01/25/21

# Friend Circle Queries

##### Short Problem Definition:

Friend Circle Queries

##### Complexity:

time complexity is O(n(logq+logn))

space complexity is O(N)

##### Execution:

This is a typical Union-find problem statement. The particular UF implementation I use does not track groups and their sizes. The best way to determine the current largest group is to assume that the group that was just merged is the largest one.

##### Solution:
```#!/bin/python

import math
import os
import random
import re
import sys
from collections import defaultdict

"""UnionFind.py

Union-find data structure. Based on Josiah Carlson's code,
<a class="vglnk" href="http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/215912" rel="nofollow"><span>http</span><span>://</span><span>aspn</span><span>.</span><span>activestate</span><span>.</span><span>com</span><span>/</span><span>ASPN</span><span>/</span><span>Cookbook</span><span>/</span><span>Python</span><span>/</span><span>Recipe</span><span>/</span><span>215912</span></a>
with significant additional changes by D. Eppstein.
"""

class UnionFind:
"""Union-find data structure.

Each unionFind instance X maintains a family of disjoint sets of
hashable objects, supporting the following two methods:

- X[item] returns a name for the set containing the given item.
Each set is named by an arbitrarily-chosen one of its members; as
long as the set remains unchanged it will keep the same name. If
the item is not yet part of a set in X, a new singleton set is
created for it.

- X.union(item1, item2, ...) merges the sets containing each item
into a single larger set.  If any item is not yet part of a set
in X, it is added to X as one of the members of the merged set.
"""

def __init__(self):
"""Create a new empty union-find structure."""
self.weights = {}
self.parents = {}

def __getitem__(self, object):
"""Find and return the name of the set containing the object."""

# check for previously unknown object
if object not in self.parents:
self.parents[object] = object
self.weights[object] = 1
return object

# find path of objects leading to the root
path = [object]
root = self.parents[object]
while root != path[-1]:
path.append(root)
root = self.parents[root]

# compress the path and return
for ancestor in path:
self.parents[ancestor] = root
return root

def __iter__(self):
"""Iterate through all items ever found or unioned by this structure."""
return iter(self.parents)

def union(self, *objects):
"""Find the sets containing the objects and merge them all."""
roots = [self[x] for x in objects]
heaviest = max([(self.weights[r],r) for r in roots])[1]
for r in roots:
if r != heaviest:
self.weights[heaviest] += self.weights[r]
self.parents[r] = heaviest

# Complete the maxCircle function below.
def maxCircle(queries):
uf = UnionFind()

largest_element = 0
result = []

for query in queries:
uf.union(query[0], query[1])
current_grouping = uf.weights[uf[query[0]]]
largest_element = max(largest_element, current_grouping)
result.append(largest_element)

return result

if __name__ == '__main__':
fptr = open(os.environ['OUTPUT_PATH'], 'w')

q = int(raw_input())

queries = []

for _ in xrange(q):
queries.append(map(int, raw_input().rstrip().split()))

ans = maxCircle(queries)

fptr.write('\n'.join(map(str, ans)))
fptr.write('\n')

fptr.close()

```

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01/24/21

# HackerRank ‘Game of Maximization’ Solution

##### Short Problem Definition:

There are n piles of stones, where the ith pile has ai stones. You need to collect the maximum number of stones from these piles

Stones Problem

##### Complexity:

time complexity is O(N)

space complexity is O(1)

##### Execution:

This challenge is fairly simple. It does not matter how those stones are arranged. The only important bit is that either the sum of all even elements is bigger than the sum of all odd elements or vice versa.

##### Solution:
```def maximumStones(arr):
return 2*min(sum(arr[0::2]), sum(arr[1::2]))
```

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