Counting Potential Attack Vectors in Network Keys Based on Queries
Problem Understanding:
We are given an array keys
and a 2D array queries
. We need to find potential attack vectors (i, j, x)
where keys[i]
and keys[j]
are divisible by x
. Additionally, each query in queries
specifies a range [a, b]
within which x
should fall.
Initial Breakdown:
 Loop through
queries
to handle each query individually.  Loop through
keys
to identify pairs(i, j)
where both elements are divisible byx
.  Filter the pairs based on the query range.
Subproblem Refinement:
 Parse a query
[a, b]
to identify the range forx
.  Generate all possible pairs
(i, j)
fromkeys
.  Check if
keys[i]
andkeys[j]
are divisible byx
, wherex
lies within[a, b]
.
Task Identification:
 Parse a query to identify the range
[a, b]
.  Generate all possible pairs
(i, j)
.  Check divisibility of
keys[i]
andkeys[j]
byx
.  Check if
x
lies within[a, b]
.
Task Abstraction:
 Parsing the query: Extract
a
andb
.  Pair generation: Loop twice through
keys
array.  Divisibility Check: Use the modulus operation.
 Range Check: Use conditional statements.
Method Naming:
 parseQuery
 generatePairs
 checkDivisibility
 validateRange
Subproblem Interactions:
 Parsing the query precedes all other operations.
 Pair generation can occur in parallel with range validation.
 Divisibility checks depend on the generated pairs and the query range.
Code:
Here is the code:


This code has a time complexity of O(n^3 * q) which is not optimal for larger arrays.
Optimization Steps:
 Precompute all pairs of divisible keys to avoid recomputation for each query.
 Use a more efficient data structure to store precomputed pairs for faster lookup.
By precomputing the divisible pairs, you could reduce the time complexity substantially, perhaps to O(n^2 + q).
Optimization Step 1: Reduce the Number of Iterations for x
In the original code, the innermost loop iterates through the range [a, b]
for each query. This is inefficient. A first step in optimization is to precompute the factors for all keys, so that you can look them up instead of calculating them during each query.
Here’s how the code can be revised:


How the Optimization Works
The
factors_dict
dictionary holds the count of keys that can be divided by each factor. This avoids having to compute the factors within each query, thus saving time.In each query, you only iterate through the range
[a, b]
once, checking the precomputedfactors_dict
.
This change significantly reduces the number of iterations, thereby improving the algorithm’s time complexity.
Optimization Step 2: Precalculate the Number of Pairs
After calculating the factors for each key, we can also precalculate the number of pairs (i, j)
that can be formed by each factor, further reducing the need for nested loops in the query phase.
Here’s the updated code:


How the Second Optimization Works
The
pairs_count
dictionary precalculates the number of pairs(i, j)
that can be formed by each factor. This is done by simply squaring the count of keys that can be divided by that factor. This avoids having to iterate through all(i, j)
pairs for each query.During each query, the code now only iterates through the range
[a, b]
once, using bothfactors_count
andpairs_count
.
With both optimizations, the algorithm should be considerably faster and more efficient.