# Algorithms Exercise 2

## Imports



In [1]:

%matplotlib inline
from matplotlib import pyplot as plt
import seaborn as sns
import numpy as np



## Peak finding

Write a function find_peaks that finds and returns the indices of the local maxima in a sequence. Your function should:

• Properly handle local maxima at the endpoints of the input array.
• Return a Numpy array of integer indices.
• Handle any Python iterable as input.


In [2]:

def find_peaks(a):
"""Find the indices of the local maxima in a sequence."""
peaks = []
for i in range(len(a)):
if i == 0 and a[i] > a[i+1]:
peaks.append(i)
elif i == (len(a)-1) and a[i] > a[i-1]:
peaks.append(i)
elif a[i] > a[i-1] and a[i] > a[i+1]:
peaks.append(i)
return np.array(peaks)




In [3]:

find_peaks([2,0,1,0,2,0,1])




Out[3]:

array([0, 2, 4, 6])




In [4]:

p1 = find_peaks([2,0,1,0,2,0,1])
assert np.allclose(p1, np.array([0,2,4,6]))
p2 = find_peaks(np.array([0,1,2,3]))
assert np.allclose(p2, np.array([3]))
p3 = find_peaks([3,2,1,0])
assert np.allclose(p3, np.array([0]))



Here is a string with the first 10000 digits of $\pi$ (after the decimal). Write code to perform the following:

• Convert that string to a Numpy array of integers.
• Find the indices of the local maxima in the digits of $\pi$.
• Use np.diff to find the distances between consequtive local maxima.
• Visualize that distribution using an appropriately customized histogram.


In [5]:

from sympy import pi, N
pi_digits_str = str(N(pi, 10001))[2:]




In [6]:

a = pi_digits_str
b = np.array([int(a[i]) for i in range(len(a))])




In [8]:

plt.figure(figsize=(12,6))
plt.hist(np.diff(find_peaks(b)), bins=50)
plt.xlim(0,18)
plt.xlabel("Distance")
plt.ylabel("Occurences")
plt.title("Distance Between Consecutive Maxima for First 10,000 Decimals of Pi")
plt.show()







In [ ]:

assert True # use this for grading the pi digits histogram