren't sure what change caused the bug we can use git bisect to figure out which change is the culprit.

When we run git bisect we provide it with a point in time that we are certain no bug existed. git bisect will then present us a change between that time and now.

1. No bug
2. ?
3. ?
4. broken?
5. ?
6. ?
7. Bug

We then have to answer; is the software at point 4 broken? If yes, the bug was introduced between 1 and 4. This rules out all points after 4. Cutting the points we have to analyse down in half. git bisect continues, running a binary search over the changes in the project until we find the change on the timeline that introduced the bug.

Essentially we are using binary search to answer the statement: The bug exists, because we are interested when the state transitions from "The bug doesn't exist" to "The bug exists".

The Boolean Function

We can now separate binary search into two distinct parts. The algorithm part, which moves a pointer either up or down the timeline and the function that computes a boolean response. With this we can ask more interesting questions for which binary search provides an answer, as an example; "When did spend increase?" or "When did the user go into debt?".

We can even provide two arrays and find the answer for "When did trend A over take trend B", without computing the delta for all observations on the timeline.

The main underlining assumption taught with the traditional explanation is that the set you're operating on should already be sorted for the search to work. The example above shows that this isn't true. Binary search only requires the underlying data to be bisect-able.

Binary search is an effective tool for these state transition problems. It is interesting that the algorithm on its own is easy to understand, the complexity and it's application seems to grow when you separate the algorithms iterator and the statement.

I'd be interested in finding out if the application of other algorithms grows when they're decomposed in a similar way.

Further reading

• Root Finding

On May 24th 2000 the Clay Math Institute announced a $1,000,000 prize for solving one of 7 maths problems. Since then only 1 has been solved. The mathematician that solved the problem rejected the money.¹

One of those problems is the P vs NP question which relates to how we estimate time and space complexity for algorithms in computer science and software engineering.

Complexity

Complexity is an estimate of the growth in either time or space given an increase in the size of the input. Some algorithms have a linear relationship between time and input, we signify this complexity as O(n).

Constant complexity is signified as O(1) this means that as we increase n (the size of the input) the time and space is consistent, the algorithm doesn't take any longer and doesn't require more RAM.

An example of an algorithm that take constant O(1) space is finding the largest number in an unordered list of n numbers. You'll notice below, as we move through the list there's only one element we need to keep track of. This element is the largest item we've seen up to that point. When we see a larger number we replace the tracked number with this new larger number.

No matter how many items we include in this list we would still only need to track a single number, so the space complexity of this algorithm is O(1). We might apply this algorithm to find the largest file on a computer.

In terms of time complexity if we were to add more files or more items to this list the amount of time it takes would increase proportionally to the number of items we add. We therefore say this algorithm has an O(n) time complexity.

If we wish to improve the time complexity of finding the largest file, we can do this by having something track the largest file in the systems and checks and updates this whenever a file is written. This would provide us with a constant time O(1) algorithm as we avoid having to check all the file sizes every time we request the largest file. The trade off here is added complexity when updating any file on our system. If we were to reduce the size of the largest file the second largest might now be the largest and we'd have to find which file that might be.

We don't often require knowing what the largest file on the system might be so this added complexity is probably a waste of resource, additionally the overhead of creating, updating and deleting files has now increased. An O(n) search is probably enough for most machines.

Sorting

Sorting is often used to explain time complexity as there are a number of ways to sort items each with their own complexity. It's also useful because we can visualise sorting. Below I have provided an example of bubble sort, this has a time complexity of O(n²).

This isn't the most efficient way to sort a list of items, but it's a good representation of an O(n²) algorithm.

Categorising complexity

So far we've been discussing polynomial time algorithms, there exist algorithms which take exponential time to produce a solution. As an example in the case we have 2 items this would take 4 operations, if we increase to 3 items it would take us 8 operations. Such an algorithm is said to have O(2ⁿ) time complexity.

An example of an exponential algorithmic problem is the N-Queens ii problem. The time complexity for an algorithm to solve this problem is O(n!). So 3 items is 6 operations, 4 items is 24 operations. These are algorithms which a computer would struggle to solve, in a sensible time, as n gets larger.

Within the realm of these exponential algorithms exist problems which we can solve in O(n!) time but we can validate the solution in polynomial time. So even if n is exceptionally large, given a solution, we are able to validate the correctness relatively quickly.

P vs NP

This class of algorithms forms the basis for one of the millennium problems, known as P vs NP. The problem asks if there's a relationship between the complexity of validating the solution and complexity in solving the problem. If P = NP then problems that can be verified in polynomial time can also be solvable in polynomial time. However P ≠ NP implies that some problems are inherently harder to solve than they are to verify.

There are a few problems which have no existing polynomial solution, which upon finding one will determine that P = NP. Problems such as the traveling salesman problem (TSP), The boolean satisfiability problem and the vertex cover problem Proving that there exists a polynomial solution to these problems will net you $1 million.

In Practice

Solving these problems have practical applications. Solving TSP would provide us optimal solutions for delivery routing and vertex cover has an application in DNA sequencing. So how do we produce solutions in these areas when attempting to find the best solution can take a computer as long as the lifetime of the universe.

We tend to rely on algorithms that are fast on average, these algorithms may involve backtracking with a large amount of pruning; this approach is used in chess. There are also approximation algorithms or heuristic based approaches such as simulated annealing.

Trade Offs

Understanding time and space allows us to make trade offs between strategies. Certain algorithms can provide us an efficient approach to solving technical problems. There is a limit to how much we can do with algorithms and sometimes even problems that sound simple on the surface could have a $1 million bounty standing for 25 years, without you realising it.