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Algorithm Notes

This project reconstructs several assignments from an undergraduate Algorithm Design and Analysis course into a consistent, testable C++17 codebase.

1. Longest Increasing Subsequence — Dynamic Programming

For each position i, define dp[i] as the length of the longest strictly increasing subsequence ending at i.

dp[i] = 1 + max(dp[j]) for every j < i with a[j] < a[i]

The implementation also stores a predecessor index so one valid optimal subsequence can be reconstructed.

  • Time: O(n^2)
  • Space: O(n)

2. Inversion Count — Divide and Conquer

A pair (i, j) is an inversion when i < j but a[i] > a[j]. During merge sort, whenever an element from the right half is placed before an unmerged element from the left half, all remaining left-half elements form inversions with it.

  • Time: O(n log n)
  • Space: O(n)

The coursework version contained a merge-step pointer bug; this portfolio version corrects it and adds regression tests.

3. 0-1 Knapsack — Exhaustive Backtracking

Each item creates two branches: include or exclude. In the worst case this explores the full binary decision tree.

  • Exact: yes
  • Worst-case time: O(2^n)
  • Search statistic: nodes visited

4. 0-1 Knapsack — Branch and Bound

Items are ordered by value density. At each search node, a fractional-knapsack relaxation estimates an optimistic upper bound for the best value still reachable. If that bound cannot beat the incumbent solution, the subtree is pruned.

  • Exact: yes
  • Worst-case time: exponential
  • Practical improvement: fewer explored nodes when the bound is informative

5. 0-1 Knapsack — Genetic Algorithm

A chromosome is a binary vector indicating selected items. The implementation uses:

  • random population initialization;
  • fitness equal to total value for feasible solutions;
  • elitism;
  • one-point crossover;
  • bit mutation;
  • deterministic seeds for reproducible experiments.

This is a heuristic method and does not guarantee the global optimum.

6. Maximum Clique — Branch and Bound

The search incrementally constructs a clique. Candidate vertices are filtered to those adjacent to every vertex already selected. If current_size + candidates_remaining cannot exceed the incumbent clique size, the branch is pruned.

  • Exact: yes
  • Worst-case time: exponential
  • Search statistics: nodes visited and branches pruned