## Google Interview Question

Software Engineers**Country:**United States

Here is a relatively simple O(n^2) solution that iterates over pairs of horizontal and vertical line segments. Not sure how to improve the solution.

```
# -*- coding: utf-8 -*-
from typing import List, Tuple, Optional, Dict
Segment = Tuple[str, int, int, int]
# line segment is represented by ({V|H}, intercept, start, end), where
# 1. {V|H} represented where the line is Vertical or Horizontal,
# 2. intercept is the x-intercept for vertical line segment, and the y-intercept
# for the horizontal line segment.
# 3. start is the y-axis start of the line segment for vertical, and the x-axis
# start for the horizontal line segment.
# 4. end is analogous to start
# sort the line segments into vertical and horizontal, and sort them by their
# intercepts.
def sort_line_segments(line_segments: List[Segment]) -> Tuple[List[Segment], List[Segment]]:
vertical_segments = list(filter(lambda s: s[0] == 'V', line_segments))
horizontal_segments = list(filter(lambda s: s[0] == 'H', line_segments))
return (sorted(vertical_segments), sorted(horizontal_segments))
# stores the line segments in a dict keyed by its intercept.
def create_segment_dict(segments: List[Segment]) -> Dict[Tuple[str, int], Segment]:
return {(s[0], s[1]): s for s in segments}
def get_intersection(v_segment: Segment, h_segment: Segment) -> Optional[Tuple[int, int]]:
# assuming the two are vertical and horizontal segments.
if ((h_segment[2] <= v_segment[1] <= h_segment[3]) and
(v_segment[2] <= h_segment[1] <= v_segment[3])):
return (v_segment[1], h_segment[1])
return None
import itertools
def count_squares(segments: List[Segment]) -> int:
square_count = 0
v_segments, h_segments = sort_line_segments(segments)
v_dict = create_segment_dict(v_segments)
v_len = len(v_segments)
h_len = len(h_segments)
for (v_index, h_index) in itertools.product(range(v_len), range(h_len)):
v_segment = v_segments[v_index]
h_segment = h_segments[h_index]
intersection = get_intersection(v_segment, h_segment)
# print(f"intersection is {intersection}")
if intersection is None:
continue
# we have an intersection point. Check if we have another horizontal segment
# that intersects with v_segments[v_index]
for h_index2 in range(h_index+1, h_len):
h_segment2 = h_segments[h_index2]
intersection2 = get_intersection(v_segment, h_segment2)
# print(f"intersection2 is {intersection2}")
if intersection2 is None:
continue
# We have a second intersection point on the vertical line segment.
# length of the line segments between these two points is the length
# of a square, it exists.
square_length = h_segment2[1] - h_segment[1]
# print(f"Square_length is {square_length}")
# The vertices of the square are `intersection` and `intersection2` on
# the same vertical line segment.
# If a square exists, then there must be another vertical line segment
# ('V', v_segment[1] + square_length, <=h_segment[1], >= h_segment2[1])
if ('V', v_segment[1] + square_length) not in v_dict:
continue
v_segment2 = v_dict[('V', v_segment[1] + square_length)]
# print(f"v2 is {v_segment2}")
if (v_segment2[2] <= h_segment[1]) and (v_segment2[3] >= h_segment2[1]):
# Found a square
# print(f" Found {square_length} square {v_segment}, {v_segment2}, {h_segment}, {h_segment2}")
square_count += 1
return square_count
segments = [('V', 0, 2, 8),
('H', 3, 0, 2),
('V', 2, 3, 7),
('H', 5, -1, 3)]
assert count_squares(segments) == 1
segments = [
('V', 10, 5, 15), ('V', 10, 15, 25),
('V', 20, 5, 15), ('V', 20, 15, 25),
('H', 10, 5, 15), ('H', 10, 15, 25),
('H', 20, 5, 15), ('H', 20, 15, 25)
]
assert count_squares(segments) == 0
segments = [
('V', 5, 0, 20), ('V', 10, 0, 20), ('V', 15, 0, 20), ('V', 20, 0, 20),
('H', 5, 0, 25), ('H', 10, 0, 25)
]
assert count_squares(segments) == 3
segments = [
('V', 5, 0, 20), ('V', 10, 0, 20), ('V', 15, 0, 20), ('V', 20, 0, 20),
('H', 5, 0, 25), ('H', 10, 0, 25), ('H', 20, 0, 25)
]
assert count_squares(segments) == 6
```

Complexity - O(mn)

M: Number of vertical lines

N: Number of horizontal lines

```
class Line():
def __init__(self, y1, y2, x1, x2, direction):
self.y1 = y1
self.y2 = y2
self.x1 = x1
self.x2 = x2
self.direction = direction
def getLength(self):
return int(((self.x1 - self.x2)**2 + (self.y1 - self.y2)**2)**0.5)
def matchLinePairs(linePairOne, linePairTwo):
lineOne, lineTwo = linePairOne
lineThree, lineFour = linePairTwo
if lineOne.getLength() == lineThree.getLength():
if (lineOne.x1, lineOne.y1) == (lineThree.x1, lineThree.y1) or (lineOne.x1, lineOne.y1) == (lineThree.x2, lineThree.y2):
return True
return False
def numberOfSquares(aListOfVertLines, aListOfHorLines):
horPairs = []
verPairs = []
for i in range(len(aListOfHorLines)):
line1 = aListOfHorLines[i]
for j in range(i, len(aListOfHorLines)):
line2 = aListOfHorLines[j]
if line1.getLength() == line2.getLength():
if line1.y1 == line2.y1 and line1.y2 == line2.y2:
horPairs.append((line1, line2))
for i in range(len(aListOfVertLines)):
line1 = aListOfVertLines[i]
for j in range(i, len(aListOfVertLines)):
line2 = aListOfHorLines[j]
if line1.getLength() == line2.getLength():
if line1.x1 == line2.x1 and line1.x2 == line2.x2:
verPairs.append((line1, line2))
numSquares = 0
for horLinePair in horPairs:
for verLinePair in verPairs:
if matchLinePairs(horLinePair, verLinePair):
numSquares+=1
return numSquares
```

Hi All,

- uj.us October 07, 2019Question is about line segments not the lines. Each line segment is represented with the start and their end points and the intercept point on the axis. And another boolean to represent if the line segment is horizontal or vertical.

Example: 10 20 5 V represents a line which is vertical and its x-intercept is 5 and start from 10 and ends at 20 ie., points are (5, 10) (5, 20) represents the line