Algorithms

title image

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 …

Written on June 1, 2019