Python dictionaries are fundamental building blocks for the Python programming language. They are essentially finite maps in the mathematical sense. We give a treatment following § 6.2 of

  • A Primer on Scientific Programming with Python by Hans Petter Langtangen, 2nd edition.

Dictionaries are also briefly described in the Python Tutorial.

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# Standard imports
import numpy as np
import matplotlib.pyplot as plt
import math as m
from mpmath import mp, iv

Example of a simple dictionary

In Python curly brackets denote a dictionary. Here is a simple one:

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d = {2: 3.14, "dog": "cat", 2.5: "bird"}
{2: 3.14, 'dog': 'cat', 2.5: 'bird'}

A dictionary represents a finite mapping, meaning there are accepted inputs in a finite set and each input has an output. For example, the notation 2: 3.14 in the curly brackets indicates that 2 is mapped by the dictionary to 3.14. We can apply the map using square brackets:

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Note that the inputs and outputs can be nearly any object known to Python and the objects can be freely mixed. (Later we will see that the inputs must be hashable.)

Iterating though a dictionary

The accepted inputs of a dictionary are called the dictionary's keys. You can iterate through the list of keys as follows:

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for key in d:

If we want to access the image of key we can use d[key] as above so we can do:

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for key in d:
    print("{} mapsto {}".format(key, d[key]))
2 mapsto 3.14
dog mapsto cat
2.5 mapsto bird

An alternate way to loop through the same information is demonstrated below:

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for key, value in d.items():
    print("{} mapsto {}".format(key, value))
2 mapsto 3.14
dog mapsto cat
2.5 mapsto bird

Improvement over lists

Often data is more naturally presented in terms of dictionaries than lists or tuples (the main other data collection types we have used).

For example, the following two lists represent times (in years) at which measurements of atmospheric $CO_2$ were made and the actual measurements (in ppm).

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measurement_time = [ 2019.042, 2019.125, 2019.208, 2019.292, 
                     2019.375, 2019.458, 2019.542, 2019.625, 
                     2019.708, 2019.792, 2019.875, 2019.958 ]
measurement = [ 410.83, 411.75, 411.97, 413.32, 414.66, 413.92,
                411.77, 409.94, 408.54, 408.53, 410.27, 411.76 ]

Here measurement[i] was made at measurement_time[i]. But it is awkward to keep this data separate, and we are forced to use an index i which is not so relevant to us.

The same information can be stored in a dictionary as follows and makes the relationship clear.

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data = { 2019.042: 410.83,
         2019.125: 411.75,
         2019.208: 411.97,
         2019.292: 413.32,
         2019.375: 414.66,
         2019.458: 413.92,
         2019.542: 411.77,
         2019.625: 409.94,
         2019.708: 408.54,
         2019.792: 408.53,
         2019.875: 410.27,
         2019.958: 411.76 }

You can get a list of keys with d.keys() and the corresponding list of values with d.values(). (Really the objects returned are not lists, but they behave like them.) So, you don't really lose any flexibility by using a dictionary.

For example, if you want to plot the data above, you can use the following:

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plt.plot(data.keys(), data.values(), "o")
[<matplotlib.lines.Line2D at 0x7f81941960f0>]

Modifying dictionaries

Recall the dictionary d is as below:

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{2: 3.14, 'dog': 'cat', 2.5: 'bird'}

Perhaps we want to map 4 to 'frog'. We can do that with:

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d[4] = 'frog'
{2: 3.14, 'dog': 'cat', 2.5: 'bird', 4: 'frog'}

This added a new key and a new value to our dictionary.

We can also modify an existing value. That 3.14 value seems out of place. To fix it we can do:

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d[2] = 'aardvark'
{2: 'aardvark', 'dog': 'cat', 2.5: 'bird', 4: 'frog'}

The 'dog' key also seems out of place. It can be removed with:

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del d['dog']
{2: 'aardvark', 2.5: 'bird', 4: 'frog'}

Building dictionaries from scratch

You can build an empty dictionary with the command:

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squares = {}

So one way to build a dictionary is to start with an empty dictionary and then add key-value pairs one at a time.

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for n in range(-2,5):
    squares[n] = n**2
{-2: 4, -1: 1, 0: 0, 1: 1, 2: 4, 3: 9, 4: 16}

Checking keys

You can check if an object is a key in a dictionary using the in keyword. For example:

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2 in d
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'squid' in d

Because 'squid' is not in the dictionary, an exception will occur if you try to access d['squid'].

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KeyError                                  Traceback (most recent call last)
<ipython-input-20-2fced4e2c6d4> in <module>()
----> 1 d['squid']

KeyError: 'squid'

You can then deal with the possibility you might use a bad key using a try-except block.

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    print("squid maps to {}".format( d['squid'] ))
except KeyError:
    print("I guess 'squid' is not a key!")
I guess 'squid' is not a key!

The except block gets run only if a KeyError occurs in the execution of the try block. So for example if we set d['squid'] and then execute the same block we see different behavior:

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d['squid'] = 14
    print("squid maps to {}".format( d['squid'] ))
except KeyError:
    print("I guess 'squid' is not a key!")
squid maps to 14

Making a copy

You can make a copy of a dictionary with:

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d2 = d.copy()
{2: 'aardvark', 2.5: 'bird', 4: 'frog', 'squid': 14}

This is useful because now you can make changes to the copy without chaing the original and vice versa. For example, changing d2 so that 2.5 maps to 'salamander' will not affect d.

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d2[2.5] = 'salamander'
print("d = {}".format(d))
print("d2 = {}".format(d2))
d = {2: 'aardvark', 2.5: 'salamander', 4: 'frog', 'squid': 14, 'list': ['item0']}
d2 = {2: 'aardvark', 2.5: 'bird', 4: 'frog', 'squid': 14, 'list': ['item0']}


The .copy() method returns a shallow copy, meaning that the command d2 = d.copy() is equivalent to the following:

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d2 = {}
for key,value in d.items():
    d2[key] = value
{2: 'aardvark', 2.5: 'bird', 4: 'frog', 'squid': 14}

The copy is shallow because the values are identical in the two dictionaries. So for instance if a value is a list and that list changes, then it will change in both dictionaries. For example:

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d['list'] = []
d2 = d.copy()
print("d = {}".format(d))
print("d2 = {}".format(d2))
d = {2: 'aardvark', 2.5: 'bird', 4: 'frog', 'squid': 14, 'list': []}
d2 = {2: 'aardvark', 2.5: 'bird', 4: 'frog', 'squid': 14, 'list': []}
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print("d = {}".format(d))
print("d2 = {}".format(d2))
d = {2: 'aardvark', 2.5: 'bird', 4: 'frog', 'squid': 14, 'list': ['item0']}
d2 = {2: 'aardvark', 2.5: 'bird', 4: 'frog', 'squid': 14, 'list': ['item0']}

Note that the above examples make a subtle distinction:

  • The command d2[2.5] = 'salamander' assigns a new object ('salamander') to 2.5.
  • The command d['list'].append("item0") makes a change to an existing value. The first type of change does not propagate to a copy, but the second does. Actually the second statement does not change d or d2. (The dictionaries just store a pointer to an object not the object itself.)


How dictionaries work

Keys in a dictionary are distinguished by their hash, an integer associated to an object that is immutable (does not change). You can access the hash of an object with hash(obj).

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The hash of two objects need to be the same if the two objects are equal. So for example:

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hash("fr" + "og") == hash("frog")

A python dictionary is implemented as a hash table and is generally very fast unless there are hash collisions (where multiple keys have the same hash value).

Internal use in Python

Python uses dictionaries to store all the variables you use in your code. Variables are stored using their variable name as a string mapping to the variable's value.

You can access these variables in an alternate way through the locals() and globals() functions. See the documentation for more detail.

So we can access our dictionary d built above using the command:

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{2: 'aardvark', 2.5: 'salamander', 4: 'frog', 'squid': 14, 'list': ['item0']}


Sets are very similar to dictionaries but store only keys instead of key-value pairs. They are useful for keeping track of objects in case you want to eliminate duplicates. For example:

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s = { 2, 3, 4, 2+2, 5-3, 1+1}
{2, 3, 4}

You can read more about sets in the Python documentation.