maps read prooved

amejiarosario · amejiarosario · commit 04b5e122bf55 · 2019-02-21T17:26:33.000-05:00
diff --git a/book/book.adoc b/book/book.adoc
@@ -79,8 +79,19 @@ include::chapters/tree.adoc[]
 // (g)
 include::chapters/binary-search-tree.adoc[]
 
-// ()
-include::chapters/map.adoc[]
+// (g)
+// include::chapters/map.adoc[]
+include::chapters/map-intro.adoc[]
+
+:leveloffset: +1
+
+include::chapters/map-hashmap.adoc[]
+
+include::chapters/map-treemap.adoc[]
+
+include::chapters/map-hashmap-vs-treemap.adoc[]
+
+:leveloffset: -1
 
 include::chapters/set.adoc[]
 
diff --git a/book/chapters/map-hashmap-vs-treemap.adoc b/book/chapters/map-hashmap-vs-treemap.adoc
@@ -0,0 +1,27 @@
+= HashMap vs TreeMap
+
+.A map can be implemented using hash functions or binary search tree:
+- *HashMap*: it’s a map implementation using an *array* and *hash function*. The job of the hash function is to convert the key into an index that contains the matching data. Optimized HashMap can have an average runtime of *O(1)*.
+- *TreeMap*: it’s a map implementation that uses a self-balanced Binary Search Tree (red-black tree). The BST nodes store the key, and the value and nodes are sorted by key guaranteeing an *O(log n)* look up.
+
+
+.When to use a TreeMap vs. HashMap?
+* `HashMap` is more time-efficient. A `TreeMap` is more space-efficient.
+* `TreeMap` search complexity is *O(log n)*, while an optimized `HashMap` is *O(1)* on average. 
+* `HashMap`’s keys are in insertion order (or random in some implementations). `TreeMap`’s keys are always sorted.
+* `TreeMap` offers some statistical data for free such as: get minimum, get maximum, median, find ranges of keys. `HashMap` doesn’t.
+* `TreeMap` has a guarantee always an *O(log n)*, while `HashMap`s has an amortized time of *O(1)* but in the rare case of a rehash, it would take an *O(n)*.
+
+== TreeMap Time complexity vs HashMap
+
+As we discussed so far, there are trade-off between the implementations
+
+.Time complexity for different Maps implementations
+|===
+.2+.^s| Data Structure 2+^s| Searching By .2+^.^s| Insert .2+^.^s| Delete .2+^.^s| Space Complexity
+^|_Index/Key_ ^|_Value_
+| Hash Map (naïve) ^|O(n) ^|O(n) ^|O(n) ^|O(n) ^|O(n)
+| Hash Map (optimized) ^|O(1)* ^|O(n) ^|O(1)* ^|O(1)* ^|O(1)*
+| Tree Map (Red-Black Tree) ^|O(log n) ^|O(n) ^|O(log n) ^|O(log n) ^|O(log n)
+|===
+{empty}* = Amortized run time. E.g. rehashing might affect run time to *O(n)*.
diff --git a/book/chapters/map-hashmap.adoc b/book/chapters/map-hashmap.adoc
@@ -0,0 +1,282 @@
+= HashMap
+
+A HashMap is a Map implementation. HashMaps are composed of two things:
+1) a hash function and
+2) a bucket array to store values.
+
+Before going into the implementation details let’s give an overview of how it works. Let’s say we want to keep a tally of things:
+
+.HashMap example
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=snippet, indent=0]
+----
+
+How are the keys mapped to their values?
+Using a hash function. Here’s an illustration:
+
+.HashMap representation. Keys are mapped to values using a hash function.
+image:image41.png[image,width=528,height=299]
+
+
+.This is the main idea:
+1.  We use a *hash function* to transform the keys (e.g., dog, cat, rat, …) into an array index. This array is called *bucket*.
+2.  The bucket holds the values (linked list in case of collisions).
+
+In the illustration, we have a bucket size of 10. In bucket 0, we have a collision. Both `cat` and `art` keys map to the same bucket even thought their hash codes are different.
+
+In a HashMap, a *collision* is when different keys lead to the same index. They are nasty for performance since it can reduce the search time from *O(1)* to *O(n)*.
+
+Having a big bucket size can avoid a collision but also can waste too much memory. We are going to build an _optimized_ HashMap that re-sizes itself when it is getting full. This avoids collisions and doesn’t spend too much memory upfront. Let’s start with the hash function.
+
+== Designing an optimized hash function
+
+To minimize collisions, we need to create an excellent hash function.
+
+IMPORTANT: A *perfect* hash function is one that assigns a unique array index for every different key.
+
+It’s no practical and memory-wise wasteful to have a perfect hash function, so we are going to shoot for a cost-effective hash function instead.
+
+.To recap:
+- A hash function converts keys into array indices.
+- A hash function is composed of two parts:
+1.  *Hash Code*: maps any key into an integer (unbonded)
+2.  *Compression function*: maps an arbitrary integer to integer in the range of [0… BUCKET_SIZE -1].
+
+Before doing a great hash function, let's see what a lousy hash function looks like. 😉
+
+=== Analysing collisions on bad hash code functions
+
+The goal of a hash code function is to convert any value given into a positive integer — a common way to accomplish with summing each string’s Unicode value.
+
+.Naïve hashing function implementation
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hashing.js[tag=naiveHashCode, indent=0]
+----
+
+
+This function uses `codePointAt` to get the Unicode value of a character. E.g., `a` has a value of 97, `A` is 65, even https://en.wikipedia.org/wiki/Emoji#Unicode_blocks[emojis have codes]; “[big]#😁#” is `128513`.
+
+.JavaScript built-in `string.charCodeAt` vs. `string.codePointAt`
+****
+The `charCodeAt()` method returns an integer between `0` and `65535` representing the UTF-16 code unit at the given index. However, it doesn’t play nice with Unicode, so it’s better to use `codePointAt` instead.
+
+The `codePointAt()` method returns a non-negative integer that is the Unicode code point value.
+****
+With this function we have the can convert some keys to numbers as follows:
+
+.Hashing examples
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hashing.js[tag=naiveHashCodeExamples, indent=0]
+----
+
+Notice that `rat` and `art` have the same hash code! These are collisions that we need to solve.
+
+Collisions happened because we are just summing the character's codes and are not taking the order into account nor the type. We can do better by offsetting the character value based on their position in the string. We can also add the object type, so number `10` produce different output than string `'10'`.
+
+.Hashing function implementation that offset character value based on the position
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hashing.js[tag=hashCodeOffset, indent=0]
+----
+
+Since Unicode uses 20 bits, we can offset each character by 20 bits based on the position.
+
+.JavaScript built-in `BigInt`
+****
+BigInt allows to operate beyond the maximum safe limit of integers (Number.MAX_SAFE_INTEGER => 9,007,199,254,740,991). BigInt uses the suffix n, e.g. 1n + 3n === 4n.
+****
+
+As you can imagine the output is a humongous number! We are using `BigInt` that doesn’t overflow.
+
+.Verifying there's not hashing code duplicates
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hashing.js[tag=hashCodeOffsetExample, indent=0]
+----
+
+As you can see We don’t have duplicates if the keys have different content or type. However, we need to represent these unbounded integers. We do that using *compression function* they can be as simple as `% BUCKET_SIZE`.
+
+However, there’s an issue with the last implementation. It doesn’t matter how humongous is the number if we at the end use the modulus to get an array index. The part of the hash code that truly matters is the last bits.
+
+.Look at this example with a bucket size of 4.
+[source, javascript]
+----
+10 % 4 //↪️ 2
+20 % 4 //↪️ 0
+30 % 4 //↪️ 2
+40 % 4 //↪️ 0
+50 % 4 //↪️ 2
+----
+
+We get many collisions. [big]#😱#
+
+Based on statistical data, using a prime number as the modulus produce fewer collisions.
+
+.Let’s see what happens if the bucket size is a prime number:
+[source, javascript]
+----
+10 % 7 //↪️ 3
+20 % 7 //↪️ 6
+30 % 7 //↪️ 2
+40 % 7 //↪️ 4
+50 % 7 //↪️ 1
+----
+
+Now it’s more evenly distributed!! [big]#😎👍#
+
+.So, to sum up:
+* Bucket size should always be a *prime number*, so data is distributed more evenly and minimized collisions.
+* Hash code doesn’t have to be too big. At the end what matters is the few last digits.
+
+Let’s design a better HashMap with what we learned.
+
+=== Implementing an optimized hash function
+
+Take a look at the following function:
+
+.Optimal Hash function
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=hashFunction, indent=0]
+----
+
+Is somewhat similar to what we did before, in the sense that we use each letter’s Unicode is used to compute the hash. The difference is:
+
+1.  We are using the XOR bitwise operation (^) to produce an *avalanche effect*, where a small change in two strings produces completely different hash codes. E.g.
+
+.Hash Code example using FVN1a
+[source, javascript]
+----
+hashCode('cat') //↪️ 4201630708
+hashCode('cats') //↪️ 3304940933
+----
+
+.Fowler/Noll/Vo (FNV) Hash
+****
+It is a non-cryptographic hash function designed to be fast while maintaining a low collision rate. The high dispersion of the FNV hashes makes them well suited for hashing nearly identical strings such as URLs, keys, IP addresses, zip codes, and others.
+****
+
+We are using the FVN-1a prime number (16777619) and offset (2166136261) to reduce collisions even further. If you are curious where these numbers come from check out this https://en.wikipedia.org/wiki/Fowler%E2%80%93Noll%E2%80%93Vo_hash_function[link].
+
+FVN-1a hash function is a good trade-off between speed and collision prevention.
+
+Now that we have a proper hash function. Let’s move on with the rest of the HashMap implementation.
+
+== Implementing a HashMap in JavaScript
+
+Let’s start by creating a class and its constructor to initialize the hash map. We are going to have an array called *buckets* to hold all the data as below:
+
+.HashMap's constructor
+[source, javascript]
+----
+class HashMap {
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=constructorPartial, indent=2]
+    this.buckets = new Array(this.initialCapacity);
+    this.size = 0;
+    this.collisions = 0;
+  }
+
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=getLoadFactor, indent=2]
+}
+----
+
+Notice that we are also keeping track of collisions (just for benchmarking purposes) and a load factor. *The load factor* measures how full the hash map is. We don’t want to be fuller than 75%. If the HashMap is getting too full, then we are going to fix it doing a *rehash* (more on that later).
+
+=== Inserting elements in a HashMap
+
+To insert values into a HashMap, we first convert the *key* into *an array index* using the hash function. Each bucket of the array will have an object `{key, value}`.
+
+There are multiple scenarios for inserting key/values in a HashMap:
+
+1.  Key doesn’t exist yet, so we create the new key/value pair.
+2.  Key already exists, then we will replace the value.
+3.  Key doesn’t exist, but the bucket already has other data, this is a collision! We push the new element to the bucket.
+
+In code, it looks like this:
+
+.HashMap's set method
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=set, indent=0]
+----
+
+Notice, that we are using a function called `getEntry` to check if the key already exists. It gets the index of the bucket corresponding to the key and then checks if the entry with the given key exists. We are going to implement this function in a bit.
+
+=== Getting values out of a HashMap
+
+For getting values out of the Map, we do something similar to inserting. We convert the key into an index using the hash function.
+
+.HashMap's getEntry method
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=getEntry, indent=0]
+----
+<1> Convert key to an array index.
+<2> If the bucket is empty create a new linked list
+<3> Use Linked list's <<Searching by value>> method to find value on the bucket.
+<4> Return bucket and entry if found.
+
+With the help of the `getEntry` method, we can do the `HashMap.get` and `HashMap.has` methods:
+
+.HashMap's get method
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=get, indent=0]
+----
+
+and also,
+
+.HashMap's has method
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=has, indent=0]
+----
+
+For `HashMap.has` we only care if the value exists or not, while that for `HashMap.get` we want to return the value or `undefined` if it doesn’t exist.
+
+=== Deleting from a HashMap
+
+Removing items from a HashMap is not too different from what we did before:
+
+.HashMap's delete method
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=delete, indent=0]
+----
+
+If the bucket doesn’t exist or is empty, we don't have to do anything else. If the value exists, we use the
+https://github.com/amejiarosario/dsa.js/blob/master/src/data-structures/linked-lists/linked-list.js[`LinkedList.remove` ]
+method.
+
+== Rehashing the HashMap
+
+Rehashing is a technique to minimize collisions when a hash map is getting full. It doubles the size of the map and recomputes all the hash codes and insert data in the new bucket.
+
+When we increase the map size, we try to find the next prime. We explained that keeping the bucket size a prime number is beneficial for minimizing collisions.
+
+.HashMap's rehash method
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/hash-maps/hash-map.js[tag=rehash, indent=0]
+----
+
+In the
+https://github.com/amejiarosario/algorithms.js/blob/master/src/data-structures/hash-maps/primes.js[prime.js] file you can find the implementation for finding the next prime. Also, you can see the full HashMap implementation on this file: https://github.com/amejiarosario/algorithms.js/blob/master/src/data-structures/hash-maps/hashmap.js[hashmap.js]
+
+== HashMap time complexity
+
+Hash Map it’s very optimal for searching values by key in constant time *O(1)*. However, searching by value is not any better than an array since we have to visit every value *O(n)*.
+
+.Time complexity for a Hash Map
+|===
+.2+.^s| Data Structure 2+^s| Searching By .2+^.^s| Insert .2+^.^s| Delete .2+^.^s| Space Complexity
+^|_Index/Key_ ^|_Value_
+| Hash Map (naïve) ^|O(n) ^|O(n) ^|O(n) ^|O(n) ^|O(n)
+| Hash Map (optimized) ^|O(1)* ^|O(n) ^|O(1)* ^|O(1)* ^|O(1)*
+|===
+{empty}* = Amortized run time. E.g. rehashing might affect run time.
+
+As you can notice we have amortized times since, in the unfortunate case of a rehash, it will take O(n) while it resizes. After that, it will be on average *O(1)*.
diff --git a/book/chapters/map-intro.adoc b/book/chapters/map-intro.adoc
@@ -0,0 +1,18 @@
+= Map
+
+A map is a data structure to store pairs of data: *key* and *value*. In an array, you can only store values. The array’s key is always the position index. However, in a *Map* the key can be whatever you want.
+
+IMPORTANT: Map is a data structure that _maps_ *keys* to *values*.
+
+Many languages have maps already built-in. JavaScript/Node has `Map`:
+
+.JavaScript Built-in Map Usage
+[source, javascript]
+----
+include::{codedir}/data-structures/maps/map.js[tag=snippet, indent=0]
+----
+
+The attractive part of Maps is that they are very performant usually *O(1)* or *O(log n)* depending on the implementation. We can implement the maps using two different techniques:
+
+* *HashMap*: it’s a map implementation using an *array* and *hash function*. The job of the hash function is to convert the key into an index that contains the matching data. Optimized HashMap can have an average runtime of *O(1)*.
+* *TreeMap*: it’s a map implementation that uses a self-balanced Binary Search Tree (red-black tree). The BST nodes store the key, and the value and nodes are sorted by key guaranteeing an *O(log n)* look up.
diff --git a/book/chapters/map-treemap.adoc b/book/chapters/map-treemap.adoc
diff --git a/book/chapters/map.adoc b/book/chapters/map.adoc

-Original file line number
+Diff line change
@@ @@ -0,0 +1,88 @@ @@
 += TreeMap
++
 +A TreeMap is a Map implementation using Binary Search Trees.
++
 +Implementing a Map with a tree, TreeMap, has a couple of advantages over a HashMap:
++
 +* Keys are always sorted.
 +* Statistical data can be easily obtained like the median, highest, lowest key.
 +* Collisions are not a concern so in the worst case is still *O(log n)*.
 +* Trees are more space efficient and don’t need to allocate memory beforehand (e.g. `HashMap`’s initial capacity) nor you have to rehash when is getting full.
++
 +Ok, now that you know the advantages, let’s implement it!
 +For a full comparison read the <<HashMap vs TreeMap>> section.
++
 +Let’s get started with the essential functions. They have the same interface as the `HashMap` (but the implementation is different).
++
 +.TreeMap class overview
 +[source, javascript]
 +----
 +class TreeMap {
 +  constructor(){}
 +  set(key, value) {}
 +  get(key) {}
 +  has(key) {}
 +  delete(key) {}
 +}
 +----
++
 +== Inserting values into a TreeMap
++
 +For inserting a value on a TreeMap, we first need to inialize the tree:
++
 +.TreeMap constructor
 +[source, javascript]
 +----
 +include::{codedir}/data-structures/maps/tree-maps/tree-map.js[tag=constructor, indent=0]
 +----
++
 +The tree can be an instance of any Binary Search Tree that we implemented so far. However, for better performance, it should be a self-balanced tree like a https://github.com/amejiarosario/algorithms.js/blob/master/src/data-structures/trees/red-black-tree.js[Red-Black Tree] or https://github.com/amejiarosario/algorithms.js/blob/master/src/data-structures/trees/avl-tree.js[AVL Tree].
++
++
 +Let's implement the method to add values to the tree.
++
 +.TreeMap `add` method and `size` attribute
 +[source, javascript]
 +----
 +include::{codedir}/data-structures/maps/tree-maps/tree-map.js[tag=set, indent=0]
 +----
++
 +Adding values is very easy (once we have the underlying tree implementation).
++
 +== Getting values out of a TreeMap
++
 +When We search by key in a tree map, it takes *O(log n)*. This is the implementation:
++
 +.TreeMap `get` and `has` method
 +[source, javascript]
 +----
 +include::{codedir}/data-structures/maps/tree-maps/tree-map.js[tag=get, indent=0]
 +----
++
 +One side effect of storing keys in a tree is that they don't come up in insertion order. Instead, they ordered by value.
++
 +.TreeMap iterators
 +[source, javascript]
 +----
 +include::{codedir}/data-structures/maps/tree-maps/tree-map.js[tag=iterators, indent=0]
 +----
 +<1> We implemented the default iterator using the in-order traversal. That's useful for getting the keys sorted.
++
 +.JavaScript Iterators and Generators
 +****
 +Generators are useful for producing values that can you can iterate in a `for...of` loop. Generators use the `function*` syntax which expects to have a `yield` with a value.
 +****
++
 +== Deleting values from a TreeMap
++
 +Removing elements from TreeMap is simple.
++
 +.TreeMap `delete` method
 +[source, javascript]
 +----
 +include::{codedir}/data-structures/maps/tree-maps/tree-map.js[tag=delete, indent=0]
 +----
++
 +The BST implementation does all the heavy lifting.
++
 +That’s it! To see the full file in context, click here: https://github.com/amejiarosario/algorithms.js/blob/master/src/data-structures/maps/tree-maps/tree-map.js[here]