feat: Add Bitmask DP TSP implementation

[YashKumar-404]

Bitmask Dynamic Programming: Traveling Salesperson Problem (TSP)

Overview

This implementation demonstrates how to use Bitmask Dynamic Programming to solve the Traveling Salesperson Problem (TSP).

The Traveling Salesperson Problem asks: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?"

What is Bitmask DP?

When solving optimization problems that involve subsets or combinations of elements (where the order/inclusion matters), tracking the "visited" state using standard arrays or objects can be highly inefficient.

Bitmask DP optimizes this by representing a subset of elements as bits in an integer:

If the $i$-th bit is 1, the $i$-th element is included in the subset.

If the $i$-th bit is 0, the $i$-th element is excluded.

Core Bitwise Operations Used:

Add an element to the subset: mask | (1 << i)

Check if an element is in the subset: (mask & (1 << i)) != 0

Because an integer is typically 32 bits, this technique is incredibly fast and memory-efficient but is generally restricted to problem sizes where $N \le 20$.

Algorithm Complexity

For a graph with $N$ vertices:

Time Complexity: $O(N^2 \cdot 2^N)$

There are $2^N$ possible subsets. For each subset, we iterate through $N$ currently visited nodes, and for each, we try to visit $N$ unvisited nodes. This is a massive improvement over the naive $O(N!)$ brute-force approach.

Space Complexity: $O(N \cdot 2^N)$

We use a 2D array dp[1 << N][N] to store the minimum cost for each subset and ending node.

Example

If we have 4 cities (0, 1, 2, 3), a subset containing cities 0, 2, and 3 is represented by the binary number 1101 (which is the integer 13).

The DP state dp[13][2] stores the minimum distance traveled to visit cities {0, 2, 3} where the last visited city is 2.

Transitions are made by checking unvisited cities (like city 1), adding them to the mask (1101 | 0010 = 1111), and updating the minimum cost.

How to Run & Test

If you have cloned the repository locally, you can run the JUnit tests for this specific implementation using Maven:

mvn test -Dtest=BitmaskDPTest

Resolves #7503

[codecov-commenter]

Codecov Report

❌ Patch coverage is 0% with 22 lines in your changes missing coverage. Please review.
✅ Project coverage is 80.16%. Comparing base (12aab70) to head (fe27f6a).

Files with missing lines	Patch %	Lines
...om/thealgorithms/dynamicprogramming/BitmaskDP.java	0.00%	22 Missing ⚠️

Additional details and impacted files

@@             Coverage Diff              @@
##             master    #7505      +/-   ##
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- Coverage     80.23%   80.16%   -0.07%     
+ Complexity     7354     7352       -2     
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  Files           810      811       +1     
  Lines         23785    23807      +22     
  Branches       4678     4686       +8     
============================================
+ Hits          19083    19084       +1     
- Misses         3942     3963      +21     
  Partials        760      760              

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