Algorithm Design Laboratory with Applications

Stefano Leucci
Academic Year 2023/2024

Schedule

Office hours: Thursday 16:30 - 18:30. Please send me an email or ask before/after the lectures.

Lectures and Material

Introduction and Prefix Sums

Course overview.
Introduction to laboratory exercises: exercises format, writing, compiling, evaluating, and debugging a solution. Programming tips, assertions.
The STL library: overview, basic types, containers (arrays, vectors, deques, queues, stacks, (muti-)sets, priority queues), iterators, algorithms: (heaps and sorting, linear search, binary search, deleting and replacing elements).
The prefix sums technique. Counting the number of contiguous subsequences with even total sum: cubic-, quadratic-, and linear-time solutions.

Material

Sorting, Binary Searching, and Sliding Window

Sorting and binary searching with predicates. Computing good upper bounds using exponential search.
The sliding window technique. Example with cubic-, quadratic-, almost linear-, and linear-time solutions.

Material

Greedy algorithms

Greedy algorithms. Interval scheduling: problem definition, the earliest finish time algorithm, proof of correctness (greedy stays ahead).
Interval partitioning: problem definition, greedy algorithm, proof of correctness (using structural properties).
Scheduling jobs with deadlines to minimize lateness. The Earliest Deadline First algorithm. Proof of correctness through an exchange argument.

Material

Divide and Conquer, Memoization, and Dynamic Programming

The divide and conquer technique and the polynomial multiplication problem.
Recursion and memoization: computing Fibonacci numbers recursively in linear-time.
Introduction to dynamic programming. A trivial example: computing Fibonacci numbers iteratively. The Longest Increasing Subsequence problem: a O(n^2) algorithm and a O(n log n) algorithm.

Material

More Dynamic Programming

Maximum-weight independent set on paths (linear-time algorithm), Maximum-weight independent set on trees (linear-time algorithm), Maximum weight independent set on trees with cardinality constraints. Counting the number of ways to distribute budget with stars and bars. Dynamic programming algorithm to optimally distribute budget.

The minimum edit distance problem: definition, quadratic algorithm, techniques for reconstructing optimal solutions from the dynamic programming table.

Material

Even More Dynamic Programming and the Split and List Technique

The Subset-Sum problem: definition and a dynamic programming algorithm. Pseudopolynomial-time algorithms.

See Chapter 3.8 of [E] and Chapter 6.4 of [KT].

The split and list technique: Improving the trivial O*(2^n)-time algorithm for subset-sum to a O*(2^(n/2))-time algorithm. Techniques for generating all subsets sums of a set, a trivial O(n2^n) algorithm, a O(2^n) algorithm (see also Chapter 17.1 of [CLRS]).

The Knapsack problem: a dynamic programming algorithm with polynomial running time in the number of items and in the maximum weight (see Chapter 6.4 of [KT]). A dynamic programming algorithm with polynomial running time in the number of items and in the optimal value. A split and list algorithm (see also Chapter 9.1 of [F]).

The 1-in-3 Positive SAT problem: definition, improving the trivial O*(2^n)-time algorithm to a O*(2^(n/2))-time split and list algorithm. See Chapter 9.1 of [F].

Material

Laboratory Problems

References