amolnayak311 / algorithms-illuminated Goto Github PK

There is a typo in Line 52 in the equation in the introduction.ipynb file in the discussion of the recursive implementation of multiplying two integer numbers. The original equation is (Line 52):
x.y = (10^ {n/2}.a + b) \times (10^ {n/2}.c + d) = 10^n.ab + 10^{n/2}.(a.d + b.c) + b.d

The correct equation should be:
x.y = (10^ {n/2}.a + b) \times (10^ {n/2}.c + d) = 10^n.ac + 10^{n/2}.(a.d + b.c) + b.d

I think the solution to problem 2.1 is wrong.
I believe the crux of the issue is that f(n) = O(g(n)) doesn't imply lg f(n) = O(lg g(n)).

A counterexample is f(n) = 2 and g(n) = 1 for all n, then f(n) lg f(n) equals 2 and g(n) lg g(n) equals 0 and we cannot bound f(n) lg f(n) from above by g(n) lg g(n).
If taking constant functions feels funny, try f(n) = 2 - exp(-n) and g(n) = 1 + exp(-n).

Hi,

I tried to decrease in a matrix a value -1 - I made it -3, and result is giving me 9 as a local minima, which is surely not local minima:

import numpy as np

# Assumption is that the matrix has unique numbers
def localminima1(matrix):
    m = matrix.shape[1] // 2  # Num cols
    i_minval = np.argmin(matrix[:, m])  # Done in O(n)
    min_val = matrix[i_minval, m]
    # Comparison done in constant time, the recursive calls would be done log n times
    left_val = matrix[i_minval, m - 1] if m > 0 else float("inf")
    right_val = matrix[i_minval, m + 1] if m < matrix.shape[1] - 1 else float("inf")
    if min_val < left_val and min_val < right_val:
        return (m, i_minval)
    else:
        return (
            localminima1(matrix[:, 0:m])
            if left_val < min_val
            else localminima1(matrix[:, (m + 1) :])
        )

# fmt:off
array = np.matrix([
    [1, 2, 5, 4], 
    [0, -2, 9, 6], 
    [7, -3, 3, 10], 
    [11, 12, 13, 14]
])
# fmt:on
print("Input is\n", array)
ix = localminima1(array)
print("\nA local minima is at", ix, "with value", array[ix])

Result:

[[ 1 2 5 4]
[ 0 -2 9 6]
[ 7 -3 3 10]
[11 12 13 14]]

A local minima is at (1, 2) with value 9

Hi,

What program did you use to make those diagrams and images?

ConventionalMultiplication.png
ConventionalMultiplication.png
MergeSortIllustration.png

Thanks

The justification to problem 1.3 isn't squaring up for me:

For n=2 and k=4, (2*4(4 - 1)/ 2) = 12. But here's what I expect:
merge 1: 2 + 2
merge 2: 4 + 2
merge 3: 6 + 2
total: 18 operations

What gives? Thank you in advance!