Algorithms

Expert Practice - Algorithms

Practice minimum spanning trees, max flow, topological sort, amortized analysis, NP-completeness, and string algorithms with hard problems.

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Questions

Multiple choice with one correct answer

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Expert-reviewed questions with clear answer keys

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Algorithms
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About This Algorithm Practice

Tackle the hardest algorithm challenges with guided solutions

These expert exercises cover NP-hard reductions, flow networks, string algorithms, and competitive programming techniques, each with a detailed solution walkthrough.

The problems are untimed and demanding, built to stretch your problem-solving and proof skills.

Working through them is excellent preparation for contests and advanced coursework.

What You Will Practice at Expert Level

Graph Optimization Problems

Exercises guide you through Kruskal and Prim for minimum spanning trees, Ford-Fulkerson max flow, and topological sort of directed acyclic graphs.

Amortized Analysis Drills

You will practice the aggregate, accounting, and potential methods, proving results like amortized O(1) appends on a dynamic array.

NP-Completeness and Hard DP

Guided problems cover reductions between NP-complete problems and dynamic programming over bitmasks and intervals for exponential search spaces.

Range Structures and Strings

You will build segment and Fenwick trees for O(log n) range queries and implement KMP matching and suffix array construction.

Sample Questions

A few real questions from this test, with answers and explanations. Take the full test above for the complete set.

The Master Theorem solves recurrences of the form T(n) = aT(n/b) + f(n). For T(n) = 2T(n/2) + n, the solution is:

Answer: O(n log n)

With a=2, b=2, f(n)=n: we compare f(n) with n^(log_b(a)) = n^1. Since f(n) = Theta(n^(log_b(a)) * log^0(n)), Case 2 applies, giving T(n) = O(n log n). This is Merge Sort's recurrence.

The Max-Flow Min-Cut theorem states that:

Answer: Maximum flow equals the minimum cut capacity

The Max-Flow Min-Cut theorem proves that the maximum flow from source to sink equals the minimum capacity that, when removed, disconnects the source from the sink. This fundamental theorem connects two seemingly different optimization problems.

A Fibonacci Heap supports decrease-key in what amortized time?

Answer: O(1)

Fibonacci Heaps achieve O(1) amortized time for decrease-key by cutting the node from its parent and adding it to the root list. This makes Dijkstra's algorithm run in O(V log V + E) instead of O((V+E) log V) with binary heaps.

To prove a problem X is NP-complete, you must show:

Answer: X is in NP and a known NP-complete problem reduces to X in polynomial time

NP-completeness requires two proofs: (1) X is in NP (solutions can be verified in polynomial time), and (2) a known NP-complete problem can be reduced to X in polynomial time, showing X is at least as hard as all NP problems.

The KMP (Knuth-Morris-Pratt) algorithm preprocesses the pattern to build a failure function. What does this achieve?

Answer: It avoids re-examining previously matched characters

KMP's failure function records the longest proper prefix that is also a suffix for each position in the pattern. When a mismatch occurs, the algorithm uses this information to skip ahead instead of restarting from the beginning, achieving O(n+m) time.

Frequently Asked Questions

Find answers to common questions about this assessment

Build a small flow network with capacities, then run Ford-Fulkerson: repeatedly find an augmenting path in the residual graph and push flow along it until none remains. Using breadth-first search to find paths gives the Edmonds-Karp variant at O(V times E^2). Tracing residual capacities by hand clarifies why the method terminates.

Pick an operation sequence, such as many dynamic array appends, and apply each method in turn. The aggregate method averages total cost over all operations, the accounting method assigns credits, and the potential method tracks stored energy. Proving that array doubling gives amortized O(1) per append is a classic drill worth repeating.

Represent a subset of items as bits of an integer and let the DP state be that mask. The traveling salesman path problem is a canonical exercise, running in O(2^n times n^2). Practice iterating over masks and their subsets, and use memoization so each state is computed once, which keeps the exponential blow-up manageable.

KMP matches a single pattern in O(n + m) time using a prefix table, which is ideal for one query. A suffix array sorts all suffixes of a text so many pattern queries run in O(m log n) each after construction. Practicing both teaches you to match the structure to whether you have one query or many.

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