• Sorting
  • Quicksort
  • Heapsort
• Data Structures
  • Red-Black Trees
  • Augmenting data structures
  • B-Trees
  • Disjoint sets
  • Trie
  • Segment Tree
• Graph
  • Topological sort
  • Minimum spanning tree
  • Single-Source Shortest Paths
  • All-Pairs Shortest Paths
  • Maximum flow
• Selected topics
  • Linear programming
  • String matching
  • Computational geometry
• Miscellaneous
  • Permutations & combinations
  • Selection rank
  • Newton-Raphson algorithm
  • PSO algorithm
• To be continued …

Sorting

Quicksort

def quicksort(self, nums):
    if len(nums) <= 1:
        return nums

    pivot = random.choice(nums)
    lt = [v for v in nums if v < pivot]
    eq = [v for v in nums if v == pivot]
    gt = [v for v in nums if v > pivot]

    return self.quicksort(lt) + eq + self.quicksort(gt)

Heapsort

• The (binary) heap data structure is an array object that we can view as a nearly complete binary tree (except possibly the lowest)

• sort in place

• runtime $O(n \lg n)$, height $\Theta(\lg n)$

• index of array starts at 1.

• def parent(i):
      return i // 2
    
  def left(i):
      return 2 * i
    
  def right(i):
      return 2 * i + 1
  
  def max_heapify(A, i):
      l = left(i)
      r = right(i)
      if l <= A.heap_size and A[l] > A[i]:
          largest = l
      else:
          largest = i
      if r <= A.heap_size and A[r] > A[largest]:
          largest = r
      if largest != i:
          A[i], A[largest] = A[largest], A[i]
          max_heapify(A, largest)
            
            
  def build_max_heap(A):
      A.heap_size = len(A)
      for i in range(len(A) // 2, 0, -1):
          max_heapfiy(A, i)
  

Data Structures

Red-Black Trees

Augmenting data structures

B-Trees

Disjoint sets

Trie

Segment Tree

Graph

Topological sort

Minimum spanning tree

Single-Source Shortest Paths

All-Pairs Shortest Paths

Maximum flow

Selected topics

Linear programming

String matching

Computational geometry

Miscellaneous

Permutations & combinations

Selection rank

• similar to quick sort

Newton-Raphson algorithm

PSO algorithm

To be continued …