Partial lesson 2019-04-09

notes.md

@@ -143,7 +143,7 @@ def right(x):
 
 **Max heap property**: for all i > 1 A[parent(i)] >= A[i]
 
-# Data structures
+# Some data structures
 
 Way to organize information
 
@@ -160,7 +160,7 @@ A data structure has data and meta-data (like size, length).
 
 ## Queue (FIFO)
 
-## Structure
+### Structure
 
 - Based on array
 - `length`
@@ -175,7 +175,7 @@ Data structure for fast search
 
 A way to implement a dictionary is a *Direct-access table*
 
-## API of dictionary
+### API of dictionary
 
 - `Insert(D, k)` insert a ket `k` to dictionary `D`
 - `Delete(D, k)` removes key `k`
@@ -183,12 +183,14 @@ A way to implement a dictionary is a *Direct-access table*
 
 Many different implementations
 
-## Direct-access table
+# Direct-access tables
 
 - universe of keys = {1,2,...,M}
 - array `T` of size M
 - each key has its own position in T
 
+## The 'dumb' approach
+
 ```python
 
 def Insert(D, x):
@@ -228,3 +230,38 @@ def Chained_hash_search(T, k):
   return List_search(T[hash(k)], k)
 
 ```
+
+Elements are spreaded evenly across the table if the hash function is good.
+
+*alpha* = *n / |T|*
+
+is the average-case average length of the linked lists inside
+the table (where n is the number of elements in the table and *|T|* is the size
+of the table.
+
+A good hash table implementation makes the complexity of *alpha* O(1).
+If *n*, the number of elements that we want to store in the hash table, grows,
+then *|T|* must also grow. *alpha* represents the time complexity of both
+insertion and search.
+
+## Growing a Chained hash table
+
+In order to grow a table, a new table must be created. The hash function (or its
+range parameters) must be changed as well.
+
+### Rehashing
+
+*Rehashing* is the process of putting all the elements of the old table in the new
+table according to the new hash function. The complexity is O(n), since
+`Chained-hash-insert` is constant.
+
+### Growing the table every time
+
+If the table is grown by a constant factor every time the table *overflows*,
+then the complexity of insertion is O(n^2) due to all the *rehashing* needed.
+
+### Growing the table by doubling the size
+
+If the table size is doubled when *overfloiwng*, then the complexity for
+insertion becomes linear again by sacrificing some memory complexity.
+